.

JOHN LANGTOM

From tKe

ESTATE OF JOHN LANGTONT

to tke

UNIVERSITY OF TORONTO
1920

A

TEXT-BOOK OF PHYSICS

HEAT

A TEXT-BOOK OF PHYSICS.

BY

J. H. POYNTING, J. J. THOMSON,

SC.D., P.R.S., M.A., F.R.S.,

Late Fellow of Trinity College, Camljriilf,".-, Fellow of Trinity College, Cambridge ; Prof, of

Professor of Physics, Limiin;;ham Experimental Physk-s in the University

University. of Cambridge.

VOLUME I. FIFTH EDITION, Revised, with Illustrations. Price 10s. 6d.

PROPERTIES OF MATTER.

CONTENTS. — Weight and Mass. — The Acceleration of Gravity ; Its Variation
and the Figure of the Earth. — Gravitation. — Elasticity. — Strain, Stresses, Relation
between Stresses and Strains. — Torsion.- -Bending of Rods. — Spiral Springs. —
Impact. — Compressibility of Liquids. — Pressures and Volumes of Gases. — Thermal
Effects Accompanying Alterations in Strains. — Capillarity. — Laplace's Theory of
Capillarity. — Diffusion of Liquids. — Diffusion of Gases. — Viscosity of Liquids. —
INDEX.

VOLUME II. — FIFTH EDITION, Revised, with Illustrations. 8s. 6d.

SOUND.

CONTENTS. — The Nature of Sound and its Chief Characteristics. — The Velocity
of Sound in Air and other Media. — Reflection and Refraction of Sound. — Frequency
and Pitch of Notes. — Resonance and Forced Oscillations. — Analysis of Vibrations. —
The Transverse Vibrations of Stretched Strings or Wires. — Pipes and other Air
Cavities. — Rods. — Plates. — Membranes. — Vibrations maintained by Heat. — Sensitive
Flames and Jets. — Musical Sand. — The Superposition of Waves. — INDEX.

VOLUME III. — FOURTH EDITION, Revised, Illustrated. 15s.

HEAT.

Remaining Volumes in Preparation —

LIGHT; MAGNETISM AND ELECTRICITY.

In Two Volumes, Large 8vo, Strongly Bound in Half- Morocco. Sold Separately.

PHYSICO-CHEMICAL TABLES

FOR THE USE OF ANALYSTS, PHYSICISTS, CHEMICAL MANUFACTURERS,
AND SCIENTIFIC CHEMISTS.

VOLUME I. Chemical Engineering, Physical Chemistry. Price 24s. net.
VOLUME II. Chemical Physics, Pure and Analytical Chemistry. Price 36s. net.

BY JOHN CASTELL-EVANS, F.I.C., F.C.S.,

Superintendent of the Chemical Laboratories, and Lecturer on Inorganic Chemistry and Metallurgy
at the Finsbury Technical College.

In Large Crown 8vo, Handsome Cloth. 15s. net.

ELECTRICAL THEORY AND THE PROBLEM OF
THE UNIVERSE.

By G. W. DE TUNZELMANN, B.Sc.

" One of the most valuable contributions to electrical literature that the year has produced.*—
Times.

LONDON: CHARLES GRIFFIN & CO,, LTD.; EXETER STREET, STRAND.

A

TEXT-BOOK OF PHYSICS.

BY

J. H. POYNTING, So.D., F.R.S. ;

FOREIGN MEMBER OP THE ACCADEMIA DEI LINCEI, ROME ; HON. D.Sc., VICTORIA UNIVERSITT ;

LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE ; MASON PROFESSOR

OF PHYSICS IN THE UNIVERSITY OF BIRMINGHAM.

AND

SIR J. J. THOMSON, M.A., F.R.S. ;

CORRESPONDING MEMBER OF THE INSTITUTE OF FRANCE ; FOREIGN MEMBER OF THE
BERLIN ACADEMY; HON. So.D., DUBLIN; HON. D.L., PRINCETOWN; HON.

D.Sc., VICTORIA; HON. LL.D., GLASGOW; HON. PH.D., CRACOW;

FELLOW OF TRINITY COLLEGE, CAMBRIDGE ; CAVENDISH PROFESSOR OF EXPERIMENTAL

PHYSICS IN THE UNIVERSITY OF CAMBRIDGE ; PROFESSOR OF NATURAL

PHILOSOPHY IN THE ROYAL INSTITUTION.

HEAT.

WITH 193 ILLUSTRATIONS.
FOURTH EDITION, REVISED.

LONDON:
CHARLES GRIFFIN AND COMPANY, LIMITED;

EXETER STREET, STRAND.
1911.

[All rights reserved.]

JAN201EB7

Printed by BALLANTYNE, HANSON <^ Co.
At the Ballantyne Press, Edinburgh

PREFACE

TO FOURTH EDITION.

WE desire to express our hearty thanks to readers of
the earlier editions of this volume who have kindly sent
us lists of errata. These have been corrected, and a
number of alterations and additions have been made.

June. 1911.

PREFACE.

THIS volume on Heat is the third of a series forming a Text-Book
on Physics. The first two volumes dealt with the Properties of
Matter and Sound, and the succeeding volumes will deal with
Magnetism and Electricity, and Light.

The Text-Book is intended chiefly for the use of students who
lay most stress on the study of the experimental part of Physics,
and who have not yet reached the stage at which the reading
of advanced treatises on special subjects is desirable. To bring
the subject within the compass thus prescribed, an account is
given only of phenomena which are of special importance, or
which appear to throw light on other branches of Physics, and the
mathematical methods adopted are very elementary. The student
who possesses a knowledge of advanced mathematical methods,
and who knows how to use them, will, no doubt, be able to work
out and remember most easily a theory which uses such methods.
But at present a large number of earnest students of Physics are
not so equipped, and the authors aim at giving an account of the
subject which will be useful to students of this class. Even for
the reader who is mathematically trained, there is some advan-
tage in the study of elementary methods, compensating for their
cumbrous form. They bring before us more evidently the points
at which various assumptions are made, and they render more
prominent the conditions under which the theory holds good.

J. H. P.

CONTENTS.

CHAPTER I.
TEMPERATURE.

PAOBS

Introductory Eemarks — Temperature — Thermal Equilibrium — Construction of
Mercury - Glass Thermometers — Fixed Points: Centigrade, Fahrenheit,
and Keaumur Scales — Marking Fixed Points — Calibration and Graduation
— Precautions in use — Limits of accuracy — Range — Scales of Temperature
given by expansion arbitrary — The Work Scale — Air and Hydrogen Scales
Platinum Resistance Thermometers — Table of Temperatures — Maximum
and Minimum Thermometer — Thermostats 1-16

CHAPTER II.
EXPANSION OF SOLIDS WITH RISE OF TEMPERATURE.

Linear Expansion of Solids — Ramsden's Method — Modern Use of the Method —
Method of Lavoisier and Laplace — Results — Fizeau's Optical Method
Applications of Linear Expansion — Volume Expansion of Solids . . 17-28

CHAPTER III.
EXPANSION OF LIQUIDS.

Volume Expansion of Liquids — U-Tube Method applied to Mercury — Dulong
and Petit — Regnault — Expansion of other Liquids by Specific Gravity
Bottle— By Dilatometer--Matthiessen's Hydrostatic Method — The Expan-
sion of Water — Hope's Apparatus — Apparatus of Joule and Playfair —
Results 29-40

CHAPTER IV.
EXPANSION OF GASES.

Expansion of Gases depends on Pressure Changes — Volume Expansion at
Constant Pressure — Gay-Lussac's Method — Regnault's Experiments —
Increase of Pressure with Constant Volume — Gas Thermometry — Regnault's
Normal Air Thermometer — Hydrogen Thermometer — Bottomley's Air
Thermometer — Callendar's Compensated Air Thermometer . . . 41-52

vil

viii CONTENTS.

CHAPTER V.

PAOB3

CIRCULATION AND CONVECTION IN LIQUIDS AND GASES.

Circulation and Convection of Heat — Hot- Water Heating Systems — Ocean
Currents — Convection in Gases — Convection Currents in the Atmosphere —
Winds— Land and Sea Breezes— Trade- Winds— Water- Vapour aids Convec-
tion Currents — Weather Forecasting in the Case of Cyclones — Convection
in Chimneys and Hot- Air Heating Systems 53-63

CHAPTER VI
QUANTITY OF HEAT. SPECIFIC HEAT.

Quantity of Heat — Unit Quantity: the Calory— Specific Heat — Water Equiva-
lent and Capacity for Heat — Method of Mixtures — Regnault's Determinations
by the Method of Mixtures — Experiments on Solids — On Liquids — On Gases
— Liquid Specific Heat by Mixture with known Solid — Method of Cooling —
Method of Melting Ice — Bunsen's Ice Calorimeter — Method of Condensing
Steam — Joly's Steam Calorimeter — Differential Steam Calorimeter — Method
of Electric Heating — Specific Heat of Water — General Results — Law of
Dnlong and Petit 64-87

CHAPTER VII.
CONDUCTIVITY.

The Passage of Heat from one Body to Another — Conductivity — Differs
enormously in different Substances — General remarks on Conductivity in
the Three States— Definition of Conductivity — Diffusivity — Emissivity —
Measurements of Conductivity — Peclet's Method — Bar Methods of Despretz,
Forbes, Neumann, and Angstrom — Gray's Method — Berget's Experiment
on Mercury — Experiments of Wiedemann and Franz — Kundt's Experiments
— Senarmont's Experiments on Crystals — Lees's Experiments — Lundquist
— Weber — Conductivity of Gases — Experiments of Stefan, Kundt, and
Warburg 88-107

CHAPTER VIII.

THE FORMS OF ENERGY. CONSERVATION OF ENERGY. MECHANICAL
EQUIVALENT OF HEAT. FIRST LAW OF THERMODYNAMICS.

Introductory Remarks — The Various Forms of Energy — The Identity of Energy
— The Conservation or Constancy of Energy — Statement of the Principle —
Mayer's Calculation of the Mechanical Equivalent — Joule's Researches —
Later Repetition — Experiments of Rowland— of Miculescu — of Reynolds
and Morby — of Griffiths — of Schuster and Gannon — The First Law of
Thermodynamics 108-128

CONTENTS. ix

CHAPTER IX.

rum

THE KINETIC THEORY OF MATTER.

Atomic Hypotheses — Solids — Liquids — Gases — Kinetic Theory of Gases — Mean
Value of the Square of the Velocity of Translation V2 — Mixture of Gases —
Relation between V and Temperature — Energy of Translation and Internal
Energy — Joule's Approximate Method of Calculating the Velocity of Mean
Square — Effusion or Transpiration through a small Orifice into a Vacuum
—Thermal Transpiration— The Mean Free Path— The M.F.P. calculated
from the Coefficient of Viscosity — Conduction of Heat in Gases — The
Diameter of the Molecules and the number of Molecules per Cubic Centi-
metre— Forces acting on unequally heated Surfaces in High Vacua — The
Gas Equation of Van der Waals 129-156

CHAPTER X.
CHANGE OF STATE— LIQUID-VAPOUR.

General Account of Evaporation — Vapour-Pressure — Boiling — Delayed Boiling
— Condensation on Nuclei — Measurements of Vapour-Pressure — Determina-
tion of Vapour Density — Density of Saturated Vapour — Measurements of
Latent Heat of Vapours — Specific Heat of Saturated Vapour — Spheroidal
State 157-184

CHAPTER XI.
CHANGE OF STATE-LIQUID-VAPOUR (continued).

Indicator Diagram — Critical Point — Critical Constants — Equation of Van der

Waals — Liquefaction of Gases 185-199

CHAPTER XII.
CHANGE OF STATE— SOLID-LIQUID.

Melting of Ice and Melting of Wax — Melting of Ice at a Definite Point and on
the Surface only — Latent Heat — Supercooling — Regelation — Effect of
Pressure on Melting-Point — Melting-Points of Solids — Explanation of
Melting on the Kinetic Theory — Resemblance of Solution to Fusion —
Evaporation from Solids 200-208

CHAPTER XIII.
WATER IN THE ATMOSPHERE.

Hygrometry — Relative Humidity — Dew - Point and its Determination —
Regnault's Researches on the Density of Water- Vapour — Cloud — Con-
vective Equilibrium — Halos and Parhelia — Coronas— Rate of Fall of Cloud
Drops— Hail— Fog— Dew 209-219

x CONTENTS.

CHAPTER XIV.

PAOM

GENERAL ACCOUNT OF RADIATION.

Radiant Energy — Radiometers — Radiant Energy and Light resemble each other
— Light is Radiant Energy to which the eye is sensitive — Radiant Energy
has a much greater range of Wave-Length than Light — Radiometers only
measure Energy Streams and do not indicate Quality — Comparison of
Emissive Powers — Radiation of different Wave-Lengths — Comparison of
Absorptive Powers — Comparison of Reflecting Powers — Diffusion — General
Results— Radiating and Absorbing Powers vary together — Illustrations —
Transparency and Opacity — Radiation and Absorption by Gases and
Vapours ..... 220-236

CHAPTER XV.
THEORY OF EXCHANGES.

Theory of Exchanges — Uniform Temperature Enclosures — Full Radiation —
Propositions regarding Uniform Temperature Enclosures — Bodies ex-
changing Radiation at different Temperatures — Bodies in the same
Physical State continue to absorb the same kind of Rays independently
of Change of Temperature — Radiation of every kind emitted by a Body
increases as the Temperature rises — Application to Special Cases . . 237-243

CHAPTER XVI.
RADIATION AND TEMPERATURE.

Variation of Rate of Radiation with Temperature — Newton's Law of Cooling —
Dulong and Petit's Law — Rosetti's Law — Stefan's Law — Constants of
Radiation — Radiation from Surfaces which absorb selectively — Rate of
Solar Radiation — Solar Constant — Pouillet's Pyrheliometer — Violle's Actino-
meter — Langley's Researches — Crova's Researches — Effective Temperature
of the Sun— Source of Solar Energy 244-257

CHAPTER XVII.
THERMODYNAMICS.

The Second Law of Thermodynamics — The Indicator Diagram — Isothermals —
Adiabatics or Isentropics — Heat Engines — Carnot's Reversible Heat Engine
— Carnot's Cycle — Conditions for Reversible Working — Examples of Rever-
sible Processes — Of Irreversible Processes — Efficiency of an Engine —
Absolute or Work Scale of Temperature — Efficiency expressed on the
Absolute Scale — Comparison of the Absolute with the Air Scale — Water-
Wheel Analogue — Reversible Cycles in general — Entropy — Entropy-
Temperature Diagram — Quantities Analogous to Entropy — Entropy tends
to increase — Dissipation of Energy — Intrinsic Energy — Available Energy —
Possible Efficiency of a Steam Engine 258-283

CONTENTS. xi

OHAPTEK XVIII.

PAOZS

THERMODYNAMICS OF ISOTHERMAL AND ADIABATIC CHANGES.

Heat taken in when a Body expands Isothermally — Heat to a neighbouring
Adiabatic the same by all paths — Change in Temperature when a Body
undergoes a small Adiabatic Change — Adiabatics steeper than Isothermals
— Specific Heats at Constant Pressure and Constant Volume — Their Ratio y
equal to the Ratio of the Isentropic and Isothermal Elasticities — Experi-
mental Determinations of y for Gases — Adiabatic Gas Equation — Decrease
of Temperature Upwards with Convective Equilibrium — Internal Energy
taken up by a Gas in Expanding — Comparison of Air Scale with Absolute
Scale — Generalisation of Indicator Diagram for any Stress and correspond-
ing Strain 284-305

.CHAPTER XIX.

THERMODYNAMICS OF CHANGE OF STATE AND OF SOLUTIONS.

First Latent Heat Equation — Volume of Saturated Steam — Triple Point and
Difference of Vapour-Pressures of Ice and Water below 0° C. — Second
Latent Heat Equation — Alteration of Vapour-Pressure with Curvature of
Liquid Surface — Connection with Change in Melting-Point by Pressure —
Solutions — Vapour-Pressure less than that of the Solvent — Osmotic
Pressure — Raising of Boiling-Point — Lowering of Melting-Point — Semi-
Permeable Membranes — Van t'Hoff's Application of Thermodynamics —
Molecular Theory of Osmotic Pressure ....... 306-332

CHAPTER XX.
THERMODYNAMICS OF RADIATION.

General Principle — The Pressure of Radiation — The Normal Stream of Radia-
tion, the Total Stream, and the Energy Density — The Pressure on a fully
Radiating Surface — The Relation between E and 6 in full Radiation, the
Fourth Power Law — Full Radiation remains full Radiation in any Adiabatic
Change — Relation between Volume and Temperature in an Adiabatic
Change — Entropy — Application of Doppler's Principle — Change of Energy
in a given Wave-Length — Change of Energy of each Wave- Length in an
Adiabatic Expansion of full Radiation — Maximum Value of Energy for
given Range of Wave-Length — Form of the Function expressing the
Distribution of Energy in the Spectrum 333-342

INDEX 343

LIST OF ILLUSTRATIONS.

no. PAGK

1. Apparatus used to show Expansion of a Heated Rod .... 3

2. Gauge into which Bar fits only when Cold 3

3. Boiling-Point Apparatus 5

4. Calibration of a Thermometer 6

5. Joule's Observations on the Alteration of the Freezing-Point in Thermo-

meters 8

6. Six's Thermometer 13

7. Constant Temperature Apparatus 14

8. Thermostat . 15

9. Early Apparatus for Measuring the Linear Expansion of a Rod . 17

10. Ramsden's Expansion Apparatus

11. Measurements made in Ramsden's Apparatus

12. Diagram of Expansion Apparatus at the Bureau International . .

13. Diagram of Expansion Apparatus of Lavoisier and Laplace . .

14. Mirror Method of Reading Deflections

15. Diagram of Fizeau's Expansion Apparatus

16. Methods of Providing for Expansion in Rods working Railway Points

17. Compensation Measuring Bar

18. Principle of Gridiron Pendulum of Iron and Brass Bars

19. Gridiron Pendulum of Two Metals '.

20. Compensating Balance for Chronometers

21. U-Tube Hydrometer

22. Diagram of Regnault's First Apparatus for the Expansion of Mercury

23. Regnault's Second Apparatus for the Expansion of Mercury . .

24. Graphic Method of Determining Results

18
19
20
21
22
23
25
26
26
27
27
30
31
32
33

25. Callendar's Apparatus for the Expansion of Mercury 34

26. Dilatometer 37

27. Hope's Apparatus 37

28. Indications of Thermometers in Hope's Apparatus 38

29. Joule and Playfair's Apparatus for Maximum Density of Water . . 38

30. Density-Temperature Curve of Water 39

31. Curve of Expansion of Water 39

32. Flask Thermoscope to show Expansion of Gases 41

33. Gay-Lussac's Apparatus for Determining the Expansion of Gases at

Constant Pressure ........... 42

34. Reguault's Apparatus for Determining the Expansion of Gases at Constant

Pressure 43

35. Regnault's Apparatus for Determining the Expansion of Gases at Constant

Pressure 43

36. Regnault's Apparatus for Determining the Expansion of Gases at Constant

Pressure 44

37. Simple Air Thermometer 49

37a. Bottomley's Air Thermometer ......... 50

38. Callendar's Compensated Air Thermometer 50

39. Circulation of Water in Heated Flask ........ 53

40. Boiling Water by Convection 53

41. Boiling Water when Circulation and Convection are Prevented . . 54

42. Principle of Hot Water Heating Systems 54

43. Diagram to Illustrate Production of Ocean Currents by Heat ... 55

44. Isobars and Winds in a Cyclone 58

45. Cyclone Prognostics 59

46. Progress of a Cyclone 60

xiii

xiv LIST OF ILLUSTRATIONS.

FIG. PAGB

47. Diagram of Circulation established by an Open Fireplace . . .61

48. Candle in a Flask 61

49. Candle in a Wide Cylinder 62

50. Tobin Ventilation 62

51. Arrangement for Warming Incoming Air 62

52. Calorimeter 67

53. Eegnault's Apparatus for the Determination of Specific Heat of Solids . 68

54. Eegnault's Apparatus for the Determination of Specific Heat of Liquids . 69

55. Begnault's Apparatus for the Determination of Specific Heat of Gases . 71

56. Bnnsen's Ice Calorimeter 73

57. Joly's Steam Calorimeter 74

58. Joly's Differential Steam Calorimeter 75

59. CaUendar- Barnes Electric Heating Method of Determining the Specific

Heat of Water 79

60. Diagram of Results of Different Experiments on Specific Heat of Water . 80

61. Experiment Illustrating that Liquids are Bad Conductors .... 91

62. Low Conducting Power of a Fibrous Solid 92

63. Principle of the Safety Lamp 92

64. Illustrating Definition of Conductivity 93

65. Ingenhousz's Apparatus for Illustrating Diffusivity ..... 94

66. Despretz's Bar Experiments on Conductivity of Metals .... 96

67. Diagram of Temperature Curve .... .... 97

68. Forbes's Bar Experiment 98

69. Diagram of Conductivity in Crystals 101

70. Lees's Disc Experiments on Conductivity ....... 102

71. Lees's Disc Experiments on Liquids 104

72. Diagram Illustrating Strain Energy - . 112

73. Joule's Expansion of Air Apparatus 120

74. Modified Form of Joule's Apparatus 120

75. Joule's Water Churning Apparatus for Determining the Mechanical

Equivalent of Heat 122

76. Diagram Illustrating the Kinetic Theory of Liquid Viscosity . . . 132

77. Diagram Illustrating the Kinetic Theory of Gas Viscosity .... 145

78. Diagram Illustrating the Kinetic Theory of Gas Conductivity . . . 149

79. Crookes's Radiometer ... 150

80. Diagram Illustrating the Explanation of the Radiometer .... 150

81. Diagram Illustrating the Explanation of the Radiometer . . . . 151

82. The Dust-Free Space above a Hot Wire 152

83. The Diminution of Path in Molecular Collision 153

84. Barometers with Water above the Mercury 158

85. Barometer with Water above the Mercury surrounded by Heating Bath . 158

86. Evaporation into an Air Space 160

86a. Apparatus for Distillation 161

87. Diagram Illustrating Kinetic Theory of Increase of Vapour Pressure with

Temperature 162

88. Apparatus used in Demonstrating that Vapour Pressure equals Atmos-

pheric Pressure at Boiling-Point 163

89. Apparatus Illustrating Reduction of Boiling-Point by Reduction of Pressure 161

90. Boiling Water by Cooling it 164

91. The Formation of a Drop 165

92. Diagram illustrating Relation between Volume and Pressure of Gas con-

tained in a Bubble in a Liquid 167

93. Diagram illustrating Relation between Volume and External Pressure on

a Bubble 167

94. Apparatus for obtaining a Dust-Free Space 168

95. Apparatus for Researches on Vapour Pressure of Water .... 172

96. Apparatus for Researches on Vapour Pressure of Ice 173

97. Diagram of ^JTater Vapour Pressure and Ice Vapour Pressure . . . 174

98. Dynamical, or Boiling-Point Method of Determining Vapour Pressure of

Water above 50" 174

99. Apparatus for Determination of Boiling-Points ...... 175

100. Vapour-Density Bulb. Dumas' Method 176

LIST OF ILLUSTRATIONS. xv

FIG. PAOE

101. Gay-Lussac and Hoffmann's Method for Vapour- Density Determination . 176

102. Victor Meyer's Method for Vapour-Density Determination . . . . 177

103. Principle of Determination of Density of Saturated Vapour . . . 178

104. Apparatus for Eough Determination of Latent Heat ..... 179

105. Berthelot's Apparatus for Latent Heat 180

106. Illustrating the Spheroidal State 183

107. Spheroidal State of Drops of Alcohol 184

108. Indicator Diagram for Water Steam 185

109. Indicator Diagram showing Prolongations for Water and Steam . . 187

110. Continuous Form of Isothermal 188

111. Andrews' Tube for Experiments on Carbon Dioxide 189

112. Diagram of Andrews' Compression Apparatus . . . . . . 189

113. Diagram showing Isothermals for Carbon Dioxide 190

114. Diagram showing Continuous Paths from Gas to Liquid .... 191

115. Diagram showing Isothermals as given by Van der Waals' Equation . . 194

116. Dewar's Vacuum Vessel 198

117. Diagram of Regenerator Process for Liquefying Gases .... 198

118. Bunsen's Apparatus for Determining Raising of Melting-Point by Pressure 204

119. Melting-Point Apparatus 204

120. Ideal Experiment Illustrating Nature of Melting ..... 206

121. Ice and Water Isothermals 207

122. Regnault's Dew- Point Hygrometer 210

123. Dines's Hygrometer . , 210

124. Wet and Dry Bulb Hygrometer 211

125. Chemical Method for Determining Dew Point 213

126. Diagram of Mixture of Equal Masses of Unsaturated Air at Different

Temperatures 215

127. Cumulus on Horizontal Base of Cloud 217

128. Thermopile 221

129. Radio-Micrometer 221

130. Bolometer 222

131. Proof of the Law of Inverse Squares 223

132. Radiation spread out into Spectrum . 224

133. Leslie's Comparison of Emissive Powers 226

' 134. Diagram illustrating Langley's Method of separating Spectra of different

orders 227

135. Leslie's Comparison of Reflecting Powers 230

136. Melloni's Method of Determining Diffusion 231

137. Apparatus for Demonstrating the Relation between Absorbing and Radia-

ting Powers 232

138. Tyndall's Experiment on Emission and Absorption of Gases . . . 234

139. Effect of the Medium on Radiation 241

140 (A and B) Diagrams illustrating Variation of Rate of Radiation with

Temperature 244

141. Dulong and Petit's Cooling in Vacuo 246

142. Radiation Curves broken up into Cooling Curves ' 247

143. Radiation Curves for Spectrum at two Different Temperatures ... 250

144. Pouillet's Pyrheliometer 251

145. Violle's Actinometer 253

146. Langley's Curves of Solar Radiation 254

147. Indicator Diagram 258

148. Work done by a Body in Expansion 259

149. Work done in a Cycle 260

150. Carnot's Engine 262

151. Carnot's Cycle 263

152. Indicator Diagram representing Carnot Cycle for a Reversible Engine . 266

153. Equal Temperature Intervals on the Work Scale 266

154. Equal Temperature Intervals on the Work Scale 267

155. Diagram of Comparison of the Absolute with the Air Thermometer Scale 269

156. Representation of a Reversible Cycle of any Form 272

157. Gain in Entropy Independent of Path on Diagram ..... 274

158. Entropy-Temperature Diagram 275

xvi LIST OF ILLUSTRATIONS.

FIG. PAGE

159. Intrinsic Energy on Indicator Diagram « 278

160. Work done greater in Isothermal than in Adiabatic Pressure decrease . 279

161. Work Obtained under Condition of Constant Entropy .... 280

162. Entropy-Temperature Diagram for Steam- Engine 281

163. Heat taken in when a Body Expands isothermally 284

164. Coefficient of Pressure Increase at Constant Volume 285

165. Change of Temperature in Adiabatic Change of Volume .... 286

166. Entropy-Temperature Diagram 287

167. Clement and Desormes' Apparatus for Comparison of the two Elasticities

of a Gas 291

168. Diagrammatic Representations of Porous Plug Experiment . . . 297

169. Porous Plug 298

170. Indicator Diagram representing the Gas in the Porous Plug Experiment . 300

171. Indicator Diagram of a Stretched Wire 303

172. Diagram of First Latent Heat Equation 306

173. Ideal Experiment to show that Vapour-Pressures of Ice and Water are

equal at 0° C 309

174. Triple Point on Temperature Pressure Diagram 310

175. Difference of Vapour Pressures of Ice and Water below 0° C. . . 311

176. Equal Pressure Lines on the Entropy-Temperature Diagram . . . 312

177. Diagram showing that Cloud is produced on Expansion of Saturated

Water Vapour 313

178. Alteration of Vapour Pressure with Curvature of Liquid Surface . . 314

179. Ideal Experiment to show the Alteration 314

180. Ideal Experiment to show Effect of Pressure on Vapour Pressure . . 318

181. Diagram representing the Alteration of Vapour Pressure .... 319

182. Vapour Pressure of Solution less than that of Pure Solvent . . . 320

183. Diagram of Raising of the Boiling-Point 321

184. Diagram of Lowering of the Melting-Point 322

185. Solvent and Solution separated by Semi-permeable Membrane . . . 328

186. Diagram of Reversible Cycle 328

187. Diagram illustrating Henry's Law applied to Osmotic Pressure . . 329

188. Diagram illustrating Vapour Pressure given by Van t'Hoff's Law . . 330

189. Pressure of Radiation on a Surface . 334

190. Energy Density in a Fully Radiating Enclosure 334

191. Diagram of Cycle for a Space filled with Radiation ..... 336

192. Change of Wave-Length in Normal Reflection 339

193. Change of Wave-Length in Oblique Reflection 339

HEAT.

CHAPTER I.
TEMPERATURE.

Introductory Remarks — Temperature — • Thermal Equilibrium — Construction of
Mercury -Glass Thermometers — Fixed Points: Centigrade, Fahrenheit, and
Reaumur Scales — Marking Fixed Points — Calibration and Graduation — Pre-
cautions in use — Limits of accuracy — Range — Scales of temperature given by
expansion arbitrary — The Work Scale — Air and Hydrogen Scales — Platinum
Resistance Thermometers— Table of Temperatures— Maximum and Minimum
Thermometer — T hermostats.

Introductory Remarks. — In the science of Heat, we investigate those
phenomena which are chiefly revealed to us by our sense of warmth or
cold. We use the words " hot " or " cold " to describe the condition of
external bodies which corresponds to the sensation we receive through
our skin on touching or approaching them, and we habitually compare
bodies with respect to the sensations so received, describing one as
"hotter" or "colder" than another. Given several vessels of water,
we could with very little trouble arrange them in order of hotness, and
we have a number of expressions in common use to describe their con-
ditions, such as "ice-cold," "cool," "chill taken off," "tepid," "luke-
warm," "warm," "hot," "boiling hot." Our primary sensations are,
therefore, those of hotness or of coldness, and we are accustomed to think
of hotness as varying in degree. If we put a hot body in contact with a
cold one — if, for instance, we pour hot water into a cold vessel — the hot
water is cooled while the cold vessel is heated. We regard this change
as the passage of something which we term heat from the hotter to the
colder body, its loss by the former being accompanied by cooling, its gain
by the latter by heating. We do not mean to imply by " something " some
kind of matter. We may fairly describe kinetic energy as " something,"
and say that when one body strikes another, setting it in motion, " some-
thing," .viz., kinetic energy, has passed from the one to the other, yet
we do not think of energy as matter.

So, here, we only describe the heat as " something," because we
believe that we can identify the heat gained by the cold vessel with that
lost by the hot water. We also think of heat as greater or less in
amount. If the hot water cools very considerably, we think of it as
giving up more heat than if it cools only slightly. Or if the quantity of
hot water cooling is comparatively large, we think of it as giving up more
heat than a smaller quantity of water cooling to the same extent.

A

2 HEAT.

These, then, are our two fundamental ideas: that bodies are comparable
as to their hotness or coldness, and that in general, on becoming hotter
they receive something which we call heat — the amount of heat received
depending both on the quantity and nature of the matter and on the
degree to which it becomes hotter.

Our first aim must be to render our conceptions more definite, by
obtaining some numerical scale to express how hot a body is.

We shall be occupied, in the earlier chapters of the book, with a
description of the mode in which such a scale is obtained, and an account
of its use in the investigation of the change of dimensions of bodies when
heated. Afterwards we shall show how numerical expressions may be
obtained for quantities of heat, and how these quantities may be deter-
mined by experiment.

Temperature and Thermal Equilibrium. — The number which
expresses on some definite scale how hot a body is, is termed its
temperature.

Any instrument, such as the ordinary mercury and glass instrument,
used to obtain the temperature of a body is termed a thermometer.

We know from common observation that bodies in contact with each
other, and not subjected to changes of external conditions, after a time
get neither hotter nor colder — i.e. heat does not pass from one to another.
They are then said to be in thermal equilibrium with each other.

For example, in a room not exposed to draughts or to sunlight, and
containing no fire, the objects lying on a table will after a time all be in
thermal equilibrium with the table and with each other. A glass of
water on the table will neither be heated nor cooled by putting into it
an iron rod which has also been lying 011 the table. The sensation
received by an observer touching the various objects may, however, be
very different. The iron will feel colder than the table on which it has
been placed, and the water in the glass will feel warmer than the iron
and colder than the table. But if, instead of the finger, we use a
thermometer, we find that it will register the same whether it is laid on
the table, placed against the iron, or put in the water — the difference of
sensation not being due to difference of temperature but to different
rates of communication of heat from the hand to the surfaces touched.

The thermometer, then, shows us that bodies in thermal equilibrium
with each other are at the same temperature, and as the thermometer is
also one of the bodies it is also at the same temperature as the substance
in which it is placed.

Conversely it is true that bodies at the same temperature are in
thermal equilibrium with each other.

This may be proved by direct experiment ; by pouring, for instance,
a quantity of mercury into a vessel containing water at the same
temperature, and noting that the temperature remains constant.

Thermometers. — In order to obtain a definite scale of temperature,
we make use of the fact that, in general, bodies expand on being heated.
Various simple experiments show this. For instance, if a metal rod
(Fig. 1) be fixed at the end V in a vice and if the end A presses down
on a small roller r on a flat plate, then a pointer P attached to the
roller will move over a scale S when the rod is heated, say by a gas
flame.

TEMPERATURE. S

Or, if a brass rod (Fig. 2), exactly fits into a space in a brass plate
when cool, it can no longer be inserted into the space after it is
heated, if the plate remains cool.

We might use the expansion of a brass rod to give us a scale
of temperature, if we had a
sufficiently delicate and simple
method of measuring its length.

We might call the tempera-
ture of the brass when placed
in ice-cold water, 0°, and de-
scribe its temperature as rising
1° for every increase in length
of TOOOOO but the difficulty

of measuring such a small in- FIG. 1.— Expansion of a Heated Rod.

crease as would occur for any

ordinary rise of temperature would render such a scale of little practical

value.

We therefore make use of the facts that the expansion of liquids by

heat is usually much greater, and that of gases enormously greater, than

that of solids.

If a glass flask is filled with water, and then closed by a cork with a

tube of narrow-bore passing through the cork, the warmth of the hand

is quite sufficient to make the water rise very appreciably in the narrow

tube. For though the glass expands, thus making the internal capacity
greater, the water expands still more, not only filling
up the additional volume of the flask, but also rising
in the tube. This is the principle which is used in
the ordinary thermometer, the bulb containing either
mercury or alcohol, and the expansion in excess of
that required to fill the increase of volume of the
bulb being indicated by the rise of liquid in the tube.
Gases expand still more than liquids. If a flask
partially filled with water be closed by a cork,

. , through which passes a narrow tube with its lower
FIG. 2.— Gauge, into , f. , „ . , . .,

which a Bar fits enc^ dipping under the surface oi the liquid, the
when cold but not warmth of the hand will make the air in the upper
when hot. portion of the flask expand and drive the liquid

rapidly up the tube. Gas thermometers on this
principle may be made far more sensitive than the liquid thermometers,
but there are, as we shall see, difficulties in their use, which render them
unsuitable for ordinary purposes.

The Construction of Mercury-in-Glass Thermometers.— To

construct a good thermometer for scientific purposes, a tube is selected
with a capillary bore as nearly uniform as possible. A bulb is blown on
the end of this, the size of the bulb being adjusted by the experience of the
glass-blower to the sensibility required in the thermometer — the greater
the sensibility the larger the bulb or the finer the bore. In order to fill
the bulb with mercury, it is heated to expel some of the air, and the open
end of the tube is inserted under mercury. On cooling, the pressure of
the remaining air diminishes, and the external atmospheric pressure
drives some mercury up into the bulb. The tube is then held bulb

4- HEAT.

downwards, and the bulb is heated till the mercury boils, the mercury
vapour rising and displacing the remaining air. On again inverting the
tube and placing the open end under mercury, as the temperature falls,
the whole space will, as a rule, be filled with mercury. Should the air
not be entirely expelled, however, the operation must be repeated. The
tube is then drawn out near the end till the bore is nearly closed, the
bulb is again heated until the mercury flows past the narrow part, and the
latter is rapidly sealed up by means of a blowpipe flame. It is found by
experience that the bulb of a thermometer undergoes contraction. The
contraction is very considerable during the first few weeks or even
months after it has been blown, but after two or three years it takes
place very slowly and is hardly appreciable over short intervals of time.
It is advisable, therefore, to wait till this stage is reached before the
graduation of the instrument is proceeded with.

Fixed Points. — If we only had to use one thermometer, to indicate a
rise in temperature in one particular case, it would be sufficient to mark
on the stem divisions of some chosen length, say 1 mm. each, to number
these from the bottom to the top of the tube, and to call each step of one
" division " a degree. If, for example, the mercury stood 50 mm. above
the lowest mark, we might call the temperature 50°. But we wish to
use different instruments for different cases and to compare their
indications. Further, we wish to compare the indications of our thermo-
meters with those of instruments used by other experimenters on other
occasions. We must, therefore, have a definite scale, as nearly as
possible the same on all instruments. This is secured by the use of
definite " fixed points," the same on all instruments, each corresponding
to a definite fixed temperature. The two fixed points universally used
are 1st, the temperature of ice when just melting under the atmospheric
pressure of 760 mm. ; and 2nd, the temperature of steam from water
boiling normally under the same atmospheric pressure. The height of
the mercury column of a thermometer placed under constant conditions
in different vessels containing melting ice will remain invariable. Placed
under similar conditions in the steam from water boiling normally at a
pressure of 760 mm. in different vessels, it will again remain invariable,
but at a point much higher in the stem. The volume of the tube — or,
if it is of even bore, the length — between these two fixed points is
divided into a number of equal parts, and each part indicates a degree.

The Centigrade Scale* — On the centigrade scale, now universally
employed for scientific purposes, the temperature of melting ice is 0°,
and that of boiling water under the stated conditions is 100°, and the
interval is divided into 100 parts. The scale is indicated by writing C.
after the temperature, as 0° C.

The Fahrerilieit Scale. — On the Fahrenheit scale, the temperature of
melting ice is 32°, and that of boiling water is 212°, the interval being
divided into 180 equal parts. A Fahrenheit temperature is indicated by
writing F. after the temperature, as 212° F.

This scale was arranged by Fahrenheit early in the eighteenth century.
He found that a mixture, of which he did not state the proportions, of
ice, water, and sal-ammoniac, or sea-salt instead of sal-ammoniac, gave
a very low definite temperature, which ho took as 0°. He found that

* Some interesting notes on the history of therinometry will be found in Bolton's
Evolution of the Thermometer,

TEMPERATURE.

the normal temperature of the human body was nearly constant, and
using a duodecimal scale he took this as 2 x 12 = 24°. He found that
ice melted at 8° on this scale. These degrees being inconveniently large,
he quartered them, so that he had 0°, 32°, and 96° as fixed points.
Having verified Amontons' statement that water boils at a constant
temperature, he found that it was at 212 of his new quartered degrees.
The estimate of 96° for the human body was then found to be more
nearly 98° than 96°.

The Reaumur Scale. — On this scale the fixed points are 0° for melting
ice and 80° for boiling water. Reaumur used alcohol diluted with one-
fifth of water as the liquid in his thermometer, and each degree marked
an expansion of To^o0 of the volume of the liquid at 0°.

Conversion of Scales. — If the same teinperatxire is indicated respectively
on the three scales by C, F, and R. then evidently the equations —

0 _F-32
100 ~ "180

JR

;80

enable us at once to convert from one scale to another.

Marking" the Fixed Points. — In this country the lower fixed point
is marked first. For this purpose the bulb of the
tube is immersed in a metal vessel containing a
mixture of small pieces of melting ice and air-free
distilled water. The vessel may also be surrounded
by melting ice to prevent the temperature of the
water in it rising above that of the ice. The ther-
mometer is placed so that the mercury rises just to
the top of the ice, and when it is steady a file mark
is made to show its position.

The thermometer is then placed in a metal vessel,
the construction of which is indicated by Hg. 3, the
bulb and the tube being entirely surrounded by
steam. The thermometer is placed so that the
mercury rises just into sight and its final level,
which is only attained after some time, say a quarter
or half-an-hour, is again marked by a fine file.

It is essential to read the barometer, as the
boiling temperature of water varies with the atmos-
pheric pressure, being only 99° at 733'2 mm.
(slightly below 29 inches). The point marked will
only be 100° on the rare occasions when the baro-
meter is at 760 mm., and, in general, allowance
must be made for the deviation.

Pressure also affects the melting point of ice,
but the ordinary variations of pressure produce no
appreciable effect. Practically it is only important ~"
in ordinary thermometers to use pure water, and FIG. 3. — Boiling-Point
to be sure that the ice is all at the melting Apparatus,

temperature. Impurities lower the melting point,

and large lumps of ice which have not been long melting may easily be
below the melting point inside.

It may be noted that 760 mm. of mercury is not an invariable

6 HEAT.

standard, but varies with the variation of gravity. At the equator,
760 mm. of mercury would only imply a pressure equal to something
less than 758 mm. in England, and water would boil therefore at
slightly above 99'9°. If great accuracy is required, the standard
pressure is taken as 760 mm. of mercury in lat. 45°. The same
barometric height at Greenwich corresponds to a pressure greater by
about 56 in 100,000, which will alter the boiling-point about '016° C.

Calibration. — If the thermometer is to be used as a standard, it must
be calibrated, i.e. the variations in the capillary bore must be deter-
mined and allowed for. There are various modes of effecting this,* the
simplest and quickest being as follows : —

The mercury in the tube is detached near the neck of the bulb,
either by warming the tube in a very fine gas flame at the point where
it is to be detached, or by manipulation of the trace of air still remain-
ing, and which collects in the vacuum left in the bulb if the thermometer
is inverted and the mercury is sent down to the end of the tube. The de-
tached thread is then run down into an enlargement of the bore provided
for the purpose at the end farthest from the bulb, leaving the bore clear.

A short thread, say about 20 mm. in length, is then detached from

A B C D .

FiQ. 4.— Calibration of a Thermometer; AB, BC,—HK,= equal
volumes of the bore.

the mercury still remaining in the bulb and is measured at various
points along the tube by a travelling microscope, or by the dividing
engine.f Let us suppose, for example, that the length AK, Fig. 4,
along the tube, represents 100°, and that we wish to know how to
adjust the intervening divisions to allow for variations of bore. Let us
bring one end of the thread to A, the other end being at B. Measure
AB. Now pass the thread along by gently tapping the end farthest
from the bulb till the thread occupies the position BC. Measure BC.
Now bring the thread to CD. Measure CD ; and so on. Let us suppose,
for simplicity, that the last position of the thread is HK, the end of the
thread exactly falling at K, and suppose that there are in all 25 lengths
of the thread between A and K. Then the lengths AB, BC, CD— HK
represent 25 equal volumes, and each of them must contain 4°. To
graduate to 100°, we must divide each of the lengths AB, BC, &c., into
four equal parts. At each of the points B, C, &c., there is, therefore, a
sudden though small change in the length of the degrees. Hence the
method is only applicable when the variation in the bore is practically
negligible through one thread-length. For still greater accuracy, the
tube is previously graduated to equal lengths, and the correction to be
applied at each point is determined by graphic methods. The mode of
employing the dividing engine for graduation is described in Stewart
and Gee's Practical Physics, p. 24.

* Report of Committee on Methods Employed in the Calibration of Mercurial Thermo-
meters, British Association, 1882.

t See Stewart and Gee, Practical Physics, vol. i. p. 16. An excellent instru-
ment of simple construction is described by Brown in Phil. Mag., vol. xiv. 1882, p. 67.

TEMPERATURE. 7

When the thermometer is not to serve as a standard, the labour of
calibration may be saved by comparing the readings with some standard
instrument, and observing the deviation from the true reading at
various points along the scale. Such comparisons are undertaken by
the National Physical Laboratory, and a certificate is issued with each
thermometer compared, stating its errors.

Precautions in Use. — In order to obtain consistent values for a given
temperature, certain precautions must be observed in using the mercury-
glass thermometer. If the instrument is first used for a low tempera-
ture, then exposed to a high one, and lastly brought back to the first low
temperature, it will give a lower indication than before. The effect is
entirely due to the glass, which does not on cooling at once contract to
its original volume. In the course of days or weeks, however, it does
return to that volume. The effect may be well observed by immersing
a thermometer in melting ice till it is at 0°, then putting it into steam
for twenty minutes or so and then returning it to the ice. The zero
point will be found to be depressed by an amount differing with the kind
of glass used, ranging from about Ol° C to perhaps 0*5° 0. With some
kinds of glass the depression is nearly proportional to the high tempera-
ture reached, but with others the relation is not so simple. If time cannot
be allowed to eliminate the effect, the thermometer, if of English make,
should have its zero point re-determined immediately before being used for
any temperature lower than the high one to which it has been subjected.
Abroad it is usual to mark the fixed point 100° before the 0° point in
graduation, and with such a thermometer the zero point should be re-
determined immediately after the reading of an intermediate temperature.

We have already mentioned the gradual contraction of the bulb and
consequent rise of zero point. Though after a few years this becomes
very small, it may still be sensible for delicate instruments, and it is
necessary, therefore, to find the zero point at intervals and subtract the
rise from the indication on the scale. Dr. Joule observed the rise of
zero point on two delicate thermometers at intervals during forty
years, and the results obtained for one of them (Scientific Papers,
vol. i. p. 358) may be represented by the curve in Fig. 5. It practically
coincides with the curve,

the height y being in arbitrary divisions of the stem, 13 divisions to
1° F., and t being the time in years from 1844.

The total rise in thirty-eight years was 1° F., and if the curve truly
represents the results, it appears to show that it had still, in 1882, about
Jjyth of a degree Fah. to rise, and that it will halve its distance from
the final value about every ten years.

In taking a temperature the whole instrument should if possible be
at that temperature. If, for instance, the bulb alone is in a hot liquid
while the stem emerges into the colder air, not only is the stem con-
ducting heat from the bulb and keeping it at a lower temperature than
the liquid, but the glass and the part of the mercury in the air have too
small volumes, and on both accounts the temperature indicated is too low.
There are formulae for correcting for the emergence of the stem, but they
are unsatisfactory, and where possible their vise should be avoided.

HEAT.

A thermometer should be used when possible in the position in which
its fixed points were marked. If, for instance, they were marked with
the stem vertical, then in the horizontal position the internal pressure
on the bulb due to the column of mercury in the stem is removed and
the bulb contracts slightly, indicating too high a temperature.

Corrections can, however, be determined by direct experiment and
can be applied to the observed reading. With the most sensitive instru-
ments it is also necessary to take into account the varying pressure on
the outside of the bulb due to change of atmospheric pressure or depth
of immersion in a liquid.

Limits of Accuracy. — With different thermometers made of the same
kind of glass and carefully graduated, the indications of a given tempera-
ture should agree to within about 0'01° C. Different kinds of glass have

1334
13

12-
II-

n-

9-

tf-

7-
6-

5-
4~
3-
i-
I-
O'F.P

Asymptote 13 54

April I8W

1844 47 SO S3 56 39 62 <35 P8 71 74 77 80 83 86

FlG. 5. — Joule's Observations on the Alteration of the Freezing Point in Thermo-

. • i±

meters, and the Comparison with the Curve «/=13'94:°-9-5lV^3

different expansions and, though their indications at 0° and 100° will
agree, intermediate indications may differ by, perhaps, more than 0'1°C.
Above 100° their indications may differ much more widely than this. In
recent years very careful attention has been paid to the qualities of
different glasses for thermometer purposes and to the methods of
correction to be employed to make the readings of different thermo-
meters give the same value of the same temperature.*

It is to be hoped that, as a result of these investigations, the quality
of glass used will be improved, and that scientific thermometer makers
will everywhere use the same glass, the same mode of marking the fixed
points, and the same mode of graduating, so that the indications of
different instruments may be immediately compared.

Range of the Mercury-Glass Thermometer. — There are limits

* In Dr. Chree's " Notes on Thermometers," in the Philosophic"/ .!/</</" -.me, xlv. 1898
p. 205, will be found a description and full discussion of the various corrections

TEMPERATURE. 9

to the use of the mercury thermometer in both directions. The freezing
point of mercury being about -39° C. it cannot be used for lower tempera-
tures, and hence, for meteorological purposes, it is usual in cold climates
to replace it by alcohol, since alcohol has a much lower freezing point.
Nor is it safe to use an ordinary mercury thermometer much above
350° 0., the pressure of mercury rising from about | atmosphere at
300° to 1 atmosphere at 356°, the normal boiling-point of mercury, and
then increasing still more rapidly to 2 atmospheres at 400° and 4
atmospheres at 450°. This great increase of internal pressure may
very seriously alter the capacity of the bulb. Mercury thermometers
made of specially hard glass and containing nitrogen above the mercury
are now, however, made with a range up to 500° 0., but they are hardly
suitable for exact work.

Scales of Temperature given by Expansion depend on the Sub-
stances used. — Beginners in the study of heat sometimes suppose that
mercury and glass are chosen in the construction of thermometers, because
their expansion is regular and equal for each successive degree. But this
regularity is simply due to our definition, that equal degrees shall be
such equal expansions. Within the short range from 0° 0. to 100° 0.
most substances which remain otherwise in the same physical condition
between these points expand nearly regularly with rise of temperature
as indicated by the mercury-glass thermometer. And so most substances,
with the same fixed points would give nearly the same scale. Thus, if the
expansion of a brass rod between 0° and 100° were used and divided into
100 equal steps, each step would have very nearly the same value as the
corresponding steps on the mercury-glass scale. But not exactly, for even
between 0° and 100° there are measurable deviations from expansion in
the same rates, and outside that range the deviations become more con-
siderable. As we have seen, even different kinds of glass expand
differently, so that two thermometers of different glasses agreeing at
0° and 100° will not agree exactly at all intermediate points. The
disagreement fortunately is very small within that range.

The Work Scale of Temperature. — There is one scale of tem-
perature, due to Lord Kelvin, which is quite independent of the particular
substance used to indicate it. We shall give a full account of this scale
in chapter xvii. Here we can only attempt a brief sketch of its nature
in order that the reader may know that such a scale exists.

The work scale depends on the amount of work obtained from a given
supply of heat to a heat engine.

We may roughly describe an ordinary steam engine as a heat engine
which takes in heat at the temperature of the boiler, and turns some of
this heat into work by the expansion of the steam in the cylinder.
Though the steam cools as it expands it does not turn all the heat
received into work but retains some of it till it passes into the cooler
or condenser where it returns to the liquid form. It can therefore at
the very most only convert into work the difference between the heat
taken into the boiler and the heat put out in the condenser.

We can imagine an ideal engine, in which any substance is used, like
steam in the ordinary engine, to do work by expansion. The substance
works between a source of heat, like the boiler in an ordinary engine,
and a cooler receiver like the condenser. The working substance takes

10 HEAT.

in heat from the source, converts some of it to work, and gives out the
balance to the receiver. When the engine is imagined to work under
certain ideal conditions first prescribed by Carnot (whence it is known
as a Carnot engine), the fraction of the heat received which is converted
into work depends solely on the temperatures of the source and receiver,
and for two given temperatures is the same whatever working substance
is used. Or, putting the statement in another way, the ratio of the heat
put in at the higher temperature to the heat put out at the lower
temperature depends solely on these temperatures. We may therefore
use the Carnot engine to give us a scale of temperature in the following
way. Let a quantity of heat Ql be put in at the higher temperature which
we will denote by Ol and let a quantity Q2 be put out at the lower
temperature 02. Then we fix the ratio of these temperatures by
putting

If we keep 0l and Ql constant, Q2 is less the lower #2. If all the
heat Qj is turned into work, none remains to be put out at 02. In this
case Q2 is zero and therefore 02 is zero. This implies that the new scale
dates from a point such that a Carnot engine working down to that
point will turn all the heat which it receives from the source into work.
We can imagine no greater degree of cold than that of such a receiver,
and its temperature is therefore termed the absolute zero.

It can be shown that if a Carnot engine works between the tempera-
tures of boiling water as source, and melting ice as receiver, then for
every 373 parts of heat put in at 100° C. it will turn out about 273 parts
at 0° C. and convert 100 parts into work. The ratio of these tempera-
tures on the work scale is therefore 373 : 273 very nearly. If we decide
to make the length of degree on the scale such that there are 100 of them
between melting ice and boiling water, then melting ice is at 273° A.
(where A denotes the work, or, as it is often termed, the absolute scale),
and boiling water is at 373° A. The absolute zero then is at — 273° C. or
273 absolute degrees below the temperature of melting ice. Though the
Carnot engine is ideal merely, and though we can only approximate to
it in practice, we shall see in chapter xvii. that we can tell how it would
work if realisable. The new scale is, therefore, a perfectly definite one,
and it is possible to determine the relation between the work expression
of a temperature and its expression on other scales.

Air and Hydrogen Scales. — By the experiments which we shall
describe in chapter iv. it has been found that different gases of sufficient
tenuity, and sufficiently above their condensing points, expand nearly
equally for equal rises of temperature when kept at the same pressure,
and that if their density is kept constant their pressure increases nearly
equally. Two gases have been chiefly used for thermometric purposes,
dry air, and hydrogen, and it is usual to employ the increase of pressure
at constant density to give a scale of temperature. Taking Yg^th of the
increase between 0° C. and 100° C. to indicate a degree the scale agrees
very nearly with the mercury glass scale within that range. A gas
thermometer is applicable through a far wider range than the mercury-
glass thermometer. Its scale has the further advantage of being nearly

TEMPERATURE. 11

coincident with the work scale. Formerly the air scale was the standard,
but now the hydrogen scale, as used at the Bureau International des
Poids et Mesures has taken its place. At the Bureau there is a hydrogen
thermometer in which the pressure at 0° 0. is 1 metre of mercury, and in
which the density is kept constant. The degrees of the scale of this
instrument are increments of pressure, each TJ^th of the increase of
pressure between 0° C. and 100° C. We shall return to the subject of
gas thermometry in chapter iv.

Platinum Resistance Thermometer. — The electrical resistance of

pure metals increases almost in direct proportion to the rise of tempera-
ture as indicated by the mercury-glass and gas scales. Siemens was the
first to employ the resistance of a platinum wire to indicate temperature,
and the method has been thoroughly investigated by Callendar ("On
the Practical Measurement of Temperature," Phil. Trans., 1887, A.,
p. 161 ; PTiil. Mag., 1899, xlviii., p. 519). He has shown that it gives
an instrument convenient in form, easy to use, and applicable through a
far wider range than any other. The platinum wire he used in his
original experiments was '017 cm. diameter, about a metre long, and
about 5 ohms resistance. This was wrapped as a spiral on a glass
tube, and the ends soldered into thicker platinum leads '073 cm. diameter,
the tube being inserted in the enclosure of which the temperature was to
be measured. The resistance of the leads could easily be allowed for,
and the resistance of the platinum spiral itself at any temperature could
be found. The platinum scale is defined as giving equal degrees by
equal increments of resistance, 100 of such degrees making the interval
from 0° C to 100° C. Let E, be the resistance at any temperature, R0
and R100 the resistances at 0° C. and 100° 0. If we denote a tempera-
ture on the platinum scale by pt, then

Callendar compared this scale with the air scale, and found that if t is
the temperature on the latter, then to a close approximation

where 8 is constant for a given wire and has nearly the same value, 1'57
for all specimens of pure platinum.

It will be seen that the difference between the two scales must by
definition vanish at 0° 0. and 100° C. At 50° 0. it is a maximum — t
being less than pt by about 0'4°. Above 100° t is the greater, at 200°
by about 3°, at 300° by about 9°, at 500° by about 31°.

One great advantage of the platinum thermometer lies in its easy
use for the determination of low temperatures.

Thermo-Electric Thermometer. — This thermometer makes use of
the fact that when a circuit consists of two different metals, A and B,
an electric current in general flows round the circuit when the two
junctions are at different temperatures. The driving E.M.F. depends
solely on the nature of the metals A and B and on the temperatures of
the two junctions. Further, if a galvanometer be included in the
circuit, with wire of another metal C, inserted, say, in the course of the
wire B, then so long as the temperatures of the junctions of C with B

12 HEAT.

are equal, the E.M.F. is the same as if the circuit consisted of A and B
only. The thermo-electric thermometer is made in many forms, and
with many pairs of metals, according to the purpose for which it is used.
It will be sufficient here to describe one form, devised by Le Chatelier,
and used by Roberts-Austen for the determination of certain high
temperatures (Nature, xlv.. 1891-2, p. 534).

The active metais in this form are platinum, and an alloy of platinum
with 10 per cent, of rhodium. The junction to be inserted in the vessel
of which the high temperature is to be measured, consists of a platinum
wire round which the platinum-rhodium wire is twisted. The two wires
are brought out of the vessel and connected up to a D'Arsonval galvano-
meter, the junctions with the galvanometer being kept at the same lower
temperature, that of the room. The E.M.F. drives a current deflecting
the galvanometer, by an amount depending on the difference of tempera-
tures. The instrument is calibrated by inserting the testing junction
into vessels of known temperatures, containing in succession, say, boiling
water 100° C., melting lead 326° C., and boiling zinc 940°, the deflection
of the galvanometer being observed for each of these, and other tempera-
tures being determined by interpolation.

Some Important Temperatures. — The following table gives a few

important temperatures determined in various ways. They are put here
merely to enable the student to realise the range of measurement possible
with the instruments and methods now available : —

TABLE OF TEMPERATURES.

(Chiefly from " Travers' Study of Gases," and from Callendar,
Phil. Mag., xlviii., 1899, p. 519.)

Temperature on

Centigrade Scale.

Absolute zero, work scale . .

.s-HZ

Melting point of hydrogen . . —

256 to - 257

Boiling , „ . . -

252 to - 253

Boiling

, oxygen

. -183

Melting

, mercury

. -38-8

„

, ice ...

0

Boiling

, alcohol . .

78-3

H

, water . . .

. 100

„

, aniline . .

. 184-1

„

, mercury . .

. 356-7

Melting

, lead . . .

. 327-7

,,

, zinc ...

. 419-0

Boiling

, sulphur . .

. 444-5

Melting

, silver . . .

. 961

„

, gold .

. 1061

„

, platinum

. 1820

Crater of electric arc, of the order .

. 3500

Sun's radiating surface, of the order

. 6000

Bodies begin to emit visible rays, about

380

Red heat, about ....

. 500 to 1000

White heat above ....

. 1000

TEMPERATURE.

13

10

10

20

40

50-

60

70

80

90-

IOO

no

120-

130-

ISO

120

Maximum and Minimum Thermometers. — It is often important

for meteorological purposes to register the highest and lowest tempera-
tures which have been attained in any
period during the absence of the observer.
For maximum thermometers, a common
device is to have a short rod of iron in the
tube above the mercury ; the thermometer
is then placed with its stem horizontal. As
the mercury moves outwards along the
stem, the rod is pushed in front of it; but
when the mercury recedes, it leaves the rod
behind, thus indicating the farthest point
reached. The iron index may be brought
back into contact with the mercury by means
of a magnet.

For minimum thermometers, a small glass
rod is put in the tube, which is horizontal.
The liquid (in this case usually alcohol) flows
past the glass rod in rising, but in falling it
pulls the rod back with it owing to capillary
adhesion. The glass index may be brought
again to the end of the liquid column by
inverting the thermometer.

Six's TJiermometer. — This instrument,
which is a maximum and minimum thermo-
meter in one, is now very commonly used.
The construction is shown in Fig. 6. It
consists essentially of a U tube with a bulb
at each extremity of the U. The bulb G
contains alcohol or other suitable liquid,
extending along the tube to a; ab is a
thread of mercury extending round the bend
to b ; above b the tube and part of the bulb
H to which it leads are filled with the same
liquid as that in G, and above the liquid is
a space V containing the vapour of the liquid,
which can be compressed or extended, and
serves as a sort of spring.

In the tubes above a and b are two
small iron rods or indices with hairs
attached to them, the hair giving just
enough friction to keep the index in posi-
tion when the mercury column retreats and
leaves it. The indices may be brought into
contact with the mercury at a and b by means
of a small magnet applied outside the tubes.

Suppose that at a given temperature FKJ. g.

the indices U are in contact with the

mercury on each side. If now the temperature rises, the liquid in
G expands, pushes down the mercury from a, and the mercury thread
ab moves round, b rises and the vapour space in the bulb above it is

80

7O

GO

50

40

10

o

10

HEAT.

decreased. The index above a is left in its initial position, while the
index above b is pushed to the furthest position reached by the mercury.
If, on the other hand, the temperature falls, the liquid in G contracts, the
point a rises and pushes its index in front of it, while the index above b
is left in its initial position. Two graduated scales are fixed behind the
two limbs of the U, that on the left running from above downwards,
that on the right from below upwards. Evidently the former gives
minimum and the latter maximum temperatures. The instrument is

Nonconducting Ltd

^§§S^M^

s
Steam

Constant

Temperatur
Chamber

Water

Manometer
To Air Pump

FlG. 7. — Constant-Temperature Apparatus, using the Boiling -Point Method.

reset after an observation by moving the indices back to the ends of the
mercury thread by the magnet.

The liquid in G is here the chief expanding liquid. The mercury is
little more than a device for moving the indices.

Thermostats, or Constant-Temperature Instruments. — In a

great number of physical experiments, it is desirable to keep a body at a
known fixed temperature for a considerable time. There are certain
temperatures which are easily maintained, as, for instance, that of melt-
ing ice. A body placed in melting ice, or in a chamber surrounded by
melting ice, will remain indefinitely at 0° C, if proper precautions are
taken to keep the temperature of the ice and water uniform either by
stirring or by surrounding the vessel with a second vessel containing a
similar mixture. There are also definite " freezing mixtures " which
give fairly constant temperatures below 0°. Or, again, the steam from
water boiling in a metal vessel is very nearly 100° at ordinary altitudes,

TEMPERATURE.

15

A

8

~l

Benzolim

and by observation of the barometer any deviation from 100° can be
accurately determined. But, as we know, the boiling-point varies in a
definite manner with the variation of pressure,
so that it is possible, by regulating the pressure,
to keep the steam at temperatures other than
100°. This method is made use of in one class
of constant-temperature apparatus. Water, or
some other liquid suitably chosen, and contained
in a closed vessel, is supplied with so much heat
that it boils. The vapour passes into a cooling
arrangement, so that it is condensed back into
liquid as fast as it is formed. The pressure is
so regulated by varying the amount of air in
the vessel, that the boiling-point is the desired
fixed temperature. To adapt this apparatus to
secure a constant temperature, the vapour is made
to surround the chamber in which the constant-
temperature operations are being carried on.

The sketch in Fig. 7 will illustrate an applica-
tion of the method.

This will be seen to be merely an adaptation
of Regnault's apparatus for determining the
pressure of water vapour at high temperatures,
described in chapter x.

Ramsay and Young have investigated the
change in vapour pressure of a number of liquids
in the neighbourhood of their boiling points, choos-
ing liquids which are suitable for use in such an

apparatus as this, and tables embodying their results will be found in
the Journal of the Chemical Society, Sept. 1885, vol. xlvii. p. 640. The
liquids investigated were : —

Approximate
Liquid. Boiling-Point.

Carbon bisulphide . . . 46°

Ethyl alcohol .... 78°

Chlorobenzene . . . . 132°

Bromobenzene .... 155°

Aniline 184°

Methyl salicylate .... 222°

Bromonaphthalene . • . 280°

Mercury ..... 358°

In another class of constant-temperature apparatus, the constant
temperature enclosure is heated by gas, and the supply of gas to the
burner or burners is regulated by means of a " thermostat," so that if
the temperature tends to rise above that required, the gas supply is
checked, while if it tends to fall below it, the gas supply is increased.
There are many devices for effecting this. The following (Nicol, Phil.
Mag., 1883, xv. p. 340) will serve as an example. The thermostat
(Fig. 8) is placed with its bulb in the constant-temperature chamber.
The gas passes from A to B and thence to the burner, partly through

FIG. 8. — Thermostat.

16 HEAT

a small hole at a and partly up from the end C of the smaller tube. The
larger tube D is filled with mercury, which extends round to the lower
part of the bulb E, the upper part of which is filled with benzoline or
paraffin, or some more expansible, and, therefore, more sensitive liquid
than mercury. The tube A can be raised or lowered, and is so adjusted
that when the thermostat is at the desired temperature, the mercury
just reaches to the end C of the tube. If the temperature now rises, the
mercury seals up the end C, and the gas only passes through the small
sidehole a, and is just enough to keep the burner lighted. If the
temperature falls, the mercury allows a free passage to the gas through
the end C, and the supply of heat increases.

CHAPTER II.

EXPANSION OF SOLIDS WITH RISE OF TEMPERATURE.

Linear Expansion of Solids— Ramsden's Method — Modern Use of the Method —
Method of Lavoisier and Laplace — Results — Fizeau's Optical Method — Applica-
tions of Linear Expansion — Volume Expansion of Solids.

As a general rule bodies expand with rise of temperature. As a general
rule, also, gases expand more than liquids, and liquids more than solids.
As regards both liquids and gases, we have only to consider change of
volume, for fluid substances have no shape of their own ; but with solids
we have also to consider change of length as well as change of volume.
We shall first deal with the change of length of solids.

Linear Expansion of Solids. — Though the increase of length of
solids with an ordinary rise of temperature is small, it is still sufficiently
considerable in many cases to be of great practical importance. For
instance, in the construction of
railways it is necessary to leave
a small interval between the
rails to allow free play for
expansion of the iron. Iron
tubular bridges, again, have to
be fitted on rollers, so that on
expansion they may lengthen
freely. Iron water-pipes have
sometimes to be provided with
telescopic joints. In tubular
boilers, the fact that copper

expands more than iron is made use of to secure water-tight joints.
The copper tubes are fitted into the iron end-plates when cold, and on
expansion they fit still more tightly, and so prevent leakage. Pendulum
clocks, especially with metal pendulum-rods, go appreciably slower in
summer than in winter, owing to the lengthening of the pendulum,
and the rate may easily change to the extent of one minute per
week. These examples all show the importance of an exact know-
ledge of expansion, while the last prepares us for the difficulty of the
investigation by showing us how small is the quantity to be measured.
A change of rate of one minute in a week is a change of 1 in 10,080,
which may be shown to correspond to a change in length of the pendulum
of 1 in 5000.

The earliest attempt to measure linear expansions appears to have
been made by apparatus resembling in principle the well-known instru-
ment given in Fig. 9, the bar being placed against the short arm of the
lever L when cold, and again when hot, and the movement of the long

FlG. 9. — Expansion Apparatus.

18

HEAT.

arm on the scale being noted. But there are two serious objections to
this apparatus. Firstly, it is difficult to maintain the bar at any desired
temperature ; and secondly, during the experiment the measuring part
of the apparatus may change in dimensions, as well as the body to be
measured. For accuracy it is necessary that the measuring apparatus
shall remain absolutely at a constant temperature, while the body to
be measured shall be varied in temperature in a known manner. The
latter condition may be fulfilled by taking the length of the body first
in melting ice, then in boiling water. To fulfil the former condition
several methods have been adopted.

Ramsden's Method. — One of the best is that of Ramsden, devised
in 1785 to determine the expansion of the rods used by General Roy to
measure the base line on Hounslow Heath, on which was founded the

original Ordnance

Survey of the
United Kingdom.
The apparatus,
of which the gene-
ral arrangement
may be gathered
from Fig. 10 and
the accompanying
description, con-
sisted of three
troughs, each over
5 feet long, placed

FlO. 10. — Plan of Ramsden's Expansion Apparatus. — (1) Wood parallel on a table,
trough at 0° containing cast-iron bar bb, with cross wires Cast-iron bars were
in uprights uu' fixed near ends ; uu' shown also in eleva- fixed in the two
tion; (2) copper trough with lamps underneath to raise i v," v,

temperature, and containing bar BB to be tested, upright e a Wougns, wmcn
U pressed against left end by adjusting screws SS, upright were always filled
U' pressed against right end by spring sp. UU' carry with melting ice,
microscope object-glasses; (3) wood trough at 0° contain- so that the bars
ing cast-iron bar b'V with uprights w/ fixed near ends , :nvar:nuift

carrying microscope eyepieces, that on v' being provided *
with a micrometer m, length. Near the

ends of the bar, bby

were fixed uprights carrying cross wires. In the middle trough were
two sliders, moving only horizontally along the trough with uprights
UU' at their more distant ends, carrying the object-glasses of two
microscopes. The bar to be tested was placed on rollers in this trough,
and its ends were made to bear always against the uprights UU'.
The upright U was kept fixed so that any expansion pushed out the
upright U'. Near the ends of the bar b'b' were fixed uprights, carrying
the eyepieces of the microscopes, each with a vertical cross wire, the wire
at v being fixed, that at v' being movable by a micrometer screw m. The
middle trough could be heated by means of lamps.

The general nature of an experiment was as follows : All three
troughs being filled with melting ice, the uprights were so adjusted that
the cross wires on uu', seen in the microscopes, were in the centre of the
field, and coincided with the eyepiece wires. The middle trough was
then filled with hot water, which was further heated by the lamps to

EXPANSION OF SOLIDS WITH RISE OF TEMPERATURE. 19

boiling. One observer at the left-hand microscope took care to keep the
upright U (and therefore the left end of the bar) in its original position,
by means of the adjusting screws SS ; while another observer at the
right-hand microscope, measured the displacement of the image of the
right hand cross wires at u' across his field of view, by means of the
micrometer m. The displacement of the object-glass U' in terms of
the micrometer divisions was determined thus. In a preliminary ex-
periment, two vertical fibres fixed at u, one on each side of the diagonal
fibres and exactly -^ inch apart, were viewed by the microscope, and
the distance of their images apart measured by the micrometer. To this
was added the value in micrometer divisions of y^-inch motion of the
micrometer thread, and the total gave the value of ^-inch motion of
the object-glass. For, if LM,
Fig. 1 1 , be the two vertical wires
at u' and Im their images, when A<>'

o is moved to o't a distance
equal to ML, the image of L
moves to Z', where LZ' is parallel
to Mm, and therefore ml' - oo' =
LM. Hence, ll' = lm + ml' = lm
+ LM; or, the distance apart FlG>

of the images of LM + the dis-
tance apart of the objects, is the displacement of the image of them by a
motion of o through a distance LM.

Ramsden's results may be put thus : —

Standard brass scale, 1,000,000 parts at 0° expanded to 1,001,855 at 100°.
Brass rod 1,001,893

Brass trough
Steel rod
Iron rod
Glass rod

1,001,895
1,001,145
1,001,109
1,000,808

Ramsden also found that on dividing the interval from 0° to 100*
into three equal steps, the expansion for each step upwards in tempera-
ture was the same, within the limits of errors of observation. This
result has been shown to be not quite true when the measurements are
made with extreme accuracy ; but assuming it as sufficiently exact for
ordinary purposes, it follows that a rod expands by the same fraction of
its length at 0° for each rise of 1°. This fraction is termed the co-
efficient of expansion, and is usually denoted by k. Hence, if 10 lt are
the lengths of a bar at 0° and t",

The method of Ramsden has been since modified by attaching a
micrometer to the object-glass o', instead of to the eyepiece. The
expansion is then measured directly by the distance through which the
object-glass must be moved back after expansion to give coincidence
again.

Modern Form Of the Method. — A very similar arrangement to
that of Ramsden is adopted at the Bureau International des Poids et
Mesures at Paris, for the determination of the expansion of metre scales.
The general plan consists in keeping one scale at a constant temperature,

20

HEAT.

and therefore of invariable length, and in measuring the difference
between this length and the length of the scale whose expansion is
sought, the temperature of the second scale being varied. For the
comparison, two microscopes are placed vertically, as nearly as possible
one metre apart, passing through projections overhanging from the tops
of two massive pillars (Fig. 12), the bodies of the microscopes being very
firmly attached to the projections. Two parallel troughs, somewhat more
than a metre in length, are fixed to a table running to and fro on rails,
one of the scales being placed in each, so that either scale may be brought
with its end-marks under the microscopes. The troughs are double-
walled, and a stream of water, kept at a constant temperature by a
thermostat, circulates in the space between the walls in each. The inner
troughs are also filled with water, one being kept at a constant tempera-
ture, while the other is heated to successive higher temperatures. The
difference in the lengths of the two scales is measured at each tempera-
ture by the microscopes, which are provided with micrometer eyepieces
for the purpose. Special arrangements are adopted for the adjustment
of the scales in position, and stirrers in the form of turbines are used to

JL JL

J I

T T

trough containing bar

FIG. 12. — Diagram of Expansion Apparatus at the Bureau International

mix up the water thoroughly before each reading is taken, so that the
temperature throughout the trough is uniform. The thermometers used
are all studied carefully, so that the temperature is ascertained with
great accuracy. Full details of the method are given in Travaux et
Memoires du Bureau International des Poids et Mesures, vol. ii.

Another method of the same class was devised by Pouillet, to
measure expansions of bars at very high temperatures, as in a furnace.
The bar to be tested was placed horizontally in a chamber, the tempera-
ture of which could be regulated, and its two ends were viewed through
windows in the chamber by two horizontal telescopes of short focal
length. Arrangements were adopted so that all expansion took place at
one end and the rotation of the telescope to keep the end in the centre
of the field at the high temperature was observed.

Method of Lavoisier and Laplace. — Shortly before Ramsden
made his experiments, an apparatus was devised and used by Lavoisier
and Laplace, though their results were not published till many years
later. The arrangement will be understood by the aid of Fig. 13.

The bar BB, of which the expansion was sought, was supported
horizontally on glass rollers rr in a trough filled with liquid. One end
abutted against a vertical, fixed, glass rod FF, suspended from a cross-
piece T supported by two firm pillars, of which only the back one is

EXPANSION OF SOLIDS WITH RISE OF TEMPERATURE. 21

shown ; the other end pressed against a lever CL, which could rotate
round the axis C, supported in bearings on two other pillars, of which
again only the back one is shown. To the axis of the lever CL was
attached a telescope about 6 feet long, directed to a vertical staff
divided to twelfths of a French inch, and distant 600 feet from the
telescope. The temperature of the trough was raised by a furnace
underneath it, but from their size and distance from the furnace the
pillars were unaffected. The axis 00, therefore, remained a constant
distance from the fixed piece TFF. Hence, an expansion of the bar
moved the end L of the lever and rotated the telescope, the length CL
being such that an expansion of the bar through one line moved the
cross-wire in the eyepiece over 744 divisions of the image of the staff.
This must not be taken to imply that the accuracy was increased 744
times, for the passage of the cross-wire over one division of the scale

T i L_f

c r

f

D

. B

B

a

Irouqh

Bade Pillar

Back Pillar

FlG. 13. — Diagram of Expansion Apparatus of Lavoisier and Laplace.
Front pillars not shown. To a crosspiece T, supported horizontally
across the two left-hand pillars, was attached the fixed vertical rod
FF ; ft were two other crosspieces to which were attached vertical
rods carrying rollers rr. Another crosspiece served as the axis of the
lever CL and the telescope.

probably could not be estimated nearly so accurately as an increase in
length of one line in the bar when directly observed.*

Modifications of the method have since been made by Miiller and
others, in which the telescope is replaced by a mirror on the axis 0. The
reflection of a fixed scale is viewed in the mirror by a fixed telescope,
and the motion of the image of the scale across the field of view through
the rotation of the mirror gives the expansion of the bar. The bar
abuts against small rounded projections attached to the vertical pieces
at each end, so that it touches the lever CL at a fixed point. The adop-
tion of the mirror method has two great advantages : — (1) It reduces the
weight of the moving parts by the substitution of a light mirror for a
heavy telescope ; (2) it economises space, for the telescope and scale may
be placed at half the distance of the scale in the original experiment,

* The details of Lavoisier and Laplace's work were not very fully published, and
it is not known whether they took sufficient precaution to maintain the length CL
of the lever invariable or not.

22

HEAT.

and the same accuracy will be obtained. This may be seen at once from
Fig. 14.

If mm be the position of the mirror which reflects the central division
c of the scale S into the telescope, when the mirror turns to mm'
through an angle mOm the division reflected into the telescope will be a
where aOc = 2mOm'. Hence, what we may call the reflected line of sight,
Oa, passes over twice as many divisions of the scale as the normal to the
mirror On, and therefore over as many divisions as the line of sight of a
telescope at 0, directed along On to a, on a similar scale S' at the full
distance. It may be worth noting that on contrasting Lavoisier and
Laplace's telescope method with the mirror-telescope method with scale
and telescope close together, and using the same telescope and scale in

each case, there is no gain in
5' sensitiveness by the use of the
mirror. For though the scale is
half the distance from the mirror
in the second method, it is still
the original distance from the
telescope, and the size of the

j m ^^f*\n^~—~~ — ' — image is the same in both

\ ^--'^--- — r methods, and, as we have seen,

the same number of divisions
correspond to the same rotation.
If, however, the telescope can be
brought quite close to the mirror,
the scale being left at the half
distance, then the number of
divisions passed over is the same,
but the size of the image is
doubled, and the accuracy of
estimation is increased. We

should, therefore, by this last arrangement, both economise space and
increase the sensitiveness.

The following are a few of Lavoisier and Laplace's results : —

0

FlO. 14. — Mirror Method of Reading
Deflections.

The length at 0° is in each case 1.
Length at 100".

Untempered steel
Tempered steel
Silver . . .
Copper . .
Brass . . .
Iron (soft) . .
Iron wire drawn .
Glass with lead
Glass without lead

1-00107912
1-00123956
1-00190974
1-00171733
1-00187821
1-00122045
1-00123504
1-00087199
1-00087572

Coefficient of expansion between
0° and 100° obtained by divid-
ing in each case the total in-
crease in length by 100.

The increments in length here given contain five or six significant
figures, but the accuracy of the methods does not justify us in placing
confidence in more than the first two or three figures. The determination

EXPANSION OF SOLIDS WITH RISE OF TEMPERATURE. 23

of the temperature by the older observers hardly warrants us in going
further. But even with accurately observed temperatures the numbers
obtained to four places would not be the same for different specimens of
a substance, so much does the expansion depend on physical condition.
This is well illustrated by the difference between tempered and un-
tempered steel in this table, and is also shown by the difference
between soft iron and iron passed through a draw-plate. Comparing
the results of Lavoisier and Laplace with those of Ramsden, prieviously
given, for substances with the same name, the difference is still more
striking. In fact, we can hardly trust the generality of the results in
the tables beyond two figures ; and if greater accuracy is required, the
expansion must be determined for the particular specimen concerned.

Fizeau's Method. — A very remarkable and accurate method has
been devised by Eizeau, in which use is made of Newton's rings. It is
well known that, if a slightly convex lens is placed on a flat glass plate,
several bright rings are seen round the point of contact, these being due
to the interference of the light reflected upwards at the curved glass
surface with that which, passing down through the film of air, is
reflected back at the lower plane glass surface. If, instead of white
light, we use the nearly homogeneous light from a sodium flame, the
number of rings is very greatly increased, and they extend, in general,
to the edge of the lens.

lei

o

i

,;,~jl

w

' 1

3

(

1

!

B

Ik

!

ffll

C

\

B/

i!

7

it

1

f— i-

'1 C

J

j

x. i!

5

i

5

x —

— NU-, V

^

•

i('iG. 15. — Diagram of Fizeau's Expansion Apparatus. — B, block of which expansion
is to be measured placed on table T ; ss, levelling screws, also supporting lens,
L just above B ; /, film of air-forming interference bands ; P, right-angled
prism throwing light down and reflecting it out again ; Na, sodium flame behind
the plane of the figure ; m, mirror in the plane of the figure reflecting the sodium
light towards P ; tel, telescope directed towards P and viewing the interference
bands ; F represents the appearance of the bands in the telescope, which is
provided with cross- wires. The expansion table is protected by a case which
can be heated. This case is not represented.

Now if the path of the rays of light is normal to the lower glass
plate, each concentric bright ring from the centre outwards cor-
responds to a thickness of the film of air - greater than the preceding

24 HEAT.

one, where A is the wave-length of the light employed ; so that even
if we do not know the absolute thickness at any point, we know the
difference in thickness of the film of air at any two points by counting
the number of rings between them. Further, it is not necessary to have
contact between the curved and plane surfaces. If the lens is gradually
raised upwards, the rings contract, disappearing as they reach the centre;
but the distances apart of the successive rings remain the same at the
same points, corresponding to the same differences in the thickness of
the film of air. In Fizeau's experiment the rings were still visible when
the air-film was over a centimetre thick. His method, as used at the
Bureau International des Poids et Mesures, is as follows * : — A flat
metal plate T (Fig. 15) is supported horizontally by three screws passing
upwards through three holes near its edge. On the three screws is
supported a plano-convex lens LL with the plane surface downwards.
The so-called plane surface is in reality slightly convex, as is the case
with most plano-convex lenses, and if allowed to touch another truly plane
surface, Newton's rings are seen round the point of contact. A plate B
of the substance of which the expansion is required is prepared with
parallel polished faces, and about 1'5 cm. thick, and this is laid on the
centre of the metal plate T. The lens is then adjusted, at some small
distance above it, so that when sodium light is thrown on it normally,
Newton's rings are seen through interference between the rays reflected,
at the lower surface of the lens and those passing through and reflected
at the upper surface of the plate of the substance. The light of a sodium
flame is thrown in and reflected out again by a right-angled prism, and
then received by a telescope. The metal plate with the substance and
lens is enclosed in a chamber maintained at a uniform temperature by a
thermostat. When the temperature is raised, the thickness of air is
altered by the difference between the expansion of the supporting screws
and that of the substance, and the rings are shifted. By counting the
number of rings passing a given point in the field of view of the tele-
scope, this difference is measured in terms of the wave-length of the
light used. Preliminary experiments are made to determine the expan-
sion of the screws, and so the expansion of the substance is known.
Since the wave-length of sodium light is about '000589 mm., the
method, as might be anticipated, is susceptible of very great accuracy,
and by it Fizeau was able to determine the difference of expansion of
crystals along their different axes with great exactness. He also
succeeded in showing the variation of the co-efficient of expansion with
the temperature, taking as the co-efficient the increase per 1° rise in
temperature of a length which is equal to 1 at 0° C.

Applications of Linear Expansions. — In many cases in which

metals are used in construction, account must be taken of variation in
their length with variation of temperature. Railway lines must be laid
with a small interval between the successive rails, otherwise in hot
weather the rails would tend to force each other out of the straight so
as to allow the necessary expansion. This is perhaps most easily
realised by calculating the total increase on some long line. For
instance, the distance from London to Edinburgh is about 400 miles. If
the rails are laid in cold weather, we may easily conceive the possibility

* Vol. i. c., Travaux et Memoircs.

EXPANSION OF SOLIDS WITH RISE OF TEMPERATURE. 25

of a rise in temperature of 25° C. Now Ramsden's value of the co-
efficient of expansion of steel is -00001145. Then 400 miles will increase
with a rise of 25° by

400 x -00001 145 x 25 miles = '1145 mile,
or over 200 yards.

Then this distance, at least, must be left in intervals between London
and Edinburgh.

In iron bridges some arrangement must be adopted to allow for
expansion without serious change of shape in the structure. The Menai
tubular bridge, of which the total length is over 1500 feet, is mounted
on rollers, and the joints between the successive tubes are telescopic,
and have a play of several inches. In the Forth Bridge rocking columns
are interposed between the ends of the central girders and the ex-
tremities of the central Inchgarvie cantilever arms, and the shore arms
of the side cantilevers are left free to slide on their abutments.

In the long lengths of iron rod used to work railway points at
a distance from the signal box, expansion or contraction through
change of temperature might seriously interfere with the working. The
rod is therefore divided into successive lengths connected by short
cross-pieces. Thus in the simplest case, represented in Fig. 16, it may

At.

ID

C

FIG. 16. — Provision for Expansion in Rods Working Railway Points.

be divided into two halves AB, CD, with a connecting piece BOO turning
about O, which is fixed. If A and D are fixed, expansion is provided for
by BC turning round as indicated by the dotted lines. But if A be
moved back by the pointsman, D will move the same distance forward.

An interesting application of the expansion of iron was first made
many years ago in order to draw together the two side-walls of a
gallery at the Conservatoire des Arts et Metiers at Paris, which were
bulging outwards through the weight of the roof. Long iron bars were
passed through the two walls, and circular plates were screwed outside
on to the two ends of each, till they came against the walls. The bars
were heated inside and expanded. The plates, having been thus pushed
out, were screwed farther on till they again touched the walls, and, on
cooling, the bars contracted and drew the walls together. By several
repetitions of the process, the alternate bars being heated in each
operation, the walls were brought to their proper position.

It is well known that a thick glass vessel is very liable to crack, if
hot water is poured into it. This is due to the sudden expansion of the
inner layers, the outer layers not at once receiving heat, since glass is a
poor conductor. The strain in the glass is relieved by the rupture.

The contraction of iron on cooling is made use of in putting hoops
on casks and tires on wheels ; the iron is put in position while hot,
and as it cools the contraction keeps it firmly in its place.

HEAT.

The contraction of iron in cooling must be allowed for in iron castings,
the pattern always having to be made slightly larger than the casting
required.

In measuring base lines for trigonometrical surveys " compensation

bars," first designed by
General Colby for the
Indian Survey, are now
used. To understand
their construction, sup-
pose that we have two
parallel bars, AB of brass
and CD iron, held at
their middle points PQ.
Let these be of equal
length at some given tem-

C"-

D

A A

B B

FIG. 17. — Compensation Measuring- Bar. — EP, of
invariable length ; AB, brass; CD, iron.

perature. Let their tem-
perature rise and let their
expansions be as 3:2.

If then AB expands to A'B' and CD to C'D', AA' : CC' = BB' : DD' = 3:2.
If A'C' be produced to cut AC produced in E then EA : EC = 3:2 and E
is a fixed point. Similarly F is fixed and the distance EF is indepen-
dent of temperature variations so long as the two bars are all at one
temperature. It is usual to make AB of brass and CD of iron. ACE,
BDF are steel pieces jointed at A and C, B and D, and marks are made
at the extremities E and F.

In a pendulum clock, since the rate depends on the length of the
pendulum, the time of swing tends, as already pointed out, to become
longer in summer than in winter. An iron pendulum, for example,
increases by about '000012 of its length for a rise of
1°, altering the rate by half this proportion, and so -A ~

tending to make the clock lose about half a second
per day. With a variation of 20° or 30°, the change
of rate becomes very serious. This effect of tempera-
ture is eliminated as far as possible by employing a
" compensating " pendulum, made up of two metals,
arranged in such a way that the distance between
the centres of suspension and oscillation — the effec-
tive length — remains constant.

The form most commonly used* is the "grid-
iron" pendulum, invented by a clockmaker named ^~^ ^~^ B---*fi
Harrison about 1720. If we make an arrangement _ ig —Principle of
of bars of iron and brass as in Fig. 18, Hi being Gridiron Pendulum
iron, bb brass, and suppose the expansions to be in of
the ratio 2:3, then the distance between A and B,
measured parallel to the bars along a/3, is invari-
able. For suppose that each iron bar expands a length t, and that
each brass bar expands a length b, then 3i=26. If we first suppose
the iron alone to expand, it is evident from the arrangement that,

* A simple form, on exactly the same principle as that employed by Colby in his
compensation bar, was devised by Ellicott and described in the Philosophical
Transactions for 1761.

Iron and Brass
Bars-

EXPANSION OF SOLIDS WITH RISE OF TEMPERATURE. 27

if A is kept fixed, B will be lowered a distance, 3i. If after this
we allow the brass to expand, B will be now raised a distance 2b, and
since 2& = 3i, B resumes its original position.

The pendulum as actually made is merely a double
arrangement of this kind as in Fig. 19, where two
metals, of expansion in the ratio 2:1, are taken for
simplicity of construction.

A mercurial compensating pendulum is frequently
employed. In this the bob consists of a glass cylindrical
vessel containing mercury. As the temperature rises,
the vessel is lowered through the increase in length
of the pendulum rod. But at the same time the
mercury expands more than the glass, and its centre of
gravity rises in the vessel. The depth of the mercury
is so adjusted that the effective length of the pendulum
is invariable.

M. Guillaume has discovered a nickel-steel, which
he terms " Invar," which has a negligible co-efficient
of expansion at ordinary temperatures. This is very
suitable for pendulum rods, and will probably super-
sede compensating pendulums. It has already super-
seded compensating measuring rods for survey purposes.

In compensating chronometers, the difference of
expansion is made use of in another way. The rate of
the chronometer depends on the resistance of the hair-
spring to coiling, and on the disposition of the mass
of the balance-wheel. As the temperature rises, the
resistance of the hair-spring decreases, and if the
wheel were of one metal it would expand and throw
its mass farther from the centre, On both accounts

IG. 19.— Gridiron
Pendulum of 2
metals, with ex-
pansion : : 1 : 2.

the time of vibration would be longer and the chronometer would lose.
But the rim of the balance is made of two metals, say brass and steel, of
unequal expansion, that with the less expansion being inside, and the

wheel is discontinuous as shown in Fig. 20.
The outer rim expands more than the inner,
and the two loaded ends therefore curl in-
wards with rise of temperature. The loads
are so adjusted that the mass is thrown
inwards by a sufficient amount to compensate
for the weakening of the hair-spring and the
expansion of the spokes. It may be noted
that the effect of the weakening of the spring
is far greater than that of the expansion.

Metallic thermometers are made in which
FIG. 20.-Compensating Bal- the Une1ual expansion of the two parts of a
ance for Chronometers, compound curved bar, formed by two strips of
Outer rim brass, inner rim different metals brazed together, results in a
steel. change of curvature. But the strains to

which the metals are subjected on being

raised to a higher temperature are usually so great in these instru-
ments that the metals on being brought back to the original lower

28 HEAT.

temperature do not recover at once from them, and do not give con-
sistent indications.

There is one case in which the nearly equal expansion of two dif-
ferent substances is of great importance in the construction of scientific
apparatus — that of glass and platinum. Referring to the table of
Lavoisier and Laplace, it will be seen that they found as the co-efficient
for glass '0000087, while Borda obtained for platinum a value nearly
•0000086. If then a platinum wire be inserted in glass, and the glass be
fused round it, in cooling the two contract nearly equally. There is
therefore little strain, and the glass does not break away from the
platinum, as it would from other metals. Platinum wires can thus be
fused through glass, and through these wires electric currents can be
led into closed glass vessels, such as vacuum tubes and eudiometers.

Volume Expansion Of Solids. — We do not very often require
to know the volume expansion, or, as it is frequently termed, the
" cubical dilatation " of solids, except in so far as it is necessary in the
measurement of the expansion of liquids and gases, when we may require
to know the volume expansion of the containing vessel. We shall
describe how this expansion is determined in connection with the expan-
sion of liquids, and shall here only state that, if the solid expands
equally in all directions, its volume expansion may be found from its
linear as follows : If a solid of cubical form has a co-efficient of linear
expansion k, the length I of the edge at 0° becomes I (1 +M) at t°. The
volume, therefore, increases from Z3 to Z3(l + Jct)& = Z3(l + 3kt + 3/^2 + ksts).
Now Jet is, for moderate temperatures, so small that we may with more
or less exactness neglect Jc2t2 and &3^3, and the volume is very nearly
P(l + 3kt). The co-efficient of volume expansion — the increase of the
volume which is 1 at 0° for each degree rise of temperature — is, there-
fore, 3/c or three times the linear coefficient of expansion.

CHAPTER III.
EXPANSION OF LIQUIDS,

Volume Expansion of Liquids — U-Tube Method applied to Mercury — Dulong and
Petit — Kegnault — Expansion of other Liquids by Specific Gravity Bottle — By
Dilatometer — Matthiessen's Hydrostatic Method — The Expansion of Water —
Hope's Apparatus — Apparatus of Joule and Playfair — Results.

Volume Expansion Of Liquid. — Since liquids have no definite
shape of their own, their dimensions depend on the containing vossel.
We are, therefore, only concerned with their volume expansion as the
temperature rises.

The most obvious mode of observing the expansion of a liquid is to
enclose it in a vessel such as a thermometer bulb, and to note the chango
in level of the liquid in the stem as the temperature rises. If we know
the expansion of the bulb and stem we can deduce the expansion of the
liquid. For suppose, at a given temperature, the liquid reaches to a
certain mark on the stem. On raising the temperature the internal
capacity of bulb and stem increases, but in general the volume of the
liquid increases still more, so that it not only occupies the increased
space in the vessel, but also rises in the stem. Its total increase of
volume is therefore : — increase of volume of bulb and stem to first level
+ additional volume of stem between first and second levels, and when
we know these we can determine the increase in volume of the liquid.

The linear expansion of the bulb and stem may be measured directly,
and the volume expansion is approximately three times as great. Chappuis
(Travaux et Mdmoires du Bureau International, 1907) has used this method
with a weight thermometer to determine the expansion of mercury. But
it is not satisfactory, as we cannot assume that the expansion of the bulb is
the same in all directions. Practically there is only one method of accu-
rately gauging the internal volume of a vessel, and this consists in finding
the weight of a liquid of known specific gravity filling it. Thus, if o> be the
weight filling it at one temperature and p is the specific gravity of the gaug-
ing liquid used, w/p is the volume. If at a higher temperature the weight is
<i>' and the specific gravity of the gauging liquid is />', the volume is u'/p'.
So that <j)/p has expanded to w'//o'. The problem of determining liquid
expansion in general, therefore, resolves itself into that of determining
the specific gravity of some one gauging liquid through the range of
temperature over which the expansion is required. This specific gravity
must be determined independently of the containing vessel.

U-Tube Method. — The best method yet devised for this purpose is
a special application of the U-tube hydrometer. As a simple illustration,
let us suppose that two glass tubes are fixed vertically with their upper
ends open and their lower ends joined by narrow horizontal tubing. A
short length of indiarubber tubing, having a pinch-cock upon it (Fig. 21),
is inserted in the horizontal tube.

HEAT.

The pinch-cock being closed, hot water is poured into one tube, and
cold water, to about the same level, into the other. On now opening the
pinch-cock, it is found that the level of the hot water is quite appreciably
higher than the level of the cold, since it takes a longer column of
lighter hot water to balance a given column of heavier cold water. In
fact, if h, ti be the heights, p, p the densities, we have, on equating the
hydrostatic pressures at the bottom,

lip = h'p',
But if Y, V be the volumes of equal masses of densities /o, p't

Hence

V V

Vh'

Hot

Cold

so that we can express the expanded volume in terms of the original volume
Though water serves very well for an illustration of the method,
mercury has been chosen as the liquid on which
to make exact experiments for the following
reasons : — it is easily obtained pure ; it expands
nearly regularly as the temperature increases, when
this is indicated either by the mercury-glass scale
or the air scale ; it has a high boiling-point (about
356° 0.) ; it does not evaporate much until the
temperature is approaching 200° ; it does not wet
the vessel in which it is placed, if this be glass,
so that its level may be read with great accuracy ;
and its great specific gravity renders it especially
suitable for gauging vessels, when its expansion
has been found.

The first experiments of this kind with mercury
were made by Dulong and Petit, who employed

FiG.21. U- u $ y ro- arrangement exactly similar in principle to that

meter, with hot and & . '

cold water. represented in iig. 21. Ihe two tubes were

enclosed, the hot one in an oil-bath, and the cold

one in a vessel filled with melting ice, the upper ends rising above the
enclosing vessels, and the upper levels of the mercury being just visible
when the height was measured. The connecting cross-tube at the bottom
was of very fine bore, so as to prevent circulation from one tube to the
other when the temperatures were steady. An air- thermometer was
used to give the temperature of the hot vessel.

If H and H0 are the two heights, hot and cold, and p and p0 are the
respective densities,

But if a volume Y0 at 0° expands to V0 (1 + At) at <°, since the mass is
the same at each temperature,

or
Hence

H

EXPANSION OF LIQUIDS.

81

and A,, or the expansion of unit volume at 0° for a rise of t° can be
found from H and H0.

As there were several details in this method open to criticism,
Regnault, in making a redetermination, introduced some modifications
in the apparatus. He carried out a very extensive series of experiments,
which have till recently been the standard ones on this subject.

He arranged the apparatus in two forms.

The general principle of one arrangement may be understood from
Fig. 22. The two vertical tubes AB, A'B', are united by the hori-
zontal crosspiece AA' with a small hole at 0, so that at that level
the mercury is exposed to
the atmospheric pressure.
The lower crosspiece is
broken in the middle by
the insertion of the in-
verted U tube BCD',
connected with the re-
servoir M, of compressed
air, the pressure being
adjusted so that the
mercury rises to the
levels 00' in the two
arms of the U. The
temperature of A'B' is
kept constant throughout
by surrounding it with
water running from the
mains, while that of AB
may be raised to any
desiredpoint by surround-
ing it with a bath of oil
heated by a furnace. The
temperature of AB is
given by an air thermo-

FIG. 22. — Diagram of Regnault's First Apparatus
for finding the Expansion of Mercury.

meter. The heights AB,

A'B', CD, and C'D' are

all measured at each

temperature by a cathetometer, and from these the expansion may be

found as follows :

The pressure at A is equal to that at A', both being equal to the
atmospheric pressure through the communication with the atmosphere at
o. Also the pressure at 0 is equal to that at 0', both being equal to the
air-pressure in the reservoir M.

Then, pressure at 0 - pressure at A = pressure at C'

— pressure at A' (1)

Expressing these differences by the usual hydrostatic formula, we
obtain an equation giving the expansion of the mercury. For, let H, H'
be the heights of AB, A'B', let h, h' be the heights CD, C'D', and let
T, T' be the temperatures of AB, A'B' ; and let us, to simplify matters,
suppose the temperature of CD, C'D', to be T'. Let p, p be the densities

HEAT.

of mercury at T, T', AT, A^ the expansions of unit volume at 0°, ou
raising the temperature respectively to T and T'.

Then Hp - hp' = H>' - tip from ( 1 )

or Hp = (H.' + h-ti)p

But if p0 is the density at 0°, the volume of the same mass being
inversely as the density, we have

P 1 P' 1

Po 1+AT

Then

H

1 + AT

or

H

(2)

Now Ajv is in practice only small, and we may take an approximate

Hot

H'

fold

FIG. 23. — Regnault's Second Apparatus for the Expansion of Mercury.

value for it without seriously affecting the value of AT. Then (2) gives
us Aj in terms of known and measurable quantities.

Regnault also used a method very nearly the same as that of Dulong
and Petit, which will be understood from Fig. 23, the U tube in the
lower crosspiece being replaced by a flexible iron tube, so that the two
halves could expand independently.

The great advantages over the arrangement of Dulong and Petit
consisted in the maintenance of known fixed temperatures in each
vertical part and in the greater accuracy of measurement of the vertical
heights, attained by bringing the two levels close together.

It will be easily seen that in the arrangement of Fig. 23
H h H' + h'

1+V

or

i+V

H

AT,).

EXPANSION OF LIQUIDS.

33

Regnault made, in all, about 130 observations in the range of
temperature from 25° 0. to 350° C., and from these he constructed a
table of expansions by a "graphic" method, i.e. by representing his
results on a volume-temperature diagram. The principle he adopted is
as follows : — Taking two lines at right angles, and marking along one
temperatures and along the other the volumes assumed by the mass
which has at 0° the volume 1000, each observation will be represented
by a point on the diagram. If the observations were perfectly free
from error, we might expect that all these points would lie on a regular
curve or a straight line, the true curve which we seek to determine.
But through imperfections in the measurements of length and tempera-
ture, the points will probably lie sometimes above, sometimes below the
true curve.

Let us suppose that the observations are indicated by the crosses in
Fig. 24. Then a curve is drawn so as to pass as nearly as possible midway
through the points, i.e.

so as to make the sum , J-/0/6

of the heights above
the curve equal to the
sum of the depths be-
low it. Of course, if
there is any systematic
error running through-
out the work and always
in the same direction,
this method will not
detect it.

If the expansion
per 1° were the same
throughout the range,
the curve would be a
straight line. Regnault
found, however, as
Dulong and Petit had

• • 1015

.JO/0

. -1005

100

1000

FIG. 24.— Graphic Method of Determining Results.

_ already observed, that the line bends

slightly upwards, that is, that the co-efficient of expansion slightly
increases.

A constant co-efficient of expansion would be represented by

AT = aT.
The slight bending upward may be represented nearly by taking

and Regnault, from his diagram, found that

a =-00017905
£ = •0000000252.

Regnault's results have since been studied by others who have sought to
obtain formula more nearly representing his observations, but the differ-

34 HEAT.

ences they have introduced are very slight. If we do not require very
great accuracy, we may take

a = -0001800
/3= -00000002.

Callendar * has modified the second method used by Regnault by
employing six pairs of hot and cold columns placed in series as repre-
sented diagrammatically in Fig. 25A, the successive columns being alter-
nately cold and hot, as marked by C and H. If the mercury when in
equilibrium stands at a in the gauge-tube connected tfl the first cold
column, and at z in the gauge-tube connected to the last hot column, the
difference of level to be measured, represented by a', z, will be six times
that due to a single pair of hot and cold columns. As the columns were

f

c d gl h.

I o

FIG. 25A.

FIG. 25B.

nearly 2 m. long in place of 1*5 m., the length of Regnault's columns,
the expansion was nearly eight times as great. In the actual apparatus
the cross tube ef was doubled back, so that fg lay behind be, and ih
behind ed, and so on, as shown in plan in Fig. 25B. All the cold columns
were placed in one tube, and all the hot columns in another, the tubes
containing oil rapidly stirred. One was surrounded by ice and the
other was electrically heated. For further details we refer the reader
to the original paper.

Callendar found that if Oat is the mean co-efficient of expansion
between 0° C. and t° C., then

1010 x Oat = 1805553 + 12444^/100 + 2539^/10000.
For approximate work we may put

108x Oa,= 18006 + 2*.

* Phil. Trans., A. 211, p. 1, 1911.

EXPANSION OF LIQUIDS. 35

We may use these values to give us the specific gravity at any
temperature. For if p0 is the specific gravity at 0° and if pt is that at t"
and A is the expansion of unit volume at 0° when raised to t°, we have

Po = Pt(l+ A).

The specific gravity of mercury at 0° is, according to Regnault,
13 '596. Then the specific gravity at 100°, for example, is

Po _ 13-596 _ . „ .._„
/>ioo-1+A-pol82-

Determination of Liquid Expansion, using the known Ex-
pansion of Mercury. — We may use our knowledge of the specific
gravity of mei-cury to gauge a vessel at different temperatures and then
fill it with a liquid of which we require the expansion. Two methods
are employed, one which we may illustrate by the use of a specific
gravity bottle, the other in which a dilatometer, virtually a thermo-
meter bulb, is used.

Specific Gravity Bottle Method. — In its original form the specific
gravity bottle is a flask of thin glass with carefully ground stopper
perforated by a very fine tube. When the bottle is filled with liquid
and the stopper is inserted, the excess of liquid is forced out through this
tube, and it is assumed that the stopper takes a definite position. There
is a newer form in which the bottle is a U-tube with two fine tubes
turning out horizontally from the ends of the U and provided with
stoppers. A mark is made in each fine tube, and the liquid occupies the
U and the fine tubes to the marks. Any excess of liquid may be easily
removed.

The first aim is to determine the expansion of the bottle. For
simplicity, we will suppose that we are going to heat it from 0° 0. to
100° C. We first fill the bottle, when surrounded with ice at 0°, with
mercury, and find the weight W0 of this mercury. Then we fill it when
surrounded with steam at 100° and we find the new weight W100 of the
mercury.

Let V0 be the internal volume of the bottle at 0° 0. and

Y100 „ „ 100° C.

Let pa be the density of mercury at 0° and

Pm " » 100°

Let G be the expansion of 1 cc. of the internal volume of the bottle
from 0° to 100°

7 „ „ mercury.

Then V10Q = V8(l + G) and Po = Ploo( 1 + y).
From the weighings

V0 = W0/Po : V100 = W100//>ioo-
Then Y0(l +G) = Wm/Pm.

w w

and w.?(l+G)=— ^

Po Pm

W n W

^f-^-^1^

Since we know y we obtain 1 + G.

36 HEAT.

We can now use the bottle to determine the expansion of any other
liquid between 0° and 100°. Let 1 cc. of the liquid ut 0° expand to 1 + A
at 100°, and let <r0, cr100 be its densities at 0° and 100°.

Then o-0 = o-100(l + A).

Filling the bottle first at 0° and then at 100° let a>0, w100 be the weights
of the liquid.

We have o-0 = <o0/V0, and trm = w100/V100.

Then 1 + A = — ° = -^ ^P.

°"ioo wioo vo

- -"(1+G).

Whence we know A.

Dilatometer Method. — The dilatometer is now usually made in the
form represented in Fig. 26. B is a bulb with a fine graduated stem, s,
rising from the bulb and open at the top. Below, the bulb is con-
nected with a fine-bore tube with a slight thickening at t and open at e ;
st, is a screw stopper which can be put over the end of this tube, being
sprung over the thickening. On screwing the stopper, a pad effectually
closes the opening. This form of apparatus is very readily cleaned and
filled.

First we must calibrate the bulb and stem, and measure its expansion.
Let W0 be the weight of mercury, density p0, filling the bulb at 0° from
e up to the zero of the scale. Then the volume V0 = W0//o0.

Let the weight of mercury filling an observed number of divisions of
the stem be found. From this the volume of each scale division in terms
of the volume of the bulb can be found. Let it be AV0. A, will be a
very small fraction.

Now start with the bulb filled with mercury at 0° up to the zero, and
raise the temperature to 100°. Suppose that the mercury rises N
divisions.

Let V0 expand to V100 = V0(l + G).

Then the total volume of the mercury is

V:00 + NAV100 - V0(l + G)(l + NX).

:00 100 0

But if y is the expansion of mercury we have

V0(l+y) = V0(l+G)(l + N

Whence

Now repeat this operation with the liquid, of which the expansion
is to be determined, and let it rise from the zero division through n
divisions when the temperature is raised from 0° to 100°. Let 1 cc.
expand to 1 + A cc. Then the total volume at 100° is V100(l +n\)

But this is also V0(l + A).

EXPANSION OF LIQUIDS.

37

, = (l + G)(l+rcA).
1+nX

Then

Whence A is determined.

Matthiessen's Hydrostatic Method. — Matthiessen* determined
the expansion of water by a hydrostatic method. For
this purpose the linear expansion of a glass rod was
measured, and its volume expansion was deduced. A
piece of the rod was then cut off and weighed in water
at different temperatures. The loss of weight gave the
weight of a volume of water equal to that of the glass,
and since the expansion of the glass was known that of
the water could be determined.

Matthiessen also applied the method to find the
expansion of mercury when that of water had been
determined, weighing a small bucket containing mercury
in water at different temperatures. He obtained results
very close to those of Regnault. Later he applied the
method to other metals.

The Expansion Of Water. — Researches on the
expansion of water have been made by many experi-
menters using one or other of the methods already
described, that with the dilatometer giving, probably,
the best results. By some the dilatometer has been
modified so as to have a constant internal capacity.
As usually employed, the rise of the water in the dilato-
meter shows only the so-called apparent expansion — the
excess of expansion of water over the containing vessel.
But the expansion of mercury is about seven times that
of glass, so that if about i of the bulb of the dilatometer
is filled with mercury, the internal capacity is constant,
and the rise of the water shows its
true expansion. Though interest-
ing, this modification probably does not give such
accurate determinations as the simple instrument
used in the ordinary way.

All the methods concur in showing that water
has a point of maximum density at about 4° 0.,
and that the volume is nearly equal for tempera-
tures equidistant on the two sides of this point.
But since the rate of change of volume is very
small near the maximum density, it is exceedingly
difficult to determine the exact position of the
point.

Hope's Apparatus. — The existence of a maximum
density may be shown by Hope's apparatus, which
consists of a cylindrical vessel of tin, surrounded
midway by a gallery. The vessel is filled with water, and two thermo-
meters are inserted, one near the top, the other near the bottom, as

* Phil. Trans., clvi., 1866, pp. 231-248, "On the Expansion of Water and
Mercury."

2(J _
meter.

FIG. 27.— Hope's

Apparatus.

38

HEAT.

indicated in Fig. 27. A freezing mixture is put into the gallery. At
first, the water is some few degrees above the point of maximum
density. The water in contact with the side near the gallery becomes
cooled, and, therefore, more dense, and sinks, while the warmer, lighter

water from below rises up
and takes its place, and
circulation of the liquid is
produced. This readjust-
ment of unequally heated
matter by its change of
density is an instance of
" convection." There is, at
first, no circulation in the
upper half of the vessel,
and the higher thermometer
only cooled by conduc-

is

Time

FlG. 28. — Indications of Thermometers in
Hope's Apparatus.

tion, which takes place but
slowly. Its fall is, therefore, very gradual. But the denser water falls
to the bottom in the lower half, and the lower thermometer falls rapidly.
The circulation goes on until all the lower part has become of the
maximum density. Then the circulation ceases, for further cooling only
makes the water lighter, and it does not descend. The cooling of the
lower layers can now only take place by con-
duction, and is very slow. When uniformity
is reached in the lower half, the further cool-
ing of the middle strata leads to circulation
upwards, and this will go on to a higher and
higher level until the upper thermometer is
reached, when its fall will be much more rapid.
The upper part may now fall below the point
of maximum density, for the lighter colder
water goes on rising upwards.

Conduction may now begin to affect the
lower thermometer, and it may fall somewhat
more rapidly. Fig. 28 represents a series of
observations on a time-temperature diagram
— the long, nearly horizontal portion of the
lower thermometer indicating a temperature
in the vicinity of that of the maximum
density.

Apparatus of Joule and Play fair. — Pro-
bably the most accurate mode of determining
the point of maximum density is that of
Joule and Playfair,* based on a principle FlG< 29.— Apparatus of Joule
similar to that applied in Hope's apparatus. and Playfair for Point of
Two tall iron cylinders A and B (Fig. 29) Maximum Density of Water.
are open above, and connected by a trough

T of about one inch cross-section. In this is placed a glass bulb only
just floating. Below is a cross pipe P, furnished with a stop-cock.

The cylinders are filled with water, one just above, the other just
* Joule's Scientific Papers, vol. ii. p. 173.

B

EXPANSION OF LIQUIDS.

below the temperature of maximum density ; the communication is cut off
at P, the contents are stirred, and the temperature of each cylinder read.
Communication is then made at P, and when the disturbance has subsided,

t'

Temperature

FlG. 30. — Density-Temperature Curve of Water.

the direction of flow is ascertained by the motion of the glass bulb along T.
It will evidently take place, below, from the heavier to the lighter,

10400

10300

Magnified
five times

10200

10100

10000

0 4 10 20 30 40 50 60 70 80 90 100
FIG. 31. — Curve of Expansion of Water vol. 10,000 at 4° 0.

and above from the lighter to the heavier column. If, for instance, the
warmer cylinder has temperature t, the colder temperature t' and the
flow through the trough is from colder to warmer, the colder is less

40 HEAT.

dense. Hence, assuming that the density curve (Fig. 30) is symmetrical
about the point of maximum density, t' is further below it than t is

above it, and — — is below the maximum density. If the flow is in

Z

t + t'

the opposite direction, — — is above the point of maximum density. In
2

this manner Joule and Playfair obtained a number of values of -

a

respectively above and below the point, and were able to fix the tem-
perature at 3-95° C. with very slight error. The alteration of density
from this to 4° is so slight as to be for practical purposes negligible, and
we may take 4° as the point of maximum density. For tables of the
density and volume of 1 gramme of water we refer the reader to
Landolt-Bornstein, Physikalisch-Chemische Tdbellen, 1905, p. 37. These
contain results beginning at — 10°, as it is not difficult to keep water
liquid even at that temperature in a dilatometer.

Results. — The following table gives the expansion of a mass having
volume 10,000 at 4° 0. for every 10 degrees up to 100°, and the results
are represented in Fig. 31 : —

Temperatures. Expansion of Mass

Degrees. having vol. 10,000

lit ~t .

- 10 18-6

0 1-32

4 0-0

10 2-73

20 17-73

30 43-46

40 78-2

50 120-7

60 170-5

70 227-0

80 289-9

90 359-0

100 434-3

It will be seen that the curve is nearly a parabola, the expansions being,
however, rather greater than in the ratios of the square of the excess of
temperatures above 4°.

The expansions of a great number of other liquids have been de-
termined. The results may be expressed by the formula

V, = V0(l + at + W + ct3 + &c.).

Usually at is the most important term, but in the case of water, as we
have seen, bt2 is very important. The terms after ct3 may probably be
neglected.

CHAPTER IV.

EXPANSION OF GASES.

Expansion of Gases depends on Pressure Changes — Volume Expansion at Constant
Pressure — Gay-Lussac's Method — Regnault's Experiments — Increase of Pressure
with Constant Volume — Gas Thermometry — Regnault's Normal Air Thermometer
— Hydrogen Thermometer — Bottomley's Air Thermometer — Callendar's Com-
pensated Air Thermometer.

The Expansion of Gases. — Since the volume of gas is very much
affected by alteration of the pressure to which it is subjected, the volume
expansion with a given rise of temperature may vary widely in different
cases through different changes in pressure, and may even be prevented
altogether, if the containing vessel is one which preserves a constant
capacity. If, for example, a flask containing a small quantity n

of liquid is closed by a cork through which passes an open
tube, dipping under the liquid (Fig. 32), a rise of tempera-
ture is accompanied by an expansion of air, which drives the
liquid up the tube. But the height to which the liquid rises,
and, therefore, the expansion of the air, will differ with
different liquids. With mercury it will be much smaller
than with water, for the gas inside the flask is exposed to
the atmospheric pressure + that due to the column of liquid
in the tube. The greater increase of pressure, due to the
column of heavier liquid, lessens the expansion of the air.

We see from this, that the rise of temperature may have
two effects : — increase of volume and increase of pressure ;
and that the effects may differ for different arrangements.
It is usual to study the two effects separately, investiga-
ting the change of volume when the pressure is kept constant, and the
change of pressure when the volume is kept constant.

It may be observed that if Boyle's Law were exactly true, the two
effects for a given rise of temperatnre would be equal. For, let a
volume V of a gas expand, by a given temperature-rise, to V, the
pressure remaining at its original value, P. Now bring back the volume
to V, increasing the pressure to P'. Then the pressure P would increase
to P', with the given temperature-rise, at constant volume V. But by
Boyle's Law

FIG. 32.

and

V

v:

F
P

Or the volume changes in the same ratio in the one case as the pressure
in the other.

42 HEAT.

But we know that Boyle's Law is not, in fact, quite true, so that it
is necessary to investigate the two cases separately.

The Volume Expansion at Constant Pressure— Gay-Lussac's
Method. — The earliest investigations having any approach to accuracy
were made by Dalton and by Gay-Lussac. One arrangement which Gay-
Lussac adopted for the purpose is shown in Fig. 33.

Dry air or other gas was admitted to a bulb A through a drying-
tube, and a small mercury index i was introduced in the tube to indicate
the volume, and to cut off communication with the external air. The
bulb was then placed, with the stem horizontal, in a vessel of water, as
in the figure, the temperature being roughly indicated by the ther-
mometer t', and, more accurately, by the thermometer t, which could be
drawn out till the column of mercury was just visible. The air tube
could also be drawn out till the index i was visible. The vessel of water
was heated from below. The position of the index was observed at various
temperatures, and the bulb and tube having been previously gauged, the
air expansion could be found. Gay-Lussac arrived at a result which may

Drying

Tube

FIG. 33. — Gay-Lussac's Apparatus for Determination of Expansion of Gases
at Constant Pressure.

be stated thus : whatever the gas, 267 volumes at 0° increase by 1
volume for each rise of 1° temperature, becoming 367 volumes at 100°.

Dalton had, a short time previously, found that different gases
expand equally ; he gave the expansion as 1 in 483 volumes at 32° F.
for each rise of 1° F. This corresponds to 1 in 269 volumes on the
Centigrade scale.

Later investigations have shown that the general conclusions of Dal-
ton and Gay-Lussac are nearly, but only nearly, true, their numerical
results giving too large an expansion, while the expansion is also found
to vary slightly for different gases. According to Regnault, to whom we
owe a very extensive series of researches, 273'1 volumes of hydrogen
at 0° expand 100 volumes for a rise of 100°, while the same expansion
is given by 272-4 volumes of air and 269 '6 volumes of carbon dioxide.
Other experimenters have found nearly the same values. It will be
sufficient if we here describe

Regnaulfs Experiments. — Some of Regnault's researches were carried
out by a method first used by Rudberg. The two parts of the
apparatus are represented in Figs. 34 and 35. The bulb A was
placed in a boiler, where it was surrounded with steam and then alter-
nately exhausted by the pump, and filled with dried air many times, till
all traces of moisture were removed. The taps tt were then turned, so

EXPANSION OF GASES.

43

that A was left in communication with the external air, and it was kept
at the boiling-point for some time. The junction j was then disconnected
and the end of the tube was sealed by a blowpipe flame, the bulb thus being

J

J!

FIG. 34. — Regnault's Apparatus for Expansion of Gas at Constant Pressure
(Rudberg's method). Bulb filled with gas at 100° C. and sealed.

filled with dry air at the atmospheric pressure, and at the temperature

of the boiling-point. It was then arranged as in Fig. 35, being inverted

with the end of the tube under mercury, the bulb being surrounded with

melting ice. The end of the tube was broken off, and the mercury rose

into the bulb through the contraction of the

air. After some time the end of the tube was

closed by a small piece of wax, which could be

pressed up against it by the arm w. The

height of the mercury in the bulb above that

in the lower vessel was then determined, and

from this could be found the pressure to which

the air was subjected. The bulb was taken

away and weighed, first when thus partially

filled with the mercury which had risen into it,

and afterwards when quite filled, and so it was

easy to determine what fraction of the volume

the air occupied at 0°. By correcting for the

reduced pressure, the fraction of the volume

which it would have occupied at the atmospheric

pressure was calculated. The air occupying this FIG. 35. — Regnault's Expan-

volume at 0° expanded at 100° to fill the whole s™n of Gas at Constant

, n /•. ic C i -i. i-u \, it- Pressure (Rudberg s

bulb (itself of larger capacity through the ex- Method). Bulb opened

pansion of the glass), and so the expansion under mercury,
could be calculated.

Regnault also devised another method represented in Fig. 36. The
bulb A communicated by a capillary tube with the manometer MM',
placed in a constant-temperature water-bath with glass sides. R was a

44

HEAT.

three-way tap, by which M could either be brought into communication
with M' or with the pipe P opening downwards. E,' was a simple tap
allowing M' to be brought into communication with the pipe P' opening
downwards ; p was a branch tube, put into communication with a
drying apparatus and pump before the commencement of measurements,
so that A could be filled with air thoroughly dried. A was then
surrounded by melting ice, mercury was poured into the manometer till
it rose to a in both limbs, p was sealed up, and the barometric height was
read, and the temperature of the water-bath observed.

A was next exposed to the steam from the boiler, some of the air
being pushed in consequence into M, driving the mercury down that
tube and up M'. The tap E.' was turned, to allow mercury to run out of
the manometer, until the levels were again the same in M and M', say
at p. The barometric height and the temperature of the water-bath
were again observed. For simplicity, we may suppose them the same

as before. The air in the bulb
and the part of the stem exposed
to the steam has, in rising from
0° to the boiling-point, filled the
increase of volume in these, and
also the volume of the manometer
a tube between a and (3. The various
volumes being gauged, it is easy
to find the expansion of the air.
V' For if Y is the volume of the bulb
and the part of the stem exposed
to change of temperature, K its mean
co-efficient of expansion, v the volume

Furnace

of M between a and (3, T the boiling-
point, t the temperature of the
bath, a the co-efficient of expansion
of air, assumed to be constant — in
other words, we use the gas scale of
temperature to be described here-

after — the volume V of air at 0° has expanded to fill V(l + *T) at T, and
v at t, at the same pressure. But if the air contained in v were also at

FIG. 36. — Regnault's Expansion of Gas
at Constant Pressure.

— -
1 + at

and the total volume at T would be

T, its volume would be v

But the increased volume of air, all at T, may also be expressed by

V(l + aT).
Equating these, we have

which determines a.

In practice, the variations of barometer and of temperature of water-
bath were allowed for, and the equation was slightly more complicated.

Increase of Pressure with Constant Volume. — Nearly the same
form of apparatus was used by Regnault to determine the co-efficient of

EXPANSION OF GASES. 45

pressure-increase when the volume is constant. The manometer was,
however, somewhat modified in its details, and the water-bath was
removed. As before, the bulb was filled with dry air through the
tube p, and it was first exposed to the boiling temperature. The
mercury being brought to the same level a in both tubes, p was
sealed. The temperature was now allowed to fall, and ultimately A was
surrounded by melting ice. The air tended to contract and draw the
mercury above A, but the pressure was diminished by allowing mercury
to run out at P', so that the level in M was maintained at a, while it fell
to y in M'. The barometric height was read for each of the two
temperatures. We will suppose it the same throughout. The tempera-
ture of the manometer is that of the surrounding air t, not being dis-
turbed by the introduction of hot air from the bulb. Let the volume of
the bulb at 0° be V, and at the boiling-point T let it become V (1 + *T).
K may be measured by using the bulb as a mercury thermometer in a
preliminary experiment. Let the volume of the connecting-tube and
manometer down to the level a be v. Let the barometric height be H
and the difference of the levels ay be h. Let j3 be the co-efficient of
pressure-increase at constant- volume — assumed to be constant, that is
to say we use the gas scale of temperature. Then, in the first part
of the experiment, we have a volume V(l +*T) of air at T° and v at t,
all at a pressure H, which would become

1 -4- aT
V(l + K!) +v-= - , if all were at the one temperature T.*

1 T" &£

In the second part of the experiment, we have a volume V of air at
0°, and v at t, all at pressure (H — A), which would become

V + , v , if all were at 0°.
1 +at

Now, these volumes are nearly equal, since K and v, are both small.
Then, without sensible error, we may use Boyle's Law to find what H
would become if the former volume were reduced to the latter. It would
obviously be

and now we have the two pressures (H — h) and H' with the same
volume.

Hence H' = (1 + /3T)(H - h)

, - orp H

and 1 + pT = x

H-fc

l+at

* We have omitted the correction for increase of volume through increase of
internal pressure as this is in general too small to come into account (Callendar,
Phil. Trans., A., 1887, p. 170).

46 HEAT.

We may use either the values of a previously found, or, since
a only appears in small terms, we may put it equal to /3 without
sensible error.

Regnault found for air at atmospheric pressure . a = -0036706
and at 0° ...... £ = -003665

From his other results we select those for hydrogen —

For hydrogen at atmospheric pressure . a = -0036613
„ atO° ..... /? = -0036678

The value of a is not quite independent of the pressure, nor is
that of /3 quite the same for different values of the initial pressure at
0° C., but the variations for small changes of pressure are inconsider-
able. Thus for hydrogen even when reduced to i atmosphere he found
a = -0036616 ; while for air at £ atmosphere a = -0036954.

Gas Thermometry. — The researches just described showed that the
relation between pressure, volume, and temperature on the mercury-glass
scale, for the less easily condensed gases such as oxygen, nitrogen, air,
and hydrogen, may, without great error, be represented for ordinary
temperatures by

where K is a constant for a given portion of gas and a = -00366 = 1/273
approximately. If we date the temperature from - 273° 0. as a new
zero and write 6 for 273 + 1 and R for KCL we have

For the same kind of gas R is proportional to the mass dealt with, and
if we deal with equal masses of different gases R is inversely as their
molecular weights.

If Boyle's Law were exactly true, R would be constant for a given
portion of gas at a given temperature, and though it would not be quite
constant for different temperatures on the mercury-glass scale, we might
arrange a new temperature scale so that R should be constant.

But as Boyle's Law is not a quite correct expression of the relation
between pressure and volume for any gas, we cannot give such a simple
definition for a gas scale of temperature. We must specify the way in
which the pressure or the volume is allowed to vary.

Two methods have been used in practice, corresponding to the two
kinds of research described above. In the one the pressure of the gas is
kept constant, say at 1 atmosphere, and equal degrees of temperature
are defined by equal increments of volume of the gas. In the other the
volume of the gas is kept constant and equal degrees of temperature are
defined by equal increments of pressure, starting, say, from 1 atmosphere
at 0° 0. In each there are 100 degrees between 0° C. and 100° C.

In the first case, if a is |y,« °f the expansion between 0° 0. and

100° C. of a volume which is 1 at 0° 0., the temperature t, measured from
0° C. is given by

EXPANSION OF GASES. 47

where v0 is the volume of the gas dealt with at 0° C, v its volume at the
temperature to be measured.

Since 1 vm - V0

a-foox ~^T

we may put

t=JLJ^..xiQQ (i)

If we choose a new zero - below 0° 0., we may term this the gas
zero, and if we put

e=t+-

a

~t +
then

or putting 00 for -

a

»

In the second case, if /? is ^ of the increase between 0* 0. and

100° 0. of a pressure which is 1 at 0° 0., the temperature t, measured
from 0° 0. is given by

when p0 is the pressure at O8 and p that at the temperature to be
measured. Evidently, as with the volume scale, we have

° xlQQ (3)

and dating from a gas zero -^ below 0° 0., on which 0° 0. is - = 90 we
have 6=- P xlQQ

PlOO - Po

and 60

_0
~

. .

There are therefore two different scales for each gas. Fortunately,
however, they are nearly coincident with each other, and nearly coinci-
dent, as certain experiments show, with the work scale.

Taking t as the temperature dating from 0° 0. on the work scale,
tv the temperature defined by (1) for air, tp the temperature defined by

HEAT.

(3), also for air, Oallendar has calculated that the differences are as in
the following table (Phil. Trans., A., 1887, p. 179):—

I

tv-t

tp-t

0

+ •0

0

100

0

0

200

0-04

0-084

300

0-09

0-20

500

0-23

0-47

1000

0-62

1-19

A gas thermometer has a very great advantage over a mercury glass
thermometer, in that the expansion of the containing vessel has a com-
paratively small effect, and an approximate knowledge of its expansion
suffices to give the necessary correction unless the temperature be high.
The gas, if pure, will always behave in the same way, and therefore
the same values should be obtained for a given temperature with different
instruments, and direct comparisons of the instruments should not be
necessary. The methods used by Regnault for the measurement of
expansion at constant pressure, and for pressure increase at constant
volume between 0° 0. and 100° 0., illustrate the two types of instrument.
A study of Fig. 36 will show at once that, though the gas scale has
advantages, its practical use has serious disadvantages. The apparatus
is bulky. It is not " direct reading," i.e. the temperature is not at once
read off, but manipulation is required, and calculation must be made
from the measurements taken. Corrections too must be made for the
expansion of the gas reservoir, and for the gas in the tube connecting
the reservoir to the manometer. As the temperature of this gas is
different at different points, the correction for it is uncertain.

Regnault's Normal Air Thermometer. — Regnault's researches
first made exact gas thermometry possible. He employed a thermometer
of the constant- volume principle, the bulb containing dry air freed from
carbonic acid, and with the pressure at 0° C., equal to 1 atmosphere.
The instrument is represented by Fig. 36. The mercury was always
brought to the same point on the bulb side of the manometer by adding
mercury on the open side, or running it out at the tap below, and
the barometric height + or — the difference of level on the two sides
of the manometer gave the pressure of the air in the bulb. He termed
this instrument " the normal air thermometer."

A simple form of the instrument, devised by Jolly, is represented
in Fig. 37.

The two limbs of the manometer are connected by a flexible tube t,
and one side, M', is movable up and down. On the other side, M, is an
index mark at a, to which the mercury is always brought before the
pressure is measured. At S is a screw by which the bulb and connect-
ing tube can be detached, and after being thoroughly dried they can
be connected up again. R is a three-way tap either putting the bulb

EXPANSION OF GASES.

M'

into connection with the manometer or closing that connection and

putting the manometer into connection with the outside air. When the

tap is turned on in the latter way, mercury

is poured into M' until it oozes out at R.

R is then turned so as to put the bulb in

communication with the manometer, and the

instrument is ready for use.

Hydrogen Thermometer. — For a long

time the normal air thermometer gave the
generally accepted scale of temperature.
Since 1887, however, it has been superseded
by the scale of a hydrogen thermometer at
the Bureau International. This instrument
is also on the constant- volume principle. The
bulb is a cylinder of platinum-iridium 110 cm.
long, and with about 1 litre capacity. It is
placed in a horizontal position in a vessel
with a window, so that mercury thermometers
may be laid alongside, and their readings
compared with those of the hydrogen thermo-
meter. -The open side of the manometer is
used as the cistern of a barometer. The
difference in level between the mercury at
the top of the barometer and that in the
manometer where the gas presses on it is the
total pressure. Thus the separate reading
of the barometer is avoided. The pressure
is adjusted to be 1 metre of mercury at
0°C.

The temperature on the gas scale is
obtained from the observations with a gas
thermometer in the following way. For the
small variations of volume allowed, Boyle's
Law is sufficiently exact, and, therefore, for a given mass of gas :

Pressure x volume/temperature from gas zero = constant.

The fact that the mass of the gas in the thermometer is constant is
therefore expressed by

2 PV/0 = constant,

PV

where we form the quantity —s- for each part of the volume, and add the

t>

results. Let V be the volume of the bulb at 6, V its volume at 0'. Let v
be the volume of the connecting tube to the manometer at 6, v' its volume
while the temperature & is being measured. The temperature of v will-
be different at different points. Let - be the mean of the reciprocals of

c

these temperatures. Let P and P' be the observed pressures.
Then. P'V PV_PV Pp

~~~"~~''=~~ ~~

FIG. 37.— Simple Air
Thermometer.

50

HEAT.

whence

&

P'Y'

An approximate value of & will suffice to determine V from V.
Unless the temperature & be very high, it is not necessary to take into
account the increase of volume through increase of internal pressure,
and the expansion, with rise of temperature, may be determined by
preliminary experiments. The greatest uncertainty is introduced by the

term — — — , for it is difficult to determine t.

Jrt

Calendar's and Bottomley's Constant-Volume Thermometers.

It is evidently advis-
able to diminish v as
much as possible ; at
the same time it is
advisable to^ have the
mercury manometer
as far removed as
possible from the
high temperature en-
closure. To reconcile
these two opposing
conditions, Callendar*
and Bottomley f have
both devised air ther-
mometers in which a
U tube pressure gauge
M containing sulphuric
acid is interposed be-
tween the air bulb and
the manometer.

Bottomley's Air
B /T Thermometer.— Fig.

37A represents the
form used by Bottom-

FlG. 37A.— Bottomley's Air Thermometer.— A, Air bulb ; ^' ^\s w^ su^"
B, Sulphuric acid gauge ; P, Air pump to make pres- ciently illustrate the
sure equal on the two sides of B ; M, Manometer. - principle.

The sulphuric acid

gauge is at B, as near the bulb A as is convenient. Between this and
the manometer M is a force pump, P, by which air can be forced in
to make the two sides of the gauge B level. The manometer M then
indicates the pressure. The volume v or v' is that of the connecting tube
from the bulb to B, and the length of tubing from B to M is immaterial,
so that M may be quite protected from any high temperature. The
enlargements in B are to allow the air to expand while the temperature is
rising without driving the sulphuric acid out of the manometer.

Constant-Pressure Gas Thermometers. — We may take Fig. 36

to represent the ordinary form of this class of thermometer. Since the
* Phil. Trans., A., 1887, p. 166. t Phil. Mag., xxvi., 1888, p. 149.

I

P

(

)

n

ft

•
^—^

EXPANSION OF GASES.

51

pressure is only allowed to vary very slightly, we may take Boyle's Law
as true, and therefore 2 PV/0 = constant, where we add up for each part
of the gas, expresses the constancy of the mass of gas in the thermo-
meter. In practice the pressure at the end of the experiment will not
be the same as at the beginning, for even if the sides of the manometer
are level the barometric pressure is likely to change slightly. And
it may be more convenient not to trouble to adjust the levels to
equality, but to make them nearly the same, measure the difference and
add it to or subtract it from the height of the barometer.

If, then, P be the pressure when the whole apparatus is at 6 ; P' the
pressure when the bulb is at 0' ; if V, V be the volumes of the bulb, v, v'
the volumes of the connecting tube, 1ft the mean reciprocal of the tem-
perature of v' when the bulb is at 6', M the volume of air expelled into
the measuring tube when the bulb is at 0', M itself being at 0, we have

FV' PV FM_PV Pv

& + t + e :" e + e

whence

e

P'V

r,

An approximate value of & will suffice to calculate V, and v' and t must
be estimated as exactly as possible.

In this class of instrument, then, it is necessary to measure P and P'
and also M, and the cor-
rections for the connecting
tube are at least as un-
certain as in the other
class. The greater number
of measurements in this
method probably led
Regnault to prefer the
constant - volume method ,
though good determina-
tions had already been
made of certain high
temperatures by Pouillet
with the constant-pressure
method.

Callendar's Com-
pensating Constant-Pressure Thermometer. — Callendar has devised

a comparatively simple form of constant-pressure air thermometer, in
which measurements of the external pressure are no longer necessary,
and in which automatic corrections are made for the connecting tubes.*
In this instrument Yx (Fig. 38) represents the air bulb. M is the
measuring tube initially filled with mercury. ^ is the connecting tube.
V2 is a bulb of volume equal to that of YI} and with equal " dummy "
connecting tube v2, following as nearly as possible the course of vl and
closed at the end. The two systems are connected by a U tube pressure
gauge G, containing sulphuric acid.

* Proc. R. S., vol. 1., 1892, p. 247.

FiO. 38. — Callendar's Compensated Air
Thermometer.

52 HEAT.

Let us suppose that at first the whole instrument is placed in melt-
ing ice at temperature 60, M being filled with mercury and the pressure
being so adjusted that the levels in G are the same. Now let the
temperature of Vj rise to 0, M and V2 being still at 00. Run out
mercury from M till the equality of pressures in G is restored. The
volume of air expelled into M is determined by the volume M of the
mercury run out. Let the initial pressure be P, the final pressure be
P'. The two are nearly equal, so that the constancy of mass of the
gas is expressed by

PV

2-z- = constant.
V

Let Vj expand to Vj', vl and v% to v^ and v2', and let - be the

t>

mean of the reciprocals of the temperatures of Vj', v2'. We have for the
bulb and manometer

FV,' F< FM PV, Pv,

!_ 4. . L j ij 1

e t e0 " e0 + e0

and for the other system Y2 and v2,

FV PV PV.

e~ ' t

Now making V2 = Vj, v2 = vlt and v% — Vj', we may equate the two
left-hand members, and P' divides out so that

or

The thermometer is easily handled and gives exceedingly accurate
and consistent determinations of temperature. One advantage of the
automatic compensation is that the connecting tube may be of consider-
able length and of flexible material.

CHAPTER V,

CIRCULATION AND CONVECTION IN LIQUIDS AND GASES.

Circulation and Convection of Heat — Hot- Water Heating Systems— Ocean Currents
— Convection in Gases — Convection Currents in the Atmosphere — Winds — Land
and Sea Breezes — Trade-Winds— Water-Vapour Aids Convection Currents —
Weather Forecasting in the Case of Cyclones — Convection in Chimneys and
Hot- Air Heating Systems.

Convection Of Fluids. — Owing to the ease with which one portion
of a fluid can glide past adjacent portions, any local change in density
due to expansion by heat, at once s-^.

results in motion. If, for example, jf( ^ ^
a flask be heated from below, as in (( n>> ty
Fig. 39, and a little bran be put in
the water to show the direction of
motion, it is very soon seen that the
heated, and therefore lighter, water is
rising up from the bottom, its place
being taken by a down-current of the
colder water from the top. Usually
the down-current is along the side ;
but if the flask be heated at one side
only, it is easy to establish the circu-
lation up that side and down the oppo-
tion of Water in s^e colder side. The circulation may
a Heated Flask, also be seen through the varying re-
fraction of the up and down currents,
which give an apparently shimmering motion to any
object looked at through the water. This process of
circulation through expansion by heat carries the heat
from one part of the vessel to another, and this car-
riage of heat by motion of the heated matter is termed
convection. Convection obviously expedites the com-
munication of heat to the liquid as a whole, for not only
are fresh portions of liquid being continually brought
into contact with the heating surface, but also the heated
liquid is continually coming into contact with colder
surroundings, with which it shares its heat much more
rapidly than with surroundings nearer to it in tempera-
ture. As an illustration of this, we may compare the
method of boiling water in a test tube by applying heat
at the bottom as in Fig. 40 with that represented in Fig. 41. In the
latter case the circulation is very local, and the hot water, being the

S3

IG ^Q Boiling

Water by Con-
vection.

HEAT.

lightest, remains near the top, only sending down heat to the lower
part of the tube by conduction. Even long after boiling takes place
near the surface, no appreciable rise in temperature can be detected

by the hand near the bottom of the tube.
*s' °^ course> to make use of convection
that boilers or vessels of water are always
heated from below.

Hot- Water Heating Systems. — Con-
vection of heat by water is used in warm-
ing buildings by hot-water pipes. Fig. 42
shows the principle on which such systems
are based. From the heated vessel or
" boiler " a pipe leads out at A, near
the top, and after circulating round the
building along B, C, D, returns into the
" boiler " at E, near the bottom. At
the highest point of the course, say 0, is
a cistern by which the whole system is
filled with water. The circulation is com-
FIG. 41.— Boiling Water when menced and maintained thus. The hot
Circulation and Convection are water rises in the boiler by convection —
Prevented. not going sideways into the pipe at E —

and soon the pressure at E, due to the

depth below 0 as traced through CBA, which is partly warmed, is appre-
ciably less than the pressure due to the depth below C as traced through
ODE, which is still all cold. Hence, the cold column presses the water
near E into the boiler, and some of the warm water is forced into the
pipe AB, and the circulation is started.
Once started, it will evidently be kept
up, for the water going out of the boiler
is always the hottest, and therefore
CBAE always gives a less pressure
than ODE. The efficacy of the system
depends, to some extent, on the ver-
tical height of the highest point of the
system above E. If this is small, the
circulation will only be slow. It is
sometimes found necessary to increase
the height artificially by putting in a
vertical pipe leading from A to some
height above the level to be warmed,
and then returning down to that level.
Some cases of ocean currents are
probably examples of convection due
to heat, arising, however, in a slightly
different manner, since the heating of the ocean is chiefly from above.
To understand the way in which the circulation is maintained, let us
imagine a long canal with a horizontal bed stretching from the equator
at A (Fig. 43) to the pole at B. If it were all at one temperature to
start with, its surface would also be horizontal as AB. But through
equatorial heat and polar cold a rise would take place in the surface

B

Boiler

S

FIG. 42.— Principle of Hot-Water
Heating Systems.

CIRCULATION AND CONVECTION IN GASES. 55

at A to A', and a fall at B to B'. The surface, however, thus sloping,
could not keep in equilibrium, and at once a surface-current would start
from A towards B. The pressure at the bottom at A would, therefore,
fall, while that at B would rise through
the removal of water from A to B. A
reverse current would then be started „ ,
along the bottom, the excess of pressure
urging the cold water from B towards A. FIG. 43.

In apparent confirmation of this ex-
planation, we know that in the Atlantic the Gulf Stream flows along the
surface from the tropics to the polar regions, while it is found, by deep
ocean soundings, that the temperature near the bottom, even in the
tropics, is not much above the freezing-point, doubtless through a return
under-current from the polar regions. But in all probability the surface-
current is almost entirely due to winds along the surface. A persistent
wind blowing along the surface in one direction will give far more
kinetic energy to the water than can be acquired from the potential
energy due to heat and expansion.

Circulation also takes place in lakes and ponds when cold weather
sets in. The surface-water cools and falls, its place being taken by the
warmer water pushed up from below, and so the water is turned over
and over as it were, successive portions of it being cooled. But at 4°
the process stops, for at that point the maximum density is attained.
When the first mass of water is cooled down to that temperature, it goes
to the bottom, and remains there, and the subsequent circulation stops
short of this layer of densest water. The non-circulating water gradu-
ally increases, the circulating part becoming shallower and shallower,
till all is at 4°. Then circulation entirely ceases, and the top layer goes
on cooling towards 0°. Hence the existence of a point of maximum
density hastens the arrival of the freezing-point by cutting short the
process of circulation.

Convection in Gases. — Convection is even more marked in gases than
in liquids, partly through their greater expansion with change of tempera-
ture, and partly through their smaller frictional resistance to motion.

A very common instance of convection in the atmosphere occurs over
the heated surface of the ground on a hot summer day. The air im-
mediately over the surface expands and rises, owing to its diminished
density, its place being supplied by downward currents of cooler air from
above. The existence of these currents is shown by the tremulous motion
of distant objects looked at through the strata of air near the ground.
The variations in density produce refraction, and the refraction of the
rays is continually altered by the rapid change in position of the up and
down currents.

We have similar effects in the tremulous motion of objects looked at
through the ascending currents over a gas-flame, and in the flickering
shadow of a flame thrown by sunlight.

A special instance of these local convection-currents is afforded by
the haze so common on some hot days in summer. If any distant object
is looked at through a good telescope on such a hazy day, its outline is
found to be continually undulating, owing to the varying refraction. Since
variation in refraction is always accompanied by reflection, a considerable

56 HEAT.

quantity of the sun's light is reflected from the surface of these currents
— especially at great angles of incidence — and this gives rise to the glare
seen, especially, towards the sun. It is very probable that in some cases
what is termed " haze" is due to convection-currents started, either by
lighter air ascending from the surface, or by heavier cold air descending
from upper currents.*

Winds. — Convection also occurs on a much greater scale in the
atmosphere, the currents formed being recognised as winds. A well-
known example is given by

Land and Sea Breezes. — It is very often noticed at the seaside
that there is, during the daytime, a sea-breeze, which changes to a land-
breeze at night. In tropical regions these land and sea breezes are even
more marked than in higher latitudes. We may explain them as con-
vection-currents. During the day the surface of the land becomes much
hotter through the sun's rays than the surface of the sea, the higher
specific heat of water, and the mixing up of the surface layers by the
waves, both combining to lessen the rise of temperature. The air over
the land is, therefore, more heated, and expanding upwards tends to
overflow above. The overflow in the upper strata takes place towards
the sea, and so the pressure at the sea surface is increased while that at
the land surface is diminished. There is therefore a tendency for the
surface layer of air to move from sea to land, the motion constituting
a sea-breeze. At night, however, the land radiates out its heat more
rapidly than the sea, the high specific heat and the agitation of .the sea
both tending to keep up its temperature. There is, therefore, a con-
traction of the air over the land, and an overflow in the upper strata
from sea to land, accompanied by an opposite flow in the surface strata
from land to sea, constituting the land-breeze.

Trade- Winds. — We may also explain in a similar way the well-
known trade-winds, which blow in certain latitudes, in our hemisphere,
from the north-east towards the equator. As in the land and sea breezes,
the equatorial heat expands the air, which overflows in the upper strata
towards the polar regions, tending to decrease the pressure at the surface
near the equator, and increase the surface pressure in higher latitudes.
The surface layers of air are therefore pressed from the north and south
towards the equator. The north-easterly direction in the Northern
Hemisphere, and the south-easterly direction in the Southern Hemi-
sphere, of these lower currents arise from the rotation of the earth.
For, taking the northern trade-winds, the mass of air moving towards
the equator continually comes into regions moving faster from west to
east than the region just left. There is, therefore, a tendency on the
part of the winds to lag behind the earth's surface in its west to east
motion ; or, the wind has a motion towards the west as well as towards
the south, making it a north-east wind.

The corresponding upper current, in its journey towards the pole,
ultimately comes down to the surface somewhere about 35° N. latitude,
and constitutes a south-west wind. But it is not nearly so constant as
the trade wind.

* An explanation of the twinkling of the stars as due to convection- currents in
the air has been given by Montigny, Exner, and Rayleigh (Phil. Mag., xxxvi. p. 129,
1893).

CIRCULATION AND CONVECTION IN GASES. 57

Water- Vapour aids Convection-Currents.— Convection arising
from heating of the lower layers of the atmosphere is greatly aided by
the increased amount of water-vapour which the air takes up, the vapour
being much lighter than the air which it displaces. As an example of
the joint-effect of water-vapour and expansion through heat to produce
convection, we may probably instance the formation of a thundercloud.
When a storm is first gathering, an observer a short distance away may
see enormous piled-up masses of cloud rising far into the upper regions.
These show that a great volume of light, damp air has risen by
convection, expanding in the ascent, and therefore cooling through the
work done in expansion until the temperature of cloud-deposition is
reached. At the same time, it frequently happens that there is a
surface indraught towards the storm area from the surrounding region,
the upper strata with the clouds, moving in the direction opposite to
that of the strata immediately below them.

These examples will prepare us for the general statement that
winds are convection-currents in the atmosphere due to local diminu-
tions of air density, either through heat, or increased evaporation, or
both. Since the weather depends so largely on the direction of the
wind, it is, of course, of the utmost practical importance that the nature
and origin of all atmospheric movements should be investigated and
explained as thoroughly as possible. But though the above general
statements may be made with confidence, meteorologists have not yet
succeeded in discovering, except in a few cases, how particular winds
arise, i.e. what share in their origin is to be assigned to heat, and what
share to evaporation. They are still further from foretelling what winds
will be formed from a given distribution of the atmosphere with known
temperature and amount of water-vapour. The problem is one of
enormous difficulty, which will probably only be fully solved in the
distant future.

Weather Forecasting in the Case of Cyclones. — At present, the
art of weather-forecasting depends largely on the fact that a given distri-
bution of weather travels onwards irf a definite course, and that certain
definite types of weather have been recognised and their movements
studied. As an example, we may give a short account of the simplest
type, that of the Cyclone, and the mode in which English weather is
forecast when a cyclone is approaching our shores.

There are about fifty meteorological stations scattered over the north-
western part of Europe, at which meteorological observations are taken
at stated times every day. These include the readings of the barometer,
the direction and force of the wind, the kind of weather, and the
temperature. The results are at once telegraphed to London. The
barometer readings are marked on a map containing all the stations,
each reading at its own station. Curves are then drawn on the map,
joining all points where the pressure is the same, one curve for every
fifth of an inch. There will thus be a curve for 29 inches, another for
29'2, another for 29*4, and so on. It generally happens that these
curves do not pass exactly through the stations, but their position may
be ascertained from the known readings. If, for example, the reading
at London is 29 '25, and at Dover 29'15, it is assumed that the 29'2 line
passes about halfway between these places. The map with these curves,

HEAT.

23

or isobars, shows the distribution of barometric pressure, and when the
direction of the wind is marked on the map, it is found that it blows, in
general, from higher to lower pressures, but not straight down the slope.
Through the rotation of the earth, the wind has a tendency to go to the
right of the area of lowest pressure, as explained already in the case of
the trade-winds. One of the commonest arrangements of the isobars is
that of a series of oval curves, round the area of lowest pressure. This
will be seen from Fig. 44, which gives the map for November 23, 1874,
in which the lowest pressure was over the Midlands. The arrow-heads

giving the direction of the wind,
show us also that it is every-
where blowing towards the right
of this low pressure area, and so
much so that the whole of the
system constitutes a whirlwind
or cyclone. The rotation in such
a cyclone is, in the northern
hemisphere, always counter-
clockwise. Through the in-
draught of air to the centre, the
depression is gradually filled up ;
but not so rapidly as we should
expect, if the air merely moved
to the centre, and stopped there.
It appears from observations
made on high clouds that, in
general, it ascends there, and
flows outwards again in the
upper strata, forming an " anti-
cyclone."

A study of cyclones has shown
that in the various parts, there
are not only characteristic winds
but also characteristic kinds of
weather. Thus, to the east of the
centre, the wind is usually from
FIG. 44.— The isobars and wind in a Cyclone. tne south or south-east, and, far
(From Abercromby's Forecasting by from the centre, the weather is
Weather Charts.) nne> Moving westward towards

the centre, the weather gradually

gets damper ; the sky becomes overcast ; and near and at the centre,
there is usually rain. Passing the centre, the wind is now from the
north or north-west ; it is colder and drier ; and some distance from the
centre the sky usually becomes clear. To the south of the centre, the
wind is usually south-west, and north of it north-east.

The distribution of weather in a typical cyclone is represented in
Fig. 45.

Cyclones usually move in an easterly direction — most commonly to
the north-east in our latitude, the centre travelling at any rate up to
70 miles an hour. But with a given cyclone the rate is often nearly
constant for some time.

November 23, 1874.

CIRCULATION AND CONVECTION IN GASES.

59

The figure on following page (Fig. 46) shows the history of a cyclone
on four successive days.

It is now easy to see how a forecast may be made. A cyclone is
shown by the isobars to be approaching as in Fig. 46, October 26. The
track of the centre is either observed or guessed at, and so the subsequent
positions of the cyclone are foretold. Each part of the cyclone carries
with it its wind and weather, and so the wind and weather may be fore-
told for the districts passed over by a given part of the cyclone. For
instance, if a cyclone-centre passes over Yalentia and is moving towards
the north of Scotland, south-west winds may be foretold for England. If
the isobars are near together, the pressure-slope is steep, and it may be
expected that the winds will be strong. If the centre moves more towards
the south, say over the Midlands, rain with varying winds may be fore-
told there, north-easterly winds for Scotland, and south-westerly for the

Blue

Blue

Windy Cirrus

FIG. 45.— Cyclone Prognostics. (Abercromby.)

southern coasts. It will be seen from this short account that it is ex-
ceedingly difficult to forecast English weather with accuracy for any
length of time beforehand, as the stations extend so little to the west-
wards. A cyclone first observed to be approaching in the west of Ireland
may already be well on its way towards England, with all its changes of
weather, before its course and nature can be accurately observed. When
communication with ships in the Atlantic by wireless telegraphy becomes
general, no doubt the proportion of successful predictions, already large,
will be further increased.

Convection in Chimneys and Hot- Air Heating Systems.— A

chimney depends for its successful working upon convection. Taking
the case of the ordinary open fireplace, when a fire is first lighted in a
room, the column of air in the chimney over the grate is heated and
expands, some of it flowing out from the top. The air column in the
chimney now weighs less, and the pressure at the grate, as traced down
through the chimney, will be less than that at the same level in the rest
of the room. The air of the room, therefore, moves towards the grate,

60

HEAT.

and if there is a sufficient communication with the outside, the circulation
thus started is maintained, the pressure due to the cold air outside the
chimney being always in excess of that due to the warm air and gas in
the chimney. In general, the communication with the outside is effected
through or under the door, and through the crevices between the window-
sashes, the air moving as represented by the arrow-heads in Fig. 47.

October 26, 1880.

October 27.

October 28. October 29.

FIG. 46.— Progress of a Cyclone, October 26-29, 1880. (From the Times.)

Sometimes this is not sufficient, and unless a passage be made for the
admission of cold air, the circulation is not established, and the chimney
smokes.

Sometimes the chimney, if a wide one, may even establish within
itself a down and up current and so maintain the needed circulation.

We may illustrate these points by lowering a lighted candle into a
flask (Fig. 48a). No circulation is established, and as soon as all the
air is used up, the candle goes out.

If, however, a thin partition be inserted down the neck of the flask

CIRCULATION AND CONVECTION IN GASES.

61

(Fig. 48&), the circulation is thoroughly established down one side and up
the other. The direction is easily shown by blowing out a lighted taper,
and holding it, while still smoking, first on one side, then on the other.

If the candle be put at the bottom of a wide cylinder, it will usually
establish its own circulation, but,
as the direction of the currents is
constantly changing, the flame is
very unsteady.

When a chimney with an open
fireplace is working properly, a
great amount of air in addition to
that used to burn the coal is
drawn into the chimney through
the open space above the grate,
and much of the heat of combus-
tion is used to warm this air and
send it up the chimney. At first
sight, this heat would appear to be
wasted, but the additional circula-
tion has one advantage, namely,

FIG. 47. — Circulation established by
an Open Fireplace.

that it brings into the room a
greater supply of fresh air, and so
aids ventilation. But in the ordi-
nary arrangement, or rather want of arrangement, by which the cold
air is left to find its own way into the room, the greater circulation
makes itself evident by the greater draughts along the floor and near the
windows. The ventilation in this case is also inefficient, for much of
the fresh air rushes straight to the chimney, and the impure air, which
is warmer and damper, and therefore lighter, rises to the ceiling, and

FlG. 48. — -Candle in Flask, showing (a) no circulation, (6) circulation.

is only renewed slowly by diffusion. It is much better to provide a
special passage for the air into the room by means of some such arrange-
ment as the Tobin ventilator, in which a pipe is brought from the out-
side somewhat as in Fig. 50.

The pipe being carried up a few feet, the stream of cold air is directed
towards the ceiling, and is warmed there before it descends to the lower
part of the room. Since there is thus a more general renewal of the
air of the room, the ventilation is much more efficient, while cold draughts
are diminished.

62

HEAT.

FIG. 49.

\

Arrangements have been made by which the incoming cold air is
warmed by some of the excess of heat which
would otherwise be wasted up the chimney. The
air is admitted from the outside into boxes placed
round the grate and chimney, and then carried into
the room — preferably to the furthest corner from
the grate, somewhat as in Fig. 51. If the entrance
into the room is sufficiently subdivided, no serious
draughts will be felt.

In large rooms used for meetings, the problem
of warming, and at the same time ventilating,
efficiently, is one of great difficulty, and one which
architects have only very partially solved. The
shape of the building and its surroundings may

greatly influence the direction in which convection-currents tend to
establish themselves, so that it is
almost impossible to foretell the special
difficulties to be surmounted. As
there is in such halls, generally,
nothing corresponding to the open
chimney, ventilators are usually pro-
vided near or in the ceiling, and the
whole room may be in this case
regarded as a chimney. The air of
the room is warmed by the heating
arrangements and by its occupants,
so that a circulation is established
through door or window, or through
other communication, with the external
air and out through the ventilators.
Frequently, however, especially in cold

weather, local circulation is established near the windows, the air in

contact with them being cooled,
and falling through its increased
density, this fall being recog-
nised as a cold down-draught.
This may be lessened either
by having double windows, or
warming the windows by gas-
jets or water-pipes placed in the
inside sills. Probably, the most
common fault in large rooms is
that, from motives of economy
in construction, the ventilation
is left too much to work itself.
Some means, such as a fan,
should always be provided, by
which the impure heated air
should be extracted at the top,

while fresh air, warmed if necessary, should be introduced at such a level
and in such a direction that it will not be felt as an unpleasant draught

FIG. 50. — Tobin Ventilation.

I i

FlG. 51. — Arrangement for Warming
Incoming Air.

CIRCULATION AND CONVECTION IN GASES. 63

Inasmuch as our comfort so largely depends on the efficient warming
and ventilation of rooms, it is much to be desired that the application
of the principles of convection should be more thoroughly studied. The
investigation on the scale necessary for its application to large buildings
is expensive as well as difficult, and for this reason, probably, it has
hitherto been too much neglected. But whatever the expense, it would
be worth incurring if we could thereby arrive at some mode which
should save us from the too common experience in large buildings, where
the means provided for ventilation and warmth are found to be totally
inadequate, and where improvement of ventilation means increase of
draught, and increase of warmth means absence of ventilation.

CHAPTER VI.
QUANTITY OF HEAT. SPECIFIC HEAT,

Quantity of Heat — Unit Quantity: the Calory — Specific Heat — Water Equivalent
and Capacity for Heat — Method of Mixtures — Regnault's Determinations by
the Method of Mixtures — Experiments on Solids — On Liquids — On Gases —
Specific Heat of a Liquid by Mixture with known Solid — Method of Cooling —
Method of Melting Ice — Bunsen's Ice Calorimeter — Method of Condensing
Steam — Joly's Steam Calorimeter — Differential Steam Calorimeter — Method of
Electric Heating — Specific Heat of Water — General Results — Law of Dulong
and Petit.

Quantity of Heat. Specific Heat. — When a cold body is put in con-
tact with a hot body, the colder rises in temperature, while the hotter
is cooled. We describe the rise of temperature of the one by saying
that it has gained heat, the fall of temperature of the other by saying
that it has lost heat. But we go further than this, and regard the heat
gained by the one as identical with the heat lost by the other. That
is, we regard the heat as something which we can identify at least in
thought, and which has been transferred from the one body to the other.
This is no doubt metaphysical ; for all that we are entitled to assert, from
actual experiment, is that one body is hotter, the other cooler ; but the
conception of the identity of heat enables us to describe the process more
shortly, and to think of it much more clearly.

Let us suppose that we have a number of equal vessels, containing
equal quantities of water at the same temperature. Let us plunge into
these vessels different hot masses, into one iron, into another copper,
into a third stone, and so on, and let us further suppose that the masses
and their temperatures are so adjusted that in each case the temperature
of the water has risen by the same amount in coming to thermal equi-
librium. Since the mass of water is in each case the same, the same
amount of heat has been gained by each, and, if we take the view that
this heat has passed from the body immersed, we see that we can speak
of equal amounts of heat lost by quite different substances, though the
effects are very different. The iron, for example, may have fallen 10°,
the copper 20°, the stone 30°. We need not then speak of iron heat,
copper heat, stone heat, but of heat simply, for all these different sub-
stances, on parting with their heat to one chosen substance, have the
same kind of effect on it.

We are thus led to the idea that we may measure quantity of heat
by its effect in raising the temperature of a given mass of some chosen
substance, and water is the substance which is usually chosen.

If we make an experiment in which we mix equal quantities of water
at different temperatures, we find, after allowing for the heat taken uj>

QUANTITY OF HEAT. SPECIFIC HEAT. 66

by the containing vessel, that the final temperature of the mixture
is very nearly the mean of the two initial temperatures. If, for
example, we mix 50 grammes of water at 15° with 50 grammes at 17°,
the resulting temperature is 16°, within the limits of errors of observa-
tion. Then the heat given out by a gramme in cooling from 17° to 16°
will raise another gramme from 15° to 16°, or the same heat will raise
a given gramme from 15° to 16°, and from 16° to 17°. Or, if we take a
wider range, and have initial temperatures of 0° and 30°, the mixture
is exceedingly near to 15° ; or the heat required to raise 1 gramme from
0° to 15° is very nearly the same as the heat required to raise 1 gramme
from 15° to 30°. Exact experiments, to be described later on, show that
the heat values of successive degrees in the rise of temperature in a
quantity of water are not quite the same, and we are therefore obliged
to specify the temperature range used.

The most convenient range is 1° at the average laboratory tempera-
ture, viz., from 15° to 16°, and we therefore choose the following
definition : —

The unit quantity of heat, or the calory, is the quantity which raises
1 gramme of water from 15° C. to 16° 0.*

But in rough work, sufficient for many purposes, we may neglect the
variation in the heat required for a rise of 1° at different parts of
the scale, and take the calory as simply the heat raising 1 gramme of
water 1° C. If m grammes of water are raised t°, the heat gained
by the water is therefore mt calories.

We are thus enabled to measure and express the heat given
up by any other substance in cooling through a definite range of
temperature.

Let us suppose, for instance, that 50 grammes of iron are heated
to 100° and then plunged into 50 grammes of water at 15°. The iron
and water will ultimately come to a common temperature of about
23-5°. Hence, 50 gi-ammes of iron, in cooling 76'5°, have given up heat
which raises the temperature of the 50 grammes of water 8 '5°; or,
the iron has given up 50x8'5 = 425 calories. On the assumption (not
quite accurate) that each degree has the same heat value for the iron,

each gramme of iron in cooling 1° has given up — =- calory

50 x 7o°o 9

nearly.

Thus the iron in rising 1° requires - of the heat which will raise the

i/

same mass of water 1°. This is expressed by saying that the specific
heat of iron is -.

Or, let us suppose that 50 grammes of lead at 100° are plunged into
50 grammes of water at 15°. The common temperature will now be
about 17-5°; or 50 grammes of lead in cooling 82*5° have raised
50 grammes of water 2*5°, and have given up 50 x 2 "5 = 125 calories.
Assuming equality of value of each degree, 1 gramme of lead, in cooling

* The unit of heat used by Regnault was defined as that raising 1 gramme of
water from 0° to 1°. Another unit is the -j-J^ part of the heat raising 1 gramme
of water from 0° to 100°. We may term the three calories respectively the 15°,
the 0°, and the mean calory.

66 HEAT.

125 1

1° gives out, or in rising 1° takes in, — — -^p = ^ calory, about. This

DO x oZ'o Ai

is expressed by saying that the specific heat of lead is — . These

o2

illustrations prepare us for the following definitions : —

The Specific Heat Of a Substance is the number of calories

needed to raise 1 gramme of the substance 1° C.

If the specific heat of a substance over a range t° is s, the quantity of

heat required to raise m grammes of the substance t° is mst calories.

Water Equivalent and Capacity for Heat. — Since mst calories

would raise ms grammes of water through the same range t°, the
quantity ms is termed the Water Equivalent of the m grammes. When
the range is 1° the quantity of heat required is termed the Capacity for
Heat. The two expressions have the same meaning in practice.

We shall now give an account of the chief methods of determining
specific heats. The details of the methods, though of the utmost-
importance in obtaining exact results, need not be fully described here.
These may be best understood from the accounts given by the original
workers. Our aim is to point out the general principles.

The method most easily applied is

The Method of Mixtures. — Suppose that we are to find the
specific heat s of a certain solid. Then a known mass M of it is raised
to a known temperature t', and dropped into a known mass of water W
at a known lower temperature t. The experimenter observes the
temperature 6 at which the mixture stands when the two have come
to thermal equilibrium. If all the heat lost by the solid could be
assumed to have gone into the water, and to remain there, then
expressing the equality

Heat lost by solid = Heat gained by water
we have M*(*' - 0) = W(0 - 1)

w e-t

8=MX*— 0

But in practice the heat does not all go into the water and remain there.
Some of it goes into the containing vessel or Calorimeter, into the
thermometer, and into the stirrer necessary to mix the water up
thoroughly. Some of it passes out through the calorimeter, where it
is partly given to the air, and partly radiated out into the surrounding
space. Corrections must be determined and applied on both these
accounts. We may understand how they are made by considering an
example. Let us suppose that we are to find the specific heat of a
specimen of brass. It is advisable to have the brass either in a spiral
roll, or in a coil of wire, or in pieces, in order that its surface shall
be large, and that it shall quickly part with its heat to the water
when immersed. The brass may conveniently be heated to the tempera-
ture of boiling water in a steam-jacketed chamber — or, for rough work,
in a test-tube immersed in boiling water — and the temperature may
be taken as that of boiling water at the atmospheric pressure at the

H — 760
time. This will be nearly 100+ -~ , where H is the barometric

it

height in millimetres.

QUANTITY OF HEAT. SPECIFIC HEAT.

67

The vessel or calorimeter containing the water (Fig. 52) should be
of thin metal polished on the outside — thin that it should not absorb
much heat, polished that it should lose little by radiation. It should
have a lid, and should be supported by badly conducting material,
with as small surfaces of contact as possible, say pointed wood or ebonite
pegs, and should be surrounded by an outer vessel, preferably a water-
bath, kept at a constant temperature. An intermediate thin metal
vessel, highly polished on both sides, may still further diminish the
radiation loss. To allow for the quantity of heat absorbed by the
calorimeter, thermometer, and stirrer, we must find their capacity or
water equivalent, and regard it as so many grammes of water in
addition to that actually contained in the calorimeter. This water
equivalent is determined by putting into the calorimeter about the same
quantity of water as "will be used in the final experiment with the brass,
and by adding some hot water so as to
produce about the same rise of temperature
as is expected in that experiment, and
noting the exact rise produced.

Thus, suppose that the calorimeter con-
tains 100 grammes of water at 15°,* and
that 40 grammes of water at 48° are poured
in. The contents are well stirred, and in
half-a-minute the temperature has risen to
23'55°. Meanwhile, however, the calori-
meter has been losing heat. To estimate
this loss the rate of fall is now observed.
Suppose that one minute after the last
observation the temperature is 23 '35°, and
one minute later still 23 -15°. The rate of
loss corresponds to a fall of 0'2° per minute,
or of O'l per half -minute. But during the
rise of temperature the rate of loss may
be taken as half this on the average ; for
supposing the temperature to rise uniformly
from the moment of mixture to the
maximum observed — only an approximation to the truth, no doubt — the
average excess above the surrounding enclosure is only half the final
excess, and the loss, which, for small excesses, may be taken as pro-
portional to the excess above the surroundings, is at only half the rate
of the final loss observed. Thus, had all the heat been kept in, the
temperature would have been £x 0*1° = '05 higher, or the corrected
temperature is 23 6°. f

Expressing that the gain of heat by calorimeter and contents = loss
by hot water, and putting w for the water equivalent to be found —
(w + 100) (23-6 - 15) = 40(48 - 23-6)

* In mere demonstration experiments it is easier to work with small quantities,
but, if exact results are required, then larger quantities, say not less than 500
grammes of water, should be used. Otherwise the corrections are too large a
fraction of the whole effect.

t In accurate work much more care must be taken with the correction for
loss of heat. A description of the mode of doing this will be found in Ostwald's
Physico-Chemical Measurements, p. 126.

M-J

r—

L

J

1

—

C^j

f\ A

7t A

"Water .Tnrkct

FlG. 52.— Calorimeter.

68

HEAT.

whence w = 1 3'5 nearly,

or we may regard the calorimeter, thermometer, and stirrer as 13'5
grammes of water extra.

To find the specific heat of the brass, again let us start with the
calorimeter containing 100 grammes of \\ater at 15°. The brass,
which we may suppose weighs 150 grammes, has been for some time
in the steam-jacketed chamber or heated vessel. For simplicity, let us
take its temperature as exactly 100°. It is quickly dropped into the
water, and the calorimeter is stirred. Let the temperature readings be —

maximum, \ minute after mixture .. . 23'8

1 minute after maximum .... 23*6

2 minutes ,, ,, .... 23'4
Final rate of loss of heat corresponds to fall of 0'2 per minute.

,, ,, ,, O'l per \ minute.

Loss during rise . . . . . . 0'05 „

The corrected maximum is therefore 23-85°.

If sf is the specific heat of brass on the average of the range from 100
to 23-85, then

Heat lost by brass = Heat gained by calorimeter
gives 150*(100 - 23-85) = (100 + 13-5)(23-85 - 15),

whence s = 0-0879.

Regnault's Determinations by the Method of Mixtures. —

Experiments on Solids. Regnault used the method of mixtures for the

D

A A

FIG. 53. — Regnault's Apparatus for Determination of Specific Heat
of Solids by Method of Mixtures.

determination of the specific heats of a large number of solids, liquids,
and gases. When working with solids he used an apparatus, the
principle of which is shown by Fig. 53.

QUANTITY OF HEAT. SPECIFIC HEAT.

The substance to be experimented on was broken in pieces, and
placed in a little metal basket hung in the steam-jacketed chamber
A, placed on a box B. A was closed above by a cork, through which
passed the thread supporting the basket, and a thermometer with its
bulb close to the basket, so as to indicate the temperature of the
contents. The steam was admitted at I and taken out at O. Below,
the chamber was closed by a trap-door T, in the top of the supporting
box. Steam was passed through the jacket surrounding A for one
or two hours, so that the substance was at the temperature of the
steam. The side D of the box was prolonged upwards to screen
the steam-chamber from the calorimeter. A trap door moving verti-
cally in this side
was drawn up
when the sub-
stance was suffi-
ciently heated,
and the calori-
meter 0 was
pushed in so as
come immediately
under A; the trap-
door beneath A
was opened, the
thread cut, and
the basket
dropped into the
calorimeter. The
calorimeter was
then withdrawn,
and the door in
D at once shut
down. The rise
in temperature of
the calorimeter
could then be
noted, and by
subsequent obser-
vations the loss of
heat to the sur-
roundings could FlG. 54.— Kegnault's Apparatus for Determination of Specific
be determined Heat of Liquids by Method of Mixtures,

and allowed for.

The capacity of the basket was, of course, found by subsidiary experi-
ments, and the heat which it gave to the calorimeter was subtracted.

Experiments on Liquids. — One form of apparatus which Regnault used
to determine the specific heat of liquids is represented in Fig. 54. The
actual mixture of the liquid with water was not permissible in many
cases, and therefore a thin metal vessel was fixed within the calori
meter. Into this the liquid was poured, and ultimately it came to
temperature equilibrium with the water in the calorimeter. Virtually,
then, the method may be described as one of " mixture." The liquid

70 HEAT.

was initially brought to some desired temperature, above or below that
of the calorimeter, in a vessel V contained in a constant-temperature
bath, placed close to the calorimeter, but screened from it by a badly
conducting partition. A pipe p, with a stopcock in it, led from this
vessel into the vessel v, within the calorimeter C, and when " mixture"
was to take place the cock was turned on and pressure applied through
the pipe P to the surface of the liquid, which was then forced into the
calorimeter. Besides the heat brought into the calorimeter by the
liquid some would be conducted by the connecting pipe, but this could
be determined and allowed for.

Experiments on Gases. — As we have seen, the expansion of a gas with
rise of temperature depends on the pressure to which it is subjected. In
the expansion, the surrounding material is pressed out, and heat has to
be given to the gas to do the work implied in this pressing out. The
heat thus required may be a very appreciable fraction of the whole heat
given, and so it is necessary to specify the pressure condition to which
the gas is subjected while its specific heat is being found. Regnault only
investigated the specific heat under one condition, viz., that of constant
pressure. His apparatus is represented in Fig. 55. The gas, carefully
purified and dried, was stored in a reservoir R, from which it was
allowed to flow through a gas- regulator worked by hand, so that its
excess of pressure over that of the atmosphere was constant. A water-
manometer M, connected to the gas channel by a very narrow tube,
indicated this excess. It was then conveyed through a spiral metal tube,
10 metres long and 8 mm. in diameter, coiled in an oil-bath, where its
temperature was raised. It then passed by a short tube surrounded
with non-conducting packing into the calorimeter, which consisted of a
series of brass boxes divided by spiral partitions inside, so as to lengthen
the path pursued by the gas ; and it finally emerged into the air.

The gas was allowed to flow for ten minutes, and the quantity flowing
during that time was calculated from the observed fall of pressure in the
reservoir between the beginning and end of the experiment. By collect-
ing the gas in a subsidiary experiment in a globe, and weighing it, the
weight was found to correspond with the observed difference of pressure
in the reservoir. The spiral in the oil-bath was so long that the tempera-
ture of the gas on emerging from it was that of the oil, and subsidiary
experiments showed that, except when the velocity of the gas was
exceedingly small, it lost no heat between the oil-bath and the calori-
meter, and entered the calorimeter at the temperature of the oil. It
left it at the temperature of the calorimeter. Its pressure at entry and
emergence was shown by subsidiary experiments to differ by not more
than 1 mm. of water. Hence, the pressure was practically constant.*
We see then that a known weight of gas at constant pressure was
cooled in the calorimeter by an observed mean amount. This was
again virtually the method of mixtures. Knowing the capacity of the
calorimeter, the experiment enables us to determine the specific heat
of the gas.

* Searle (Proc. Camb. Phil. Soc., xiii. Ft. V. p. 244) has shown that the heat
given up by unit mass of gas would be equal to specific heat at constant pressure
x temperature fall, even if there were a considerable difference of pressure between
entry and exit.

QUANTITY OF HEAT. SPECIFIC HEAT.

71

Corrections had to be made, however, for gain or loss of heat in
other ways during the experiment. On the one hand, the calorimeter
gains heat by conduction and radiation from the heating part of the
apparatus. This Regnault * assumed to be the same per minute through-
out the experiment. On the other hand, as the calorimeter rises in
temperature, it parts with heat by radiation and by conduction to the
surrounding air, the quantity lost being proportional to the excess of
its temperature. By observing the change of temperature for ten
minutes before the gas flows, and for ten minutes afterwards, the
quantity of heat conducted and radiated from the heating part of the

Stirrer

T Outflow
Stirrer

FIG. 55. — Regnault's Apparatus for the Determination of Specific Heat of
Gases at Constant Pressure.

apparatus, and the quantity lost to the surroundings per 1" excess of
temperature, is calculated, and so the result is corrected.

Neglecting corrections, if W = the weight of gas flowing, T = tempera-
ture of the oil-bath, tv t2 = initial and final temperatures of calorimeter,^? =
water-equivalent of calorimeter and contents, s — Specific Heat of the gas,

Determination of the Specific Heat of a Liquid by Mixture

with a known Solid. — An obvious mode of determining the specific
heat of a liquid consists in heating a known weight of a solid of known
specific heat, and immersing it in the liquid contained in a calorimeter.
An equation like that of p. 66 (modified, of course, by the necessary
corrections) then serves to give us W, the water equivalent of the
liquid. Knowing its weight, we have at once its specific heat.

Method of Cooling. — In this method the liquid to be experimented
on is placed with a thermometer in a highly polished metal vessel, serving
as a calorimeter, and this is suspended by silk threads in a colder enclosure,
the walls of which are kept at some constant temperature. This may

* Swann (Phil. Trans., A. 210, p. 231) has pointed out that Regnault's assumption
that the conduction from the heating apparatus to the calorimeter is the same during
the flow as before and after is an overestimate. The flow of hot gas through the
connecting tube will tend to raise the temperature of the part of the tube more
distant from the heater, and will so reduce the gradient and lessen the flow of heat.
Regnault therefore put down to conduction heat which was really carried by the
gas, and so he underestimated the specific heat (see below, p. 86).

72 HEAT.

conveniently be made 0° C. by suiTounding the outside of the enclosure
with melting ice. In some experiments the space between the calorimeter
and the enclosure has been exhausted. The quantity of heat given out
by the surface of the containing vessel in a given time depends not on
the nature of its contents, but on the temperature of its surface alone.
If, for example, in two different cases, the temperature is observed to
fall at twice the rate in one case that it falls in the other when the
mean temperature is the same, the heat given out in the same time is
the same in the two cases, therefore the capacity of the calorimeter and
its contents in the first case is only half as great as in the second case.
Hence we have

W + TWjSj =

where w is the water equivalent of the calorimeter and thermometer, and
m^rn^s^ are respectively the masses and specific heats of the liquids in
the two experiments.

The method was originally applied both to solids and liquids, but it
was found that with solids it did not give good results, owing to vari-
ations of temperature within the solids. The circulation in liquids
during their cooling maintains their temperature more nearly uniform
at each instant, and so the objection is much less in their case.

The Melting Ice Method — Bunsen's Ice Calorimeter. — In this
method the heat given out by a body in cooling down from some higher
temperature to 0° 0. is measured by the quantity of ice which it will
melt. It was first used by Black and afterwards by Lavoisier and La-
place. They collected and weighed the water resulting from the melting.
But as it was practically impossible to collect the whole of the water, the
method failed to give very good results. It was, however, modified by
Bunsen in such a way as to make it of very great service. In his calori-
meter, instead of collecting the water resulting from the melting, the
contraction which takes place in the change from solid to liquid is ob-
served. The amount of this change was measured by a separate experi-
ment in which a known weight of ice at 0° was contained in a bulb, the
rest of the space being filled with mercury. The ice was then melted
to water at 0°, mercury being drawn into the bulb to occupy the space
left by the ice in melting ; the additional weight of mercury gave the
contraction. He found that a gramme of ice at 0° contracted from
1-09082 cc. to 1-00012 cc. of water at 0°.*

The construction of the calorimeter is illustrated by Fig. 56.

A is a test-tube fused into the glass vessel B, which is continued into
the narrow tube 0. B is nearly filled with water which has been
previously deprived of air by boiling, and the remaining space is
occupied by mercury, which also fills the tube 0 round the bend and
some way along the horizontal part which lies against a graduated scale.
To prepare the calorimeter for use, a stream of alcohol, cooled by a

* Barnes (Physical Constants of Ice, Trans. Roy. Soc. Canada, Ser. III.,
vol. iii., Sect. III., 1909-10) gives a summary of the work on Ice, and concludes
that the best value for the density of natural ice is 0-91704, while that of
artificial ice is 0-91676. He takes as the best value of the latent heat
of fusion a determination by Professor A. W. Smith in 1903 corrected to
79-818.

QUANTITY OF HEAT. SPECIFIC HEAT.

73

freezing mixture, is passed through the test-tube A until a cap of ice ia
formed round it in the vessel B. The calorimeter is now placed in melt-
ing ice or snow, and left, preferably for some days, till the ice cap is all
at 0°. A layer of wool is placed at the bottom of A to prevent breakage,
and some distilled water is added. When all is at 0°, the calorimeter —
still surrounded by melting ice — is ready for use. It is better, as
pointed out by Professor Boys, Phil. Mag., xxiv., 1887, p. 214, to sur-
round the calorimeter by an air-jacket by enclosing it in an outer tube,
which is immersed in the melt-
ing ice. A small piece of the
substance to be experimented
on is raised to any desired
temperature, and then dropped
into A, giving its heat to the
water. But as the water rises
in temperature it becomes more
dense, and therefore remains at
the bottom of the tube round
the substance. Gradually heat
is conducted out into B, where
some of the ice is melted by
it, and the melting continues
till all is again at 0°. The
tube C having been calibrated,
the recession of the mercury
along it gives the weight of
ice melted, a contraction of
1-09082 -1-00012 = -0907 cc.,
corresponding to the melting
of 1 gramme. For comparative
measurements it is only neces-
sary to observe the contraction
in different cases, but for the
determination of the amount
of heat actually yielded, it is
necessary to know the latent
heat of water or the number of
calories taken up by 1 gramme
of ice at 0° in melting to
water at 0°. By using a known
weight of water at a known

boiling temperature, that is, by adding a known number of calories
to A, Bunsen determined the latent heat of water as 80-025.* The
calory, in terms of which this is expressed, is y^ of the heat required
to raise 1 gramme of water from 0° to 100°. It will be observed that
in this method the radiation correction disappears, for the substance
taking in the heat always remains at the same temperature, and is always
in equilibrium with its surroundings.

Joly's Steam Calorimeter. — Professor Joly has introduced and

* In Ostwald's Physico- Chemical Measurements, p. 136, will be found a description
of modifications of the calorimeter to make it more convenient in use.

/

A

\

3

\^J

Water

: t

FIG. 56. — Bunsen's Ice Calorimeter.

74

HEAT.

n — i.

perfected a calorimeter in which the latent heat of steam is used some-
what as the latent heat of water is used in Bunsen's ice calorimeter.*
The instrument is represented in Fig. 57. The calorimeter C is a light
metal vessel, double walled and covered with cloth, placed beneath a
delicate balance which is not represented in the figure. A wire hangs
from one arm of the balance, and passing through a very small hole at
the top of the calorimeter sustains a light wire platform wp within it.
Beneath and attached to the platform is a catchwater cw of platinum
foil. The substance of which the specific heat is to be found is placed
on the wire platform and counterpoised. There is a wide pipe S enter-
ing at the top of the calorimeter through which steam can be introduced.
During the counterpoising, the steam pipe is not connected to the calori-

meter and the entrance hole S is plugged
up. There is a large exit hole at the
bottom of the calorimeter, which is also
stopped at this stage by a cap c, shown
in the figure out of position. Before an
experiment a thermometer is inserted in
the calorimeter, and left till all has
come to a steady temperature tr Mean-
while steam is got up in the boiler, and
during the experiment it must be coming
off very freely. It is passed through the
coupling tube for some time before con-
nection to the calorimeter, to drive out
all air. Then the thermometer in the
calorimeter is read and withdrawn, and
the hole for its insertion is plugged up.
The exit at the bottom is unstopped, the
entrance for the steam is unplugged, and
the steam pipe connected. For 30 or 40
seconds the steam rushes freely down the
calorimeter, driving out all the air, and
FIG. 57.— Joly's Steam Calorimeter, condensing on all the surfaces. Then the

exit hole is nearly closed, and in from one

to four minutes the whole of the inside of the calorimeter rises to the
temperature of the steam, when no more should condense. While the
steam is still slowly passing through the calorimeter, the balance is
again counterpoised, and the gain in weight gives the weight of steam
condensed by the platform, catchwater, and substance in rising from t^
to the temperature t2 of the steam. A previous experiment has given
the weight condensed by the platform and catchwater, and the excess
above this is due to the substance alone.

If W is the weight of the substance, s its specific heat, w the weight
of steam condensed by it in rising from ^ to t2, \ the latent heat of steam,
then » is found from

Loss or gain of heat by radiation during the rise in temperature is
almost entirely eliminated by the sudden inrush and condensation of

* The perfected form is fully described in Proc. R. S., xlvii., 1890 p. 218

QUANTITY OF HEAT. SPECIFIC HEAT.

75

steam on all the surfaces, which very rapidly attain the temperature tv
There is, however, a slight continuous radiation from the suspended
part, and a corresponding continuous condensation and gain of weight
which must be determined and allowed for. In estimating w, correction
must be made for the difference in buoyancy of air at tl and steam
at #2.

To keep the wire free from the sides of the hole through which
it enters the calorimeter, and yet to keep that hole sufficiently small to
prevent serious leakage of steam, the following construction is adopted.
The top of the calorimeter is conical, and is ground flat so as to
make a circular hole. On this rests a little copper disc weighing
about 22 mgm., and drilled centrally with a hole about § mm. in
diameter. The suspending wire of platinum, which may be about
O'l mm. in diameter, passes down through the central hole in the
disc, and as the wire swings from
side to side it pushes the disc
about with it until the swings
have diminished to less than the
diameter of the hole, when the disc
is left and the wire finally hangs
centrally. Its weight suffices to
prevent lifting by the steam.
A platinum spiral in an electric
circuit surrounds the wire, and
during the experiment this is
made to glow just visibly. The

o

o

FlG. 58.— Joly's Differential Steam
Calorimeter.

heat it gives to the suspending
wire suffices to prevent the con-
densation of steam on the wire.

The Differential Steam
Calorimeter. — Dr. Joly has de-
vised a form of calorimeter repre-
sented diagrammatically in Fig.
58. In this a platform or holder
and catchwater depends from each

arm of the balance into a common steam chamber. The two holders
are made to have equal thermal capacities, so that if the substance to be
tested is placed on one, the excess of condensation on that side is due
entirely to the substance. The corrections for the holder and catchwater
arid also for the slow radiation from them are eliminated.

This apparatus is the only one which has as yet been used to measure
the specific heat of gases at constant volume quite directly.* For this
purpose Joly used two equal copper globes, each 6'7 cm. in diameter and
about 160 cc. capacity. In the experiments on air, one of them was
filled with air which, at the lower temperature, was at normal pressure,
while the other was filled with air at a much higher pressure. In some
cases the mean higher pressure during an experiment was 26 atmo-
spheres. The excess of condensation on the one side was due to the
excess of weight of air on that side, the volume of which was constant,
except for the expansion of the copper with rise of x temperature and its
* Phil. Trans., A., 1891, p. 73.

76 HEAT.

extension with increase of internal pressure. These could be determined
and allowed for, and the heat needed to raise a known weight of air at
constant volume from about 15° to the temperature of steam was deter-
mined. We shall give some of Joly's results later.

The Method of Electrical Heating.— This method was first

suggested and used by Joule * after he had discovered the law of heat
development in a wire carrying an electric current, viz., that the heat is
proportional to C2R,£ where C is the current, R the resistance, and t the
time for which the current runs.

Joule passed the same current through two similar platinum spirals,
one immersed in a standard calorimeter containing water, and the other
in the calorimeter containing the substance to be experimented on.
The heats developed were proportional to the resistances of the spirals.
These were measured beforehand. The heats could also be expressed in
terms of the heat capacities of the calorimeters and their contents and
the observed changes of temperature, and the equation of the two sets
of measurement gave the specific heat.

We may take as an example an experiment described by Joule to
show the possibility of the method. The standard calorimeter, with
stirrer and thermometer, had water equivalent 280 grains and it con-
tained 35,000 grains of water. The immersed spiral had resistance
taken as 100. The other calorimeter had water equivalent 260 grains ;
it contained 28,000 grains of water and 80,500 grains of lead. The
immersed spiral in it had resistance 106. A current was passed through
the two in series for 20 minutes, when the increase of temperature
was in the first 3'575° and in the second 4'35°. Expressing the fact
that the heat developed is proportional to the resistance, and denoting
the specific heat of lead by s, we have

(8050Qg + 28260)4-35 106
35280 x 3-575 ~ 100
whence s = 0'03073

This method has since been used by many experimenters, and has
been brought to great perfection by Griffiths in his research on the
mechanical equivalent of heat, to be described hereafter. It is not
necessary to employ two calorimeters. For, under proper conditions,
both the current and the resistance can be measured with very great
accuracy, and the value of C2IW is thus known in different experiments,
which may be carried out successively with the same calorimeter.
Equating the ratio of the heats developed in two experiments as ex-
pressed in terms of heat capacity and rise of temperature to the ratio of
C2IU in the two experiments, we have the data for the determination
of specific heat.

The Specific Heat Of Water. — Nearly all determinations of specific
heat have been made in terms of a water unit, such as the heat req'vired
to raise 1 gramme of water through 1° 0. at some named part ot the
scale. It is, therefore, of the utmost importance to determine whether
the choice of the particular degree affects the value of the unit, that is,
to determine whether the specific heat of water itself varies as the
temperature changes.

* Scientific Papers, vol. i. p. li)2.

QUANTITY OF HEAT. SPECIFIC HEAT. 77

It has long been recognised that, at any rate, there is no evidence
for its constancy, but determinations of the nature and magnitude of the
variation have been so conflicting till recently that experimenters have
been too often content to leave it out of account. But measurements
of temperature and quantity of heat have so greatly advanced in accuracy,
that now the results of different workers begin to show agreement, and
there is no longer any doubt as to the existence of a variation, and even
its magnitude at ordinary temperatures is probably fairly determined.

Before giving an account of more trustworthy work, we may illustrate
the difficulties of exact calorimetry by briefly describing two earlier
researches.

In 1847 Regnault published an account of experiments in which he
sought to determine the specific heat of water at different temperatures
by the method of mixtures. A large boiler was so arranged that the
water in it could be boiled under different pressures, and therefore at
different tempera tures, these ranging from 107° to 190°. When the
water was at the boiling temperature a quantity of about 10 kgm. was
rapidly forced into a large gauged calorimeter containing a known
quantity — about 100 kgm. — of cold water, at an observed temperature
ranging from 8° to 14°. From the temperature of the mixture the ratio
of the mean specific heat over the rise of temperature in the one case
to that over the fall in the other could be determined. A series of
experiments led Kegnault to express the specific heat at t°, taking the
specific heat between 0° and 1° as 1, by

Specific heat at t° = 1 + '00004* + -0000009*2.

From this the specific heat at 15° is 1-0008, while the mean specific heat
from 0° to 100° is 1-005.

In 1870 Jamin and Amaury described an electrical method. The
calorimeter was of thin copper and contained 350 grammes of water.
It was surrounded by a spiral of insulated German-silver wire of known
resistance, through which a known current could be passed, to supply a
known quantity of heat. Outside the spiral was a layer of swansdown
of such low conductivity that nearly all the heat generated was con-
ducted inwards to the water. Outside the swansdown packing was a
thin polished copper vessel. This was placed in the middle of a double-
walled enclosure, containing water between the walls. In this water
was another spiral through which a current could be passed, so as to
make the temperature of the enclosure rise at the same rate as that
of the outside of the calorimeter, and thus eliminate any radiation
correction. The results obtained were expressed by

Specific heatat e° = l + -0011< + -0000012<2.

We shall realise how widely this differs from Regnault's value by
noting that it gives the specific heat at 15° as 1-0168, and the mean
specific heat from 0° to 100° as T06.

The first results to which any value can now be attached were pub-
lished by Rowland in 1879 * in an account of experiments on the
mechanical equivalent of heat, to be described later. Here it is sufficient

* Physical Papers, p. 343. A recalculation of the results in terms of the scale of
the Paris hydrogen thermometer is given by Dr. W. S. Day, Phil. Mag., July 1898.

78 HEAT.

to say that the heat was supplied to water in a calorimeter by violent
stirring, the energy put in by the stirring being transformed to heat
proportional to the work done. Rowland found that the same amount
of work produced a different temperature rise in the same quantity of
water at different parts of the scale. The range was from 5° to 36° C.,
and there was distinct evidence for the existence of a minimum specific
heat about 30°. The recalculated results are represented in Fig. 60.
Rowland verified the variation by experiments by the method of mixtures,
from which he found the coefficient of decrease between 0° to 30° to be
•000236.

In 1893 a most careful determination of the variation between
15° and 25° was published by Griffiths,* who used the method of electric
heating. The research was really on the mechanical equivalent of heat,
but we may regard it here as one on the specific heat of water.

Taking the specific heat at 15° as 1, Griffiths found that if t lies
between 15° and 25° then

Specific heat at t° = I - -000266(« - 15)

where t is the temperature on the hydrogen scale. This is represented
in Fig. 60.

In 1894f some experiments on the latent heat of steam led
Griffiths to suppose that the mean specific heat of water between 0°
and 100° is exceedingly near to that at 15°. Since the specific heat is
decreasing at 15°, this implies, of course, that it must reach a mini-
mum before 100°, and then increase again. Griffiths' supposition was
confirmed by Dr. Joly by an experiment with his steam calorimeter, in
which a known weight of water was raised from about 12° to 100°. so
that its mean specific heat between 12° and 100" was determined.
Griffiths found from Joly's work that

Mean specific

Specific heat at 15°

instead of — — - =1'004 as found by Regnault.

1*0008

Meanwhile Bartoli and Stracciati | had been carrying out a very
extensive series of researches on the specific heat of water by the
method of mixtures. They made three sets of experiments. In all
the calorimeter contained a known weight of water initially at the
temperature of the surroundings. In the first set, different metals of
specific heat known through the range used were heated to 100° and
dropped into the calorimeter. In the second set, a known weight of
water at 0° was dropped in ; and in the third set, a known weight of
water at a temperature above that of the calorimeter was mixed. The
three sets showed results very much of the same nature. The mean
is shown in Fig. 60. It should be noted that the metal set gave a

* Phil. Tram., 1893, A., p. 361.

t Phil. Trans., 1895, A., p. 320. Dr. Joly's experiment referred to in the text ia
described in the paper.

J i\uovo Cimento, 32. A brief account is given in Beiblatter, xvii., 1893, pp.
542, 638, 1038.

QUANTITY OF HEAT. SPECIFIC HEAT. 79

gentler slope down from 0° to 20°, and a steeper slope up after that than
the water sets. Thus, at 0° the metal mixtures gave 1 '00551, and the
water mixtures 1 -00777 ; while at 31° the metals gave 1 '00337 and the
water 1 '00145. The mean of these results gives a slope from 0° to 15°,
very nearly the same as those obtained by Rowland and Griffiths (see
Fig. 60), but Bartoli and Stracciati found a minimum at 20°.

Liidin in 1895 (Beibldtter, 1897) also used the method of mixtures,
and his results are represented on Fig. 60.

Dr. Barnes has made the most complete series of experiments up to
the present (Phil. Trans., A. 199, 1902, p. 149). The method, that of
electric heating, was suggested by Prof. Callendar, and the apparatus was
devised by him (Phil. Trans., loc. cit., p. 55). But as Prof. Callendar,
with whom Dr. Barnes was associated at first, was unable to continue
the experiments, they were carried out by Dr. Barnes. In this method
a stream of water is led through a narrow tube t (Fig. 59), through which
passes a fine platinum wire. This wire carries an electric current intro-
duced and taken away by the thick wires cc. The temperatures of the
water on entering and leaving t are taken by the platinum thermometers

c

Water Jacket

Vacuum Jacket

v£_^= _

-^pt/i

FlG. 59. — Callendar- Barnes Electric Heating Method of Determining the
Specific Heat of Water.

pth, pth. t is surrounded by a vacuum jacket to diminish loss of heat by
cooling, and this again is surrounded by a water jacket. Let Q be the
quantity of liquid flowing through t per second, and let 60 be the tempera-
ture at entrance, 6l that at exit. Let s be the mean specific heat of the
liquid between 00 and Ov and J the mechanical equivalent of heat ; then
the work value of the heat gained by the water is JQs(Ol - 60). But if
E is the potential difference between the ends of the fine wire, and C
is the current in it, and if, for simplicity, we suppose all the heat given
by the current to remain in the water, the heat is EC in work measure.
Hence we have

JQs(0!-00) = EC,

and measuring Q, 0p 00, E and C, we can determine s. We must refer
the reader to the original papers for the account of the various
corrections and their determinations. There is a minimum value
at about 40° C. Callendar (Phil. Trans., A. 199, p. 142) gives the
following formula for the specific heat: From 0° to 20° C., s='9982
+ -0000045^ -40)2 + -00000005(20 -t)*. From 20° to 60° C. the last
term is omitted and s= '9982 + '0000045(2 - 40)2. From 60° to 200° C
s= -9944 + -00004* + -0000009*2 (Regnault's formula corrected).

80

HEAT.

Fig. 60, representing all the results, is taken from his paper.

General Results. — Probably in all substances the specific heat
changes with the temperature In general it increases as the tempera-
ture rises, so long as the substance does not change its state. A specific
heat increasing uniformly with rise of temperature, would be repre-
sented by

s=a+fit

but it is only over small ranges that such a simple formula represents
the results of experiments. Over larger ranges they may be better
represented by

and no doubt, if the results were accurate, they would be still better

<K>9. „„- .

O 20 40 60 SO' I0<f

— * Temperature

FIG. 60. — Results of Different Experiments on the Specific Heat of Water.

represented by the addition of terms containing higher powers of t. But
at present the errors of experiment are so great that it is useless to
trouble about these higher powers.

We probably have the most accurate knowledge of the variation of
specific heat in the case of aniline over the range from 15° to 52°. This
was determined by Griffiths (Phys. Soc., 13, 1894) by the method of
electrical heating. He selected aniline in place of water, on account of
its great suitability as a calorimetric liquid, for it is easily obtainable in
a fairly pure state, it has a low vapour pressure at ordinary tempera-
tures, and with its specific gravity, 1'02, and specific heat, '52, its
capacity for heat per unit volume is only half that of water. The
specific heat at t* is given by Griffiths in terms of the calory at 15° as

s = -5156 + (t- 20)-0004 + (t - 20)2'000002.

Regnault has obtained the specific heat of a number of liquids. From
these we may select alcohol, for which he gives, in terms of the calory
atO°,

8 ="54755 + -002242*+ -000006618*8

QUANTITY OF HEAT. SPECIFIC HEAT. . 81

Alcohol may be used in the inner tube of the Bunsen calorimeter in
place of water, for the determination of specific heats below 0°.

The change in specific heat of a number of metals with change of
temperature has also been found. Among the most important of these
is platinum, on account of its use to determine high temperatures.
Violle,* using the method of mixtures, has obtained for this metal over
the range 0° to 1200"

s = -0317 + -000012*.

Naccari f has found the value for various metals by the method of
mixtures, between ordinary temperatures and 320°, in the form

s = a(l+bt)

The following results are given by him. The unit is the calory at
0°:—

a 1066

Copper .... -09205 230'8

Silver .... '05449 392'9

Aluminium . . . '21116 449'3

Lead .... '02973 456'9

Zinc .... -09070 489-5

Nickel .... '10427 907-0

Iron .... -10442 1029'!

Tilden J has investigated the specific heat of iron, nickel, cobalt,
aluminium, silver, gold, and platinum, over the range from — 182° C. to
+ 100° C., extended in some cases to 630° 0., and his results show that the
specific heat decreases as the temperature of determination decreases.
In the case of platinum the decrease is regular, or the relation between
specific heat and temperature is linear. But in other cases the decrease
is more and more rapid as the temperature falls.

The most remarkable changes of specific heat are those which occur
with carbon, boron, and silicon. These were investigated by H. F.
Weber.§ He used the Bunsen Ice Calorimeter from — 50° to + 250°,
employing alcohol in the inner tube below 0°, and for carbon he extended
the experiments to the range between 600° and 1000°, when he used the
method of mixtures. His plan at this higher range consisted in heating
a known weight of platinum and the carbon to the same temperature
and dropping the two simultaneously, one into each of two water
calorimeters. The platinum gave the temperature, while the rise in the
other calorimeter, when the temperature was thus known, gave the
specific heat sought. His results for diamond between — 50° and + 250°
are nearly represented by

though the rate of increase is appreciably diminishing as the temperature

* Phil. Mag., vol. iv., 1877, p. 318.

t Atti R.A. de Torino, 23, 1889; Beiblatter, xii., 1888, p. 326.
J Phil. Trans., A. 194, p. 233, 1900, and A. 201, p. 37, 1903. References to other
work will be found in these papers.
§ Pogg. Ann., cliv., pp. 367 and 553.

F

82 • HEAT.

rises. In the range from 250° to 600", in which he did not experiment,
the diminution must have become much more rapid, for he found that
the specific heat in the range from 600° to 1000° only very slowly
increased towards -46.

With graphite, the specific heat about 0° may be represented nearly by

«= -152 + -0007*.

The rate of increase falls off somewhat more rapidly than with diamond,
and the values for the two are not very different over the higher range,
as the following table shows : —

SPECIFIC HEATS OF DIAMOND AND GRAPHITE.

606-7° 806-5° 985 '0°

Diamond . . -4408 '4489 -4589

Graphite . . -4431 -4529 -4674

With crystallised boron the specific heat rises in a similar way, and
between 0° and 250° it is fairly, but not exactly, represented by

8 ='22 + -0007 It.

Here again the rate of increase falls off as t rises, and from the exact
results over the range Weber deduced a limiting value at high tempera-
tures of 0'50.

With crystallised silicon the limiting value was nearly reached
in the lower range of experiments. We may take the following
values: —

Specific heat of crystallised silicon at - 50° . '13

„ 0 . -16

„ „ 50 . -18

„ „ 100 . -195

„ „ 200 . -202

and the value to which the results tend as 0'205. We shall see the
bearing of these values directly.

Dewar,* working at the same time as Weber, also found that the
specific heat of gas-carbon increased very considerably with rise of
temperature. Between 20° C. and 1040° 0. he obtained a mean value of
0-32, and between 20° C. and 2000° a mean value of 0'42, concluding
that at 2000° it must be at least 0*5.

Dewar t has also found the specific heat of diamond, graphite, and a
number of other substances at low temperatures by means of a liquid air
or a liquid hydrogen calorimeter. The calorimeter consists of a vacuum
vessel of 25 to 5l) c.c. capacity, containing liquid air or liquid hydrogen,
and it is immersed in a large vacuum vessel containing the same liquid.
From the calorimeter a long narrow tube rises up, and to its end a small
test tube containing the substance to be experimented on is attached by
a short rubber tube The rubber tube acts as a valve cutting off the test
tube from the calorimeter, except when the experiment is to be made.

* Phil. May., xliv., 1872, p. 461. t Print. R.S., A. 76, 1905, p. 325.

QUANTITY OF HEAT. SPECIFIC HEAT.

83

When the test tube is held vertical, the substance, which has been
brought to any desired temperature, drops into the calorimeter, and some
of the liquid boils off. A branch tube leads from near the upper end
of the vertical tube, and the evolved gas, passing through it, is collected
in a receiver and measured. The arrangement is similar in principle to
V. Meyer's Vapour-Density Apparatus (Fig. 102, p. 177). A gramme
calory was found to evolve 13-2 c.c. of oxygen, 15-9 c.c. of nitrogen,
and 88 -9 c.c. of hydrogen, measured in each case at 0° C. and 760 mm.
The observations were reduced by comparison with lead, of which
the specific heat had previously been found to increase very nearly
uniformly from 0-0280 at - 220-5° C. to 0-0295 at - 85° C. After an
experiment had been made with the substance under investigation, a
similar experiment was made with a quantity of lead, so chosen that
about the same quantity of gas was evolved.

The specific heats of a number of substances at various ranges down
to — 188° C. were determined, and the following results were found for
diamond, graphite, and ice : —

SUBSTANCE.

18° to - 78°.

- 78° to - 188°.

-188° to -252-5°.

Diamond .

0-0794

0-0190

0-0043

Graphite . .
Ice

0-1341
0-463

0-0599
0-285

0-0133
0-146

Referring to the original paper for details, we may note here that
Dewar was able to determine with the calorimeter the latent heats
of oxygen, nitrogen, and hydrogen. He also found that the specific
heat of liquid hydrogen is 3 -4, the value which it has in the gaseous
condition.

Influence of Change of State on Specific Heat.— The difference

between the specific heats of diamond and graphite is an illustration of
the fact that the particular condition of a substance, crystalline or
amorphous, softened or hardened, affects its specific heat, though, as a
rule, the variation with such condition is not great. But as a substance
changes from the solid to the liquid or from the liquid to the gaseous
state, the specific heat may change very considerably. As typical, we
may take water and lead. We have : —

Ice. . -5 Water . 1-0

Solid lead -0314 Molten lead -0402

Steam (constant pressure) -34

As a rule, the change is in the same direction for other substances,
but there is no known relation between the specific heats of the same
substance in the different conditions.

Atomic and Molecular Heats. — There is undoubtedly some
relation between the specific heat of a substance and its atomic or
molecular weight. Thus for the less condensable simple gases the product

84

HEAT.

specific heat x atomic weight is very nearly constant for different
gases and independent of the temperature, when the specific heat is
taken in all cases at constant pressure, or in all cases at constant
volume. Again, for a very large number of solid elements the pro-
duct is not far from constant, though the constant is not the same as
for gases.

These relations imply that if we take quantities of different ele-
mentary substances proportional to their atomic weights, and therefore
presumably containing the same number of atoms, the heat capacities of
these quantities will be nearly equal. Or the heat capacity per atom is
the same for different elements, the gas atom, however, having a
different capacity from the solid atom. The product atomic weight x
specific heat is termed the " atomic heat."

In many classes of compounds, too, the product specific heat x
molecular weight, the " molecular heat," is not far from constant, each
class having as a rule its own constant, though in some cases each
constituent atom may be regarded as having its atomic heat — as a gas if
the compound is gaseous, as a solid if it is solid.

Without entering into minute detail, we shall give examples of the
evidence on which these statements are founded.

Taking first the case of gaseous bodies. Regnault found by the
method already described that at constant pressure : —

1. The specific heat is nearly independent of the temperature.

2. It is nearly independent of the pressure so long as this is constant

during an experiment.

3. The capacities for heat of equal volumes of different gases at

equal pressures are nearly equal — whence on the molecular
hypothesis the capacity for heat of different molecules is the
same.

But the further a gas departs from the behaviour indicated by pv = ~Rd
the less nearly does it fall in with these laws.

The following are some of Kegnault's results. The value for nitrogen
was calculated from those for air and oxygen : —

Simple Gases.

Atomic Heat or

GAS.

Atomic
Weight.

Specific Heat.

Product of
Atomic Weight
x Specific
Hear.

Hydrogen .
Air ....

1

3-409
0-2374

3-409

Oxygen ....

Nitrogen
Chlorine

16
14
35-5

0-2175
0-2438
0-1210

3-4800
3-4132

4-2955

Bromine.

80

0-0555

4-4400

QUANTITY OF HEAT. SPECIFIC HEAT.

85

Compound Gases.

Molecular Heat

GAS.

Molecular
Weight.

Specific Heat.

or Product of
Molecular
Weight

x Specific Heat.

Type AB—

CO ....

28

0-2450

6-8600

NO ....

30

0-2317

69510

HC1 ....

36-5

0-1852

6-7598

Type AB2—

C02 .

44

0-2169

9-5436

N2O ....

44

0-2262

9-9456

S02 . . . .

64

0-1554

9-9528

More easily condensable —

H20 . . .

18

0-4803

86454

H2S .

34

0-2342

7-9628

The last column confirms the result that for less condensable gases the
capacity for heat of equal volumes at equal pressures is constant, for
the weights of equal volumes of these gases are then very nearly pro-
portional to their atomic weights if elements, to their molecular weights
if compounds. The heat capacities of these equal volumes are therefore
proportional to the numbers in the last column. Dividing the " Mole-
cular Heat" for gases of the type AB by 2, and that for gases of the
type AB2 by 3, we get atomic heats 3-3 or 3-4, nearly the same as those
found for hydrogen and oxygen, a result indicating that tht- atoms
have the same capacity for heat even in combination. It will be noticed
that the atomic and molecular heats for the more easily condensable
gases are somewhat widely different from those of the more permanent
gases.

E. Wiedemann * has simplified Regnault's method of experiment on
gases, and has obtained results in close agreement. His value for air
at 0° was 0-2389, and he found no change between 0° C. and 200° C. for
air, hydrogen, and carbon monoxide. The capacities for heat of equal
volumes of these gases at the same pressure were nearly the same, but
the more condensable gases gave very different values from the others,
and as their temperature rose the specific heat increased.

No doubt the specific heat at constant volume is the more appro-
priate quantity for comparison, inasmuch as at constant pressure a
certain amount of the heat goes to push out the containing surface
against the external pressure. As we shall see later, the ratio of
the two specific heats is nearly constant for air, hydrogen, oxygen, and
nitrogen, so that we may regard the results obtained in the one case as

* PhU. Mag. (5), 2, p. 81, 1876. The remarks in the footnote on p. 71 apply also
to Wiedemann's experiments.

86 HEAT.

proportional to those which may be expected in the other. But only in
very few cases have the specific heats at constant volume been worked
out, and as yet only by Joly with the steam calorimeter.* He has found
that the specific heat alters with the density. Denoting the density by
p, he found that for air about a mean temperature of 50° specific heat
at constant volume = 0'17 151 + 0'02788/D, giving 0*17154 as the value
at 0° and 760 mm. For carbon dioxide about a mean temperature of
55° and a range of pressure up to 80 atmospheres, he found 0-1650
+ 0-2125/3 + 0-340p2. For hydrogen he found 2-40, with decided in-
dications of a decrease with increasing density. By varying the initial
temperature he was able to determine that there was no change in the
specific heat of carbon dioxide at constant volume between 10° 0. and
100° 0. if the density did not exceed -08, but that there was a rapid
increase at higher densities as the temperature decreased below 30°.

Swannf has employed the method of electrical heating to determine
the specific heats of air and carbon dioxide at atmospheric pressure. A
measured quantity of the gas was driven past a coil heated by an electric
current, and the energy given by the coil to the gas was measured
electrically. The rise in temperature of the gas was determined by
platinum resistance thermometers placed in the stream before it came to
and after it left the heating coil. Swann found that in terms of the
calory at 20° 0. the specific heats were

Air at 20° 0., 0-24173; at 100° C., 0-24301.
CO2at 20° C., 0-20202; at 100° C., 0-22141.

Extrapolating, we obtain for air at 0° C. the value 0-24141, a result
nearly 2 per cent, higher than that of Regnault.

Dulong and Petit'S Law. — From their researches on specific heat
Dulong and Petit were led to conclude that for many solid elements the
product

Atomic weight x specific heat = constant,

a law now known by their name. As pointed out already, it implies
that if we take a weight of each element equal in grammes to the
number expressing the atomic weight, and therefore, on the atomic
theory, containing the same number of atoms, the capacity for heat is the
same. Or the heat capacity per atom is constant.

Regnault made a very extensive series of researches to test this law,
and found that for most solid elements the product is nearly, but only
nearly, constant. This might be expected. For, as we have seen, the
specific heat changes with condition and temperature, and we could only
expect to find any exact relation for the different elements when we had
them in corresponding condition and at corresponding temperature, and
we do not yet know in what correspondence consists. On page 87 we
give a selection from Regnault's values of the specific heat, adding
carbon, boron, and silicon, with the limiting specific heats as given by
Weber. The product, Atomic weight x specific heat, the " Atomic Heat,"
is given in the last column.

* Phil. Trans., A., 1891, p. 73, and A., 1894, p. 943.
t Phil. Tram., A. 210, p. 199, 1910.

QUANTITY OF HEAT. SPECIFIC HEAT.

87

ELEMENT.

Atomic
Weight.

Specific
Heat.

Atomic
Heat.

Sulphur . .
Phosphorus .
Zinc ....

32
31
65

•1776
•1887
•0955

5-6832
5-8497
6-2075

Aluminium ,

27-5

•2143

5-8932

Iron ....

56

•1138

6-3728

Nickel . .

58-5

•1091

6-3823

Tin .

118

•0562

6-6316

Copper . .
Lead ....

63-5
207

•0951
•0314

6-0389
6-4998

Mercury (solid) .
Platinum .

200
197

•0319
•0324

6-3800
6-3828

Iodine . . .

127

•0541

6-8707

Bromine (solid) .
Sodium . .

80
23

•0843
•2934

6-7440
6-7482

Silver . . .

108

•0570

6-1566

Gold ....

196

•0324

6-3504

Carbon . .

12

•46

5-52

Boron * .

11

•50

5-50

Silicon . . .

28

•205

5-74

Neuman first pointed out that in certain compounds of similar con-
stitution the product, Molecular weight x specific heat, or " molecular
heat," is nearly constant. Regnault investigated the molecular heats
of a very large number of constants, and found that each class had its
own molecular heat. The results for the members of a class differ,
however, several per cent, from the mean.

Later, Kopp took up the investigation, and found that by assigning
to each atom its own atomic heat — not the same for all elements — a very
great number of quite different compounds in the solid state come under
the following rule (known as Kopp's Law) : " The molecular heat of a solid
compound is the sum of the atomic heats of the constituents."

The following are the atomic heats assigned : * —

C ... 1-8 Be ... 3-7 P ... 5-4

H ... 2-3 Si ... 3-8 S ... 5-4

B ... 2-7 O ... 4-0 Ge ... 5-5

and for other elements, 6"4.

Thus for ice the molecular heat is, by Kopp's Law,

2x2-3 + 4 = 8-6
whereas experiment gives 0*474 x 18 = 8*5.

For calcium carbonate, CaCO3, the calculated molecular heat is —

while experiment gives 0'203 x 99-9 = 20'3.

For sodium chloride, Nad, the calculated molecular heat is — •

6-4 + 6-4 = 12-8
while experiment gives -214 x 58*4= 12-5

* Nernst, Theoretical Chemistry, p. 154.

CHAPTER VII.
CONDUCTIVITY.

The Passage of Heat from one Body to Another — Conductivity — Differs enormously
in different Substances — General Remarks on Conductivity in the Three States
— Definition of Conductivity — Diffusivity — Emissivity — Measurements of Con-
ductivity— Pe"clet's Method — Bar Methods of Despretz, Forbes, Neumann, and
Angstrom — Gray's Method — Berget's Experiment on Mercury — Experiments of
Wiedemann and Franz — Kundt's Experiments — Senarmont's Experiments on
Crystals — Lees's Experiments— Lundquist — Weber — Conductivity of Gases-
Experiments of Stefan, Kundt, and Warburg.

Transference of Heat by Conduction.
The Passage of Heat from one Body to Another.— There are

two modes in which heat is transferred from one portion of matter to
another — conduction and radiation. In conduction, the matter receiving
the heat is in contact with the matter from which it receives it, and the
temperature falls continuously along the course by which the heat is
flowing. If, for example, I put one end of a poker in the fire and hold
the other end, the heat is conducted along the poker from the warmer
to the colder portions, the heat passing down the slope of tempera-
ture, warming the iron as it travels, so that all the intervening portions
of the poker are intermediate in temperature between that of the fire
and that of my hand.

In radiation, the matter receiving the heat is not in contact with
the matter from which it receives it. If I warm my hands before a fire,
I do so by radiation, the heat received by my hands passing through the
intervening air without warming it. In fact, in the case of radiation,
any matter through which the radiation passes may be colder or hotter
than either or both of the bodies between which it is passing. We
cannot, therefore, suppose that the energy passes from one to the other
as heat, but that, on leaving the sender, it is converted from heat-
energy into another form which we term radiant energy, to be recon-
verted into heat on reaching the receiver.

Conductivity. — We have already noted as the chief characteristic
of heat conduction, that the heat always travels from hotter to colder
matter. The greater the slope — that is, the difference of temperature
between neighbouring points a given distance apart — the greater the
amount of heat conducted. If a thermometer at the temperature of the
room be placed in a vessel of hot water, it rises much more rapidly at
first, when the temperature slope between the water and the mercury is
great. The rate of rise gradually slackens till, ultimately, when the
mercury and the water are at the same temperature, there is no further

88

CONDUCTIVITY. 89

passage of heat. Without at present giving a precise significance to the
term, we may call this power of conducting heat " Conductivity."

The conductivity varies enormously in different substances. We
may, e.g., light a wooden match, and let it burn down nearly to the
fingers, without receiving any appreciable quantity of heat through the
wood, while an iron wire held by the side of the match rapidly becomes
uncomfortably hot, and a copper wire can only be held for a few seconds.
One end of a glass-rod may be melted in a flame while the rod is held in
the fingers two or three inches away from the melted part, while a copper
rod of the same diameter, with one end in the flame, will be too hot to
handle at a point many inches from the flame. The difference in con-
ductivity may be illustrated by smearing the two rods with wax, and noting
the difference in the times taken by the wax to melt along the two.

The apparent coldness to the touch of metals, as compared with other
solids, is explained at once by their greater conductivity. The skin is
generally at a higher temperature than the metal, rapid conduction
ensues on contact, and the hand loses much heat. If we touch a piece
of wood at the same temperature, the amount conducted for the same
slope is very much less, and the hand loses heat much more slowly. If
the hand, however, is colder than the metal and wood, then it receives
more heat from the metal, which feels much hotter than the wood.

If we paste thin pieces of paper on to two blocks, one of wood and
one of iron, and hold them with the paper exposed to the flame of a
Bunsen burner, the piece on the wood rapidly chars, while that on the
iron remains unburnt. The flame may be regarded as supplying nearly
the same quantity of heat in each case. Though the paper is itself a
bad conductor, it is thin, and a comparatively small difference of tem-
perature between the two sides will establish a sufficient slope to carry
away all the heat supplied. If then, on the farther side, there is a good
conductor, such as iron, it will rapidly convey away all the heat supplied
to it, and will not allow the temperature of the side of the paper in
contact with it to rise very high. The other side of the paper, not being
very much higher in temperature, does not rise to the charring-point.
If, however, on the farther side of the paper, we have a block of wood, it
does not rapidly convey away the heat supplied, and so the temperature
of the side of the paper in contact with the wood rises, that of the other
side rising still higher, and soon the charring-point is reached.

We may illustrate this in another way. If a thin paper cup or tray
is constructed so as to hold water, the water in the cup may be boiled
over a flame without burning the paper. The water prevents the inner
surface of the paper from rising above 100°, while a sufficient slope will
be established in the paper by a not very much higher temperature for
the outer face, to carry through all the heat supplied by the flame, and
the paper, therefore, remains unburnt. If we replace the paper by a
thin copper vessel, the copper probably does not rise more than a fraction
of a degree above the temperature of the water. As an effect of this,
it may easily be observed that the flame does not come in contact with
the heated surface. The temperature of the surface remains too low
to permit of combustion, and there is a layer of unburnt gas, of greater
or less thickness, according to the temperature of the surface with
which it is in contact.

90 HEAT.

In steam boilers we have a somewhat similar effect. A temperature
slope is established from the flame to the water within the boiler, but
the outside temperature of the metal may be not greatly above that
of the water, if the thickness is not very great. If, however, there is
a badly conducting incrustation within the boiler, the conditions are
altered. In order that the water may receive the same quantity of
heat, the outer side of the incrustation must be much hotter than the
inner ; the metal must therefore be much hotter than the water. It
may therefore rise to the temperature at which it is easily burned by
the flame, or it may even be softened through heat so much as to give
way to the internal pressure.

Another effect of the incrustation may be noted. In order to get
the same quantity of heat into the water through the badly conducting
wall, a higher temperature may be necessary within the furnace, and
then there will be a greater quantity of heat passing through every
outlet at which there is waste.

On the other hand, if we surround the outside of a boiler with a
badly conducting layer of boiler " clothing " to keep its heat in, a very
large difference of temperature between the two sides of the layer
will correspond to only a small flow of heat from the boiler, so that
though the inside may be at the temperature of the water, and the
outside at that of the air, no very great quantity of heat escapes.

Of course, the same principle is used in our own clothing. We put
on non-conducting substances of such thickness and quality that the
temperature slope from our bodies to the outer air shall not carry off
more heat than we are willing to part with.

General Remarks on Conductivity in the Three States.—
Common observation tells us that solids differ enormously in their con-
ductivity, the metals being the best conductors. They range from copper
and silver, which are the best, downwards through minerals and wood to
furs and feathers, which are probably the worst.

Liquids are Bad Conductors. — Though we may readily heat liquids
they are in general bad conductors. When we boil water in a vessel,
we always supply heat from below, so that the heated water may rise
and give place to colder water from above, which is heated in its turn.
The hot water rising, comes in contact with much colder water, so that
the temperature-slope is very steep, and the heat is soon shared between
the hotter and colder portions, even though the liquid is a bad conductor.
This process — termed " convection " — is therefore really only conduction
aided by a transportation of the hotter matter — as it were, an artificial
deepening of the temperature-slope. If, instead of heating the water
from below, we heat it from above, then it takes a very much greater
time to get heated throughout. It is quite easy to boil water on the
top of a test-tube, holding the bottom part in the hand, and receiving
no sensible amount of heat. The experiment is still more striking if a
small quantity of ice is kept at the bottom of the tube by a little wire
gauze (Fig. 61).

If convection is prevented, we find that liquids are generally to
be ranked with the worse conducting solids.

Gases are also bad conductors, worse even than liquids. Here again
the badness of conduction is masked by convection. The air gets hot

CONDUCTIVITY.

91

on a summer day, not solely by conduction from the warm ground, but
through the joint effect of convection and conduction.

The tremulous motion of objects at a distance, seen through hot air,
is due to this convection, as already explained in Chapter V. Unequally
heated masses of air are moving up and down irregularly, and refracting
the light, making it appear to the eye to come now from one point now
from another.

When convection is prevented, it is found that air is a bad conductor,
and probably, on this bad conduction of air, depends the non-conducting
power of woollen clothes, blankets, and other loosely woven textures.
The wool is in itself a very bad conductor, while, through its being
matted together, it entangles the air, which is a still worse conductor,
so as to prevent convection currents, and thus the whole layer of wool
and air conducts badly.
Even were the wool it-
self a good conductor,
the same effect might be
produced, for the path
by which the heat must
get from one side to the
other, travelling through
the wool, is an indirect
one, and so the wool
slope of temperature is
very gradual. * This
probably explains the
efficacy as a non-con-
ductor of slag wool,
a material consisting of
blast-furnace slag, blown
out into fine fibres by
steam. The slag is itself
not so bad a conductor,
but, when loosely packed,

the path from one side to the other of a layer of the wool is very
much longer than the thickness of the layer. For example, the path
from A to B (Fig. 62) through the wool is much greater than the direct
path. Then, again, the cross-section of the material is in reality
only a fraction of the whole cross-section of the layer, both conditions
combining to diminish the conduction through the wool, while the

* From the table given later, it will be seen that wool and cotton are nearly equal
in their conductivities, yet undoubtedly woollen clothes are the warmer. This is in
great measure due to the fact that woollen cloth is more open in texture, and there-
fore holds a thicker layer of air for the same weight. Probably, also, the more open
texture of the wool allows a rather freer exchange between the air within the
meshes and the air outside, so that the vapour-laden air can pass from the skin to
the outside. The more closely woven cotton hinders the passage of the air, and it
is more likely, in cooling down, to deposit its moisture in the cloth, this deposition
being aided by the hygroscopic property of cotton. If now evaporation takes
place freely from the outer surface of the moistened cloth, the temperature is
lowered there, and heat is more rapidly conducted from the skin, which may thus
be chilled.

FIG. 61.

HEAT.

air imprisoned in the fibres cannot form convection currents to aid the
transport of heat.

The low-conducting power of air thus entangled by a loosely packed
solid may be illustrated by filling several test-tubes respectively with
wool, slag wool, loose sand, and coarse copper filings, and inserting
a thermometer in each. When all are plunged suddenly in boiling
water, there is no very great difference in the rate of
rise of the thermometers, for the heat comes chiefly
through the air in each case.

Safety Lamp. — We have already mentioned that a
flame does not actually come in contact with a thin
vessel containing water, or, rather, that the metal
conducts the heat away so readily that the gas in
contact with it is lowered below combustion point. This
is made use of in the Davy safety lamp. A very simple
experiment will serve to illustrate the principle of the
lamp. If a burner is placed a short distance below a
piece of iron gauze, and the flame is lighted, it usually
burns under the gauze only (as in Fig. 63A). Or, if the flame is lighted
above the gauze, it burns over it only (as in Fig. 63fi). The gauze is a
fairly good conductor, and carries the heat away rapidly to the surround-
ing parts, and the metal thus sharing the heat has a large surface, which
can radiate the supply away without rising to the combustion tempera-
ture. The gas in contact with the gauze on the other side from the
flame is thus kept from igniting. But if the gauze has not a large
area, or if the flame is very hot~ the supply of heat to the gauze may
not be got rid of with sufficient rapidity, and the gas on the other side of
the gauze is ignited.

The Davy Lamp consists of a brass base containing the oil-reservoir,

B

FIG. 62.

FIG. 63A. FIG. 63B.

Showing Principle of Safety Lamp.

and an iron gauze chamber, in which the flame burns. If fire-damp is
present in the air in any quantity, it burns inside the lamp over the oil
flame with a bluish light, but the gauze is at first able to radiate off the
heat, or carry it away to the body of the lamp sufficiently rapidly to
prevent the gas outside from rising to ignition point. If, however, the
flame inside the lamp gets very large, there is great danger of the lamp
becoming overheated. The lamp is, therefore, not by any means
absolutely safe, but its indications are sufficient to give warning of
danger, and on such warning it should be extinguished.

CONDUCTIVITY. 93

Definition of Conductivity. — We may now give a precise signifi-
cation to the term as follows : — The conductivity of a substance is the
quantity of heat conducted per second through a square centimetre in
the substance, when the temperature changes in a direction perpendicular
to the area at the rate of 1° C. per centimetre. If then AB (Fig. 64) is
a plane over which the temperature is 0° ; CD, EF two parallel planes
at £ cm. distance, one on each side, over which the temperatures are 6 + |
and Q -\ respectively, the quantity of heat passing in one second through
1 square centimetre of the plane AB is the conductivity of the sub-
stance at the temperature 0. We shall denote this quantity by k. If
the area be A square centimetres and the time t seconds, the quantity
passing through must be If At, for, the circumstances are the same for
each square centimetre and for each successive second.

It is not easy to make direct exact experiments on the flow of heat
with different slopes of temperature, but general experience might lead
us to expect it to be proportional to the steepness of slope, or the fall of
temperature per centimetre. We may roughly verify this by immersing
a thermometer in warm water, and noting the rate of
rise. If we keep the thermometer moving in the liquid
all the time, the outside layer of the glass has probably
nearly the same temperature as the liquid, and the rise
per second is nearly proportional to the distance of the
top of the column from the final point reached — that is,
the quantity of heat received per second is nearly pro-
portional to the difference of temperature between the
mercury and the water outside.

This is evidently complicated by the expansion of

3

the glass, a thermometer giving true indications only FIG. 64.
when both glass and mercury are at the same tempera-
ture. A still better verification consists in immersing a small calorimeter
containing water in an outer vessel also containing water, but at a different
temperature, keeping the contents of both vessels well stirred, and it will
be found that the quantity of heat passing from one vessel to the other is
roughly proportional to the difference of temperature. We shall, there-
fore, assume that the quantity of heat conducted is proportional to the
slope of temperature. Such experiments as these bear to the verification
of the law the same relation that experiments with Atwood's machine
bear to the verification of the laws of motion. We must regard them as
suggesting, rather than verifying. The more exact verification is in the
agreement of experiment with calculations based on the assumption of the
truth of the law.

If the temperature-difference between two neighbouring points, d apart
at equal distances, one on each side of the area A through which heat is

flowing, is equal to r, then -. is the rate of variation per centimetre or

61

the slope of temperature, and assuming that the flow of heat is propor-
tional to this, the total quantity flowing through A in time t is given by

a
where Jc is the conductivity at 6, the temperature of A.

HEAT.

Diffusivity. — A well-known piece of apparatus used to illustrate
conductivity, was devised by Ingenhousz. In this a number of equal
metal bars are placed in a row, with their lower ends in a vessel into
which hot water can be put (Fig. 65). The rods may be smeared with
beeswax, and the rate of melting along the different rods compared.
But it is clear that the propagation of the high temperature along the
rods depends not only on their conductivity, but also on their specific
heats. If, for example, a cubic centimetre of one rod has twice the
heat capacity of the same volume of another, it requires twice the
quantity of heat to raise its temperature to the same extent, and if the
melting extends equally quickly along the two rods, the conductivity
must be twice as great in the one case as it is
in the other : that is, we must consider, not
only the conductivity, but the ratio —

Conductivity K

FIG. 65. — Ingenhousz's
Apparatus.

Heat capacity per cubic centimetre ps'

where K is the conductivity, p the density, and s
the specific heat. This quantity is termed the
diffusivity of the substance. We may regard it
as the conductivity for temperature as distin-
guished from the conductivity for heat.

Another quantity which plays an important
part in researches : on conductivity is the
emissivity of a surface. We may define this as
the heat lost by the surface per square centimetre
per second, per degree of excess of temperature
of that surface above the surroundings.
Measurements Of Conductivity. — From the definition of conduc-
tivity we have

quantity of heat passing through a sq. cm. per sec.
slope of temperature perpendicular to the area

and if we can separately determine numerator and denominator, we
obtain It.

It would appear then that the simplest way to measure conductivity
would consist in catching the heat passing through some known area
and measuring at different instants its rate of flow, and at the same
instants observing the temperatures at two points near the area to give
the temperature-slope. But while it is easy to measure the total quan-
tity of heat which a body has gained in a given time, i.e. the difference
between its gain and loss, it is quite another matter to measure the rate
at which it is gaining at any instant. Indeed, the difficulty is not unlike
that which a merchant might experience in trying to estimate his actual
rate of profit at any instant. He may succeed in finding his exact profit
or loss during a year, buo he can hardly trace all the transactions and
estimate the profit or the loss which is accruing in any one minute. And,
again, while various methods of measuring the temperature at a point
may be successful, it is not always easy to carry out the measiirement
without affecting and even diverting the flow of heat. Through these
difficulties there is no measurement in which more widely different

CONDUCTIVITY 95

results have been obtained by different observers ; the value of the con-
ductivity for a substance as given by one kind of experiment being
perhaps several times the conductivity as given by another kind of
experiment on the same substance. The chief difficulties have only
comparatively recently been so far surmounted that different methods
give fairly accordant results. We shall not attempt to give here a com-
plete account of the subject,* but rather seek to illustrate it by describing
a few typical experiments which have given good results.

Solids— Pfalet's Metlwd. — Pioneer work was done in the measurement
of conductivities by Peclet. He used a calorimeter, the bottom of which
consisted of a plate of the substance to be tested, while the sides had
very low conductivity. The calorimeter contained a known quantity of
water at a known temperature, and when it was plunged into a vessel
of warm water at a known temperature, the rate of rise of temperature
of the water in the calorimeter was observed. If A was the area of the
plate, K its conductivity, d its thickness, Q the quantity of heat coming
through per second, 0 the temperature difference of its two faces, then
if the heat-flow was normal to the surface

whence Qd

= A0

The chief difficulty consists in measuring 6. Let us suppose that the
water is at rest and that the heat flows steadily, i.e. that the same quan-
tity flows across each cross-section of plate and water alike. Let the
temperature in the water on the two sides at distances dv d2 from the
nearest faces differ from the temperature of the faces by 9V #2, and let
the conductivity of water be /c1. We have

KA0
Q" d =: dl

If we observe the water temperatures at the distances dv d2 from
the plate, and assume that these are the temperatures of the two faces
of the plate, we get for the conductivity

Qd Qd 6

or the value we find ought to be multiplied by

to give the true value.

Peclet was well aware of this source of error. With the cooler vessel
above, convection would come into play, and so reduce the difference of
temperature between the plate surfaces and the water round the thermo-
meters and virtually reduce dl + d2, but not sufficiently. Peclet aided
convection and virtually reduced d^ + dz still farther by brushing the

surfaces all the time. He supposed that thus ~r~ was rendered
negligible, i.e. that the water against the plate, being continually and
* An account will be found in Winkelmann's Handbuck der Physik, vol. ii.

HEAT.

violently renewed, had the same temperature as that further off round the
thermometer. But though, no doubt, * 2 was small, it is to be re-

f£

membered that with metals -j might be great. Thus for copper and

water, we now know that it is about 700 ; so that a layer of water which
remained against the plate on each side of it, and TTVff of its thickness
would make the result about double the true value. That P^clet failed to
remove such a layer is shown by the fact that for copper he obtained
a value about five times that now accepted. In fact the layer of water
of sloping temperature was practically 2/700 of the thickness of the

plate on each side. If - 1 were not large the layer might be still so

thick, and yet the results would not be far wrong. Accordingly, Pellet's
work for low conductors is fairly confirmed by later work.

Bar Methods of Finding the Conductivity of Metals.— There are

FIG. 66. — Despretz's Bar Experiment.

two types of experiment in which a long metal bar is heated at one end,
and the conductivity is deduced from temperature observations along the
bar. In one — the steady-flow method — one end is kept at a constant
high temperature until the temperature at each point has come to a
steady value. As no part of the bar is now gaining or losing heat,
the heat conducted per second through any cross-section must equal that
emitted per second from the surface beyond that section, and if the rate
of emission can be determined, an observation of the slope of tempera-
ture at the cross-section will give the conductivity necessary to supply
heat at the rate at which it is emitted beyond. In the other type, one
end of the bar is subjected to periodic variations of temperature, and
consequently waves of varying temperature travel down the bar. The
conductivity is calculated from the march of these waves.

To the first method belong the researches of Despretz, Forbes,
Wiedemann and Franz, and, more recently, Tait and others.

Despretz's Experiments. — Despretz used bars of different metals
of the same dimensions, and with the surfaces varnished in the same
way, so that the loss of heat to the surroundings should be the same for
the same excess of temperature. Thermometers 1, 2, 3, 4, 5, 6, 7 (Fig. 66)
were placed in holes in the bar filled with mercury.

Fourier had calculated the flow of heat along such a bar, assuming that

CONDUCTIVITY. 97

the law of conductivity is true, and that the loss to the surroundings from
any part of the bar is proportional to the excess of temperature above
the surroundings. The curve of temperature ab enabled Despretz to
verify Fourier's calculation, and at the same time to obtain comparative
results for the conductivity in the different bars. The following special
case may sorve to show that it is possible to compare conductivities by
means of the temperature-curves.

Suppose that two bars have the same cross-section, and that their sur-
faces are similarly treated, so as to lose the same amount of heat per sq.
cm. for the same excess of temperature above the surroundings. Let the
hot end of each be at the same temperature, but let the temperature-curve
in the first case slope n times as quickly as in the second, so that if T, T'
(Fig. 67) represent equal temperatures on the two bars, and the first is at
a distance OM from the end of the bar, then the other is at a distance
ON = ra.OM. Hence, we may divide the
bars into corresponding elements at the
same temperature, those of the first bar

having - the length of those of the
n

second. Then the heat lost beyond any

point of the first bar is — of the heat
n

lost beyond the point of the second bar
having the same temperature.

If, now, the conductivities are 7^ and FIG. 67.

first bar is to the heat conducted across N on the second as &x x slope at M is
to&2 x slope at N. But the total fall of temperature at M in length 1 is equal
to the total fall at N in length n, or the slope at M is n times that at N.
Then the quantities of heat conducted across M and N are as k^n to &2.

The heat conducted across a section is equal to the heat lost beyond
that section, and we have just shown that the ratio of the heats lost beyond
corresponding points is as 1 to n.

Therefore, 1 : n = k^n : kz

or, &2 = k-^n2.

Forbes's Experiment. — Forbes modified the method so as to make
it give absolute results for the conductivity of an iron bar. His
apparatus is shown in Fig. 68.

The end of the bar is immersed in a constant high temperature bath,
and after some hours' heating, the steady state is reached, the temperature
at various points of the bar being shown by thermometers, as in Despretz'
experiment, inserted in small cavities in the bar containing mercury.

If the temperature falls through T in a small distance d at any point
of the bar, then, as we have seen, the quantity of heat conducted across
the section of area A in t seconds is

Q-Ml.

a

The thermometers give r ; A is the cross-section of the bar, and if we
can find Q, this equation gives k.

In order to find Q, Forbes made a subsidiary experiment with a
second bar of the same material, but of much smaller dimensions, also

G

HEAT.

furnished with a thermometer. The bar was raised to a high tem-
perature and then placed near the other, which was now cool, so that
the surface of the small bar was exposed as nearly as possible to the
same conditions, as to loss of heat for the same temperature, as the large
bar, and the rate of cooling was observed. Knowing the capacity for
heat of the bar and the area of its surface, the rate of cooling at any
temperature gave the quantity of heat lost by each unit of area in a
given time for that temperature, and thus the quantity of heat lost per
second by every part of the larger bar in the first part of the experiment
could be determined, and so Q could be calculated.

Determining the conductivity at different sections of the bar at
different temperatures, Forbes was further able to show that the con-
ductivity of iron decreases with rise of temperature.

Tait repeated Forbes's work on the same bar, and used also bars of
copper and other metals in the same manner.* Stewart f has also
carried out experiments on the conductivities of iron and copper by the

FIG. 68.— Forbes's Bar Experiment.

method of Forbes, using in place of thermometers a thermo-electric
junction inserted in small holes in the bar under observation.

Lees, in the course of some experiments to be described below, has
also used the same method for brass.

Neumann and Angstrom's Method.— The second or temperature-
wave method of measuring conductivities was used by Neumann, who
heated one end of a bar of the substance experimented on, and then
cooled it. Observation of the march of the cooling along the bar enabled
him to determine the conductivity.

Angstrom alternately heated and cooled one end of the bar, and after
a time waves of temperature-disturbance travelled regularly along the bar.
The rate of progress at different points and the rate of diminution in the
amplitude of the disturbance enabled him to calculate the conductivity.!

Gray's Experiments. — A simple mode of determining con-
ductivity of metals has been used by J. H. Gray.§ A wire a few
centimetres long, and about 0'2 cm. diameter, was hung vertically with
its upper end soldered into the bottom of a copper vessel containing
boiling water and its lower end into a copper ball, 5-5 cm. in diameter,
drilled with a hole in which a thermometer was inserted. The wire was
surrounded by cotton wool, and the loss from the sides was negligible.

* Tait's Heat, chap. xiv. t Phil. Trams.,- 1893, A., p. 569.

t "Heat," Ency. Brit., 9th ed., or Tail's Heat, chap. xiv.
§ Phil. Tram., A., Part I., 1895, p. 165.

CONDUCTIVITY.

99

A thermometer in the boiler gave the temperature of the upper end of
the wire, and that in the ball gave practically the temperature of the lower
end. The mean slope of temperature along the wire was thus known.
The quantity of heat conducted per second was determined by the rate of
rise in temperature of the copper ball. Assuming that the conductivity
varies uniformly with the temperature, the experiments gave the con-
ductivity at the mean temperature of the wire, about 53°. The value of
the method lies in its applicability to metals which cannot easily be
obtained in large masses.

Berget's Experiments on Mercury.— Berget* determined the

conductivity of mercury, using a column of the liquid in a vertical glass
tube surrounded by a much wider column of the same liquid, both being
heated at the top by steam or mercury vapour, and conducting the heat
downwards to a base kept at 0°. The central column terminated in an
ice calorimeter, the central column and the wider surrounding column
being at the same temperature at the same level. The heat flow was
vertically downwards, a conclusion confirmed by the temperature-slope,
which was uniform down the column. The quantity of heat arriving
at the lower end was determined by the amount of ice melted. The
temperature at various points along the column was determined by
thermo-electric junctions, and thus the temperature slope was known. He
found that the conductivity was constant between 0° and 100°, and that
it then diminished.

The following table gives some of the results obtained for metals,
the different values for the same metal by the same observer being
obtained with different specimens.

Metals.

Conductivity.

Observer.

Iron ....

•209(1- '00147*)

Forbes.

•197(1- -00002*)

Tait.

•175(1- -0015*)

R. W. Stewart.

•164

Neumann.

11 •

•199(1 - -00287<)

Angstrom.

Copper (1)

1-08(1 + -0013(1)

Tait.

,, (2)

•71(1 + -00141)

»>

1-108

Neumann.

(1)

1 -027(1 - -00214*)

Angstrom.

(2)

•983(1 - -00152«)

11

1-12(1 --0010

E. W. Stewart.

(1)

•9594 ^

J. H. Gray.

(2)

•8884 between 10°

(3)

•8612 > and 97°

Very impure

copp

er(4

•3497 ( say at 53°

>j

»

(5

•3198 ;

Silver .

•9628

Gold .

•7464

Platinum

•1861

Mercury

•02015 between 0" and 100°

Berget.

>i •

•02( 1 - -000454) between 0° and 300°

ii

»

•01516 at 4-5°

H. F. Weber.

•01620 at 17°

Brass . ...

•27 between 25° and 35°

Lees.

Journal de Physique, viii., 1888, p. 503.

100

HEAT.

Experiments of Wiedemann and Franz. — Relation between Heat

Conductivity and Electric Conductivity* Wiedemann and Franz experi-
mented on the relative conductivities of metal bars working by the same
general method as Despretz, but using a thermopile instead of thermo-
meters to give the temperatures along the bar, the one junction of the
pile being brought by a suitable arrangement in contact with any desired
point of the bar. They found afterwards that the same results were
given by using thermometers. The bars were silver-plated, and polished
so as to have the same emissivity, and the chamber in which they were
placed could be exhausted. The results obtained were comparative.
They appeared to show that there is some connection between conducting-
power for heat and for electricity. For the metals were found not only
to follow the same order for the two conductivities, but in many cases
the numbers bore nearly the same ratio to each other.

More recent work has confirmed this supposition. The following
are some of the values for metals and alloys of the ratio thermal con-
ductivity/electrical conductivity or k/c at 18° C. as determined by Jaeger
and Diesselhorst (Phys. Tech. Reichsanstalt Wiss., Abh. 3, 1900), together
with the temperature co-efficient of the ratio.

Copper, pure (1)
(2)

Silver
Gold
Zinc

Cadmium
Lead
Tin

The electron theory of conduction for heat and for electricity (J. J.
Thomson, Corjwscular Theory of Matter) gives an explanation of the
connection between the two quantities. According to that theory the
ratio should be proportional to the absolute temperature, i.e. should have
a temperature co-efficient 0-00367, and at 0° 0. its value should be
6 '3 x 1010. The table shows that for many metals the values are not
very different from those given by the theory. With alloys considera-
tions of thermo-electric effects probably come in to add to the effective
resistance, and so to diminish the conductivity and increase the ratio.

Kundt's Experiments on the Relation between the Velocity of Light in
Metals and their Electric and Heat Conductivities. — Kundtt determined
the refraction of light by exceedingly thin prisms of various metals and
taking the velocity in each metal as being inversely as the refractive
index, he found for red light —

lO-10Jfc/c,

Temperature.
Co-efficient.

10-«fc/o.

Temperature.
Co-efficient.

6-65
6-71
6-86
7-09

0-0039
0-0039
0-0037
0-0037

Platinum, pi
Iron (1)

» (2)
Steel .

ire . 7-53
. 8-02
. 8-38
. 9-03

0-0046
0-0043
0-0044
0-0035

6-72

0-0038 Bismuth

. 9-64

0-0015

7-06

0-0037

Constantan

. 11-06

0-0023

7-15
7-35

0-0040 Manganin
0-0034

. 9-14

0-0027

Silver

Gold

Copper

100
71
60

Platinum
Iron

15-3
14-9

Nickel
Bismuth

12-4
10-3

values not very different from the relative values for heat or electrical
conductivity. The actual refractive index from air into silver was
0'27, and there was in this case but little dispersion. In other

• Pogg. Ann., Ixxxix., 1853, p. 497. f Phil. Mag., xxvL, 1888, p. 1.

CONDUCTIVITY.

101

metals there was considerable dispersion. Probably the refractive index
for very long waves should be compared to bring out any true physical
relation, and red light is only the best approximation which could be
made to such long waves. At present we can only say that Kundt's
results point in the direction of some connection between the three sets
of phenomena, light velocity, heat conductivity, and electric conductivity.

Solids of Low Conducting Power and Crystals.— Many experi-
ments have been made to determine the conductivity of non-metallic
solids. With amorphous solids usually some method similar to that of
Peclet has been used, and the range of conductivity has always been
found to be far below that of metals. Taking first the conductivity of
crystals, we might expect that this would be different along the different
axes, an expectation verified by experiment. The subject was first
studied by Senarmont, who used a very simple method, preparing a
plate of the crystal with a small hole through it. The plate was covered
with a film of beeswax, and a silver wire passing through the hole was
heated. The heat was conducted through the crystal, and the beeswax

FlG. 69 (a and 6). — Showing Conductivity in Crystals.

was melted. When the conduction was the same in all directions in the
plane of the section, as in a plate of quartz cut perpendicular to the axis,
the figure of the melted wax was circular, as in Fig. 69a. When it
differed in different directions the figure was elliptical as in Fig. 69&.
The conductivities along the two axes of the ellipse may be shown to be
proportional to the squares of the axes.

A method of experiment developed by Lees * is especially suitable for
the determination of the conductivity of crystals. A long brass bar with
diameter 1'93 cm. was used as in Forbes's experiment, and its conduc-
tivity was determined by his method as -268{1 + -002(£ - 17°)}. It was
then cut in the middle and a plate of the crystal, of the same area
as the cross-section of the bar, was inserted between the cut faces, these
being amalgamated to give good contact with the crystal. Temperature
observations along the bar, made by means of thermo-electric junctions,
gave the temperature of each face of the plate and therefore the tem-
perature slope through it, while the known conductivity of the bar and
the temperature slope in it adjacent to the plate gave the rate of passage
of heat into and out of it. From these data the conductivity could be
determined.

Lees found the following conductivities between 25° and 35° : —

Quartz, along the axis .... '0299

,, perpendicular to the axis . . '0158

Iceland spar, along ,, ,, . . '0100

,, perpendicular to ,, ,, . . *0084

Mica, ,, cleavage planes . "0018

* Phil. Trans., A., 1892, p. 481.

102

HEAT

The method was used for various other solids and the results obtained
were in close agreement with those obtained by another and more exact
method which he devised later, now to be shortly described.

Lees's Disc Experiments* — The substance to be tested was formed
into a disc X (Fig. 70), say 4 cm. in diameter and 2 or 3 mm. thick. This
was placed between two copper discs Gl C2 of the same order of thickness,
continuity being ensured by a negligible layer of glycerine. Against
one copper disc was laid a flat coil through which a current could be
passed so as to supply heat at a determinate rate, and on the coil was a
third copper disc 03. Fig. 70 represents diagrammatically the pile of
discs thus made, suspended in a constant-temperature enclosure. The
surfaces of the pile were varnished to give them the same emissivity,

h say, so that the rate of heat
emission from a square centi-
metre v degrees above the
enclosure would be hv. The
copper discs would each be of
uniform temperature through-
out to a near approximation,
and their temperatures were
taken by thermo-electric junc-
tions inserted in small holes
drilled in at their edges. Let
their temperatures, measured
as excesses over the tempera-
ture of the enclosure, be as
indicated on the right hand

S.t

v.

Copp er

c,

s/
s

S2

Copper

c,

",

Vj+Vz

e
v.

Substance X

Copper

c.

Uniform temperature enclosure

FIG. 70. — Lees's Disc Experiments on Conduc-

tivity. — CiC2C3, copper discs ; X, disc to be of the figure, and let their

tested; Viv2v3, temperatures above the en- emitting surfaces have areas

closure; ., ,lWfc emitting surfaces. ag indicated on the left hand>

Let the rate of heat supply by

the coil be H. Then taking the mean temperature of X as the mean of
the temperatures of

and 02, we have

+ Jis2v2

whence h could be determined, since H was known electrically.
The heat flowing through X may be put as

and it may be taken as equal to the mean of the heat flowing into it
from Gj and that flowing out of it to C2. But the heat flowing from Ca
into X is equal to that emitted from X and C2, and that flowing from X
into 02 is equal to that emitted from 02. Whence we have

n^=l!

d 21

-.

v, +

* Phil. Trant., A., 1898, p. 399.

CONDUCTIVITY.

103

Knowing h from the equation for the heat supply H, we can hence find
K. We have omitted all account of corrections. For these the original
paper may be consulted.

The method gave the conductivity at different temperatures, and
assuming that it might be put in the form

the values which Lees found for the conductivity at 35°, and for the
co-efficient a are

Window glass
Sulphur .
Ebonite
Shellac .

•00245
•00067
•00042
•00058

+ •0025

- -0036
-•0019

- -0055

We add some conductivities which he found by the divided bar
method for a mean temperature of 30° : —

Marble .

Slate .

Paraffin

Paper .

Silk

Mahogany

Cork

•00709

•00475

•00061

•000315

•00022

•000465

•000129

The following conductivities are taken from Lord Kelvin's article on
Heat in the Encyclopedia Britannica (9th ed.). They probably need
revision : —

Substance. K

Authority.
'Forbes and Lord Kelvin ob-
tained by observations on
underground thermome-
ters, noting the delay of
the heat of summer and
the cold of winter.

Ayrton and Perry.
Peclet.

Sandstone of Craigleith Quarry -01068
Underground strata . . '005
Trap rock of Calton Hill . -00415

Porphyritic trachyte . . '0059

Oak across fibres . . . -00059

Fir along fibres . . . -00047

Fir across fibres . . . -00026

Cork -000029

Writing paper . . . -000119

Carded wool .... '000122

Finely carded cotton wool . -000111

Eider-down .... '000108

Conductivity Of Liquids. — Usually conduction in a liquid is
greatly assisted by convection. If, however, convection be prevented
by heating the liquid from the top, it is found that except in the case of
mercury the conducting power is low. The first quantitative experiments
appear to have been made by Despretz, who heated a column of water
from above, and observed the temperature at various depths when the
steady state was reached. He found that the slope of temperature was

104

HEAT.

in accordance with Fourier's calculation founded upon the law of con-
ductivity, and thus proved that the law was the same as for solids.

Other experimenters have since made determinations of the con-
ductivity of liquids. Thus Lundquist, employing Angstrom's method,
obtained for the conductivity of water at 40'8° K = -00156. H. F. Weber *
used a disc method, the liquid layer, '231 cm. thick, lying between two
copper discs 16 cm. diameter, and being kept in position by capillarity,
or by a glass rim to the lower plate. The upper plate was about 1 cm.
thick, and was assumed to be at one temperature throughout, the
temperature being given by a thermo-electric junction attached to it.
The lower plate was half the thickness of the upper. When the whole
apparatus was at one temperature it was suddenly placed on a block of
ice at 0°, in an enclosure at 0°, and the rate of cooling of the upper
plate was observed. The mathematical deduction of the conductivity

Copper

Heating

Copper C,

Glass G

Copper Ca

Ebonite] Liquid X \Ebonite

Copper C3

FlG. 71. — Lees's Disc Experiments on Liquids. — CiC3C2, copper discs ;
G, glass disc ; X, liquid surrounded by ebonite ring.

from the rate of cooling is not simple, and Weber's calculation has been
criticised and amended by Lorberg, whose recalculation gives a value for
water nearly 10 per cent, greater than that of Weber. But, where so
much uncertainty exists, it is sufficient to state Weber's results for
water in the form

*t= -0012(1 + -0080

where the assumption is made that the conductivity changes uniformly
with the temperature.

Other experimenters have obtained values of the same order of
magnitude. We shall describe only the method used by Lees,f a
modification of his disc method already described.

The liquid tested filled the cavity made by an ebonite ring between
two copper discs. The principle of the method consisted in sending a
known quantity of heat down through the upper of these discs, and from
the temperature of the lower disc determining the conductivity of the
liquid layer. The heat thus sent down was determined by putting a

* Chree, Phil. Mag., xxiv., 1887, p. 1, gives an account of this and other work.
t Phil. Trans., A., 1898, p. 399.

CONDUCTIVITY. 105

glass disc of known conductivity (as in Fig. 71) on the upper disc, and
then a copper plate on the glass, a heating coil on this, and a final
copper disc on the coil. The pile thus built up was varnished to give it
uniform and known emissivity, and placed in horizontal position in a
uniform temperature enclosure. The temperature excesses of the three
discs v:v2vs were determined by thermo-electric junctions inserted in
small holes, and were taken as uniform in each disc. Let the conducting
area of each face of the glass disc be ag, that of its emitting surface sg ;
let its conductivity be Kg, and let its thickness be g. We may take the
heat passing through its middle surface as

K avi ~ V2

9
The heat emitted by its surface is

and that emitted by the lower half is approximately

hsvi + v2

M9±^

Hence the heat passing through the lower face is

which is determinate.

Now, passing to the copper disc with temperature excess #2, let its
emitting surface be sc, the heat passing from this with the liquid and
ebonite is equal to the heat passing in minus the heat emitted by the
disc, or is determinate as

Q2 = Q! - hseva

Assuming that the now is vertically down through the liquid, that its
conductivity is K, its area a, and its thickness I, the heat passing through
it is

The heat certainly does not flow vertically through the ebonite, but
we may put it as

A(v2 - »8)

where A is some constant, and we may then put

fa V3)

If, then, we can determine A, we can find /c, since Q0 is known.

To find A a separate experiment was made, in which the liquid was
replaced by air, of which the conductivity is approximately known.

The results obtained, between 25° and 45°, agreed with the formula

106 HEAT.

The following table is taken from Lees's paper : —

Weber's values of K
Liquid. K& a between 9° and 15°.

Water . . . -00136 -'0055 -00136

Glycerine . . . -00068 -'0044 -00067

Methyl alcohol . . -00048 -'0031 '000495

Ethyl alcohol . . -00043 -'0058 -000423

At first sight the results obtained by Lees appear in fair agreement with
those given in the last column as obtained by Weber. But while Lees's re-
sults for water, for example, may be put in the form Kt = -00155(1 — -0048£)
where, it must be remembered, the actual range of observation lies
between £ = 25° and tf = 45°, Weber's results give Kt= '0012(1 + -008£)
where the range of observation lies between t = 4° and t = 24°. They agree
nearly at t = 25°, but have opposite signs for the temperature co-efficient
Further experiment is urgently needed to find whether there is any
reality in this change of sign. Indeed, until different methods give
closer agreement, all the results must be regarded as uncertain.

Conductivity Of Gases. — The investigation of the conductivity of
gases is complicated, not only by the ease with which convection occurs
but also by their transparency to radiation. It is necessary, then, to dis-
entangle the three effects of convection, radiation, and conduction, and
to find how much is due to conduction alone. Passing over earlier work,
which was merely qualitative, we shall describe briefly the experiments
of Stefan,* Winkelmann, Kundt and Warburg, and Todd.

Maxwell calculated the conductivity of gases from the kinetic theory of
gases (chap, ix.), and showed that over a wide range of pressure the con-
ductivity should be nearly independent of the pressure. The experiments
to be described were made with the view of testing Maxwell's results.

Stefan used two coaxal cylinders of thin copper. The inner one
served as an air thermometer, its tube passing through the end of the outer
and dipping into a vessel of water. The outer vessel was everywhere
separated by the same distance from the inner one, and the space
between them was occupied by the gas to be experimented on. The
outer cylinder was then surrounded by a mixture of snow and water,
and the rate of fall of temperature of the air in the inner cylinder
was observed. Hence, the conductivity could be calculated. For Air
Stefan found k =-0000558. Maxwell had already calculated the value
Jc = -000054. The result for Hydrogen was seven times as great, in
accordance with Maxwell's result The conductivity was also found to
be independent of the pressure, in accordance with Maxwell's prediction.
Stefan, however, took no account of the heat radiated.

Winkelmann f experimented both with cylinders and spheres in a
method closely resembling that of Stefan, and obtained with different
apparatus a value of the conductivity of air very close to that of Stefan.

Kundt and Warburg J sought to eliminate convection and radiation.
They noted the cooling of a thermometer in an enclosure containing
the gas, the heat capacity of the thermometer being known. As the
pressure of a gas is reduced, there is a limit beyond which the conduc-

* Journal de Physique, ii p. 147, 1873. t Wied. Ann., xlviii., p. 181.

J Pogg. Ann., civ. and clvi.

CONDUCTIVITY. 107

tivity is no longer independent of the pressure but falls very rapidly
with it. In order, then, to determine the effect of radiation they re-
duced the pressure far beyond this limit, so that conduction was negligible,
and at such low pressure convection also vanished. This was borne out
by the fact that the loss of heat was the same wherever the thermometer
was placed in the enclosure. Radiation would be the same, while con-
duction, if it existed, would depend on the distance of the thermometer
from the walls. Gas was then admitted, but only to such a pressure,
that while conduction was active, convection had not yet come into play.
The radiation effect could now be allowed for, and it was found that be-
tween an upper limit of 150 mm., and a lower limit of 1 mm. for air and
9 mm. for hydrogen, the rate of cooling was quite independent of the
pressure, and they assumed that, within this range, convection did not
exist and that conduction was constant. Their result for hydrogen was
seven times that for air, as Maxwell had expected, and they estimated
the conductivities as

Air .... -000048

Hydrogen . . '000341

Todd * experimented on the heat conducted through a layer of gas
between two horizontal metal plates. The upper one was the base of a
steam chamber, and was so maintained at 100° 0. The lower was main-
tained at a constant temperature, about 10° C., by a stream of water
flowing against its under surface. The amount of water flowing in any
time and the difference of temperature at inlet and outlet gave the heat
absorbed by the lower plate. This heat was partly conducted, partly
radiated. Convection was eliminated by having the hot plate uppermost.
To understand the principle of the method let us suppose that the plates
are of indefinitely large area to eliminate edge effects. Let Q be the heat
received below per sq. cm. per second. Let K be the conductivity of the gas,
6 the temperature difference of its two surfaces x apart, and let R be the
heat received by radiation per sq. cm. per second. Then Q = R + K&/X.
But if x be varied R is constant, so that we have (Q-R)z = K0.
Plotting Q against x, we have a rectangular hyperbola with Q = R as
asymptote, and the curve will give this, and therefore K0 and hence K.

Todd found for air K = 0-0000571, for carbon dioxide K = 0-0000411,
and for nitric oxide K = 0-0000888, all at 55° C.

The subject of conductivity is one of which the mathematical develop-
ment— based on certain assumptions — has outstripped experimental
verifications. Fourier, the founder of the mathematical theory, in his
Theorie Analytique de la Chaleur, discussed many problems, such as
that of the motion of heat in bars, which have been made use of by
Despretz and succeeding experimenters, and of the motion of heat in
spheres, of which we have a special case in the earth.

The reader will find an account of the problem presented by the
penetration of the sun's heat into the earth in Tait's Heat, p. 218, and a
sketch of Fourier's theory in Maxwell's Theory of Heat, 5th ed., p. 288.
* Proc. Roy. Soc., A., vol. Ixxiii. p. 19, 1909.

CHAPTER VIII.

THE FORMS OF ENERGY. CONSERVATION OF ENERGY.
MECHANICAL EQUIVALENT OF HEAT. FIRST LAW OF
THERMODYNAMICS.

Introductory Remarks — The Various Forms of Energy — The Identity of Energy —
The Conservation or Constancy of Energy— Statement of the Principle — Mayer's
Calculation of the Mechanical Equivalent — Joule's Researches — Later Repetition
— Experiments of Rowland — of Miculescu — of Reynolds and Morby — of Griffiths
— of Schuster and Gannon — The First Law of Thermodynamics.

Introductory Remarks. — The investigation of the conditions under
which heat appears in a system, or disappears from it, leads us to
regard heat as one among various forms of energy, and when it appears
we find that some other form disappears, and vice versa. The investiga-
tion of the relation of heat to other forms of energy constitutes the
subject of Thermodynamics. In this chapter we shall set forth the
evidence which leads us to recognise various forms of energy, and to
adopt the principle known as the Conservation of Energy. This
principle, as applied to heat, is known as the First Law of Thermo-
dynamics.

The study of mechanics leads to the recognition of two great principles
of conservation or constancy — the conservation of mass and the constancy
of momentum in a given line. These principles have been recognised
from the time when Newton placed the science of dynamics on a firm
foundation. We have now to add a third great principle of conservation,
the Conservation of Energy. This principle was, naturally, only recog-
nised at a much later date, for its recognition depended on a much wider
knowledge of physical phenomena and their mutual relations than was
possible when the other principles were first enunciated.

The fundamental idea that there is some identity underlying the
apparently different phenomena of mechanics, heat, light, and electricity
only assumed prominence at the beginning of the nineteenth century,
though it was sometimes vaguely perceived in earlier times. At first it was
expressed merely qualitatively, by saying that there was some mutual
relation between the various " forces of nature," or, as we should now
say, between the various forms of energy, so that one "force" was
convertible into other " forces." But as modes of measurement im-
proved, and numerical relations accumulated, it was gradually perceived
that quantitative relations held between the various correlated "forces,"
and in 1845 we find Faraday saying : "I have long held an opinion,
almost amounting to conviction, in common, I believe, with many other
lovers of natural knowledge, that the various forms under which the

ted

THE FORMS OF ENERGY. 109

forces of matter are made manifest have one common origin ; or, in other
words, are so directly related and materially dependent that they are
convertible, as it were, one into another, and possess equivalents of
power in their action. In modern times the proofs of their converti-
bility have been accumulated to a very considerable extent, and a
commencement made of the determination of their equivalent forces"
(Exp. lies., iii. p. 1). *

This is a full statement in the language of the time of the principle
of the conservation of energy, made just when the principle was
struggling into general recognition, and before it was placed on
a firm experimental foundation by the work of Joule and others.
Faraday's statement is divisible into two parts, the first asserting the
existence and mutual convertibility of " forces," the second asserting this
convertibility in definite ratios. Our account will naturally divide into
two corresponding parts. We shall give

(1) An account of each of the forms of energy hitherto recognised,
and a statement of the evidence which leads to the belief that they are
all forms of one " something " which we term energy, t

(2) An account of the modes of measuring the amount of each form
in a system, and an examination of the evidence which leads to the
belief that one form changes into another in a definite ratio or at a fixed
" rate of exchange." We shall then see in what sense we can hold that
the total quantity of energy is constant.

The Various Forms Of Energy. — We say that a man possesses
"energy" when he can do work in overcoming obstacles, either mental
or physical. By analogy, the same word is used in physics, and we say
that a body possesses energy, when by virtue of its motion or condition
it can do work in moving either itself or other bodies against resistance.

When the body can do work by virtue of its motion, it is said to
possess

Kinetic Energy. — Tf a body of mass m starts with velocity v, and
moves through a distance s against a uniform force which would pro-
duce in it acceleration a, and therefore be measured by ma, we know
that its velocity v at the end of s is given by

If we multiply each side of this equation by m we get —

mv2 - mv'2
- =mas

= force x distance travelled against it,
= work done against the force.

If v' = 0 the whole motion is exhausted, and

n

- = work which the body does against the force in coming to rest.
jL

* An excellent history of the growth of the doctrine of energy is given in Hel tn's
Die Energetik.

+ The footnote on p. 116 may be read here.

110 HEAT.

TTIV^

The quantity — is defined to be the Kinetic Energy or Energy of

2i

Motion of the body. It measures the work which the body can do in
exhausting its motion. Since it is equal to work it can be measured in
foot-pounds, foot-poundals, kilogramme-metres, ergs, <fcc., according to
the units chosen.

Potential Energy. — Let us consider the special case in which a
body of mass m is projected straight upwards against its own weight with
velocity v. For simplicity let us suppose that there is no air-resistance,
so that the weight alone acts. As the body rises its kinetic energy
gradually disappears, and at the highest point reached the body is for
an instant at rest and without kinetic energy. But we do not suppose
that this energy is gone out of existence. For, as the body falls it
regains energy, and when it has come back to the starting point its

tYLIl

velocity is again v and its kinetic energy again—— .

2i

We suppose that the energy did not cease to exist, but that it took
a new form no longer manifested in motion but in change of position
or change of configuration with respect to the earth. This new form we
term Potential Energy or Energy of Position.

We recognise the existence of kinetic energy by our sense of
sight — we see the body moving, but we think of potential energy in
terms of the muscular sense as well as in terms of the sense of sight.
For constraint is always needed to preserve at rest a configuration
involving potential energy, and we think of ourselves as upholding by
pull or push a body possessing such energy.

It is convenient to measure the gain in potential energy of a body
when rising against its weight, by the work done in moving it from
its original to its new position. The gain in potential energy is then
equal to the loss in kinetic energy, and the sum of the two energies5
potential and kinetic, remains constant.

It might seem at first sight that this constancy is merely a result of
definition and does not involve any observation or experiment. But it
is to be remembered that the force acting is the weight of the body,
which is the same whatever the velocity or the direction of its motion,
and whenever the motion takes place. Hence the velocity and the
kinetic energy are the same at any given point, both in the rise and
fall, and the possibility of regaining in the fall all the kinetic energy lost
in the rise depends entirely on the nature of the force acting. Were
the force dependent on the velocity or its direction, or did it change
with time, then the kinetic energy at a given point would be no longer
the same in the rise and fall. Indeed, in reality, the air-resistance is
always opposed to the motion and the kinetic energy lost against this is
not regained in the fall, so that at re-arrival at the starting-point there
is a diminution. The loss of kinetic energy is still, of course, equal to
the work done against the forces, but this work can no longer be
regarded as measuring potential energy stored.

This simple case will serve as an illustration of the general principle
that the kinetic energy of a system is only wholly regained on return
to the original configuration, when the forces depend solely on the con-
figuration and not on the motion of the parts, or on the time at which

THE FORMS OF ENERGY. Ill

the motions occur. When and only when this condition is fulfilled, the
work done against the forces in moving from one configuration to another
will be independent of the mode in which the change is effected, and for
each configuration there will be a . definite amount of kinetic energy.
The loss of kinetic energy in any change is thus recoverable on changing
back again, so that we can assume that potential energy is gained equal
to the kinetic energy lost. A system of this kind is termed a " con-
servative system," and the forces are termed "conservative forces."
The experimental basis, then, of the assertion that the sum of the
potential and kinetic energies of a system is constant is the observation
that the forces depend solely on the configuration.

Probably no system is exactly conservative. But the planets and
the sun form a system in which we have not as yet been able to detect
any departure from conservation of kinetic + potential energies, in
the comparatively short time over which astronomical observations have
extended, and it gives us the best illustration of such a system. Even
in the solar system, however, there are tidal effects which depend on
the velocity of the bodies producing them, and we know that they must
decrease the sum total of the kinetic and potential energies though we
have not actually observed the decrease. If we include comets in the
system, there appear to be some cases of observed decrease and diminu-
tion of orbits.

We continually use stores of potential energy in practical life to
obtain work. Thus we use a head of water to turn a turbine or a water-
wheel. We wind up a clock-weight to keep a clock going, and so on.
But we can never use the potential energy directly. We must always
allow the body or system to move in the direction of the forces acting,
and convert the potential in the first place into kinetic energy, and
thus get work from the kinetic energy. It would be absurd to attempt
to get energy from a reservoir of water without allowing the water to
run down hill, or to attempt to keep a clock going from the potential
energy of the weights without allowing them to fall.

Heat Energy. — In almost all cases of motion with which we are
concerned on the surface of the earth, the forces are not conservative.
We generally find friction of some kind coming into play, some force
opposed to the motion. If a body is projected along a horizontal table
its kinetic energy gradually disappears, work being done against the
friction. When the body comes to rest, the whole of the kinetic energy
is lost without any gain of potential energy, for there is no tendency
on the part of the body to return to its original position. But a new
phenomenon is observed. The body and the table are both slightly
heated. This appearance of heat is illustrated by the familiar experi-
ment of rubbing a button on a table and then applying it to the skin,
when the heating is found to be quite appreciable. In Sir Humphry
Davy's celebrated experiment, ice was melted by rubbing together two
pieces in a vacuum. Kinetic energy was continually supplied to the ice.
It disappeared in doing work against friction and heat appeared in the
melting of the ice. Many other examples of the appearance of heat on
the loss of kinetic energy by friction, will occur to the reader. In all
these cases the kinetic energy has disappeared, and there is no means
of regaining it by allowing the body to retrace its path. It has, there-

112 HEAT.

fore, not been converted into potential energy. But we suppose the heat
which has appeared to be itself a form of energy, and that the energy
which was formerly kinetic has taken this new form. We are strengthened
in this conclusion by the fact that the appearance of kinetic energy is
frequently accompanied by loss of heat. For example, in the steam-
engine, the steam cools in moving the piston and in setting the
machinery in motion.

Strain Energy. — Often, when kinetic energy disappears, we observe
the appearance of strain in matter.

If an indiarubber cord, of unstretched length OA (Fig. 72), is fixed at
0, and has a mass attached to the end A, this mass has potential energy.
If it is now allowed to fall it acquires kinetic energy, and will move down
until it comes to rest for a moment at B, a point below the final position
of equilibrium. If it is detained in this position it has lost both potential
and kinetic energy, but the indiarubber is stretched, and in this stretched
condition we must recognise the cord as still possessing energy.
For, when the mass is released, it begins to move upwards, and,
as the cord contracts, the mass acquires both kinetic and potential
energy. We may term this new form of energy, recognised in
the stretched or strained condition of the cord, Strain Energy.
On allowing the mass to oscillate up and down, we have continual
interchanges between the three forms of energy — kinetic, potential,
and strain. At B the energy is all strain. As the body moves
upwards, the strain energy is converted into kinetic and then
into potential, so that at an intermediate point C all three co-
I | I exist. When the body returns to rest at A, the strain and
kinetic energies have disappeared, and we have only potential
energy left, and so on.

We have another case of strain energy when one end of a
,_JS_, wire is fixed and the other end is twisted. Thus, if the upper

' J end of a vertical wire is fixed, and a mass is attached to the

FIG. 72. lower end, when the mass is twisted round and released it
oscillates to and fro round the wire as axis. We again recognise
the wire as possessing strain energy when in this twisted condition. On
the molecular theory of matter we may probably describe this strain
energy as, at least in part, potential energy of molecular grouping — i.e. as
energy depending on the relative positions of the molecules to each other.
But we cannot give a complete account of it on this theory until we
can say how the molecules are arranged in the strain, for the energy
does not depend merely on the strain as a whole, but on the mode in
which it was effected and the time which has elapsed since it was made.
In potential energy due to gravitation, the forces acting are the same
for the same grouping at all times, and the potential energy cannot be
obtained in any other form, unless the members of the system are
allowed to move. But if a wire is strained by twisting, and kept
strained, the energy of the strain in general slowly decreases, though
the twist of the wire remains the same. If, then, the energy is
potential energy of molecular grouping, we must suppose that the
molecules do not remain in the same distorted position, but gradually
undergo rearrangement of some kind.

Sound Energy. — We frequently find that strain energy disappears,

THE FORMS OF ENERGY. 113

giving rise to sound. If a wire is stretched between two points and
pulled transversely, we have strain energy ; but, on letting the wire go,
it gives a musical note. Or, if the prongs of a tuning fork are pinched
together, and then freed, the fork sounds its note. The strained body
is observed to oscillate, and an interchange of strain and kinetic energies
takes place in it. The oscillations gradually die away, and their energy
is transferred to the surrounding air or other bodies, where it produces
the peculiar condition which is capable of affecting our sense of hearing.
We know that this condition consists of a combination of motion and
strain travelling out in wave form from the strained source through the
surrounding medium. There is, therefore, in the waves a mixture of
kinetic and strain energies, forms already recognised. But since a
special sense — that of hearing — is affected by the combination, we give
it a special name, that of Sound Enenjy.

Light Energy, a Particular Case of Radiant Energy.— When

a body has been raised to a sufficiently high temperature, it becomes
incandescent, i.e. it is sending out light through the surrounding
medium. Meanwhile there is a loss of heat in the body, for if the
supply of heat is cut off, the body is observed to cool. Conversely, if
light is allowed to fall on an opaque surface, the surface is found to
be heated. We are familiar with this heating effect in the warmth
derived from sunlight, or from the light of a glowing fire. We know
that light is some modification of the medium through which it passes,
propagated out in waves from the source with a definite velocity, and
this modification on reaching our eyes affects our sense of sight. We
also know that the waves must be of lengths within certain narrow
limits, in order that they may affect us as light, somewhere between
300 and 800 millionths of a millimetre. But there are other waves
sent out from bodies, similar in kind to light waves, of lengths both above
and below these limits to which the retina is insensitive. We include the
whole series of waves under the general term of " radiation." Since their
propagation from a source accompanies loss of heat, and their absorption
by a body accompanies gain of heat, we must regard them as possessing
a form of energy which we term Radiant Energy. Formerly this energy
was described as radiant heat, but it is much more convenient to reserve
the term heat for the heat energy in the radiating body, or in the
heated receiving body, and to use "radiant energy" for the form into
which the heat energy changes when associated with the wave motion
which travels out into the surrounding medium.

Electrical Energy.— Many bodies on being rubbed are observed
to possess new mechanical properties. If a piece of sealing-wax is
rubbed with fur, or if a piece of very dry paper is stroked with the
finger-nails, it is found that small bodies, such as small pieces of paper,
are urged towards the rubbed surface. In other words, we have a
development of kinetic energy not accounted for by the immediate
disappearance of potential, strain, or heat energy. We suppose, there-
fore, that we have here a new form of energy, which we term Electrical,
and by making this supposition we may connect the kinetic energy of
the moving small bodies with the kinetic energy of the arm when
rubbing the electrified body, the electrical energy being the intermediate
form. It is slightly more difficult to rub the body which is being

H

114 HEAT.

electrified than it would be were there no electrification of its surface.
The resistance to rubbing absorbs kinetic energy, which, we suppose,
takes the form of electrical energy, and finally reappears as kinetic
energy in the particles moving to or from the electrified surface. The
increased resistance when electrification is taking place, i.e. when
kinetic is being turned into electrical energy, is well illustrated in
friction and induction electrical machines. It is a common experience
that they are more difficult to turn when they are in good order and are
generating electricity. The existence of electrical energy is further
supported by the occurrence of light and heat in the electrical discharge,
i.e. by the appearance of light-energy and heat-energy, which would
otherwise have to be thought of as arising de novo.

In certain cases we have also the appearance of electrical phenomena
as the result of heating or straining bodies, and we must then suppose
that we have conversions of heat or strain energy into electrical
energy.

Magnetic Energy. — If a bar of steel is stroked with one pole of a
magnet in one direction, the motion of the magnet is resisted and
kinetic energy is lost ; it is then found that small pieces of iron are attracted
by the ends of the bar, and that attractions and repulsions are ex-
perienced between the ends of the steel bar and those of the magnet.
We have here another case of the development of kinetic energy not
accounted for by the immediate disappearance of potential, strain, or
heat energy. We have, then, to suppose another form of energy —
Magnetic Energy — differing from electrical energy in its mode of
production, and in the phenomena accompanying its existence, but
similar to it in that we have to suppose its existence in order to con-
nect the kinetic energy lost in magnetisation with the kinetic energy
appearing in the motion which we term magnetic action.

The last two forms of energy, electric and magnetic, are closely
associated with each other, so that, in general, when we have trans-
formation of the one, we find that the other is present. In the electric
current, for example, we have every reason to suppose that there is
a transformation of electrical energy into heat taking place within the
wire ; and, accompanying this transformation, we have magnetic actions
round the wire, revealing the presence of magnetic energy there. Or,
in the dynamo-machine, we -have the transformation of kinetic energy
into electric and magnetic energies, both being associated to form the
energy of the current. Both electric and magnetic energies are forms
whose existence, like that of gravitational potential energy, we assume
as connecting links between energy disappearing when the electric and
magnetic conditions are produced, and energy appearing when those con-
ditions change. Some form of energy disappears during the production
of the conditions which we describe as electrification or magnetisation,
and we find that the same or other forms of energy may be made to re-
appear by the electric and magnetic actions. We have no special senses
which are affected by the electric or magnetic modifications of matter,
and they are therefore in some degree more hypothetical than Kinetic
energy, or Heat, or Light.

Chemical Energy. — Lastly, we have to recognise a form of energy
in Chemical Separation. Heat is necessary to decompose many sub-

THE FORMS OF ENERGY. 115

stances into their constituent elements, the heat disappearing with no
other result than the separation of the elements ; while, on the other
hand, there are numerous familiar examples of the appearance of heat
on the union of elements or the chemical combination of previously
separated substances. In the voltaic cell, chemical combination occurs
when the cell is in action, while in and round the circuit heat is
produced, or magnetic actions occur, or light issues from sparks, and
these energies we suppose to have come from the constituents in the cell
through the intermediate forms of electric and magnetic energies. Con-
versely, the electric current decomposes substances — the electric and
magnetic energies being transformed now into the energy of chemical
separation.

The following list includes all the distinct forms of energy which
have yet been recognised : —

1. Kinetic Energy.

2. Potential „

3. Heat „

4. Strain ,,

(1 and 4 are also combined in Sound.)

5. Light and Radiant Energy.

6. Electric „

7. Magnetic „

8. Chemical „

We have already given many examples in which the disappearance
of one of these forms is accompanied by the appearance of one or more
of the other forms, and our observations and experiments justify us in
regarding such accompaniment as the universal rule. There is no example
in which one kind of energy is absolutely annihilated without the appear-
ance of another, nor in which one kind of energy appears de novo without
the loss of another. Indeed, so far convinced are we of this, that an
apparent exception would lead us to suspect, not the truth of this state-
ment, but the completeness of our list of energies. We should at once
look out for a hitherto unrecognised form which appeared or disappeared,
and endeavour to obtain some independent evidence of its existence and
only after most careful research and failure to find such a form could we
suppose that energy was annihilated or came into existence de novo. It is,
of course, quite possible, that there are forms of energy yet unrecognised.
If we happened to observe the disappearance of energy at one time or
place, and the appearance in connection with it of energy at another time
or place, and yet could not suppose the existence of any yet known form
as connecting link between the two, we should be driven to suppose that
some form existed, hitherto unknown, so that the disappearing energy
took this form and that on its reappearance it emerged from it, and we
should at once look out for the conditions implying the new form. If, for
example, the cases of so-called "telepathy" were placed beyond question,
we should probably have disappearance of energy from one individual
accompanied by its appearance in another at a distance, and it is quite
conceivable that we should have to suppose that the energy in the inter-
vening space was in a form not yet known. But up to the present the
evidence hardly warrants the assumption that our list is thus incomplete.

1 16 HEAT.

It is usual to go somewhat further than the statement that the forms
of energy may replace each other — this being all that experiment alone
would warrant us in saying — and to regard these replacements as rather
transformations of one and the same thing — energy ; which we suppose
to be identical with itself, though varying in its phenomenal appearance.
With this belief we may make the following statement : —

The Identity Of Energy. — There is something which we term
" energy," and which may be recognised in various forms. When it dis-
appears in one form it appears in one or more other forms.*

The Conservation or Constancy of Energy.— So far we have

considered energy only in its qualitative aspect, and have not considered
any mode of measuring it except in the kinetic and potential forms. We
now proceed to consider in what sense we can assert the constancy in
quantity, as well as the permanence of existence of something which
appears to us in various forms not directly comparable with each other,
i.e. not directly measurable in terms of the same unit. An illustration
may assist us here. If a man possesses some bank-notes, some gold, and
some silver, he may use them all for purchase, since they are all money ;
and he may exchange any one form for one or more of the others. He
may change a note for a mixture of gold and silver, and he may change
silver for gold or notes. We may, therefore, assert qualitatively that
money appears in various forms. But we may go further, and make a
quantitative statement. There is a fixed rate of exchange between the
various forms, so that he may exchange a £5 note for five sovereigns
and each sovereign for twenty shillings. With this fixed rate of
exchange, the total amount reckoned in terms of any one form is
constant. If, for example, he begins with four £5 notes he may
exchange all or any of them for gold or silver, or both, and the total
quantity reckoned in terms of any one coin will be constant, and he will
possess twenty pounds or four hundred shillings, even though the actual
money may be a mixture of paper, gold, or silver. Were there no fixed
rate of exchange, this could not be asserted. If, for instance, his money
were partly in English, partly in French notes or coins, the amount
reckoned in any one form would, of course, vary at different times
according to the rate of exchange. The question then arises, does
energy undergo its various transformations according to a fixed rate of
exchange in each case ? If so, and only on this supposition, the total
quantity reckoned in terms of any one form is constant.

To answer this question, we have, in the first place, to consider how the
various kinds of energy are to be measured. Kinetic energy we have

TUlfi

defined as measurable by -~— , in terms of a perfectly definite unit, and we
know how the mass of any body may be found, at least in theory, so

* The belief in the identity of energy is no doubt metaphysical, as metaphysical
as is our belief in the continued existence of any portion of matter, and its identity
under various modifications. As, however, the metaphysical addition somewhat
simplifies the form of the statement, and is never likely to lead us wrong in our
experimental interpretation, we see no reason to exclude it. Were we to do so we
should have to speak of the correlation of the energies, not of the constancy of
energy. Instead of describing the conversion of, say, kinetic energy into heat, we
should have to say that kinetic energy disappeared, and that at the same time heat
energy appeared.

THE FORMS OF ENERGY. 117

that this expression applies to any kind of matter moving with any
velocity.

In the case of Heat, too, we can fix on a perfectly definite unit — say that
raising 1 gm. of water from 15° to 16° 0. — in terms of which may be ex-
pressed any quantity of heat in whatever kind of matter, and whatever the
temperature of tha't matter. Common experience of the working of the
steam-engine, or of other modes of transforming heat into kinetic energy,
raises a strong probability that if heat energy is measured simply by
the quantity of heat, that is, as so many calories, and not by any more
complicated expression, then there is a fixed rate of exchange between
it and kinetic energy ; for the greater the amount of work to be done,
the greater the amount of fuel required, and roughly in the same
proportion. This probability is converted to certainty by the investiga-
tion on the rate of exchange between kinetic energy and quantity of heat,
carried out by many physicists and especially by Joule, whose celebrated
researches on the value of the rate of exchange we shall presently describe.

But in other forms of energy we have no such general units. In
Light, for instance, we may compare the light given by one mono-
chromatic red flame with that given by another red flame of the same
quality, and we may, more or less easily, fix on a unit of red light of
some particular quality. But another, say a green light, cannot be
transformed to red to effect comparison with the unit, and so it cannot
be expressed in terms of a red unit. But for each quality or wave-
length of light, we may show that the quantity of illuminating power in
terms of a unit of its own quality is proportional to the quantity of heat
developed when the light is absorbed by a surface on which it falls.
For both the heat developed on absorption in a given time, and the illumi-
nating power vary together, inversely as the square of the distance from
the source. Hence, each kind of light, which we may assume to be light
energy, is transformed into heat according to a fixed rate of exchange.
Again, in Chemical energy we have no general unit. We may say that
in each special kind of chemical action, the amount of action is pro-
portional to the quantity of substance formed ; but we have no direct
common unit connecting the action in one case with that in another,
where quite a different compound is formed. But in each case, separately,
the quantity of heat developed is proportional to the quantity of the
substance formed, so that here also, if we assume that chemical energy
is proportional to the amount of substance which may be formed by
union, the rate of exchange between chemical energy and heat is fixed.

In the case of Strain energy, we may always think of the strain as
produced by the transformation of kinetic energy, and since the kinetic
energy can be regained on allowing the body to unstrain, with, in
general, a small loss accounted for by the heat developed in the pro-
cesses, we may fairly suppose that the strain energy gained is equal to
the kinetic energy lost, and we usually measure it by the amount so lost.
Hence, our mode of measurement, itself, assumes a fixed rate of exchange,
as in the case of potential energy, and the fact that any discrepancy can be
accounted for by other energies appearing, justifies the assumption.
We might, perhaps, measure strain energy in terms of that possessed
by a standard body with a standard strain, comparing the energy in any
other strained body by transferring its energy in a suitable manner to

118 HEAT.

the standard body, and noting the strain produced. Were this done
without any consideration of work performed in the transfer, we should
then have to measure the kinetic-energy-equivalent of the standard
strain of the standard body. But this has not been as yet attempted,
and we, therefore, content ourselves with considering the kinetic-energy-
equivalent of each case separately.

Electric and Magnetic energies, as we have already remarked, give
very little direct evidence of their existence. They are connecting links
imagined to come between some recognised form of energy disappearing
and others appearing under circumstances such that we cannot imagine
a direct transfer. Hence we look for a fixed rate of exchange between
the known forms lost on the one hand in electrification or magnetisation,
and the known forms gained on the other hand when the electric and
magnetic conditions cease. We shall see when we come to consider the
energy relations of the electric current that this fixed rate of exchange
is shown to exist. We suppose then that it holds also for the trans-
formation into, and out of, the intermediate electric and magnetic condi-
tions, which, indeed, we measure by the supposition of this fixed rate.

Since the other energies are ultimately measured on their trans-
formation either into kinetic energy or into heat energy, and since we
have fair evidence for fixed rates of transformation into these two it
remains to examine the rate of exchange between heat and kinetic
energy. We shall proceed to give an account of the experimental
evidence, all of which goes to show that here also the rate is fixed. And
anticipating this result we may sum up the whole discussion in the
following

Statement of the Principle of the Conservation of Energy.—

Energy is recognised in various forms, and when it disappears in one
form it appears in others, and in each case according to a fixed rate of
exchange. The total quantity of any energy, measured in terms of any
one form, is therefore constant whatever forms it may assume.

The Rate of Exchange between Mechanical or Kinetic Energy
and Heat Energy, or the Mechanical Equivalent of Heat.— The

determination of the rate of exchange depends on the measurement of the
work done on some system in which that work results only in a develop-
ment of heat, and the simultaneous measurement of the heat so developed.
The rate was first determined by calculation from the specific heats of
air. This method was first set forth clearly by R. Mayer in a paper
published in Liebig's Annalen in 1842. We shall therefore give it first,
though a more direct method was shortly afterwards carried out by
Joule.

Mayer's Calculation of the Mechanical Equivalent from the Specific
Heats of Air at Constant Pressure and Constant Volume. — The Specific Heat
of air, as of other gases, at constant pressure exceeds that at constant
volume, and if we can assume that the excess is due entirely to the
work done in pushing out the surrounding air in expanding, that is if
we can assume that no energy is absorbed in merely separating the
particles of air, that they possess no appreciable cohesion, this external
work is the mechanical equivalent of the difference between the two
Specific Heats. Making this assumption, let us suppose that the
volume of air is 272'5 cc., which, according to Regnault, increases 1 cc.

THE FORMS OF ENERGY. 119

for each 1° rise at constant pressure. Let this volume be at 0° and
760 mm. pressure in a vertical cylinder 1 sq. cm. section, and let the
atmospheric pressure be represented by a piston loaded with a column
of mercury 76 cm. high and 1 sq. cm. in section, and so weighing

76x13-596 = 1033 grammes weight.

When the air is heated from 0° to 1° the volume expands 1 cc. and the
piston is pushed out 1 cm., so that work is done equal to 1033 cm.
gms.

Now let us turn to the heat measurements. The density of air at 0°
and 76 cm. is, according to Regnault, 0 '001 293, so that the mass of air
heated is

272-5 x 0-001293 = 0-3523 gm.

At constant pressure the specific heat of air C.,, is, according to Regnault,
0'2375, and according to E. Wiedemann, whose value we shall take,
0'2389, and this is the heat put in per gramme of air in raising its
temperature from 0° to 1°.

But it is only the excess of this over specific heat at constant volume
Or, which is the equivalent of the work done. Now Cfl may be found
from Op from the relation (chap, xviii.)

Gp Adiabatic Elasticity.
Cv Isothermal Elasticity.

This ratio has been determined in various ways, and we may take its
value as very near 1-405,* whence

0-2389
^"TioT

Then Op -0, = 0-2389 - 0'1700 = -0689.

Multiplying by the mass heated, we find

0-0689 x 0-3523 = 0-02427 calory

as the heat equivalent of the 1033 cm. gms. of work done. Then the
mechanical equivalent of the calory is 1033 -r 0*02427 = 42560 cm. gms.,
and this is the mechanical equivalent in centimetres and grammes
weight. Mayer, using the data available in 1842, found 36500 cm. gins.
The calculation depends entirely on the assumption that no work is
done in the mere separation of the particles of air. In a second paper
published in 1845 j Mayer supported the assumption by quoting an
experiment by Gay-Lussac (Memoires d'Arcueil, 1807 : Gilbert's Annalen,
xxx., 1808, p. 249), which went to show that if a gas expands from one
vessel into another equal vessel previously empty, the first loses just as
much heat as the other gains. Mayer saw that the cooling in the first
vessel is due to the work done by the remaining gas in pushing out that
which passes into the second vessel, and the heating in this vessel is due

* Meyer, Kinetic Theory of Gases, p. 123.

+ This is slightly less than Joly's direct determination (p. 85), which gives C« =
0-17154. The uncertainty in the Specific Heat of Air at constant pressure also
makes I he result uncertain.

t Helm, Die Energetik, p. 24.

120

HEAT.

FIG. 73. — Joule's Expansion of
Air Apparatus.

to the work done on the gas already in it as successive portions enter
and compress it. But since, if the heat developed in one were trans-
ferred to the other the original temperature would be regained, the

experiment shows that mere expan-
sion of a gas when no external work
is done does not produce a change of
temperature. Thus Mayer's mode of
calculation was justified.

Joule's Researches. — Already Joule
had begun his series of researches and
sought to determine the mechanical
equivalent by compressing or rare-
fying air and equating the heat
developed or lost to the work done
on or by the air. Here again the
calculation depends on the absence
of cohesion of the air, and Joule de-
vised an experiment apparently inde-
pendently, but on the same lines as
Gay-Lussac's. Two copper vessels R
and E (Fig. 73) were connected by a

pipe provided with a stop-cock. R was filled with air at about 22
atmospheres pressure, E was exhausted, and the two were placed in a
vessel containing water. The stop-cock was then opened to allow the
air to expand and fill both vessels. On stirring the water and taking its
temperature, no appreciable alteration was found.

When R and E were arranged as in Fig. 74 in separate calorimeters
a notable cooling was observed in
the vessel surrounding R and a
very nearly equal heating in that
surrounding E. This last form of
experiment was almost identical
with Gay-Lussac's.* The method
is not capable of great exactness
owing to the large heat capacity
of the vessels and calorimeter com
pared with that of the contained
air. But by a method devised later
(chap, xviii.) Joule and Thomson
showed that there are slight absorp-
tions or evolutions of heat in a gas
on mere change of volume, though
the amounts are too small to affect FIG. 74.-
the foregoing calculation of the
mechanical equivalent.

Joule's Researches on the Mechanical Equivalent of Heat.—

At the Cork meeting of the British Association in 1843, Joule gave an

* Gay-Lussac used two globes, each about 12 litres, and each containing a
sensitive spirit thermometer. One was exhausted and the other contained air at
atmospheric pressure. On opening the connecting-cock one thermometer rose 0-58°,
the other fell 0-61°.

-Modified Form of Joule's
Apparatus.

THE FORMS OF ENERGY. 121

account of his earliest experiment on the mechanical equivalent (Scientific
Papers, i. p. 123). In this he measured the work done in turning a small
machine, which we should now call a dynamo, and he also measured the
current produced. From his law of heating effect due to current, discovered
shortly before, he was able to determine the total heat evolved in the
circuit of the dynamo from a measurement of that evolved by a known
current in a known resistance, and hence he found the mechanical
equivalent to be 836 foot-lbs. per pound of water heated 1° F. In a
postscript to the paper describing this experiment he says that by
working a piston perforated by small holes, forming narrow tubes, in a
cylindrical glass jar holding about 7 Ib. of water, each Ib. of water was
heated 1° F. by the work equivalent to raising 770 Ib. 1 foot. In the
following year he gave as the results of experiments on the work done in
the compression of air and the heat generated, 823 (Scientific Papers, i. p.
171), and in a continuation of this work he was led to devise the experi-
ment on the expansion of air described above. The result of this con-
tinuation was 798. In 1845 he first described a method in which
falling weights were employed to churn water in a calorimeter, and the
mechanical energy lost by the weights was taken as equivalent to the
heat developed in the water. The result was 890. He then proceeded
to improve the conditions of this last and most direct experiment, and in
1850 a full account was published in the Philosophical Transactions
(Scientific Papers, i. 298). The general nature of the apparatus will be
seen from Fig. 75. Two masses, each either 10 Ibs. or 29 Ibs., were
attached by strings each to the axle of a wheel and axle wa, iva. From
the wheel strings passed horizontally to the drum d, attached to a spindle
on which were fixed paddles of brass, 8 in number, revolving in a calori-
meter C of copper and containing a known weight, about 7 or 8 Ibs. of
water. The drum d could be detached from the spindle so that the
masses could be wound up to a height of 5 feet from the floor without
rotating the paddles. When they were wound up the drum was re-
attached to the spindle, and as the masses fell the paddles spun round.
In the calorimeter were fixed four brass vanes, cut out like the wards of
a lock. These allowed the paddles to pass through them, but prevented
any continuous circulation of the water and therefore any permanent
acquisition of kinetic energy by it. The water was only churned up by
the motion of the paddles, and its kinetic energy was rapidly transformed
to heat through fluid friction. The arrangement inside the calorimeter
is shown by the horizontal and vertical sections in the figure. The
masses had a terminal velocity on reaching the floor varying from 1'4 to
3'1 inches per second. This was noted and allowed for as, in effect,
diminishing the height of fall. The wheels were mounted on bearings
with as little friction as possible, but the residual friction was calculated
by subsidiary experiments, and the amount of kinetic energy absorbed
by it was allowed for. The calorimeter was on a wooden stand with
transverse slits, in order that the calorimeter should rest on a few points
of the wood only. Loss by conduction was thus reduced to a very small
quantity.

In each experiment the masses were wound up and allowed to fall
twenty times, the duration of an experiment being somewhat over half an
hour. The rise of temperature of the water in the calorimeter was then

122

HEAT.

noted, and the effect of radiation to or from the surroundings deter-
mined by subsidiary experiments. Knowing the capacity for heat
of the calorimeter, the total quantity of heat generated could be
determined.

The result obtained was that the kinetic energy due to the fall of
772 Ibs. through 1 foot at Manchester would, on transformation to heat,
raise 1 Ib. of water between 55° and 60° F. through 1° F.

Joule also made experiments with mercury instead of water, using an
iron calorimeter, and he obtained as a result 774.

In another series of experiments, he made the masses, in falling, turn

Horizontal* Section of
Calorimeter.

Vertical Section, of
Calorimeter

FlG. 75. — Joule's Water-Churning Apparatus for Determining the
Mechanical Equivalent of Heat.

a bevelled iron wheel against another fixed bevelled iron wheel, both
being placed in the calorimeter and surrounded by mercury. The result
was nearly 775. It is worthy of note, as illustrating the care with
which he experimented, that he allowed for the loss of energy by the
sound given out by the vibration of the apparatus, estimating it by
the work required to produce an equal sound as heard at the same
distance from a violoncello. But neither of the modes of experiment
with mercury was quite as satisfactory as that with the water calori-
meter, and they were disregarded in favour of the water-friction
experiment.

Later Repetition. — Many years after the publication of Joule's paper
giving an account of this work, the British Association, in framing a

THE FORMS OF ENERGY. 123

system of electric units, took as the primary unit of resistance that in
which unit rate of working would maintain unit current, and a committee
of the Association prepared resistances determined in terms of this unit.
Now, in this electro-magnetic system the unit rate of working is 1 erg per
second, and the current, 0, is measured in terms of dynes on unit pole at
1 cm. distance by 1 cm. length, i.e. the resistance, R, is really measured in
mechanical units, and when we put for any circuit

Rate of working = C2R,

the product C2R is in mechanical units, if 0 and R are in terms of the
above units.

But Joule showed that the heat developed in any resistance is
proportional to C2R, which leads us to suppose that the work required
to maintain C in R is all transformed to heat in the wire having that
resistance. If, then, we measure the heat in calories, the current in
electro-magnetic measure, and the resistance in terms of the new unit,
we shall have on the one hand the heat, and on the other its mechanical
equivalent in C2R. As soon as the new resistance standards were fixed,
Joule himself made an experiment to obtain the mechanical equivalent
by this method (Scientific Papers, i. p. 542). Converting to foot-lbs.
and degrees Fahrenheit, the result was 782 instead of 772, so different a
result that Joule was induced to undertake a repetition of his water-
friction experiment to find whether the error lay in his work or in that
of the committee. The result of his second great experiment was
published in the Phil. Trans., 1878, pt. ii.

The method adopted generally resembled that of his earlier experi-
ment, but he employed a different contrivance for doing work, one which
had already been used by Him. The set of paddles was rotated at a
uniform speed by a handle. If the calorimeter had been free to move
round the axis of the paddles, it would have spun with them, but it was
kept fixed by applying a couple which could be measured. This couple
was equal and opposite to that which was exerted on the water and
calorimeter by the paddles, and equal to the couple applied to the
handle since the rotation was steady. Now, the total work done by a
couple G in n revolutions is

so that the value of the mechanical energy was known at once from n
and G. For other details the original paper may be consulted (Scientific
Papers, i. 632). Joule's final result was that 772'55 Ibs., falling 1 foot
at the sea level in the latitude of Greenwich, would acquire kinetic
energy which, transformed to heat, would raise 1 Ib. of water from
60° to 61°. This confirmed in a remarka,ble manner the accuracy of the
earlier work, and showed that the determination of the electric standard
of resistance was faulty. But the fault had been made plain from other ex-
periments, and a redetermination showed an error of more than 1 per cent.
It will be convenient to express Joule's value of the mechanical equiva-
lent here in various units, for the sake of comparison with later results.
These later results are all somewhat larger than Joule's, chiefly through
the adoption of the nitrogen or hydrogen scale of temperature, with slightly
larger degrees about 15°, than those of the mercury-in-glass scale used
by Joule.

124 HEAT.

Writing after the result the units in which it is expressed, we
have

772-55 foot-lbs. at Greenwich per Ib. of water per 1° F. at 60° F. ;

g

multiplying by - for the Centigrade scale,
5

1390'6 foot-lbs. at Greenwich per Ib. of water per 1° C. at 15° 0 ;

multiplying by 3O48 to convert to centimetres, and remembering that
we may use any unit of mass, if we use it on both sides of the transac-
tion :

42385 centimetre grammes at Greenwich per gramme of water
per 1° 0. at 15° C. ;

multiplying by 981*17 the value of g at Greenwich,

4'155 x 107 ergs per gramme of water per 1° 0. at 15° 0.

Later Researches. — Joule's work has been followed by other
researches on the value of the mechanical equivalent of heat. We
shall give a brief account of some of the more important of these.

Rowland's Experiment. — In 1877-78 Rowland repeated Joule's water-
friction experiment on a large scale at Baltimore (Proc. An>. Ac., xv.,
1879, p. 75). The calorimeter and its contents had a water equivalent
about 9 kgm. The paddles were worked by a steam engine, and the
spindle to which they were fixed passed up through the bottom of the
calorimeter. The calorimeter was hung on a wire, which of course tended
to twist when the paddles revolved, but a measured couple was put on
to bring the calorimeter back to its undisturbed position. When the rate
of working was about 2200 kgm. metres per minute, the rise in tempera-
ture was about 25° in 40 minutes. The temperature was measured by
mercury-in-glass thermometers, which were standardised in terms of
the air thermometer, and the air thermometer temperatures were then
reduced to absolute temperatures by means of Thomson and Joule's
experiment (chap, xviii.).

More recently Dr. Day has determined Rowland's temperatures in
terms of the hydrogen scale, which is now generally used as the standard
(Phil. Mag., xlvi., 1898, p. 1). In the table below are given Rowland's
values in ergs per gramme degree of water on the absolute scale, and
the values corrected to tlie hydrogen scale, by Dr. Day : —

Absolute Kowland's Values Corrected ergs

Temperature in ergs and in degrees on

above 0° 0. absolute degrees. hydrogen scale.

5 4-212 xlO7

6 4-209 4-203 x 107

7 4-207 4-201

8 4-204 4-199

9 4-202 4-198

10 4-200 4-196

11 4-198 4-194

12 4-196 4-192

13 4-194 4-191

14 4-192 4-189

15 4-189 4-188

THE FORMS OF ENERGY. 125

Absolute Rowland's Values Corrected ergs

Temperature in ergs and in degrees on

above 0° C. absolute degrees. hydrogen scale.

16 4-187 4-186

17 4-185 4-185

18 4-183 4-184

19 4-181 4-182

20 4-179 4-181

21 4-177 4-180

22 4-176 4-179

23 4-175 4-178

24 4-174 4-177

25 4-173 4176

26 4-172 4-176

27 4-171 4-175

28 4-171 4-175

29 4-170 4-174

30 4-171 4-174

31 4-171 4-174

32 4-171 4-174

33 4-172 4-174

34 4-172 4174

35 4-173 4-175

36 4-173 4-175

These values bring out very clearly that there is a variation in the
work required to raise 1 gramme of water 1° at different parts of the
scale — in other words, that there is a variation in the specific heat of
water with temperature. They also point to the existence of a minimum
value in the neighbourhood of 30°. Rowland's work was the first to
indicate the minimum value, and his conclusion has since been con-
firmed by others.

Miculescu's Experiment. — In 1892 Miculescu presented to the Paris
Faculty of Sciences a thesis, in which he gave an account of a water-
friction experiment carried out on new lines. The stirring power was
supplied by a 1 h.-p. electric motor which was balanced on knife edges
in the horizontal line of the axis of the armature. The spindle of the
armature was prolonged through a stuffing-box into the water-stirring
vessel, where the paddles were attached to it. The couple exerted on
the water was measured by hanging a weight on to a horizontal arm
projecting at right angles to the axis of suspension of the motor sufficient
to keep the motor from turning round. The water in the stirring-vessel
was kept at a constant temperature by circulating round the outside of
it a current of colder water passing at such a rate that it took heat from
the calorimeter as fast as it was developed by the stirring. The heat
was measured by the quantity of cooling water sent round and the
difference of temperatures of entrance and exit. This difference was
measured by a theirnocouple which was graduated in terms of a Tonnelot
thermometer, and ultimately in terms of the hydrogen scale. The tem-
perature of the water varied between 10° and 13°. We may take the
result as making 4-187 x 107 ergs equivalent to the heat required to raise
1 gramme of water at 11 -5° through 1° on the hydrogen scale.

126 HEAT.

Experiment of Reynolds and Morby. — Yet another mode of carrying
out the water-friction experiment has been described by Osborne
Reynolds and Morby (Phil, Trans., A. 190, 1897, p. 301). Having at
command a 100 h.-p. engine provided with a hydraulic brake, the idea
occurred to Professor Reynolds that this might be used to determine the
amount of work needed to raise the water in the brake from 32° F. to
212° F. The brake itself might be regarded as consisting of paddles
working in a water-stirring vessel so arranged that the couple exerted
in the stirring could be varied at will and measured at any value. The
water was delivered into the brake at 32° F., and was raised in it to
about 212° and then passed out, the rate of flow being regulated so that
the rise should be through about 180°. The quantity flowing while a
given amount of work was done was measured.

The time of running was 62 minutes with a speed of 300 revolutions
per minute, and various horse-powers and various quantities of water
were used, the total quantity of water rising in some experiments to
nearly half a ton.

The final value obtained is that the mean specific heat of water
between 32° F. and 212° F. measured in foot-lbs. at Manchester is
776'94. In ergs and degrees centigrade it is 4'1832 x 107.

Griffith's Experiment. — In 1883 E. H. Griffiths gave an account of
an experiment to determine the mechanical equivalent of heat by the
method of electrical heating (Phil. Tram., 184, A., 1893, p. 361). This
research was carried out with the greatest care in every detail, and the
original paper should be consulted for particulars and especially for the
method of temperature regulation.

A coil was immersed in a calorimeter 8 cm. deep and 8 cm. wide,
containing various quantities of water up to about 250 gms. The
calorimeter was closed by an air-tight lid through which passed the ther-
mometer, the stirrer, and the wires to the coil. It was suspended in an
exhausted enclosure and the walls of this enclosure were double, the
cavity between being filled with mercury. A graduated tube led out
from the cavity so that it formed, practically, the bulb of a big ther-
mometer in the middle of which the calorimeter was suspended. Any
variations in the temperature of the enclosure could thus be detected.
The double-walled vessel was surrounded by water, and was kept as
nearly as possible at a uniform temperature in order that radiation loss
could be exactly allowed for.

When a current was passed through the coil, if E was the fall of
potential in it (determined by comparison with a Clark's cell), and if
R was its resistance, both in electro-magnetic units, the rate of energy
supply was E2/R ergs per second. Measuring the heat developed in
any time the number of ergs per calory could be determined.

The thermometer — mercury-in-glass — was compared with a platinum-
resistance thermometer and the indications of this were found first in
terms of the air thermometer and later in terms of the hydrogen scale.
Different temperature ranges were used between 15° and 25°, and the
final result corrected to the hydrogen scale was very nearly

4-2 x 107{1 - -000266(* - 15°)} ergs per gramme of water heated

1° of the hydrogen scale,
where t is the temperature on the Centigrade scale.

THE FORMS OF ENERGY.

127

Experiment of Schuster and Gannon. — Another determination by the
method of electrical heating was described by Schuster and Gannon in the
Phil. Trans., 185, A., 1895, p. 415. They immersed a coil in a calori-
meter containing about 1500 gms. of water, and passed a current through
it. The energy supplied was EC£ ergs in time t. By passing the
current through a silver voltameter, and using the known electro-
chemical equivalent of silver the value of Gt was determined, while E
was measured by comparison with a Clark's cell. The temperature was
measured by a mercury thermometer, which was standardised in terms
of the nitrogen and the hydrogen scales. It is interesting to observe
the difference in the result expressed in ergs per calory at 19*1° on
these three scales —

4*1804 x 107 on mercury thermometer of hard French glass.
4-1905 x 10r on nitrogen scale.
4' 19 17 x 107 on hydrogen scale.

The results below, taken from Schuster and Gannon's paper, give the

099 . .

0' 20' 40' 60' SO' 100'

—+• Temperature

FIG. 60. — Kesults of Different Experiments on the Specific Heat of Water.

values obtained by different observers in foot-lbs. at Greenwich, and
Fahrenheit degrees of the nitrogen thermometer at 15° 0. ; the values
in ergs and Centigrade degrees of the same thermometer are added :—

Foot-lbs. °F

Joule (1878) ... 775

Rowland .... 778*3

Miculescu .... 776*6

Griffiths . . . . 780-2

Schuster and Gannon . 779*2

And in terms of the mean calory :

Reynolds and Morby . . 776*94

Ergs. °C.
4*178 xlO7
4*1895 xlO7
4*180 xlO7
4-199 xlO7
4*194 xlO7

4-1832 x 107

The values given in terms of the nitrogen scale must be increased by
1 in 2500 to bring them to the hydrogen scale. In Fig. 60 the results

128 HEAT.

obtained by different experimenters are exhibited in terms of the varia-
tion of the specific heat of water.

An inspection of these numbers shows that the results by the two
electrical methods are somewhat closely in agreement, and higher than
those given by water-friction methods. It appears probable that there
is some error in the electrical relations assumed, and some recent ex-
periments suggest that this is in the value of the E.M.F. of the Clark's
cell. Meanwhile, Rowland's corrected value at 15° may be taken, and
the first three figures of this may be put as 4'19 x 107 ergs. Hence we
have the mechanical equivalent —

4-1 9 x 107 ergs;

4'27 metre grammes ;

778 foot-lbs at Greenwich ;

in terms of degrees on the hydrogen thermometer about 15* 0., the last,
of course, on the Fahrenheit scale.

The First Law of Thermodynamics.— When exchange occurs

between work and heat, the researches we have been describing all
tend to show that the ratio of exchange is fixed. Hence if W is the
work in ergs equivalent to heat H measured in calories, we have

W-JH

where J is the mechanical equivalent, taken provisionally as equal to
4' 19 x 107. This relation is known as the First Law of Thermo-
dynamics.

CHAPTER IX.
THE KINETIC THEORY OF MATTER.

Atomic Hypotheses — Solids— Liquids — Gases — Kinetic Theory of Gases — Mean
Value of the Square of the Velocity of Translation V3 — Mixture of Gases —
Relation between V and Temperature — Energy of Translation and Internal
Energy — Joule's Approximate Method of Calculating the Velocity of Mean
Square — Effusion or Transpiration through a small Orifice into a Vacuum —
Thermal Transpiration— The Mean Free Path— The M.F.P. calculated from the
Coefficient of Viscosity — Conduction of Heat in Gases— The Diameter of the
Molecules and the number of Molecules per Cubic Centimetre — Forces acting on
unequally heated Surfaces in High Vacua — The Gas Equation of Van derWaals

THE belief in the identity of energy throughout its apparent transforma-
tions naturally leads us to attempt to explain these various manifestations
as really identical in form, though affecting our senses differently. As
examples of hypotheses of this kind, we may instance the electro-
magnetic theory of light, in which light (or rather radiant energy) is
supposed to consist of a mixture of electric and magnetic energies ; and
the electrical theory of chemical energy, which identifies chemical action
with electric energy. But we obtain our primary idea of energy from
the power of doing work possessed by a moving body, and we are able to
study most thoroughly the energy-transactions in a system which we need
only consider mechanically — i.e. with regard to its kinetic and potential
energies. Our aim, then, must be to frame hypotheses which shall
reduce other energies to these forms, for such hypotheses can be most
completely worked out, and their consequences most thoroughly compared
with the facts of observation.

Atomic Hypotheses. — The basis of these attempts must almost
necessarily be the hypothesis of an atomic or molecular constitution of
matter. This hypothesis we owe to the Greek philosophers, who
possibly arrived at it from considerations of the compressibility, the
disintegration, and the diffusion of matter. If matter is composed of
small particles separated by interspaces, it is easy to explain compression
as an approach of these particles, solution as an entry of the particles
of one substance into the spaces separating the particles of the other,
cleavage as the forcing apart of the particles by the insertion of a wedge
of some other substance, diffusion as the scattering of the particles, and
so on. If we suppose that matter is continuous, then we must accept
such facts as compression and solution as simple facts — i.e. facts incapable
as yet of explanation.

Considered only as explaining the general and most obvious properties
of matter, the atomic hypothesis is merely qualitative. It became
quantitative when used by Dalton and other chemists to explain the
laws of chemical combination, and the simplicity of this explanation has

130 HEAT.

no doubt largely contributed to its universal acceptance. The hypothesis
in its original form may be stated thus : A given mass of an element
contains a definite number of exactly similar indestructible particles,
termed atoms, these particles being exceedingly minute and exceedingly
numerous in any volume which comes within the range of our senses.
Each atom of a given element has a definite mass, which is always the
same for the same element.

Though this form of the hypothesis is sufficient for many purposes,
the relations which exist between the atomic weights and the other
properties of the various elements lead to the supposition that the atoms
of different elements are not simple indestructible particles unrelated to
each other, but are built up of some common material, the differences
being in the quantity and arrangement of this common material. Were
an atom of iron, for instance, entirely unrelated to an atom of platinum,
had they existed as they are through all time, it would be difficult to
account for such a relation as the equality of their atomic heats. It is
now generally believed that such relations may ultimately be deduced
from the supposition that the atoms are complex bodies, containing still
smaller particles or corpuscles, all alike in all elements, but different in
number and arrangement in the different atoms. Recent experiments,
especially certain experiments on the discharge of electricity in rarefied
gases, have given great weight to this supposition, since they may be to
some extent explained if the atoms are regarded as being split up in the
discharge, giving out corpuscles, the same for different gases. In this
chapter, however, we shall not go beyond the supposition that the atoms
are simple bodies.

In chemical compounds the dissimilar atoms of the constituent
elements are supposed to be grouped together to form molecules similar
to each other, the masses of the components being in the same propor-
tion in the molecules as in the compound as a whole, and the laws of
chemical combination find a simple explanation by supposing that the
mass of each kind of atom is definite.

The atoms, even in an elementary substance (in which, so far as we
know, they are all similar), are probably to be regarded as grouped,
perhaps in pairs, perhaps in greater numbers, and we may fairly extend
the term " molecule " to describe a group so formed.

When we seek to explain the various forms of energy on this atomic
or molecular hypothesis, we have further to suppose that the atoms are
held together by forces, for work is required to effect chemical decom-
position.

Work, too, is required to alter the volume of any liquid or solid body
in its normal condition. Hence, we must regard the molecules also as
held together by forces. We may have energy of motion both of atoms
and of molecules, and energy of position both of atoms with regard to
each other, and of molecules with regard to each other.

We will now consider the explanation which the atomic hypothesis
gives of the phenomena of heat.

When two solid bodies are pressed together and rubbed, one on the
other, work is done against friction and ordinary kinetic energy dis-
appears as fast as it is supplied to the bodies. The surfaces in contact
are found to rise in temperature. We may suppose that the friction

THE KINETIC THEORY OF MATTER. 131

means that the surface-particles of one body catch against the surface-
particles of the other, and that there are forces between these particles
against which the work is done. The collision will give the colliding
particles greater motion than that which they possessed before, and so
we get the idea that the heat developed is, at least in part, due to the
increase in kinetic energy of these particles.

When a body is heated it generally expands — i.e. its molecules are on
the average at a greater distance apart. This again implies that motion
has been given to them to move them apart : but it also probably implies
the existence of energy of position, the equivalent of the work done in
increasing the separation of the molecules against the forces between
them.

We also know that a hot body radiates energy to the surrounding
bodies — energy which affects our sense of sight, if the source be hot
enough. We know, too, that this energy is associated with waves,
implying a vibratory motion at the source. The waves are so small, and
the frequency of vibration producing them is so great, that we are led to
suppose the individual molecules or atoms to be themselves the vibrating
bodies. As to the mode of vibration we can only guess. It may be that
each separate atom vibrates in itself as a bell when struck — or each atom
may travel to and fro about some mean position, as a bell on a spring
might do — or both modes may co-exist. We may therefore have energy,
existing in various ways, associated with the atoms and molecules. We
may have vibrating energy of the atom, energy of motion of the atom as
a whole, energy of motion of the molecule as a whole, energy of separa-
tion of the atoms from each other, energy of separation of the molecules
from each other, and we must now regard heat as a mixture of some or
all of these. In other words, heat consists in the kinetic and potential
energies of the ultimate particles of matter.

We must make further assumptions as to the forms and constitutions
of the atoms and the actions between them if we wish to go further and
work out their motions by ordinary mechanics. One attempt of this
kind was made in the ring-vortex theory of Lord Kelvin,* which sup-
posed that the atom was a ring-vortex of fluid existing in an infinite
fluid. According to this theory, the energy of any system is entirely
kinetic, either energy of motion of the vortices, or energy of motion of
the surrounding fluid. It would therefore reduce all forms of energy to
the single form of kinetic energy, and on this account it is worthy of
mention, though recent electrical work has led to the imagination of a
very different type of atom.f

Without entering into any details as to the construction of the atoms
or molecules, we may still be able to form some general notions of their
arrangements and motions in the various states of matter.

In solids, the atoms and molecules are probably only agitated about some
mean position, for we know that solids keep their shape for indefinitely
long periods if not exposed to external action. Carefully preserved jewels
engraved by the ancients still possess all their original sharpness of out-
line. We cannot, then, suppose that the molecules are travelling about,
but only moving to and fro to a very limited extent. The molecules

* Maxwell's Scientific Papers, vol. ii. p. 470 ; " Atom," from Encyo. £rit., 9th ed.
f J. J. Thomson, Electricity and Matter, p. 90.

132 HEAT.

are sufficiently close together to act very considerably on each other, and
work has to be done to alter their arrangements in any way, as is shown
by the elasticity of solids.

In liquids, we must suppose that the molecules are not only agitated
but are travelling about, though only progressing slowly, for the pheno-
mena of liquid-diffusion show that the molecules take some time to
travel any considerable distance in a liquid. We may suppose that the
molecules possess on the average sufficient energy to do the work needed
to get away from their neighbours, but that they are still so near
together that they readily become entangled again. Their great resist-
ance to compression and their cohesion show, too, that the molcules
are near enough to act very considerably upon each other.

The existence of viscous solids, such as pitch, intermediate between
solids and liquids — substances which will flow like liquids if a sufficiently
long time is given to them — seems to show that in these solids there
is a certain amount of travelling about of the molecules. We may,
perhaps, suppose that a given molecule will, in its excursions about its
mean position, come within the sphere of action of its neighbours in
such a way as to receive from them continual supplies of kinetic energy
which enables it to increase the extent of its excursion until it can
break away and travel on to some new position of less constraint. It
is quite possible that exactly the same process takes place in every liquid,
only on a much greater scale, many of the liquid molecules vibrating
about a mean position like solid molecules for a time, but ultimately,
by the action of neighbouring molecules, becoming detached and travelling
about till they become entangled by new surroundings.

We may probably explain, by the aid of this supposition, the viscosity
of a liquid. If a liquid is in motion in one direction, but in such a
way that each layer moves slightly faster than the one below it, then
there is a tangential action between the layers proportional to the change
of velocity per unit length perpendicular to the layers.

If, for example, the successive layers A,B,0,D, Fig. 76, are all moving
in the direction AX, but A moves v per second more than D at a

distance d from it, the tangential

. V force per square centimetre

* exerted by each layer on the

next is «-, where n is the co-
c d

efficient of viscosity. Now each

D element of the liquid is being

FIG. 76. sheared by the relative motion

of the successive layers, and we

may suppose that for a short time after the shearing the element resists
the shear just as a solid element would, but that the resistance rapidly
dies away, owing to the breaking loose of the molecules from their old
positions and their adjustment in new positions. The strain producing
the tangential stress is not the whole strain of the element since the
beginning of motion, but only that part of it in which the particles have
not yet had time to rearrange themselves. It is easy to see that if rate
of decay of strain is proportional to the strain, then if the relative
velocity per unit length v is doubled, this effective strain is doubled, and

THE KINETIC THEORY OF MATTER. 133

the force is doubled, or TJ is a constant. Anything which hurries the
rearrangement will lessen the viscosity. For instance, we have every
reason to suppose that rise of temperature increases the energy of the
molecules, and therefore enables them to get free from each other more
frequently. This agrees with our knowledge that the viscosity of a
liquid decreases as the temperature rises.

Kinetic Theory Of Gases. — In applying the kinetic theory to
gases we can go into much greater detail. The molecules in a gas are
much farther apart on the average than in solids or liquids. A
cubic centimetre of water at 100° C. forms about 1600 cubic centi-
metres of steam at 100° C. and 760 mm. pressure. Hence, the molecules
of the steam are about \/1600, say 12 times as far apart as those of the
water. And this increased distance results in an almost complete
absence of cohesion, as is shown by the experiments of Gay-Lussac and
Joule (p. 119), in which a gas is allowed to expand without doing
external work. The change of temperature is only infinitesimal, showing
that practically no work is done against the mutual actions between the
molecules. The extreme rapidity of gaseous diffusion shows that the
molecules are in very rapid motion. But as there are an enormous
number of molecules, even in a very small space, and as they are not
mere points, but have some volume of their own, they must continually
be colliding with each other, and under the term "collision" we must
include every case in which two molecules come sufficiently within each
other's sphere of action to influence each other's motion. We need not
necessarily suppose that in a collision the force between two molecules is
repulsive. We may illustrate this by the motion of a comet which comes
into our system from outer space. Drawn by the sun's attraction it
rushes inwards, travels round the sun, and then rushes away to outer
space again. From our present point of view this is a " collision," and
the collision of molecules may perhaps be similar. The collisions in gas at
ordinary pressure cannot, however, occupy a very appreciable fraction of
the time of travelling about, for this would be inconsistent with the
absence of cohesion. When, in Joule's experiment, there are 22 times
as many molecules in a given space as in ordinary air, there are many
more collisions in a given time for each molecule ; in fact, the number
for a given gas per second is nearly proportional to its density. If these
collisions occupied an appreciable fraction of the time in ordinary air
they would, therefore, occupy a still larger fraction in the com-
pressed air. At any one instant, then, an appreciable fraction of
the total number of molecules would be in collision, and hence work
would have to be done to separate them, that is, to lessen the number
within each other's sphere of action. In other words, there would be
cohesion.

We shall, in the first place therefore, regard the collisions as instan-
taneously altering the velocities and directions of motion of the colliding
molecules. We shall not attempt to inquire what goes on in a collision,
and in speaking of the velocities of the molecules we shall regard only
the velocities when free from each other's actions.

When a collision occurs, the velocities and directions of motion of
the two colliding bodies are changed ; but we shall suppose that the
total enei'gy of the motion of translation remains the same after the

134- HEAT.

collision is finished as it was before it began. We know that the
energy of translation does not in reality always remain constant, for
a glowing gas, when cooling, loses some of its energy of translation,
and loses it in part by radiation. This implies that in some cases some
of the energy of motion is converted into vibratory energy of the mole-
cules and atoms in the collision, and is then radiated out. In other
cases the collisions may convert energy of vibration into energy of trans-
lation, for radiant energy is also being absorbed from surrounding bodies
to some extent. When, therefore, the temperature is maintained con-
stant we must suppose that there is a balance between loss and gain,
and that the mean kinetic energy of the whole group remains the same,
even though it may change for individual pairs of colliding molecules.

The assumptions that the time occupied in collision is very small, and
that the number of molecules in even a small space is very large, taken
together, imply that the dimensions of the molecules are exceedingly
small compared with their distance apart.

The most characteristic features of a gas are its diffusibility through-
out any vessel in which it is contained, however the volume of the vessel
may be increased, and the uniformity of the pressure which it exerts over
the containing-walls in accordance with Boyle's Law. On our theory, the
diffusion is due to the rushing of the molecules into any space open to
them, while the pressure on the containing-walls is due to the can-
nonade of the molecules against them. Each molecule as it comes up
against the walls rebounds, and imparts momentum to the walls equal
and opposite to that which it receives. Though on an area comparable
with the dimensions of a molecule the cannonade in a very small time
may vary, on any sensible area an enormous number of impacts will
occur in any sensible time, and the average will be practically constant
If some molecules lose energy in the collision, others gain, and on the
average we must suppose the loss and gain equal. Hence the average
momentum imparted per second (that is, the pressure exerted) will be
the same everywhere. We can see, too, in a general way how Boyle's
Law is explained, for doubling the number of molecules in a space
doubles the number colliding in a given time against the containing-
walls, and so doubles the momentum imparted per second. In other-
words, the pressure is proportional to the density.

The Mean Value of the Square of the Velocity of Trans-
lation.— Knowing the pressure and the density of a gas, it is possible
to calculate the mean value of the squares of the velocities with which
the molecules are moving.

If we follow any one molecule in imagination, its velocity will be
continually changing through collisions ; but if we consider a large
number of molecules, say those in a cubic centimetre, we may safely
assume that, so long as the conditions exhibited by the whole are the
same, the velocities are distributed in such a manner that a definite
and constant fraction of the whole will be moving with a given velocity —
or with a velocity within narrow given limits, though the individuals
may be continually changing. This assumption * is justified by our

* More advanced theory than we can give here shows how the velocities of the
molecules must be distributed in order that the collisions may not affect that
distribution. We may refer the reader to O. E. Meyer's Kinetic Theory of Giises,
from which much of the theory given in the text is derived.

THE KINETIC THEORY OF MATTER. 135

experience of statistics of population dealing with large groups of in-
dividuals. Whenever the group as a whole shows constant features, we
find that it can be subdivided into, smaller groups, also showing constant
features, even though the individuals in these groups change; for example,
in a large town where the circumstances remain pretty nearly the same,
the percentage of the population whose age lies between given limits
will remain constant, though fresh individuals are coming into the
group and others are moving out of it. Or, to take another illustration
more nearly resembling the case of a gas. With similar external circum-
stances as to day, hour, and weather, probably a certain fraction of the
people in a town will be in the streets on different days, with so many
moving at four miles per hour, so many at three miles per hour, so many
in collision, stopping to talk to each other. The individuals forming
each of these groups will change, but the number in the group is probably
nearly constant.

Similarly, if we consider a sufficient number of molecules, we may
assume that a constant number move with a given velocity in a given
direction.

Let us take, for simplicity, a vessel whose interior is cubical and one
centimetre each way, and let ABO be three perpendicular faces meeting
at an angle. Consider a molecule impinging with velocity VT in a cer-
tain direction against the face A of the vessel, and resolve this velocity
into three components uv vv wv perpendicular respectively to A, B, and 0.

Then Vx2 = u^ + v* + 10*

We shall suppose that the walls of the vessel are perfectly plane and
with coefficient of restitution unity, so that a molecule impinging on a
side has its velocity perpendicular to that side exactly reversed, while
the other components are unaffected. This, as we have seen, is probably
not true for individual impacts — in some there may be a gain of energy, in
others a loss ; but, so long as the gas and wall are at the same temperature,
the average energy is the same before and after impact, and the number
of molecules moving away with a given velocity is the same as if in each
impact the above supposition were true. That this is the right view is
shown by considering the case in which it is no longer true, that in
which the temperature of the gas differs from that of the wall. Then
the energy of the gas is different after impact, and from this difference,
as we shall see later, such motions as that of the radiometer can be
explained.

If we take the mass of the molecule as m, it moves up with the
velocity u^ perpendicular to the wall, and has this changed to - uv or
there is a total change of momentum, 2mur This, therefore, is the
momentum given to the wall by the impact.

Through the velocity MX the molecule would move to and fro wx times

per second if there were no collisions, and impinge against A -^ times.

2i

Now, though no actual molecule does this, the effect is the same as if it did,
for when it has its velocity altered by a collision, some other molecule at the
same distance from A takes its place, and moves on with the same velocity,
and no appreciable time is, by our supposition, lost by the collisions. In
one second then the total momentum imparted to A by one molecule will

136 HEAT.

be Zmuj^ x -^ = mu^. If we have in all n molecules, their velocities

a

perpendicular to A being uv u2 . . . un and if the total momentum
imparted per second to A is equal to p,

Considering the face B we similarly obtain

p = m(v* + v2z +
and on 0

p = m(w12 + w.?+
Adding these three together,

. . . . + m(un* + wn2

If V2 is the mean of the squares of the velocities, then we may write
this in the form

3p = mnV2
But mn is the total mass of gas in 1 cc., that is, is equal to the density p.

Hence 3» = />V2 or V2 = 5?

P

V is not the mean velocity, but the square root of the mean of the
squares of the velocities. It is termed " the velocity of mean square,"
and it may be shown that it is somewhat greater than the mean
velocity.

Maxwell investigated the law of distribution of velocities about the
mean which would justify the supposition of constancy of distribution,
and he showed that it is of " exactly the same mathematical form as the
distribution of observations according to the magnitude of their errors,
as described in the theory of .errors of observation. The distribution of
bullet-holes in a target according to their distances from the point aimed
at is found to be of the same form, provided a great many shots are fired
by persons of the same degree of skill" (Maxwell's Theory of Heat,
p. 309, ed. 5). It can be shown that the mean velocity U is given by

or very nearly U = ^ V

19

If the gas is hydrogen at 0°, then if p — 1014000 (the value of 76 cm.
of mercury in dynes per sq. cm.) p is very nearly -00009.

Whence V= 184000 cm./sec. nearly;

and U= 170000 cm./sec. nearly.

If we take any other gas at the same temperature and pressure the

velocity is inversely as the square root of the density. Hence for oxygen

we must divide by 4, and

V = 46000 nearly,
U = 42500 nearly.

THE KINETIC THEORY OF MATTER. 1S7

For nitrogen we must divide by Vl4, and
V = 49000 nearly,
U = 45400 nearly.

If we assume that V2 is the same for a given temperature, whatever
the pressure — and this is the natural assumption, seeing that the
temperature of a gas only alters very slightly if its volume is changed

¥)

without doing external work — then at once we have — = constant for a

given temperature, or Boyle's Law follows.

It must, however, be observed that our investigation is based on the
assumption that the collisions of each molecule take up only a negligible
fraction of the time. If through a crowding up of the molecules this
ceases to be true, we can no longer- assume that a molecule of type
having velocity u perpendicular to a face of the cube will travel u cm.
per sec., for some of the time will be wasted in collisions. The more
time thus wasted the fewer returns will the molecules make to the wall,
and it will contribute to the pressure a less amount than that calculated.
But we shall return to this point later and consider how we may obtain
a more correct relation between pressure and volume than that given by
Boyle's Law.

Mixture Of Gases. — Maxwell showed that in a mixture of gases
the different kinds of molecules will exchange energy with each other till
the average kinetic energy of a single molecule of each kind is the same,
and that this is the condition for steadiness of distribution.

Now, assuming this, suppose that we have equal volumes of two
different gases at equal pressures and of densities p1 and /a2, the equality
of pressure gives

If the two gases be allowed to mix there is no work done externally,
and no change of temperature, and the pressure remains the same.
This is accounted for if we suppose that the velocity of mean square of
each gas is the same after mixture as before.

But if m^ is the mass of a molecule of the first gas, and m2 that of a
molecule of the second gas, Maxwell's investigation shows that

or _

rax ra2

But if Nj N"2 be the numbers of the two kinds of molecules in unit
volume before mixture,

whence Nj = N2

or two different gases at the same temperature and pressure contain the
same number of molecules per c.c. This is known as Avogadro's Law.

Relation between V and Temperature. — If we take a volume of

138 HEAT.

a single gas and keep its density constant while altering its temperature,
we have the relation between the pressures at t" and at 0° 0.

V2 v

But ^s=- if p is constant.

2

Therefore, V,2 = V02(l +at) or V,2

where 6 is the temperature on the gas scale. Hence the energy of trans-

"\T2

lation of the molecules in a c.c., P— ',is proportional to the gas temperature.

a

Energy of Translation and Internal Energy.— If our investi-
gation applies to real gases it is e"asy to show that the energy of trans-
lation is not for most gases the only energy possessed by the molecules.
When a gas is heated we must suppose that in general some of the
energy goes to increase molecular potential or molecular vibrational
energy. For the energy of translation of 1 c.c. at 0° 0. is

and if the volume is constant the increase in translational energy, fcr a
rise of 1° is

where a is the coefficient of pressure increase.

The total increase of energy is Ke, the work measure of the specific
heat at constant volume.

The difference of the specific heats at constant pressure and constant
volume is given by Kp- K^ = op0, the work done in expansion.

Putting =? = y , we have (y - 1 )Ke = ap0

&»

TT increase in translational energy _ 3op0_ 3(y - 1)KB 3/ _ , .

increase in total energy ~ 2 K0 ~ 2KB ~ 2

This is unity, or the total energy given is converted wholly into transla-
tional energy when and only when

r-l = _ ory = - = l-66

This is the value of y found by experiment for mercury vapour, argon,
and helium. For these gases, then, we must suppose the collisions
to be of such kind that the internal conditions of the molecules
are not appreciably affected by the collisions ; in fact, that there is no
interchange between the energy of translation and the energy of position
or vibration of the constituent parts of a molecule.

THE KINETIC THEORY OF MATTER. 139

According to a theorem due to Boltzmann, if a molecule be regarded
as a mechanical system having n " degrees of freedom " or such different
modes of motion that it requires the knowledge of n different quantities
to specify its position and configuration at any instant, the energy must
be equally shared between the different modes when the distribution
of velocities and of internal energy is permanent. A purely translational
motion has three degrees of freedom, say the motions parallel to three
perpendicular axes. If there are other degrees of freedom implying
possibility of change of internal arrangement, making with the three
translational degrees n in all, we have

translational energy 3 3/ , x , -. 2

-°J = - = ~(y —I), whence y = 1 +
total energy n 2 n

If n = 8 we have the case of mercury vapour, helium, and argon.
We may, merely for illustration, picture the molecules in these gases as
small perfect spheres, perfectly smooth if they come in contact at collision,
or else in an encounter never actually touching. Then their mutual
actions always pass through their centres, so that there never can be any
interchange of energy between the tvanslational and the rotational forms,
and we need not consider the co-ordinates necessary to specify the con-
figuration of the sphere. Of course this is a very crude illustration, but
it serves to show that it is conceivable that there may be degrees of
freedom like those expressing the rotation of the spherical molecules
which may be omitted from consideration, since the forces producing or
destroying that rotation do not come into play in encounters.

For oxygen, hydrogen, and nitrogen,

y = 1 *4 very nearly.
Then n = 5

We may here picture the molecules as pairs of atoms rigidly attached
and forming, as it were, dumb-bells. If the actions at encounters always
pass through the axis of symmetry the co-ordinates expressing the
orientation round that axis may be omitted and the position and con-
figuration of a molecule is sufficiently given by the three co-ordinates of
its centre of gravity and the two angles giving the direction of the axis
of symmetry, thus accounting for w = 5.

If the distance between the atomic pair is variable, n = 6 and y = 1 -33,
a value possessed by some gases.

As n increases, y approaches 1, and observation shows that for more
complex gases this is true.

It may be observed that if the omission of any degree of freedom is
admissible through its force not coming into play at collision, then if
motion in that degree is produced by the absorption of radiation of a
given wave-length passing through the gas, sucli radiation will not affect
the energy of translation but only the molecular configurations. Thus
we may suppose that the pressure of argon against the side>s of the con-
taining vessel will not be appreciably altered if it is exposed to radiation
which it can absorb.

But the questions here discussed or rather indicated are still
open. It is held by many that Boltzmann's theorem does not really

140 HEAT.

apply to molecular systems ; that, in fact, the conditions assumed by
Boltzmann in the proof of his theorem are not realised in actual
molecular systems.

Joule's Approximate Method of Calculating the Velocity of

Mean Square. — The first calculation of the velocity of the molecules was
made by Joule by a method which is obviously only approximately correct,
but which is valuable in that it enables us to obtain easily results of the
right order and which gives us at any rate insight into the principles of the
theory. Considering, say, a cubic centimetre, let us think of the molecules
as divided into six streams moving perpendicularly to the six faces of the
cube, one stream towards each face, and let us omit all consideration of the
collisions between molecules. Then the stream moving towards one face

at any instant has mass ^ where p is the density of the gas. Let the

velocity of the stream be Y. Then the total mass moving up to the face in

pV

one second is that which would be contained in V c.c., or is £— . But its

o

velocity is changed by impact against the face from + V to - V, so that the

pV pV2

momentum imparted to the face in one second is ~ x 2V = u_, Equating

b 3

this to the pressure, we get V2 = — .

P
Effusion or Transpiration through a small Orifice into a

Vacuum. — The phenomena of " transpiration " through a small orifice
may be generally explained by the aid of Joule's method, though, of
course, the method cannot be expected to give a complete account. If a

gas of density p is contained in a vessel with a small orifice, area S, we

PY

may think of mass -t-— as moving up per second towards the face con-
6

VS
taining S, and the mass escaping through S will be p-~- per second.

The mass escaping when the distribution of velocities according to
Maxwell's Law is taken into account can be shown to be

US 12PVS 3
'-4— 131TT-1

where U is the mean velocity and V that of mean square, so that the
approximation in Joule's method gives the numerical coefficient too
small in the ratio 13 : 18, and this example may serve to show the kind
of error introduced by that approximation.

If we have two vessels containing different gases at equal pressures
escaping through equal orifices, the masses escaping per second will be
in the ratio plVl : P2V2, and the volumes escaping in the ratio Vl : V0 ;
or since Vx2 : V22 = p2 : pv when the pressures are the same, the volumes
are as

\/P2 : \/Pr
Hence the times of efflux of equal volumes are as

THE KINETIC THEORY OF MATTER.

141

This formula is in accordance with Graham's experiments. Below are
given some of his results (Meyer, Kinetic Theory, p. 84) :

Time of Efflux of Given Volume

Gas.

,/Specific
Gravity.

Drawn-out

Perforated

Glass Tube.

Brass Phite.

Air ....

1

1

1

Hydrogen . .

0-263

0-277

0-276

Oxygen

1-051

1-053

1-050

Nitrogen . .

0-986

0-984

0-984

Carbonic Acid

1-237

1-218

1-197

If the gas flows out, not into a vacuum but into a space containing
another gas, the rate of efflux is still nearly the same so long as the
pressure of the outside gas is small, say less than half that of the inside
gas. The stream of issuing gas is to be regarded as all moving in the same
direction, sweeping the external gas away, and its velocity will depend
chiefly on the velocity of the molecules within the vessel.

But if the pressure outside is nearly equal to that inside, the external
molecules will collide with those issuing, and the time of efflux will be
lengthened out. Still, with the two gases issuing under the same differ-
ence of pressure, the masses moving up to a very short orifice will be

given by - — and the same fraction of each will issue if the external gas

is at the same pressure in each case, and still the ratio of the times of
efflux will be as

Thermal Transpiration. — If we have the same gas, but at two
different temperatures, 6l and $2, on the two sides of a small orifice, the
masses moving up to the orifice on the two sides will be proportional to
/DjVj and /32V2 respectively. If the pressures are equal to begin with,

and
whence

-VI

or if #2>#p p1V1>/)2V2, and more gas moves in from the cold side than
moves out from the hot side and the pressure of the hot gas increases,
the increase will go on until there is a balance between the two streams

or until' pl~Vl = p2V2

or putting V2 = —

P
until

142 HEAT.

But &„£

p2 P

therefore the balance is attained when

The pressures are then as the square roots of the temperatures on the
gas scale.

The phenomena of thermal transpiration have been investigated by
Osborne Reynolds (Phil. Trans., part ii. 1879). He maintained the gas
at constant temperatures of 100° 0. and 8° C. respectively, in chambers on
the two sides of a plate of biscuit, meerschaum, or stucco, and determined
the difference of pressure in the steady state. He found that at low
pressures the formula is nearly verified. As the pressure on both sides
increases, however, the difference of pressure is nearly inversely propor-
tional to the mean pressure. In his paper he gives a full investigation
of the theory, which agrees with his observations.

The Mean Free Path. — The very great velocity of the molecules
as calculated on p. 136 might lead us to expect that gaseous diffusion
would be extremely rapid, so that if, for instance, a gas-tap were turned
on in a room the coal-gas, with a molecular speed comparable with half
a mile a second, would almost instantly spread all over the room. But
this does not agree with observation. If the air in the room is free
from draughts it may be quite a considerable time before the coal-gas
is in sufficient quantity to be perceived, even a few feet from the tap.
The diffusion is hindered by the collisions of the molecules with each
other. If we could follow a given molecule we should see it continually
colliding with, or being interfered with by its neighbours, pursuing a
given direction only for a very short distance and a very small time,
then colliding with another molecule and changing its direction of motion
for another short distance, then colliding again, and so on. The mean
distance travelled between successive collisions is termed the mean free
path. We shall denote this by M.F.P. The number of collisions per
second made by a molecule is termed the Collision Frequency. Evi-
dently the collision frequency x the M.F.P. is equal to the mean velocity.

We shall see later that we may estimate the M.F.P. in air at atmos-
pheric pressure as of the order of a hundred-thousandth of a centimetre,
and the collision frequency as more than a thousand million. During a
second, then, a molecule changes its direction of motion thousands of
millions of times, moving now forward, now backward, now up, now
down, now to this side, now to that. The different displacements will
to a very large extent neutralise each other, so that at the end of a
second a molecule will generally only be a very short distance from the
point it occupied at the beginning of the second.

Molecular Dimensions. Molecular Sphere of Action.— We

cannot at present form a working hypothesis, useful for the kinetic
theory, of the structure of the molecule, or of the field of force around it.
We cannot, therefore, say how the molecules act upon each other when
they approach and are in collision- We must be content to take what

THE KINETIC THEORY OF MATTER. 143

is at the best merely an approximate representation, by supposing that
the centres of the molecules approach till they are, on an average, a dis-
tance s from each other, and that then they recede. It is often con-
venient, for calculation, to think of a sphere of action surrounding one
molecule of a colliding pair, and to concentrate our attention on the
centre only of the other molecule. We then regard the centre of the
first molecule as surrounded by a sphere of radius s, within which the
centre of the second molecule cannot penetrate, and we term « the
radius of molecular action. We ought really to picture two spheres of
action, one round each molecule, and each of radius s/2, but the result is
the same. Since the molecular centres do not get within distance s, we
may regard s as the diameter of each molecular system. If the gas is
reduced to the liquid or the solid condition, we think of each molecule as
being just within the spheres of action of its neighbours all the time, and
we therefore regard s as indicating approximately the distance of a solid
or liquid molecule from its immediate neighbours.

Dependence of the M.F.P. on Molecular Dimensions and on

the Density Of the Gas. — If the molecules were mere points, and if
they exerted forces upon each other only at infinitely small range, the
M.F.P. would be infinitely great so long as the number of molecules in
a finite space was finite. For consider a single molecule projected from
a point. The spheres of action of the surrounding molecules within any
finite distance would fill up an infinitely small fraction of what we may
term its field of vision, since the total solid angle subtended by any finite
number of molecules at the point would be infinitely small. If then
a line were drawn in any assigned direction the chance that it went
through another molecular point within a finite distance would be in-
finitely small. But if we assign a finite value to the radius of molecular
action and now think of a molecule as projected from a point, in what-
ever direction we draw the line of projection it is practically certain that
within some finite distance it will impinge on the sphere of action of
another molecule. This is illustrated by a shower of rain which has only
to be of sufficient breadth to hide entirely objects beyond it.

We may form a mental picture of the M.F.P. by imagining that all
the molecules but one are fixed in the configuration which they have at
a given instant. We may then project the one from its position in turn
in all directions till it comes into collision with another molecule, and
take the average distance traversed before collision as equal to the M.F.P.
The fixing of the molecules, which we have assumed for simplicity, gives
us, as Clausius and Maxwell showed, too great a value for the M.F.P.
It is easy to see that the motion of the molecules increases the chance of
collision, for imagine a spherical shell 2s thick drawn at a distance from
the point of projection. The projected molecule will travel through this
in time 2s/V, and meanwhile a molecule within the shell, and having the
same velocity, will travel a distance V x 2s/V = 2s, and sweep out an area
4s2. The effective area subtended at the centre of projection by the
molecule will be ?rs2 + a fraction of 4s2, the fraction depending on the
inclination of the motion to the direction of travel of the projected mole-
cule. Thus the chance of collision is increased. It can be shown to be
\/2 times as great as the chance when only one molecule moves.

If the number of molecules per c.c. remains the same, while the cross

144 HEAT.

section of each is increased, evidently the chance of hitting one within a
short distance is increased. We shall show below that, as we might
perhaps expect, the collision frequency with a given velocity is propor-
tional to the cross section of the molecular systems. If the cross section
remains the same while the number of molecules is increased, the fre-
quency of collision increases in the same proportion. This again might
perhaps be expected. For if we put into a given space a second equal
number of molecules, we might expect that the projected molecule
would collide with the added molecules as often as with those pre-
viously in the space if the packing was so open that the second set
were not appreciably screened by the first, and thus it would have
double the collision frequency. These results may be obtained as
follows. Imagine a straight line AF drawn from a point occupied by
a molecule A, and passing in succession through the spheres of action

-B C D E

of molecules at B, C, D, &c. Then the average of the lengths AB, BO,
CD, &c., is the M.F.P. For it is the average distance which a molecule
projected in the direction AF will travel before it collides with another
molecule, and this average distance will be the same whether we project
always in one direction or whether we project in all directions in turn.
Now let us suppose that round the molecule at A is a sphere of action of
radius s and cross section Trs2, and let us represent the other molecules
at B, C, D, <fec., by points. Let Trs2 sweep forward in the direction AF

through I cm. If it impinges on n molecular points in this distance the
M.F.P. , which we shall denote by L, is equal to l/n. But the volume
swept out by ?rs2 is Trs2Z. Let the number of molecules per c.c. be N.
Then if irsH is large enough the number of molecules in it will be Trs2ZN.
Equating this to n or Z/L, we have Trs2ZN = l/L, or L = l/irs2N = m/irs^p
where Nm — p, m being the mass of one molecule and p the gas-density.
L, then, is inversely as the molecular cross section and inversely as the
density, if Trs2 is constant, and we may probably assume that it is constant,
at a given temperature.

The Mean Free Path calculated from the Coefficient of

Viscosity Of a Gas. — If a gas is moving in a given direction, but
faster on one side of a given plane containing that direction than on the
other side, then the slower-moving gas exerts a dragging action on the
faster-moving gas, which in turn tends to hurry on the slower-moving
gas. This tangential force is termed the force of viscosity. It is ex-
plained in the kinetic theory as due to the interchange of molecules
between the two portions of gas across the plane. If we think of the
plane as horizontal and the upper part of the gas as moving the faster,
then the molecules moving downwards through the plane have on the
average a greater momentum in the given direction than those moving
upwards through the plane to replace them, and therefore the upper
portion tends to lose momentum in the given direction and the lower
portion tends to gain it. In other words, there is a force on the lowei

THE KINETIC THEORY OF MATTER. 145

gas parallel to the plane and in the given direction and an opposite force
on the upper gas.

Let us consider a gas contained between two plane parallel boundary
walls AB, CD (Fig. 77) a distance d apart, the lower plane CD being
fixed and the upper plane AB moving with constant velocity v from left
to right. We shall assume that the layer of gas in contact with each
wall has no motion relative to that wall, and that the velocity increases
uniformly as we pass up from CD to AB, so that at a distance x from

CD it is — . Let the viscous tangential force of one layer on the next
d

layer per square centimetre in the direction of motion be F. The motion
of each layer relative to the one below it being uniform, the force F is
the same on each layer and ultimately acts on each boundary.

Direct experiments on the vibration of a plane disc close to another
plane disc fixed parallel to it show that for a given gas in a given con-
dition F is proportional to — , and so tend to justify our assumptions.

ct

It may be mentioned, however, that when the pressure of the gas is

0

FIG. 77.

very much reduced we are no longer able to assume that the layers in
contact with the boundary planes are fixed relative to them. There is,
in fact, side slip. This, however, is inversely proportional to the pressure
of the gas, and is negligible at ordinary pressures.

Let us now imagine a plane EF, 1 cm. square, parallel to the boundary
planes and consider the transfer of molecules across it. Let u be the
average velocity of the layer at EF from left to right. The molecules
moving down through EF will come from various distances and so carry
with them various amounts of momentum parallel to u. Let us suppose
that they carry on the average the momentum possessed by the mole-
cules in the plane indicated by the upper dotted line and distant from
EF by the M.F.P. = L say. According to Joule's method, the mass

moving down through EF in one second is p— .

6

The velocity parallel to u, at the distance L above EF, is u + — '

d

Hence the momentum parallel to u transferred across the plane in one

second is p— ( u -[ — - )

6\ d J

146 HEAT.

y

But an equal mass p— comes up through EF from below, and if we

suppose that this has on the average the momentum of the layer indi-
cated by the lower dotted line, distant L below EF, the momentum parallel
to u, brought up through EF, is

Hence the gas above loses momentum equal to the difference of these,
viz. : —

V Lv
P3 ' d

while the gas below gains an equal amount. But this transfer measures
the tangential force F. Then

3 ' d

But we also have F = r^-, where 77 is the coefficient of viscosity,

d

whence

pVL

•— —

Maxwell's more exact investigation gives r) = '30967/3LU, where U is
the mean value of the velocities.

We have already seen that L/> is probably constant for a constant
temperature. V is also constant for a constant temperature, so that 77
should be independent of the density of the gas. This result, first obtained
from theory by Maxwell, was afterwards verified by direct experiment.
Since V increases with the temperature, 77 should also increase, a result
borne out by experiment.

Maxwell's method of obtaining 77 consisted in principle in allowing
a horizontal circular disc suspended by a wire to vibrate about its axis,
another horizontal disc being fixed close below and parallel to it. The
layer of gas between the two discs was, therefore, sheared by the motion,
and the viscous resistance of the gas gradually " damped " the vibrations
of the moving disc. From this damping 77 could be calculated, and it
was found to be independent of the pressure within a wide range. If,
however, the pressure is sufficiently reduced the method fails through
slip of the gas on the surface of the discs. And even if there is no
slip, it the pressure is so far reduced that the M.F.P. is comparable with
the distance between the discs, evidently the method of investigation is
no longer admissible.

The following values of t\ at 0° C. are taken from the extensive tables
given by Meyer (Kinetic Theory of Gases) : —

Air ........ -00017

Hydrogen ....... -00008

Oxygen ....... -00019

Water vapour ...... -00009

While the viscosity increases, as the theory indicates, with the tempera-
ture, it does not increase merely in proportion to V or in proportion to

THE KINETIC THEORY OF MATTER.

147

the square root of the absolute temperature, but more rapidly, apparently
indicating that at a fixed density the free path also increases with the
temperature (Meyer, I.e., p. 216).

In the equation ry = ^— — , substituting for V from the equation
8

1 = 3p, we obtain

L = TK / —

whence L can be found when rj is known.

Since the number of collisions per second, or the collision frequency,
is equal to mean speed/mean free path, we can also calculate the collision
frequency, i.e., the number of collisions per second,

Below we give the values obtained for several gases at 760 mm.
pressure from our approximate numbers, which are sufficiently near the
truth to show the order of the magnitudes involved. We give the value
for water vapour on the supposition chat it could be compressed to I
atmosphere at 0° C. Air is regarded as a simple gas.

Gas.

Density at
0° and 760.

Coefficient of
viscosity ij.

M.F.P.

L=i7\/l

PP

Collision
frequency
V/L-P/*

Air .

129 -105

17 -105

82-=-107cm.

59 x 108

H .

9-105

8-105

146-107 )}

125xl08

O .

143 -105

19 -105

87 -107 „

53 x 10s

H2O

81 -105

9-105

55 -107 „

111 x 10s

Had we used the exact formulae we should have obtained values for
the M.F.P. about ^ greater, and of course the collision frequencies would
be proportionately reduced.

Conduction Of Heat in Gases. — The conduction of heat in gases
can be explained in a manner similar to that in which viscosity has been
explained. If there is a temperature slope in a gas, there is a continual
passage of more energetic molecules from the hot side across any given
plane, and a continual passage of less energetic molecules from the
cold side, with the net result that there is a transfer of energy down
the slope. The following investigation, though very incomplete, gives
an estimate of the amount of conduction.

Let EF, Fig. 78, be 1 square cm. in a layer of which the tempera-
ture is 0, and let the slope of temperature perpendicular to EF be -j--

Y

Let us take mass p— as passing through EF per second from the

upper sicle and as having on the average the temperature of the layer AB
a distance from EF equal to L, the M.F.P. — i.e. a temperature 0 + L-r-.

148 HEAT.

In the opposite direction let mass ^— pass upwards at the temperature

dO
of the layer CD a distance L below EF, that is, at temperature 6 - L— .

The excess of energy carried through EF from above over that re-

0V
turned from below is therefore the same as if mass -5- were cooled

d.9

through 2L — ' and if KV is the specific heat of constant volume, this will
dx

be equal to

pv de_KvPvi. do

"6 clx~~3~ dx

If K is the conductivity for heat of the gas, the quantity of heat

dO oVL pVL

passing down per second is K -,-, whence ~K. = —z—KV) or putting — TT~ =">}

the coefficient of viscosity, JL = ^KV) which is independent of the density

FIG. 78.

of the gas, a result predicted from the kinetic theory by Maxwell and
subsequently verified by experiment.

The Diameter of the Molecules and the Number of Molecules

per Cubic Centimetre. — The value of the molecular diameter, or rather
the radius of the sphere of action, was first calculated by Loschmidt in
1865.

Taking s as the radius of the sphere of action as defined on p. 143,

- may be taken as the radius of each molecular system when two
2

molecules are at their nearest approach with their centres s apart. Then
the volume of a singular molecular system is of the order

The total volume of the molecules in 1 c.c. is therefore . .

If we could suddenly destroy the translatory motion of the molecules
in 1 c.c., still keeping the molecular systems unchanged in themselves,
they would simply fall to the bottom of the space and occupy a volume

N7T.S3

of the order — ^-. If we imagined them all exact spheres, piled up like

THE KINETIC THEORY OF MATTER.

149

shot, the volume would be 3 »j2/ir times greater. Probably we have an
approach to such a condensation in the liquid and solid states, where the
molecules may be regarded as each within the spheres of action of its
next neighbours all the time, and we shall therefore assume, as an
approximation of the right order, that the volume of a gas containing N

molecules is 7rs when liquefied.

If, then, 1 c.c. of gas at 0'
volume v of liquid of density 8,

and 760, and density A, condenses to

But our equation for the free path is

0)

1 (2)

From (1) and (2) we have s = 6vL.

We know v and L in a number of cases, and can therefore find s. Then,
substituting this value of s in (2), we can find N. The following table
shows the results obtained for hydrogen, oxygen, and water vapour. It
is hardly necessary to point out that, seeing the assumptions made, the
results are only to be taken as indicating the order of magnitude involved.
The value of N, for instance, differs for different gases, whereas by
Avogadro's Law it should be the same.

Gas.

Gas Density
A.

Liquid
Density 8.

A

"•£

M.F. Path.
L.

S = 6>'L

N in 1 c.c.

Mass of one
Molecule.

H

9-105

7-=-102

130-106

146— 107

11-4— 108

3-4xl018

2-7-MO23

O

143- 10s

1-24

115- 105

87-107

6 -10s

10-2 x 1018

14 -^-lO23

H20

81 -105

1

81- 105

55- 107

2-7-108

8 x 1018

10-hlO23

Using the more correct formulae, Maxwell calculated that 1ST for gases
at 0° and 760 should be 19 x 1018, say 2 x 1019 (Papers, ii. p. 372),
whence m for hydrogen comes out 4'5 -4- 1024. The value of N can also
be found by an entirely different method — from the consideration of
the electrical properties of gases. The value so found is 3'9xl019
(Thomson, Condndion of Electricity through Gases, p. 130). Maxwell
further obtained for s with hydrogen the value 5 -f 108. If we assume
that this is about the value of the molecular diameter in other cases, an
assumption perhaps warranted from the nearness of the values of v for
different substances, then this would imply that in solids and liquids the
centres of the molecules are a distance apart comparable with 5 -=- 108.

Forces Acting on unequally heated Surfaces in High Vacua.—

The motions of unequally heated surfaces in rarefied gases were first
investigated by Crookes, who was led to the invention of the radio-
meter— a beautiful illustration of such motions. In a simple form, it
consists of four mica vanes, each lampblacked on one face and attached
to the four arms of a cross, pivoted to move with very slight friction
about the centre (Fig. 79). The lampblacked faces are so placed as all
to move round forward or all backward.

150

HEAT.

The vanes are enclosed in a glass bulb, which is exhausted by a
mercury-pump till the pressure is exceedingly small, and then sealed.
On exposing the bulb to a source of radiation,
the lampblacked surfaces of the vanes are more
heated than the bare surfaces and move away
from the source, and so rapid rotation results.

The complete explanation of the action is
due to Osborne Reynolds (" On Certain Dimen-
si°nal Properties of Matter in the Gaseous
State," Phil Trans., part ii., 1879). The action
was also sufficiently explained very shortly after
by Maxwell (" On Stresses in Rarefied Gases
arising from Inequalities of Temperature," Phil.
Trans., part ii., 1879). The theory is altogether
beyond our scope, but the following account of
what occurs may give some idea of the action.
It is to be remembered that it is an account,
and not an explanation. *

Let us imagine that a plane is suddenly
introduced into a gas, one side of the plane
being hotter than the gas, while the other side
is at the same temperature with it. Consider
a small area on the plane far from the edges.
The molecules which come up on the hot side
are raised in temperature by contact with the
plane and rebound with a greater velocity than
that with which they arrived, while those on the
cold side go off with the same velocity. For a
moment after the introduction of the plane the number coming up to the,
area is the same on the two sides, since it is conditioned by the tempera-
ture and pressure of the sur-
rounding gas, and these are not
yet affected by the presence of
the plane. Hence the extra
kick off of the molecules re-
bounding from the hot side
implies a greater pressure on
that side. But very quickly
this excess of pressure will fall
off, the rebounding molecules
on the hot side sharing their
extra energy with the mole-
cules with which they collide,
and as soon as a uniform
temperature slope outwards is
established, the decrease in
density, and therefore the de-
crease in the number of mole-
cules coming up to the hot side,

compensates for the extra impulse at each collision, and so the pressure
falls to the same value as on the cold side.

* Lord Rayleigh (Nature, July 15, 1909, p. G9) considers the extreme case of a
M.F.P. large compared with the dimensions of a vane, and shows that the pressure
ia proportional to area of vane x density of gas.

FIG. 79.

Fio. 80.

THE KINETIC THEORY OF MATTER.

151

Fio. 81.

But near the edges this compensation will not be complete. For a
distance inwards comparable with the M.F.P. the lines of flow of heat
will diverge from each other as shown in Fig. 80, and the temperature
falls at an increasing rate from the hot plate outwards. The density at
the average distance from which we
may suppose the molecules to come
to the plane is greater than well
inside the edge, and the change in
temperature and in velocity of the
molecules at their impacts is greater.
This results in a greater pressure
against the area near the edge on the
hot side, which continues as long as
that side is hotter. On the cold
side there will be a similar edge
effect, for the hotter molecules will
to some extent come round the edge
and carry heat into the cold surface.
The temperature at the average dis-
tance from which we may suppose the
molecules to come will be higher and
the density less than well inside the
edge on the cold side. There results
a defect of pressure near the edge.

Reynolds and Maxwell both showed that the excess of normal pressure
on the surface is proportional to

rate of increase of temperature slope outwards
pressure.

Hence, since the temperature slope with a plane decreases outwards
only near the edges, and for a width of surface comparable with the
M.F.P., both the numerator and denominator contribute to make the
total force greater the less the pressure of the gas ; hence the necessity
for the very high vacua with vanes of the ordinary size.

If the vanes are curved, as in Fig. 81, the convex side being hot, the
lines of flow are somewhat as represented in Fig. 79, and the divergence
on the hot side is more marked, while it is hardly noticeable on the cold
side. In this case the excess of pressure on the hot side will be greater
than with a plane surface.

At ordinary pressures the edge acted on by the pressure excess is
exceedingly narrow — of the order of the M.F.P., which, as we have seen,
is comparable with 10"5 cm., while the pressure excess can be shown by
the theory to be inversely as the pressure. Hence the total force is very
minute. But if the body acted on is reduced to exceedingly narrow
dimensions, and its mass correspondingly reduced, the effect may still be
noticeable at ordinary pressures. This was pointed out by Reynolds,
and he succeeded in detecting the action on spider lines and silk fibres
at pressures comparable to the atmospheric pressure, one side of the fibre
being exposed to radiation (I.e., p. 768).

We have, perhaps, an example of the action, which we may call

152

HEAT.

" radiometer action," on exceedingly small surfaces, at ordinary
pressures, in the dust-free region surrounding a heated body.

If a thick copper rod is placed so as to project into an enclosure filled
with dusty air or smoke, on heating the part outside so that the rod inside
the enclosure becomes heated by conduction, the rod looked at endwise,
when the light is properly directed, may be seen to be surrounded by a
dust free space as in Fig. 82, where the rod is supposed to come end-on
against a window in the side of the enclosure, the light being thrown
through the space towards the observer.

Lodge explains this by supposing that the heated body heats the
nearer face of the dust particles, and that there is a backward movement
if the dimensions of the particles are comparable with the mean free
path. They retreat on all sides from the heated body, leaving a space
of clear air. This ascends by convection, as
illustrated in the figure, being replaced by fresh
air, which in turn is cleared of its dust.

If the body be cooler than the surround-
ings, the converse happens, the dust being forced
on to the cooler surface. Many illustrations of
the deposit of dust and smoke on cooler surfaces
may be found. Plaster ceilings very frequently
show the course of the laths and rafters behind
the plaster. Where the plaster is backed by
wood, it is probably kept warmer, and the dust
is not so freely deposited as on the neighbour-
ing cooler parts. Walls above hot-water pipes
are often very soon blackened, the hot dusty air
depositing its dust against the cooler wall.

The Gas Equation of Van der Waals.— The

gas equation, expressing the laws of Boyle and of Charles, viz. :

pv = H8

is only approximately in agreement with observation. In a celebrated
paper " On the Continuity of the Liquid and Gaseous States of Matter "
(English translation, Physical Memoirs of the London Physical Society,
vol. i. pt. 3), Van der Waals deduced from theory the equation

FTG. 82.

(jp + £)(«,- 6)- R0

where a and b are certain constants. This equation, though still not in
agreement with observation, represents the relation between pv and 6
much more exactly than the original gas equation. The following method
of obtaining the equation may serve to show how the kinetic theory
accounts for the failure of Boyle's Law.

Returning to the investigation of the pressure exerted by a gas on the
sides of the containing vessel, let us now take into account (1) the size of
the molecules as affecting their length of path ; (2) their cohesive forces at
collision as lengthening out the time of collision. These are really two
aspects of one transaction, but it is convenient to consider them separately.

Let us take first the size of the molecules. Suppose a molecule to
start normally from one face of a unit cube vessel. If it meets another
molecule in direct collision, moving with equal and opposite velocity, there

THE KINETIC THEORY OF MATTER.

153

will be simply an interchange of velocities, and the second molecule will
take its place on the journey across the cube, and we may now fix atten-
tion on the second molecule. If this comes into direct collision with, and
interchanges velocities with a third molecule, this third molecule takes the
place of the second, and so on. But at each collision the substituted
molecule starts with its centre a distance s farther on, where s is the radius
of molecular action or the diameter of a single molecular system. If, then,
there are v collisions per second of this kind, the total distance covered
will be, not Y but, V + vs, and the molecule, or its representative, will
return to the original starting-point oftener than if the molecules were
mere points in the ratio V + vs : V, and on this account the pressure will
be greater than that originally calculated in the same ratio. But this is
on the supposition that all the collisions are direct, whereas they must
be regarded as of all degrees of
obliquity.

To find the average increase
of path, let us suppose one of the
pair of molecules in collision to
have its centre at O, Fig. 83.
Let ACB be a section of the
hemisphere of molecular action
with radius 00 = s. If the second
molecule comes along LO, in direct
collision, the path omitted is
CO = s ; but if the molecule comes
along MP, and rebounds along
PQ, producing MP to N, the path
omitted is PN. For, if the radius s were indefinitely small, the .second
molecule with the same obliquity of collision would move off along NE,
parallel to PQ.

Now the number of molecules making collision of a given type over
a given area of the hemisphere will be proportional to the projection of
the area on the diametral plane through AOB, for we may suppose the
numbers moving up towards that plane evenly distributed over it. If
the number moving up in given time to unit area is n, the number moving
towards a given ring of radius ON = r and breadth dr is n x Zwdr, and
these each add path PN, or total path

n x lirrdr x PN.

Then all the molecules coming up in unit time add path riSlirrdr x PN,

2

where we sum up for the whole hemisphere. But ^irrdr x PN = ^m?, the

o

volume of the hemisphere, and the total path added by all the n x vrs2
molecules coming to the hemisphere is

27TS3

The average path added is, therefore,

27TS3

W_^W = |<

and we must use this instead of s.

1/54 HEAT.

In place of

we shall, therefore, put jo

V
where .v is the number of collisions per second.

But v =

whence **£

N7TS3

Here - = total volume of the molecular systems per c.c., and since this

is small we may put

V2

where v = - is the volume of unit mass.

P
If we put

,N7TS3 , , ,x V2

4-— — w = o we get p (v - 6) = _.

O o

Now let us turn to the second consideration, that of time lost in
collision. We may think of a slackening of the velocity on close approach
and then an acceleration, so that in each collision there is a time greater
on the average by r than if one free path had instantaneously changed
to the next. If v is the number of collisions in one second, the time
occupied in traversing a given path will be greater than that obtained
on the supposition of instantaneous change in the ratio

l+vrrl,

and a molecule or its representative will return less often to a face of
the containing vessel in the ratio 1 : 1 + VT. On this account then, the
pressure will be reduced to

J_ Z? _L

P v-b 3 I+VT
or, instead of p, we must

Now v = =- = VNvrs2 and Nwi = p where m = mass of a molecule.
L

therefore v = V/> — .

m

THE KINETIC THEORY OF MATTER. 155

pV2
Putting for p its approximate value ?-—- in the small quantity pvr we get

9

6m
a

I

where v = - and a = -

p om

and the equation becomes

The assumption that a is constant implies that V37rs2T is constant. We
have no data to justify this, and herein lies a weakness in the investi-
gation.

It may be mentioned that Van der Waals has calculated b from
observations on the compression of gases. This gives another method of
finding s, and the results are of the same order as those already found.

The equation of Van der Waals has been modified in various ways
to make it represent the facts of observation more closely. But the
modifications are probably suggested rather by the lack of accordance
with fact than by the kinetic theory, and must be regarded as somewhat
empirical. For an account of some of these modified equations we refer
the reader to Meyer's Kinetic Theory of Gases, p. 100. Another modifi-
cation has been proposed by Callendar (Proc. R. S., Ixvii., 1900, p. 266).
We shall consider the equation of Van der Waals further in chap. xi.

We have already seen that some of the energy given to a gas on
raising its temperature goes, in all probability, to increase the internal
energy of the molecule. This brings us to the point where the theory
for the present stops. No satisfactory hypothesis as to the construction
of the molecules, and the mode in which they possess energy, has as yet
been devised.

In order to find out what goes on in a molecule we should, of course,
like to isolate and study it in the method by which alone we could do so,
viz., by examining the nature of the energy it gives out. But this is
impossible. The nearest approach to isolation is in a rare gas, where
the molecules have a long free path, so that, for a great part of their
time, they are not interfering with one another. When such a gas is
heated under certain conditions, it sends out waves of light of definite
refrangibilities, showing that in all probability the molecules or their
parts are vibrating, but we are quite unable to do more than guess at the
mode of vibration giving rise to these light waves. It may be that the
different atoms in the molecule are vibrating about the common centre
of gravity, as the earth and moon vibrate about their common centre of
gravity ; or, it may be that the separate atoms vibrate as a bell vibrates.
If the latter is the mode of vibration giving rise to light-waves, then we
must suppose that the contiguity of other dissimilar atoms alters the
character of the vibrations, for we know that the light sent out by
incandescent compounds is different in character from that sent out by

156 HEAT.

the constituents before combination. As a heated gas, which is emitting
a line spectrum, is compressed the character of the light sent out is
found gradually to change. On examining the spectrum it is found to
consist at first of a greater or less number of bright lines. As com-
pression goes on these first widen out into bands, and then gradually a
continuous spectrum appears, showing that light of every refrangibility
is being sent out. This is explained by the gradual shortening of the
free path. The molecules interfere more and more with each other,
their time for uninterrupted natural vibration becomes less and less,
while the times of clashing and general disturbance in collisions become
greater and greater, and this general disturbance gives rise to the
continuous spectrum. If, instead of a gas, we take a glowing liquid or
solid, we cease to have any appearance of bright lines or bands. There
is merely a continuous spectrum, showing, apparently, that the molecules
interfere with each other too much to allow the natural vibrations to
have play. We might, perhaps, give an illustration of this. If a
number of bells are hung in a room by strings from a ceiling, and are set
swinging, they will occasionally collide, and the energy of translation
will partly be converted into energy of vibration. If the bells are far
apart the notes given out will be the natural notes of the bells, for the
free paths will be long and the time of free vibration long, compared
with the time of clashing. But the closer the bells are together the
more prominent will the clashing be. If they are packed loosely in a
box and rattled about the natural vibrations will be quite overpowered
by the clashing.

Recent researches on the emission of radiation by gases appear to
show that each gas only emits a line spectrum when chemical change is
proceeding.* Thus sodium vapour when raised to a high temperature
in a neutral gas, one with which it does not combine, does not give the
characteristic D lines, but probably a continuous spectrum. But if it
combines with the surrounding gas the D lines appear. This would
appear to show that the waves corresponding to the D lines are not
emitted by the molecules as a whole, but by the parts which are
changing places in the chemical actions proceeding. At present, how-
ever, our hypotheses as to the nature of these actions are hardly definite
enough to warrant us in giving an account of the origin of the radiations.

* An account of our present knowledge will be found in the Reports of the
Congres International de Physique, 1900, vol. ii., p. 100, by E. Pringsheim.

CHAPTER X.
CHANGE OF STATE— LIQUID— VAPOUR.

General Account of Evaporation — Vapour- Pressure — Boiling— Delayed Boiling —
Condensation on Nuclei — Measurements of Vapour-Pressure — Determination of
Vapour Density — Density of Saturated Vapour — Measurements of Latent Heat
of Vapours — Specific Heat of Saturated Vapour — Spheroidal State.

Change Of State. — We are accustomed to find each substance occur-
ring most commonly in one particular state of matter. Iron, for ex-
ample, is usually solid, oil liquid, and air gaseous. But we are also
familiar with the change of the same kind of matter from one state to
another, which is effected by supplying or withdrawing heat, and experi-
ment leads us to believe that, unless chemical change intervenes, every
substance may be made to assume any one of the three states, even
though our present experimental arrangements may be insufficient to
accomplish this. The change as we add heat is, in general, from solid to
liquid, and from liquid to gas, though it may be from solid to gas without
the intervening step. There are also apparent exceptions, as in the case
of the preparation of solid red phosphorus by heating ordinary molten
phosphorus. But we probably have here an absorption of energy
accompanying quite a different arrangement of the molecules — a change
of state of another kind. We may take it, therefore, as a rule that the
three states — solid, liquid, gas — are in ascending order as regards the
quantity of energy possessed, and the quantity required to effect the
change from one state to the next is usually large. If, for example,
we take a quantity of ice below 0° C. and supply heat to it, the tempera-
ture rises steadily to 0°. There is then a pause while melting takes
place, a very considerable quantity of heat being absorbed merely to
effect the change from ice to water while the temperature remains
steady. This heat is said to be latent, a term which was given on the
supposition that the general effect of heat was to raise the temperature
of bodies, whereas this heat is not affecting the temperature. From this
point of view the term is a very good one, though, of course, the heat is
not latent in any other sense, as its presence is quite evident in change
of physical state.

The ice being melted, the water again rises steadily in temperature
till it begins to boil, turning rapidly into steam or water-gas, when there
is another pause, and a still larger quantity of " latent " heat is required
merely to effect the change from water to steam without rise of tempera-
ture. But besides this rapid change at boiling, when the water is in an
open vessel a gradual change into steam takes place at the upper surface
even at ordinary temperatures, a change which is more rapid as the
temperature rises. This change, which is termed evaporation, even takes

158

HEAT.

place from the surface of ice — a fact which is familiar to us in the
disappearance of snow in a dry east wind, though the temperature does
not rise to the melting point. Evaporation also
absorbs heat, a fact which may be easily illustrated
by pouring a few drops of ether on to the hand,
when the cooling of the skin as the ether evaporates
shows that heat is being rapidly absorbed.

In studying change of state in detail, we shall
consider first the change into the gaseous condition
and the converse, because, with the aid of the kinetic
theory of gases, we are able to give a more com-
plete account of it than is as yet possible in the
other cases.

Liquid-Gas Change. — Let us suppose that an
ordinary barometer-tube, held with the closed end
downwards, is nearly filled with mercury, the re-
maining space being filled with water. On invert-
ing the tube in the ordinary way, and unclosing the
end in a cistern of mercury, the water floats up to
the top, and some of it evaporates into the Torri-
barometer; EandG, cellian vacuum. Allowing for the pressure of the
barometers with small small quantity of water, and comparing the height
quantities of water of the column thus corrected with that of a baro-
above the mercury. meter made with dry mercury, it is found to be
somewhat less, or a certain pressure is exerted on
the top of the water Testing on the mercury column.
This pressure is due to the water-gas, or, as it is usually
termed under such circumstances, the water - vapour,
present in the space which would otherwise be a Torri-
cellian vacuum. If the temperature is maintained
constant, and if there is sufficient water, this pressure
rapidly attains a definite value, quite independent of the
volume of the tube above the mercury. This may easily
be shown by raising this tube or depressing it in the
cistern, or by tilting it, or by using different tubes.
Thus in Fig. 84 A is an ordinary barometer, B and 0
other barometers, each with a small quantity of water at
the top. The height of the columns in B and 0 is the
same, and less than that in A.

In an ordinary tube, the evaporation is very rapid,
so that the steady pressure of the vapour is soon attained.
In this state the space above the mercury is said to be
saturated with vapour, and the pressure exerted is termed
the " maximum vapour-pressure " or " vapour-tension " for
the given temperature ; maximum, because any decrease
in the volume only leads to condensation of the vapour,
not to increase in its pressure.

Use of the Terms " Gas " and " Vapour."— In

ordinary language a gas is usually described as vapour
when thought of as given off by a liquid or solid,
especially when it is not far from the maximum pressure at which it

. — Baro-
meter with
Water above
the Mercury
and surround-
ed by a Heat-
ing Bath.

CHANGE OF STATE— LIQUID— VAPOUR.

159

begins to condense. We shall see later that there is probably for every
gas a temperature below which it can always be condensed to liquid by
sufficient pressure, and above which such condensation is impossible. It
has been suggested that the gas should be described as a vapour below
this temperature, and as a gas above it. This restriction of the term
" vapour " is in accordance with general usage, but it is probably more
convenient to use the term " gas " in the general sense, including "vapour"
as a particular case.

If the barometer-tube B of Fig. 84, containing the water be sur-
rounded with a water-bath, as in Fig. 85, so that its temperature can
be raised, then it is found that the pressure of the vapour rises with the
temperature, more and more of the water evaporating into the space
above ; and at each temperature there is a definite maximum pressure,
which is more or less rapidly attained.

In the case of water, the following are some of the values of the
maximum pressure. "We shall describe later how the accurate values
are obtained.

Temperatures.

Pressure in mms.
of mercury.

0°

4-6

20°

17-4

50°

92

100°

760

150°

3581-2

200°

11689

For other substances the values are very different.

Pressures in mms. of mercury.

Temperatures.

Alcohol.

Ether.

Mercury.

0°

12-7

184-4

•020

20°

44-5

432-8

34-97°

760-0

50°

219-9

1264-8

• • •

797°

760-0

100°

1697-6

4953-3

•746

200°

. ..

* . .

19-9

300°

...

• •«

242-15

357-25°

• • •

760

400°

...

...

1588

If the space into which the evaporation takes place contains air or
any other gas the evaporation goes on, but at a diminished rate. The
pressure exercised by the vapour and, therefore, the total vapour present

160

HEAT.

in a given space when the steady state is attained, is still very nearly
the same, though it appears to be always slightly less than in a space
containing only the vapour. The practical equality may be shown by
an arrangement similar to that in Fig. 86.

A is a piece of wide glass-tubing, several inches long, closed at its
upper end by a tap, above which is a small funnel. A is connected to
the open tube B by a piece of strong flexible tubing. The tap t is
opened, and mercury is poured into B until it reaches a certain mark in
A, standing, of course, at the same level in B. The tap is then closed,
and ether is poured into the funnel. B is lowered so as to reduce the
pressure in A, and when t is opened, a small quantity
of the liquid is pressed into A without the escape of
r~~1 any air ; t is now closed, and when the steady state is
$^\t reached — an excess of liquid being still present — B is
raised till the level of the mercury in A is at the original
mark. The air occupying its original volume, still exerts
the atmospheric pressure. The difference of levels of
the mercury in the two tubes is, therefore, due to the
pressure of the ether-vapour, and this is found to be
equal to its value when the air is absent. The same
arrangement may easily be used to find this value by
opening t and raising B till all the air is driven out of
A, a small quantity of liquid being still left in the tube
above the mercury and below the tap. If we close t and
lower B, then as soon as the surface in B is a depth
below that in A equal to the difference between the
atmospheric pressure and the pressure of ether vapour,
evaporation takes place. When this begins, even if B
is slowly lowered still further, A also falls so that the
difference between the levels in A and B will remain
constant.

This evaporation into % space already containing air

„ R6 E is, of course, continually occurring from water on the

ration into an surface of the earth. The pressure of the vapour present
Air Space. in the air, in fine weather, is always less, below the cloud
level, than the maximum for the existing temperature.
Evaporation, therefore, takes place from the surface of any water, or from
any damp material present, and tends to bring about saturation. But fresh
drier air is continually being brought over the water or damp material.
This may take place by winds, or it may be brought about by convection,
for the water-vapour is lighter than the air it replaces, and the vapour-
charged air tends to rise, colder air from above, containing less vapour,
taking its place. The evaporation, therefore, continues, but probably
owing to continual renewals of the air, the maximum pressure is never
produced by evaporation from the surface of the earth alone.

As a converse to evaporation, we continually have condensation
occurring in the atmosphere in the formation of clouds and fogs. These
are, probably, always produced by cooling. A mass of air containing
water-vapour, not very far from the maximum pressure, becomes cooled,
and normally as soon as the temperature falls to that for which the
maximum pressure of water- vapouris equal to the existing pressure, conden-
sation begins in the form of cloud> which is well described as water-dust.

CHANGE OF STATE— LIQUID— VAPOUR.

161

Distillation. — Condensation is used in the process of distillation to
obtain pure water.

The simplest form of still is one in which the steam rises from a
vessel, leaving the impurities behind in the water. The steam is con-
veyed through a jacket, through which cold water is kept circulating, so
that the temperature of the steam falls below the condensing-point, and
the condensed water trickles down into a receiver, as in Fig. 86a.

We may summarise the results already described in the following
statement : —

A liquid with a free surface and a space above it will evaporate, or
change into gas, into that space, until a pressure is reached definite for
each temperature. This limiting pressure is called the maximum vapour-
pressure or vapour- tension. Its value is only very slightly affected
by the presence of other gases, though these retard evaporation. It rises

FlG. 86a.— Apparatus for Distillation. A, still ; B, receiver.

with the temperature, and differs greatly for different liquids at the same
temperature.

It may assist us in arranging these facts in our minds, if we seek
to give an explanation of them on the kinetic theory of matter.

Let us first consider the case of a liquid partially filling a closed
vessel, the space above being a vacuum to begin with.

The molecules in a liquid are, in general, within each other's spheres
of action, and are entangled with each other ; but there are a great many
of the molecules which have sufficient kinetic energy to escape from the
groups which they may be near at a given instant, and these move off to
become entangled with other groups. We might, perhaps, represent
the liquid at any instant as a network of attached molecules, form-
ing a loose solid with numbers of gas molecules moving about in the
interspaces. But the members composing the network are continually
changing places with the freely moving molecules, so that the network
is not permanently solid. If, now, one of the freely moving molecules
happens to be close to the surface, and moving from the general body of
the liquid, it will entirely escape and move into the space above the
liquid, that is, it will evaporate. Escapes will take place all over the
surface of the liquid in the same way, and very soon the space above will
contain a great number of these molecules, which from their kinetic

L

162 HEAT.

energy, will behave to each other as gas molecules, colliding, and
escaping again from each other's spheres of action, cannonading the
sides of the vessel, and so exerting pressure. Through their collisions
with each other and with the sides, many of the molecules will have
their direction of motion reversed, and will return to the liquid. Some
of these will become entangled, and again become liquid molecules. The
greater the number of molecules in the space, the greater is the number
thus returning to the liquid, and as evaporation continues a point is at
last reached when the number returning is equal to the number escaping.
This corresponds to the "maximum vapour-pressure," the steady state
being due — not to a cessation of evaporation- — but to a balance between
evaporation and condensation. If, through diminution of volume, the
pressure tends to exceed the maximum vapour-pressure, the condensation
exceeds the rarefaction until the steady state is again arrived at.

We may note that the molecules escaping are the most energetic in
the liquid ; their escape therefore lessens the average energy of those
remaining, and this is the meaning of the fall of temperature in a liquid,
produced by evaporation from its surface.

If the temperature of a liquid rises, the average energy of the mole-
cules increases, and the number of molecules with velocity sufficient to

escape also increases. Hence evapora-

£ ! ^ tion goes on more rapidly. The number

of molecules in the space required to

*. .". . ' . ". . . . .*. produce a balancing condensation must,

..'...'..'. .'..'..'..'..'. .'..'..'..'..'..'.. v therefore, also be greater — in other

..*.... words, the maximum vapour-pressure

increases.

| ' ' The rapidly rising rate of increase

0 ' may, perhaps, be explained, or at least

FIG. 87. illustrated, as follows : — Let us plot on a

diagram the energy possessed by each

molecule of a given mass of liquid by putting a point at a distance
from OO' (Fig. 87) proportional to its energy. We may suppose that
the points are chiefly crowded about the line AV, whose distance from
OO' represents the average energy. But there will be numbers of
molecules possessing both more and less than the average, though the
further the distance from the average, the less the number of points.

Let the energy which a molecule must possess in order to escape, be re-
presented by the distance between ES and OO'. All molecules, therefore,
represented by points above ES will escape if they have the opportunity.

As the temperature rises, the average energy rises, so that the whole
diagram of points may be supposed to be stretched upwards and the
number above ES increases. But the Specific Heat being nearly
constant, the total energy, and therefore the average energy, increases by
nearly the same amount for each rise of temperature of 1°, that is, AV
approaches ES by nearly equal steps. But as the points are crowded
more and more, the nearer we approach to AV, each successive degree-
rise brings a greater and greater number above ES. The number able
to escape and the vapour-tension, therefore, increase much more than in
proportion to the rise of temperature.

So far we have supposed the space above the liquid to contain
only the vapour of the liquid. But if some other gas — say air— is

CHANGE OF STATE— LIQUID— VAPOUR.

163

Stirrer

present as well, its chief effect is to lengthen out the process by
which the steady state is reached. The air molecules will hinder the
evaporation by knocking back into the liquid many of the molecules
trying to escape, but they will equally hinder the condensation by get-
ting in the way of those returning to the liquid. The two processes
being equally interfered with, the same number of molecules must be
present in the space for a balance, whether the air is there or not. We
may, in fact, compare the effect of the air to a screen perforated with
holes and laid upon the surface. Such a screen would equally interfere
with both processes, and merely lengthen out the time required to attain
the maximum vapour-pressure.

Boiling. — Besides the quiet transformation from liquid to gas, which
we have hitherto considered, there is the more rapid conversion which
occurs in boiling. If we heat some
water in an open glass vessel, so that
we can notice what occurs, evapora-
tion of course goes on from the top-
surface. This evaporation is indicated
by the clouds forming where the
vapour mixes with the colder air,
and condenses through cooling below
the temperature at which its pres-
sure is the vapour-pressure. As the
temperature rises, bubbles, which
gradually increase in size, appear on
the sides of the vessel. These consist,
partly, of gases driven from solution
— the higher the temperature, the
less the quantity of gas which a
liquid will dissolve — and, partly, of
water-vapour, for evaporation takes
place into the bubbles. Perhaps
these bubbles have been formed by
the swelling out of bubbles already
existing, but too small to be seen ;
or they may have been formed in
cavities free from liquid, for however smooth the surface is, there
are still probably minute irregularities. These cavities may serve
as nuclei into which the expelled gas and the water vapour can
pass. As the bubbles increase in size, their upward buoyancy at last
detaches them and they float to the top, and with this stage is associated
the well-known " singing," which occurs shortly before boiling. Soon
afterwards, boiling commences, that is, bubbles of steam form at
points on the containing-walls and rise up to the surface. These
bubbles always rise from definite points, each point supplying a constant
succession of bubbles, and as the boiling continues these points diminish
in number. The bubbles are never formed in the middle of the liquid.
Even if they appear to be thus formed, closer examination always shows
a particle of foreign matter forming a boundary to the liquid. The
temperature now ceases to rise and it is found that the boiling-point is
that temperature at which the maximum pressure of the vapour is equal
to the atmospheric pressure. This may be illustrated by the barometer

FiG. 88. — Vapour Pressure equals
Atmospheric Pressure at the Boiling-
Point.

164.

HEAT.

experiment of Fig. 85, p. 158. For if the barometer column be jacketed
throughout with steam, the pressure of the vapour of the water

above the mercury just
depresses the top of the
column to the level of
the cistern outside, that
is, just exerts the atmos-
pheric pressure. Or if
a U tube A B Fig. 88,
closed at the top of the
short limb and open at
the other limb, be filled
with mercury round the
bend to a as in the
figure, and if then a
small quantity of liquid
— say, for convenience
alcohol boiling at a lower
temperature than water
— be passed round the
bend to the top of the
mercury, if the tube
is immersed in a water
bath and gradually
heated, the liquid be-
gins to evaporate when
its boiling-point is closely
approached, and when
ft js just reached the

FIG. 89. — Reduction of Boiling-Point by
reduction of pressure

mercury stands at the
same level in the two limbs; that is, the vapour just balances the
atmospheric pressure.

If the external pressure be diminished, boiling may take place at a
lower than the ordinary boil-
ing temperature, for the
vapour-pressure will equal the
atmospheric pressure at a
lower temperature. This may
be illustrated by connecting a r.
flask of hot water to an air-
pump (Fig. 89). After a few
strokes of the pump, boiling
begins. Or, on boiling water
in a flask till all the air is
expelled, corking the flask up,
and holding it under a stream
of cold water (Fig. 90), the
cooling is accompanied by con- Fia" *>.-Boihng water by cooling it.

densation, and therefore diminution of the pressure on the water and
rapid boiling sets in. Frequently the flask collapses in this process,
owing to the excess of external pressure.

CHANGE OF STATE— LIQUID— VAPOUR. 165

On high mountain levels the low atmospheric pressure produces a
serious lowering of the boiling-point, that on the summit of Mt. Blanc
being about 84° C. At such a temperature, boiling water has not its
usual cooking qualities.

On the other hand, increase of pressure is accompanied by rise of
boiling-point. In high pressure steam-boilers, the boiling-point rises
many degrees above the normal. This rise of boiling-point is employed
to obtain the gelatinous matter from bones, water above 100° dis-
solving this matter more freely than water at or below 100°. The
bones are, therefore, heated in a closed vessel of water, fitted with a
safety-valve, so that the water is exposed to the pressure of its own
vapour, and this increases as the temperature rises. There is therefore
no definite boiling-point short of that corresponding to the pressure at
which the safety-valve is forced open. Such an arrangement is called a
Papin's digester from its inventor, the discoverer of the rise of boiling-
point with pressure.

Delayed Boiling. — When boiling has been going on for some time
in a glass vessel, the temperature of the water begins to rise above the
normal boiling-point, and at the same time the steam is given off in larger
bubbles and from fewei points. If the vessel be very clean (it may be
cleaned by rinsing with hydrofluoric acid and then with water) and the
water very pure, by alternate boiling and cooling it is quite easy to raise
the temperature of the water ultimately to 105° or 106° 0. With still
greater precautions, the temperature may be raised many degrees higher.
In this state when boiling does occur it is almost explosive in its character,
and the phenomenon is termed " boiling with bumping." On putting
into the vessel sand, iron filings, or any rough material containing
crevices or air bubbles, boiling again becomes normal at the normal
temperature.

A still more striking experiment was made by Dufour, who prepared
a mixture of oils of about the density of water and of high boiling-point,
and' then placed in it small drops of water, which remained suspended.
They could be heated without boiling to at least 178° 0.

The observation that normal boiling is resumed when air bubbles are
present gives the clue to the nature of boiling. It is probably always
associated with the presence of bubbles or cavities.1

We have seen that as the temperature rises towards the boiling-
point, the dissolved gases are expelled, and collect in bubbles. As these

are detached, each bubble probably leaves a

small portion behind, just as a drop of water in "\^/ 7T ^-^
breaking off from a surface and falling down,

always forms a neck at which rupture takes FIQ. 91. — Formation of a
place and leaves behind the part above the neck drop by pinching in at
as in Fig. 91. These minute bubbles do not a neck,
grow so largely as before, since most of the dis-
solved gas has been expelled in their first formation. Evaporation,
however, goes on into them, and their size will be such that the vapour-
pressure for the existing temperature + the pressure of gas or air in

1 An excellent account of the phenomenon of melting and boiling, with experi-
ments illustrating the explanation, is given by Aitken in the Transactions of the Royal
Scottish Society of Arts, vol. ix, 1874-75.

166 HEAT.

them = atmospheric pressure. Now, when the temperature rises to the
point at which the pressure of the vapour equals the atmospheric
pressure, there can no longer be equilibrium, since the internal pressure
exceeds the external by the pressure of the gas or air in them. The
bubbles grow, their buoyancy increases, and finally they break away and
float to the top.

The small portion of each still remaining serves as a fresh nucleus,
and the process is repeated indefinitely, as we see from the constant
stream of steam-bubbles from the same point in the containing vessel.
The heat supplied to the liquid is taken up as the latent heat of the
steam formed.

If the heat is supplied very rapidly the temperature of the liquid
tends to rise above the normal boiling-point, and the evaporation into
any bubble present tends to increase. Thus the bubble grows rapidly.
It is true that evaporation and condensation are always going on together.
But suppose that we are considering water at 101°. The vapour-pressure
at 101° is about 787 mm., and only with that pressure of vapour in a bubble
would the evaporation and condensation balance. The growth of the
bubble, however, keeps the pressure within down at 760 mm., so that the
rate of condensation is hardly affected by the rise of temperature, while
the rate of evaporation has grown considerably. The unbalanced evapora.-
tion increasing the size of the bubble increases the evaporating surface.
Steam is more readily formed, more latent heat is taken up, and if the
evaporating surface is sufficient the temperature may be brought down
to the normal boiling-point. If, however, the points at which steam is
formed decrease in number — if there are not sufficient bubbles — the
steam given off may not be sufficient to carry away all the heat supplied,
and the temperature may rise appreciably above the normal boiling-point.
In fact, with glass vessels some rise above it almost always takes place.

The presence in steam bubbles of gas other than steam — an observa-
tion due to Grove — supports this explanation.

After a time, through the removal of the foreign gases, the portions
of the bubbles left behind probably get smaller — so small that their
surface tension seriously affects the pressure within them.

The surface tension alone exerts a pressure p = T( - + — V where r

and r' are the principal radii of curvature of the stretched surface, and
T is the tension per centimetre. In order, therefore, that a bubble may
grow, we must have (the vapour-pressure + the pressure of the contained
gas) greater than (p+ the atmospheric pressure), and if p becomes
sensible, through the diminution of the residual bubble, the temperature
must rise sensibly above the normal boiling-point before this condition
will hold. When the bubble once begins to grow, r and ?•' increase, and
p diminishes, so that the pressure within the bubble diminishes ; but the
evaporation into the bubble is still at the rate corresponding to the
higher temperature of the liquid, while the condensation is only at the
rate corresponding to the diminished pressure, now tending rapidly
towards the atmospheric pressure. The bubble, therefore, grows with
very great rapidity, almost explosively ; much latent heat is taken up,
and the temperature of the liquid falls, though not necessarily to the
normal boiling-point. As the bubble rises up, the process is repeated

CHANGE OF STATE— LIQUID— VAPOUR.

167

0 • v volume X

FIG. 92. — Relation between Volume and Pressure
of Gas contained in a Bubble in a Liquid.

with the nucleus left behind, the growth of the new bubble being delayed
till the condition for its growth obtains, when there is another sudden
swelling out. The intro-
duction of sand or iron
filings puts an end to this
state of delayed boiling,
since it presents a large
number of air-bubbles as
nuclei for evaporation,
and then ordinary boiling
occurs.

Stability of Bubbles

in a Liquid.— The stabi-
lity of bubbles formed on

the side of a vessel may be

discussed by the aid of

diagrams. The pressure

within a bubble is that

of the vapour and that of

the contained gas. This

may be represented by

Fig. 92, the different

hyperbolas representing

the relation between pressure and volume for different quantities of

contained gas. Lifting the curves above the zero pressure a distance

equal to the vapour pressure, then the relation between the internal

pressure and the volume
of a bubble containing
a given quantity of gas
will be represented by the
abscissa and ordinate of
the corresponding curve.

The external pressure
is that of the atmosphere
+ the hydrostatic pressure
due to depth below the
surface + that due to sur-

2T

face tension - — We shall
r

include the hydrostatic in
the atmospheric pressure.
Supposing the bubble
spherical, the latter term
FIG. 93.-Relation between Volume and External « Aversely proportional
Pressure on a Bubble. The surface tension curve to tne cube root ot \ne
slopes down more gradually than the hyperbolas volume. Hence, for in-
in Fig. 92. stance, to halve this part

of the pressure, we in-
crease the volume eight times. Then the pressure curve representing

2T

— slopes down much more gradually than any of the hyperbolas in

T

168

HEAT.

Fig. 92. Fig. 93 represents the curves raised a distance equal to the
atmospheric pressure above the line of zero pressure.

Superposing the two figures, so that ox is the same for each, the
points where the curve of Fig. 93 cuts those of Fig. 92 represent positions
of equilibrium for bubbles containing different quantities of gas. As long
as the vapour-pressure is less than the atmospheric pressure, the curve of
Fig. 93 cuts the curves of Fig. 92, where they have a greater slope than it,
and any increase of volume, therefore, means an excess of external over
internal pressure, and, hence, equilibrium is stable. The bubbles, there-
fore, grow only with rise of temperature and addition of gas. But when
the vapour- pressure exceeds the atmospheric it is possible for the surface
tension curve of Fig. 93 to cut a gas curve of Fig. 92 at a less slope, and
equilibrium is then unstable, as in boiling with " bumping." We leave

.,
A if*

TO pump

FlG. 94. — Apparatus for obtaining a Dust-Free Space.

the reader to consider how the diagrams would lead us to expect that
after the first " singing " discharge of bubbles no more large bubbles will
be found till the boiling-point is reached.

Condensation on Nuclei. — We have seen that the change from
the liquid to the gaseous condition is essentially a surface phenomenon.
It takes place at the top level surface in ordinary quiet evaporation, and
in all probability at the surface of already existing bubbles or cavities
in ebullition. The converse change in condensation is also generally
a surface phenomenon, the condensation occurring on nuclei of dust or
suspended matter other than air or vapour. This was first discovered
by Coulier (Journal de Pharmacie et de Cliemie, xxii., pp. 165 and 254,
1875), and later again by Aitken (Trans. E.S.E., xxx., pt. 1, p. 337),
to whom we owe extensive investigations on the subject.

When a beam of sunlight crosses a room, the visibility of its track is
due to particles of " dust," many perhaps large enough to be seen, others
quite beyond our range of sight. If the air be filtered through cotton-
wool this dust is filtered out. The clean air can then be quite appre-

CHANGE OF STATE— LIQUID— VAPOUR. 169

ciably supersaturated with vapour without formation of cloud or fog,
that is without condensation.

To show this a large clean glass globe may be arranged as in Fig. 94.

The globe G is, to begin with, filled with the air of the room, and
contains some water which can be washed round the sides so as to ensure
complete saturation of the air.

The taps ^ t% being turned off, the pump exhausts R. If now <2 is
turned on for a moment, the air in G expands, cools, and becomes super-
saturated at the new temperature. The excess of moisture at once
condenses in the form of fog which, usually evident enough, is still
more evident if a light, such as a candle flame, be viewed through the
globe when beautiful diffraction rings may be seen round the flame, the
rings being larger the smaller the drops. Then turning on both ^ and
<2 let clean dust-free air be slowly drawn through the cotton-wool filter.
After a short time <x and tz may be both turned off, and the air in G
be again expanded and cooled by turning on t2 for a moment. There
will now be a much smaller number of dust nuclei, and as the vapour
condenses on these alone, the drops are much larger and the fog is less
dense. For though the same amount of water may be present as liquid
in the air its concentration into larger drops diminishes its surface, and
there is much less hindrance to the passage of light. The diffraction
rings round the flame at the same time grow less.

If this process be repeated several times the drops soon become easily
visible, like those in a Scotch mist, and move at an appreciable speed
downwards in a shower of fine rain. When all the drops have fallen
down the air is clear and a fresh small expansion does not produce a
cloud or rain at all.

Aitken (Nature, March 1, 1888, p. 429, and Feb. 27, 1890, p. 394),
has devised a " dust counter," an apparatus to count the number of
dust particles or, at any rate, condensing nuclei per c.c. in any specimen
of air. This consists essentially of a small chamber 1 cm. deep with
a glass floor ruled in square millimetres and a glass top through which
the glass floor can be seen. The chamber is connected to a pump, and
by an inlet either the surrounding air can be introduced or any known
proportion of filtered dust-free air. The air in the chamber is kept
saturated. In general it is necessary to dilute the air to be tested with
a large proportion of clean air to reduce the number of dust particles,
and so increase the size of the drops. Suppose, for instance, that nine-
tenths of the air is dust-free. If an expansion is suddenly made, drops
are formed on the dust particles, and these should be heavy enough to fall
at once on to the ruled floor. There they are evident as little specks.
The numbers on several square millimetres are counted by the aid of a lens
and averaged. Say that the average is 5. Then on a square centimetre
there would be 500. This is the total number falling down out of a
cubic centimetre. But the air has been diluted to one-tenth its original
dustiness. Therefore, in its original condition it contained 5000 dust
particles.

Aitken has examined the air in various localities, and finds that the
number of dust particles per c.c. is greatest in rooms, greater in dry
town air than in the country, greater at lower altitudes than on moun-
tains, and that there is a great diminution after rain.

170 HEAT.

Thus (Nature, March 1, 1888) Aitken found the following numbers

Source of Air.

No. per c.c.

Outside air — raining .
„ „ fair.
4 feet from floor of room in which
gas was burning
Near ceiling ....
Air above Bunsen flame

32,000
130,000

1,860,000
5,420,000
30,000,000

Again (Nature, Feb. 27, 1890) he found at Kingairloch, on the shore
of Loch Linnhe, numbers ranging from 205 to 4000 per c.c. ; at the top
of Ben Kevis, 335 to 473 per c.c. In London and Paris, the numbers
were counted by the hundred thousand. Fridlander (Quarterly Journal
of the Royal Meteorological Society, xxii., July 1896) has tested various
specimens of air in a journey round the world, the numbers at sea
varying from 200 per c.c. in the Indian Ocean to 4000 in the Atlantic.

The explanation of the thickness of town fogs as compared with
those in the country or at sea is now evident. The number of dust
particles is always far greater in town air, and if the smoke keeps near the
ground, as it does when the upper air is much warmer than the ground
air, the number may be enormously greater than in country air. When
condensation occurs the fog is, therefore, exceedingly " dense " through
the minute subdivision of the water deposited.

The action of the nuclei in condensing the vapour probably depends
on a principle first pointed out by Lord Kelvin, to which we shall return
later (chap. xix.). The principle is that the equilibrium or saturation
vapour pressure of a space in contact with a liquid surface depends on
the curvature of the surface. If P be the normal saturation pressure at
a given temperature, it can be shown that in contact with a spherical
surface with radius of curvature r, for moderate values of r the saturation
pressure is

ff = P + ^ (1)

where T is the surface tension ; p the density of the liquid, and cr that
of the vapour.

If the surface is convex then r is negative.

When r is exceedingly minute so that jo/P differs largely from 1

2Tcr
it can be shown that log p/~P = — (2)

if we assume that Boyle's law will hold for the vapour.

It follows that small drops in a space just saturated over a flat
surface will find the space under-saturated for their own curved surface,
and will evaporate ; while if, by any accident, there is a concave surface
formed the space will be over-saturated and condensation will go on.

Now suppose that in some saturated dust-free air a few vapour mole-

CHANGE OF STATE— LIQUID— VAPOUR. 171

cules collide, and with so little energy that they form practically a liquid
particle. The radius is so minute that the saturation pressure for the drop
is far above the pressure of the vapour round it, and the drop instantly
evaporates. But if dust nuclei are present, a liquid layer may form on a
nucleus with radius sensibly equal to that of the nucleus, which may be
so large that the saturation vapour pressure is only very slightly in
excess of the normal pressure, and a very slight supersaturation may
make the liquid increase. There is another way in which soluble dust
may aid condensation. We shall see later that the saturation vapour
pressure over a solution is less than that over a pure liquid, so that
a normally saturated space may condense on to a liquid surface if the
liquid contains salt in solution. If, for instance, a particle of common
salt is floating in vapour-laden air, any accidental formation of a liquid
particle on it may lead to solution and the formation of a drop with
lowered vapour pressure, which may tend to grow even if the space is
not quite saturated for a normal plane surface of pure water. Aitken
believes that such condensation frequently occurs and largely accounts
for the haze in air.

The conditions of condensation of water-vapour in air and other gases
have been very carefully investigated by 0. T. R. Wilson (Phil. Trans.,
A., 1897, p. 265, and A., 1899, p. 403). It is sufficient here to consider
the case of air. He found that if all dust be filtered out, then the
saturated air at any temperature about 20° may be suddenly expanded in
the ratio 1 : 1'25 without condensation. But that, if the expansion ex-
ceeds this, condensation on a few nuclei (not more than a few hundred per
c.c.) does occur up to an expansion of 1 : 1'375. After this a dense fog
appears, the denser the greater the expansion. Now this means that if
we start with air, say ab 20°, the sudden adiabatic expansion 1 : 1-25
cools it to — 6°, when the density of the vapour saturating the air at 20° is
about 4*2 times the density of the vapour saturating the air at - 6°. The
expansion 1 : T375 cools it to — 16°, when the density is about 7*9 times
the density of vapour saturating the air at - 16°. Then Wilson's experi-
ments show that there are always nuclei present in small numbers
sufficient to condense the vapour when it has about four times its
normal saturation density, and in large numbers if it has about eight
times its normal saturation density. We can hardly suppose that the
nuclei are of foreign matter. Wilson shows that drops of radius of the
order 10~8 cm. would, by the formula (2) above, be in equilibrium at the
8-fold density, and such drops would contain but few molecules according
to the calculations of molecular dimensions in chap. ix. We may
suppose, then, that in the molecular collisions water particles of such
order are continually being formed, and if the vapour density is 8 times
the normal they will tend to grow instead of evaporating. As to the
formation at the 4-fold density, Wilson supposes that there is chemical
action occurring ; for instance, in some of the collisions the oxygen,
nitrogen, and water might combine to form nitric acid. The vapour
at 4-fold its normal saturating density might be saturated for even the
minutest particle of acid formed by collision, and so the particle would
grow. Such combination may always be occurring to some slight extent,
the compounds being made and unmade in successive collisions.

Wilson also found that ultra-violet light is active in producing con-

172

HEAT.

densation nuclei and that condensation occurs under its action even with
slight expansions, the fog produced being of a blue colour. Tyndall had
previously found this formation of blue fog in many other vapours denser
than water, under the action of ultra-violet light. The colour implies
that the drops formed are comparable in size with the waves of blue
light and so scatter that constituent when white light falls on them.

Exact Measurements of Vapour-Pressures, — Various observers
have made researches on the vapour-pressures of liquids at different

temperatures, the most complete as
well as the most important being those
in the case of water.

Two methods have been employed,
the "statical" method, which is illus-
trated in its simplest form by the
experiment described on p. 158, in
which the pressure of the vapour
depresses the column of mercury in
a barometer, and the " dynamical "
method, in which advantage is taken
of the fact that a liquid normally boils
when the pressure of its vapour is
equal to the external pressure.

In the first method, the temperature
is varied at will, and the corresponding
maximum pressure measured ; and in
the second method, the pressure is
varied at will, and the corresponding
maximum temperature, that is the cor-
responding boiling-point, is observed.

These methods were carried out in
Regnault's classical researches on the
vapour-pressure of water, which may
be taken as typical. He adopted three
arrangements, corresponding to three
ranges of temperature : — (a) From 0°
to 50° 0. ; (&) below, and up to 0° 0. ;
(c) from 50° upwards.

(a) From 0° to 50°. — Two barome-
ters, AB, A'B', with a common cistern
E (Fig. 95) are placed side by side —
their upper ends being in a vessel containing water which serves as a
constant-temperature bath. The front of this vessel is glazed, so that the
levels of the mercury in the two tubes may be observed by a cathetometer.
D is a spirit-lamp to raise the temperature of the water when required,
and F is a stirrer to keep the temperature uniform throughout. A small
quantity of water having been introduced into the tube AB, the vapour
from it depresses the column, and after correcting for the pressure due to
the small excess of water, for the effect of the water on the capillary
depression, and for the temperature of the mercury, the difference in
level in the two tubes gives the vapour-pressure at the temperature of the
bath. Since the vapour-pressure at 50° is only about 92 mm., the batb

FIG. 95. — Vapour- Pressure of
Water from 0° to 50°.

CHANGE OF STATE— LIQUID— VAPOUR.

173

was not very large and it was found possible to keep its contents at a
uniform temperature. But above 50° a much larger bath would have
been required, and the difficulty of maintaining a uniform temperature in
a large mass of water is so great that Regnault abandoned this method for
the dynamical.

(b) For temperatures below 0° he still employed two barometers, side
by side (Fig. 96) ; but one barometer was bent round at the upper end,
and terminated in a bulb E, containing ice. This was surrounded by a
" freezing mixture," the temperature of which was taken by a thermo-
meter. The temperature of the rest of the apparatus being above that
of the freezing mixture, the pressure exercised by the vapour on the
mercury corresponded to the temperature of the

freezing mixture — for any tendency to rise above
this pressure would be checked by condensation
in E. The freezing mixture was liquid, and by
stirring, its temperature was kept uniform through-
out. Regnault was unaware of the fact that the
vapour-pressure of ice below 0° differs from that
of water below 0° at the same temperature, a fact
deduced subsequently from theory by Kirchhoff,
and later verified experimentally by Ramsay and
Young. He, therefore, made no attempt to measure
the vapour-pressure of water below 0°. In order
to construct tables giving the most probable values
of the vapour-pressure for each degree of tempera-
ture, Regnault employed the graphic method,
plotting his results on a temperature and pressure
diagram, and then getting rid of irregularities or
discrepancies between different series by drawing
a continuous curve most nearly representing the
results. The curve, as drawn by Regnault, goes
continuously through 0°, where the evaporating
substance changes from water to ice. This throws
doubt on the accuracy of his numbers about and
below 0°. The water-vapour curve would no doubt
be continuous, but the ice-vapour curve meets it at

a small angle at 0° as represented in Fig. 97, theory showing that the
ice-vapour curve diverges from the water-vapour curve about -^ mm. per
degree. Ramsay and Young (Phil. Trans., ii., 1884, p. 461) have proved
that the difference exists by maintaining ice and water in two separate
connected vessels at a very low constant pressure. The temperatures
of the two vessels became steady at the points at which their vapour-
pressures were equal to this constant pressure. The ice was found to
be at a higher temperature than the water.

(c) For temperatures above 50° Regnault employed the dynamical
method. The arrangement of the apparatus will be seen from Fig. 98.
A copper boiler, heated by a small furnace, is in connection with the
reservoir of air G, the pressure of which is indicated by the manometer
HK. Round the connecting tube at A is a wider tube, in which water
circulates as in an ordinary still. The boiler is heated till the water in
it boils, and the steam rises into the tube A. Here it is condensed, and

FIG. 96.— Vapour-
Pressure of Ice.

174

HEAT.

the water trickles back into the boiler, so that there is no diminution of
its contents. The pressure to which the water in the boiler is subjected

FlG. 97. — Water- Vapour Pressure and Ice-Vapour Pressure. They differ
by 0-04 mm. at - 1°.

can be changed either by withdrawing air from, or forcing air into, the
reservoir G by the pipe F. Four iron tubes closed below were inserted

FlG. 98. — Dynamical, or Boiling Method of Determining Vapour-Pressnre
of Water above 50°.

through the top of the boiler, two reaching nearly to the bottom of the
water, and two shorter tubes being surrounded by the steam. These

CHANGE OF STATE— LIQUID— VAPOUR.

th

tubes were filled with oil or mercury, and thermometers were inserted
which gave the temperature. As the indications of the four agreed, the
boiling was normal and not " with bumping."

It was found that on altering the pressure in G, the temperature
soon reached a steady state, the boiling-point at the pressure indicated
by the manometer. The manometer, therefore, gave the pressure of
the vapour at the observed temperature.

For temperatures above 150°, the apparatus employed was exactly
the same in principle, but much larger and stronger, and Regnault
was able to work with it up to 230° at a pressure above 27 at-
mospheres.

The vapour-pressures of various other liquids have also been deter-
mined for a series of temperatures by Regnault and other observers
using similar methods.

An account of the various methods will be found in
Winkelmann's Handbuch der Physik, 2nd ed., vol. iii. p.
903. The values obtained for the vapours of water and
various other liquids will be found in Landolt-Bbrnstein,
Tdbellen, p. 119 et seq.

Determination of Boiling-Points.— The boiling-
points of liquids are often required, the knowledge of the
vapour-pressure at other temperatures not being needed.

If the liquid whose boiling-point is in question is
plentiful, it may be heated to boiling in a flask, or the
apparatus used in determining the 100° point on the
thermometers may be used, care being taken that the
boiling is normal. A very simple arrangement, using the
statical method, is that illustrated in Fig. 99.

AB is a U tube. The shorter limb is closed, and is
filled with mercury, which extends just round the bend.
A small quantity of liquid is then introduced, and by
careful manipulation may be floated up through the
mercury without any air, so as to occupy the end a of the shorter limb-
The tube is then immersed in a bath, which is heated until the vapour
forms and depresses the mercury. When the level is the same in both
limbs, the vapour-pressure is equal to the atmospheric pressure, and the
temperature gives the boiling-point. This method, of course, admits of
simple correction for variation of atmospheric pressure. The following
are a few boiling-points : —

FIG. 99.

Ether
Alcohol
Water
Mercury

34-87

78-4

100-0

358-5

Determination of Vapour-Density. — The method of determining
the density of gases at 0° and 760 mm. pressure consists in filling a
globe with the gas to be experimented on, at the atmospheric pressure H,
while it is surrounded with melting ice ; then closing it and weighing it.
The globe is then put into the ice again, exhausted to a low pressure h,
closed, and again weighed. Suppose the diminution in weight is W.

176

HEAT.

Assuming that Boyle's law holds, the weight of a volume of gas at 0*
and 760 mm. equal to that of the globe is

Wx760

FIG. 100.— Vapour-Density Bulb :
Dumas' Method.

Various corrections are of course needed, and we may refer the reader

to Regnault's experiments, of which
a short account is given in Jamin,
Cours de Physique, torn, ii., or to the
series of papers describing the ex-
periments which led Lord Rayleigh
to the discovery of argon (Proc. R.S.,
Feb. 1888, Feb. 1892, March 1893;
Phil. Trans., A., 1895, p. 187).

Of course, this method is not
applicable to substances which are
liquid at ordinai-y temperatures and
pressures. In their case, the weight
of a known volume of the vapour

must be determined at a known pressure, and at such a temperature

that there is no condensation at that pressure, and the ratio of

this to the weight of the same

volume of air at the same tem-

perature and pressure is termed

the " Vapour-Density."

Various arrangements have

been adopted for the purpose.

In Dumas' method, a bulb (Fig.

100) ending in a fine open tube,

is first filled with dry air and

weighed, the temperature and

pressure being observed. Some

of the substance is then put into

the bulb in the liquid state, and

the bulb is immersed in a water-

or oil-bath, the temperature of

which is increased somewhat

above the boiling-point of the

liquid, the open tube projecting

above the liquid in the bath.

The liquid boils, and, if a suf-

ficient quantity is present, its

vapour drives out the air. When

all the liquid has evaporated, and FIG. 101.— Vapour-Density Determination by

the vapour is at the tempera- Gay Lussac and Hoffmann's Method.

ture of the bath, the end of the

fine tube is sealed. The bulb is now taken out, and when cool is

weighed again. The point of the tube is now broker under mercury,

the vapour condenses, and the bulb is practically filled with mercury.

From its weight so filled, its volume can be determined. From this we

may find the weight of the air filling it in the first weighing, and thence

CHANGE OF STATE— LIQUID— VAPOUR.

177

the weight of the bulb empty. Subtracting this from the weight in the
second weighing, we have the weight of the contained vapour at the
temperature of the bath and the atmospheric pressure. Finding the ratio
of this to the weight of the same volume of air at the same temperature
and pressure, we have the vapour-density.

In Gay-Lussac's method as modified by Hoffmann, a graduated
barometer is enclosed in a wider tube (Fig. 101) through which circulates
steam or other suitable vapour from a boiler, the barometer thus being
maintained at a known temperature. A very small stoppered bottle is
then quite filled with the liquid, and
weighed, so that the weight of the liquid
used is known. This is floated up through
the mercury column, and on arriving in
the Torricellian vacuum, the stopper is
forced out, and the liquid evaporates.
Such a quantity of liquid is taken, and
such a temperature is maintained, that
the whole of the liquid evaporates, its
pressure, of course, depressing the mercury
column. The volume occupied by the
vapour is observed, and its pressure is
given by the difference between the
original and the new height of the baro-
meter column. Hence we have the data
for determining the vapour-density.

Two methods have been devised by
Victor Meyer. In that most commonly
used, which we shall describe, a known
weight of liquid is introduced in a small
stoppered bottle into a heated flask
containing air. The vapour formed dis-
places its own volume of air, and this is
collected and measured. The method is
applicable to vapours denser than air.
One form is shown in Fig. 102.

The flask A contains air, and is heated
in a constant -temperature bath. The
neck rises above the bath, and a narrow
side-tube leads down to the trough B, where the expelled air may be
collected. The neck is closed by the slanting hollow stopper 0, in which
is placed the small bottle containing the liquid. When the temperature of
the different parts of the apparatus is steady, C is turned round and the
bottle falls down to the bottom of the flask on to some asbestos packing,
put to prevent breakage. The stopper is forced out, and evaporation
takes place, the vapour driving the air upwards. If the displaced air is
raised sufficiently slowly, as each layer rises it takes the temperature and,
therefore, the volume of that which it displaces. Hence the weight of
the air forced out and collected at B is equal to that of the air displaced
by the vapour in A. The vapour-density, as compared with that of air
at the same temperature and pressure is, therefore, the weight of liquid
introduced divided by the weight of air expelled.

3. 102. — Victor Meyer's Method
for Vapour-Density Determi-
nation.

178

HEAT.

Density Of Saturated Vapour. — The densities of various saturated
vapours, that is, of vapours in equilibrium with their liquids, have been
studied by Hering. The principle of his method will be understood
from the following imaginary experiment. Let A (Fig. 103) be a
graduated tube closed at the top, and connected by a flexible tube with
B open to the air at the top. Let A be entirely filled with mercury,
which passes round the bend into B. Let a small weighed quantity of
the substance be floated round through the mercury into A. By depres-
sing B sufficiently, the pressure in A may be so reduced that the liquid
begins to evaporate. At first the vapour in A is saturated, and exerts
the vapour-pressure so that, as B is lowered still farther, the two mercury
levels continue to fall equally. The vapour in A merely increases in
quantity at the constant vapour-pressure. But when all the liquid has
just disappeared, the volume occupied by the vapour in
A is that of a known weight of vapour saturated at the
observed temperature. The point of disappearance may
be determined from the fact that, as soon as it is passed,
the level in B begins to fall more quickly than in A.
In practice it was found better to work in the opposite
direction, since, through adhesion to the glass, evapora-
tion was not complete till after the saturation-pressure
was passed. The point at which condensation begins
was therefore found. Hering used a much better but
more complicated manometer. He found that for dif-
ferent substances the density might be nearly repre-
sented by

Density of saturated-vapour = Density as compared
with that of hydrogen at the same temperature
and pressure, calculated from the molecular

weight x »y temperature measured from - 273° x

constant,

the constant being the same for all the substances
experimented on.*
FIG. 103. Measurements of Latent Heat.— We have already

seen that, when a liquid boils, its temperature normally
remains constant, the heat supplied going to change the liquid into gas
without rise of temperature, and we have called this " latent heat." But
the term has a precise signification which may be expressed as follows : —

The latent heat of a vapour at a given temperature is the quantity of
heat required to convert 1 gramme of liquid into vapour at its maximum
pressure at that same temperature. Thus the latent heat of steam at
100° is the heat required to convert 1 gm. of water at 100° into 1 gin.
of steam at 100°, and 760 mm. pressure.

The general principle of the method of measurement of latent heat
will be best understood by considering a rough experiment with steam : —

A flask containing water, with a tube passing through the cork and
nearly reaching to the bottom, is inverted, as in Fig. 104, and the water

* A series of researches has been made by Fairbairn and Tate on the density of
saturated steam. For an account of their method the reader is referred to Baynes's
Thermodynamics, p. 181.

CHANGE OF STATE— LIQUID— VAPOUR.

179

boiled by a ring-burner, or otherwise. The steam issues from the lower
end of the tube. A calorimeter, containing a known weight of water
at a known temperature, is then brought under the flask, so that the
steam passes into the water and is condensed there. Allowing the con-
densation to proceed for a short time, the calorimeter is then withdrawn
and the temperature noted. Its increase in weight gives the quantity
of steam condensed. Then, on the one hand, we have the gain of
heat by the calorimeter and its original contents, and, on the other
hand, the latent heat given up by each gramme of steam in condens-
ing, together with the heat given up in falling from the temperature of
boiling to the final temperature of the calorimeter.

Equating these, we can determine the latent heat.

For example, a calorimeter whose equivalent was 10 gms. contained
140 gms. of water at 15° tempera-
ture. Steam at 100° was passed
into it until the temperature rose
to 75°, that is, through 60°. The
contents of the calorimeter weighed
now 157 gms., so that- the total
quantity of steam condensed was 17
gms. If the latent heat given up
by each gramme is L, it may be
regarded as giving up L and then
25 more calories in falling to 75°.
We therefore have the equation

150 x 60 = (L + 25)17,
whence L = 504.

Boiler

Burner

Calorimeter

FIG. 104. — Rough Determination
of Latent Heat.

This experiment, of course, is affected
by very serious errors. Heat is in-
troduced not only by the steam but
also by the conducting pipe, and the
amount so introduced must be deter-
mined and allowed for. Heat is, on

the other hand, lost by conduction or radiation from the calorimeter, and
its amount must also be determined and allowed for. In the case of
water, we may allow the steam to mix with the water, though unless we
measure the temperature of the steam directly, a small error comes in
through the additional pressure of the steam and the consequent rise of
the boiling-point, owing to the mouth of the steam-pipe being below the
level of the water in the calorimeter. But in the case of other liquids,
we cannot have mixture with the water in the calorimeter. It is better,
therefore, to have a spiral condensing worm in the calorimeter, ending in
an enlargement to collect the condensed liquid. The arrangement
adopted by Berthelot (Fig 105) will illustrate this.

F is the vessel containing the boiling liquid, T the vapour-pipe, OSR
the condensing worm, I a ring-burner under a gauze, the calorimeter
being protected against radiation from the flame by a non-conducting
cover faced with metal nn. With this apparatus Berthelot found the
latent heat of steam at 100° 0. to be 536*2 (the mean of three determina-
tions, 535-2, 537*2, 536-2), agreeing nearly with Regnault's determination,

180

HEAT.

536-6, and therefore he concluded that the results obtained from it with
other liquids were trustworthy.

Regnault made a series of determinations of the latent heat of steam
at various temperatures, using at higher temperatures a boiler in which
the pressure could be maintained at any desired value, so that the boiling-
point could be regulated. From this the steam, still at the same tem-
perature, was led into a spiral in a calorimeter, where it was condensed
and the heat given out was measured (Jamin, ii. 243). By subsidiary
experiments he determined the corrections for conduction and radiation.

At lower temperatures the water
was contained in a spiral immersed
in a calorimeter, and the pressure
was reduced to the point at which
the water boiled. The vapour was
led off and condensed in a vessel
surrounded by ice, and the heat
given up was there measured.
Regnault put his results in the
following form : — Calling the heat
required to raise 1 gm. of water,
from 0° to *°, and then to convert
it into saturated steam at f the
" total " heat of the steam at *°, and
denotingit by Q,Regnaultfoundthat
Q = 606-5 + 0-305*.

Taking no account of the alteration
in the specific heat of water with
rise of temperature, we have

Hence,

L = 606-5 + 0-305*-*
= 606-5 -'695*.

FIG. 105. — Berthelot's Apparatus for
Latent Heat.

From this we see that the latent
heat decreases as the temperature
rises, another indication that the
liquid and gaseous conditions are
approaching each other.

Regnault's formula for the total heat of steam gives the latent heat
at 0° 0., L0 = 606-5. Winkelmann (Wied. Ann., 9, 1880), re-examining
Regnault's work, showed that the values obtained from the formula at
low temperatures are probably higher than his experimental results
warrant, and Dieterici (Wied. Ann., 37, 1889) made a direct determina-
tion of L0 by boiling water at 0° in a Bunsen calorimeter and measuring
the volume of ice formed by the subtraction of a given weight of
vapour. He obtained L0 = 596 -8.

Griffiths (Phil. Trans., A., 1895, Part I., p. 261) devised a new method
of determining the latent heat of steam, and used it for the two tempera-
tures of 30° and 40° 0. A small glass tube, open at one end and con-
taining a known weight of water, was fixed in an exhausted silver flask
immersed in a calorimeter containing oil. The water issued from the

CHANGE OF STATE— LIQUID— VAPOUR. 181

tube drop by drop, and, falling on to the silver, evaporated at the pressure
corresponding to the vapour pressure for the temperature of the calori-
meter. The vapour was pumped out as fast as it was formed, its latent
heat being supplied by the oil in the calorimeter. The oil, of course,
tended to fall in temperature, but its temperature was maintained
constant by an electric heating coil and by very rapid stirring. The
heat equivalent of the energy given to the oil by the current and by
the stirring was determined, and this gave the latent heat of the steam
formed. Griffiths' results at the two temperatures agree with the
formula

L = 596-73 --601*,

where the unit of heat adopted is the 15° 0. calory, which agrees very
nearly with the mean calory from 0° 0. to 100° 0. This gives L0 — 596'73,
closely agreeing with Dieterici's value, and L100 = 536'6, closely agreeing
with Regnault's value L100 = 537, and at this higher temperature Reg-
riault's work is probably very accurate. Griffiths' value for the total
heat of steam is

Q = 596-73 + -399*.

Henning (Ann. d. Physik, xxi., 5, 1906, p. 849) measured the steam
generated from water kept boiling continuously at constant temperature
by an electric heater, in which the heat supply was measured. Over
the range used, 30° to 100° 0., L = 598'8 - 0'5994£ fairly represents the
results, but L = 94*21(365 - £)°'31249 gives a closer agreement on the
whole. The 15° calory was taken as equal to 4' 188 joules.

By superheating the vapour, that is, raising its temperature above
the condensing-point, while maintaining the pressure the same, Regnault
used the higher temperature apparatus devised for the determination of
latent heat of a vapour to find also its specific heat at constant pressure.
For, suppose that the latent heat of the vapour at t" is L, the specific
heat a; and let the vapour be superheated in two different cases to t + 0l
and t + #2. Then the heat given up in condensing is L + d^ and L + #2cr.
We may determine both of these quantities, and knowing 6l and $2, we
may obtain both L and <r.

Regnault found in this way that the specific heat of steam at constant
pressure is constant within the limits of errors of observation, and the
value he found is '4805. He also determined the specific heat of other
vapours. But it is to be remarked that in the expression L + 6^0-, L is
much the larger quantity, and the value of L + d-p will be affected
by the errors of L. Hence, o- cannot be determined in this way with
very great accuracy.

It is important to distinguish this specific heat of vapour at constant
pressure from the specific heat of saturated vapour. In the former, the
pressure, of course, remains constant during the rise of 1°. In the latter,
the pressure varies, being always that at which the vapour is just on the
point of condensation. For instance, the specific heat of saturated steam
at 100° is the heat required to raise the temperature from 100° to 101°,
the pressure being increased meanwhile from 760 mm. to 787-63 mm.,
the latter being the vapour-pressure at 101°.

It is easily shown from the second latent heat equation (chap, xix.),
that the specific heat of saturated steam at 100° is negative, but we may see

182 HEAT.

how it is so without any appeal to that equation. If we take a quantity
of steam at 100° and 760 mm., and gradually alter the pressure to 787 "63,
not allowing any heat to escape, the work done in the compression goes
to heat the gas, and its specific heat is such that the temperature rises
above 101°. To bring it back to 101°, heat must be abstracted. Or,
putting it in another way, when in the compression the temperature
is 101°, the pressure is less than 787-63, the condensing pressure.
During the rest of the compression, to keep the temperature at 101°, we
must let heat escape. The specific heat, therefore, of saturated steam is
negative. The total energy supplied is not negative, for the positive
work done on each gramme is in excess of the mechanical equivalent of
the negative specific heat.

If, therefore, we have a quantity of steam near its condensing point,
and we compress it without allowing heat to escape, its temperature rises,
and it gets further and further away from the condensing point. On the
contrary, if we reduce the pressure it comes nearer to its condensing point,
and, with sufficient reduction, we have a cloud formed. This cooling ac-
companying diminution of pressure explains a common formation of clouds
in the atmosphere. This mode of cloud-forming is especially noticeable
near the summits of mountains, when a current of air blows up the moun-
tain side into regions of diminished pressure. It there expands, parting
with energy to push out the surrounding air, and so cools. The water-
vapour in the air at the same time expands, cools, and, at last, condenses.
The same effect is also seen during the formation of thunder clouds. Large
masses of damp and, therefore, light air rise, and cool through expansion
against diminished pressure. The temperature falls below that for which
the new pressure of the vapour is the maximum, and condensation occurs,
forming the large cumulus masses high up in the air so characteristic of
thunder clouds.

Other saturated vapours, among them alcohol, agree with water in
having this so-called negative specific heat, so that condensation will
occur with rarefaction. Some saturated vapours, such as that of ether,
possess a positive specific heat, and these on sudden compression tend to
condense. For, taking saturated ether-vapour at t°, and raising the
pressure to that corresponding to saturation at t+ 1°, the temperature is
not so high as t+ 1°, but say t + 6°, and heat is necessary to raise the
temperature from t + 6tot+I. But if this heat is not supplied, the
pressure is above the maximum at t + 6, and condensation occurs and
continues until the latent heat yielded raises the whole to t + 1°.

Spheroidal State.

When a metal plate is heated nearly to rednesss, and a few drops of
water are thrown upon it, the drops rolls about on the plate without
spreading out, and without boiling. They are then said to be in the
spheroidal state. If the plate is a flat one, by looking at it edgeways it
is possible to see that the drops are not in contact with the plate.

It is easy to show this also by bringing a wire nearly in contact with
the plate, and connecting the wire with the plate through a battery and
galvanometer as in Fig. 106, where the plate is the top of an inverted
platinum crucible. Allowing a little liquid to trickle down the wire
it assumes the spheroidal condition when it approaches the plate,

CHANGE OF STATE— LIQUID— VAPOUR.

183

and the galvanometer remains unaffected. On removing the source of
heat, and allowing the plate to cool, a point is reached at which contact
takes place, the galvanometer is deflected and the liquid boils violently.
When in the spheroidal state, the liquid never reaches the temperature
of boiling, the highest temperature for water being at ordinary pressure
about 98°. The plate must be above 140°.

Perhaps we may give a general explanation of this somewhat as
follows : — If the two surfaces were both non-volatilising solids, the air
between them would tend to get into a steady state at a pressure equal
to the atmospheric, and the upper solid would settle down into contact
more or less rapidly. But even here some little time would be taken in
the adjustment. This may be illustrated by allowing a small, very hot
plate of glass to fall flat on a smooth, cold surface, when it moves freely
about for a short time, evidently on a cushion of hot air. The layer
of air between the two becomes heated, its pressure is increased, and it
only slowly escapes out through the
narrow space round the edge of the
heated glass. The excess of pressure,
meanwhile, sustains the weight of
the glass.

In the case of the drop, evapo-
ration comes in to maintain the
excess of pressure. For by the heat
received from the plate the surface
of the drop is rapidly heated, and
evaporation takes place at the rate
corresponding to this higher tem-
perature. Let us imagine that the
drop is a large flat one, and that it is
held in position in some way a short FIG. 106.— Experiment to show that in
distance above the plate. First the Spheroidal State a Drop is not in
suppose both at the same tempera- Contact with the Plate,
ture. The space will tend to be

filled with vapour at the pressure corresponding to that temperature, and
evaporation from the drop will be balanced by condensation on to it if
we neglect the escape round the edges. Evidently in this case the drop
will have to be held up otherwise than by the pressure of the vapour. For
at the maximum temperature of the liquid, the boiling-point, the vapour-
pressure only equals that of the atmosphere. But now make the plate
much hotter than the boiling-point of the liquid. Let us suppose that
the pressure of the vapour between it and the drop is still the vapour-
pressure at the temperature of the liquid. The contact of the vapour
with the hot plate superheats it, i.e. increases the momentum of the
molecules, and if they are as a whole still exerting the same pressure on
their return to the liquid there must be a diminution in the number
returning to compensate the more violent impacts. Hence the number
condensing is lessened, or the value of the pressure is not an equilibrium
value, since the condensation is not equal to the evaporation. The
pressure will, therefore, increase until the escape round the edges balances
the excess of evaporation over condensation. Now this pressure will, in
general, be above the atmospheric pressxire, for the drop is itself near

184

HEAT.

the boiling-point, where the normal vapour pressure is atmospheric, so
that the excess may suffice to sustain the weight of the drop and we need
not think of it as held up by any outside mechanism. If the pressure
does not suffice, the drop comes lower down, nearer the source of heat,
the superheating is greater, while the escape round the edges is less, and
the pressure rises till the drop is sustained.

A curious case of the spheroidal state may often be noticed when a
stream of water falls on a water surface. Of the drops which splash up

and fall down again many will be seen
to remain some little time as drops
without coalescing with the general
body of the liquid. The effect often
occurs when, in rowing on still water,
the oars are held steady, and the drops
are allowed to fall on the hitherto un-
disturbed surface. And it is beautifully
illustrated when a fiddle bow is drawn
across the edge of a round glass vessel

containing methylated spirit. As soon as the vessel vibrates regularly
a shower of drops is thrown by the vibrating segments of the glass
towards the centre, where they remain on the surface for a short time,
forming a most exquisite pattern.

But though by sudden stretch of surface, and by evaporation in passage
through the air, the drops are doubtless colder than the general body of
the liquid, the temperature difference can hardly be enough to account for
the effect, which in this case is probably due, in some way not yet explained,
to surface tension.

FIG. 107.

CHAPTER XL

CHANGE OF STATE. LIQUID— VAPOUR (continued).

Indicator Diagram — Critical Point — Critical Constants — Equation of Van der
Waals— Liquefaction of Gases.

The Indicator Diagram for Water-Steam.— The indicator pressure

volume diagram gives us a convenient graphic method of representing
many of the facts already mentioned in connection with the water-
steam change of state, and it also serves as au introduction to other facts

200000 912000 1100000 1300000

FIG. 108.— Indicator Diagram for Water-Steam.

now to be described. In this diagram volumes are measured from left to
right, and pressures from below upwards, starting from two lines OP, OV
at right angles, Fig. 108.

Suppose we take 1 gramme of steam at 0°, and, say, 1 mm. mercury
pressure, it will have a volume of rather more than 900,000 c.c. This
will be represented by a point a on the diagram, a distance proportional
to rather more than 900,000 from OP, and a distance proportional to
1 mm. from OV. If we gradually increase the pressure, decreasing the

185

186 HEAT.

volume, but keeping the temperature at 0°, the steam will be represented
by successive points along a curve al>, which is nearly a hyperbola, since
i;he pressure multiplied by the volume is nearly constant. But when the
pressure reaches 4'6 mm., and the volume is 200,000 c.c.,the state being
represented by b, condensation normally begins and goes on at the same
pressure till all is condensed to water at 0°. The curve, therefore,
changes at b into a horizontal straight line be, points on this line repre-
senting different proportions of the mixture of steam and water ; c will
represent the volume of the condensed water, just over 1 c.c. The
pressure may now be increased, but the water diminishes only very
slightly in volume, so that subsequent points lie along a line cd, only
slightly leaning towards OP. In the figure it is impossible to represent
the curve on proper scale. It is, therefore, only drawn so as to show its
general nature. If we now start again with 1 gramme of steam at 50°
and 1 mm. pressure, its volume will be about 1,100,000 c.c., represented
by the point a'. As the pressure increases, the temperature remaining
the same, the volume diminishes, the relation between volume and
pressure being represented by the nearly hyperbolic curve a'b', till at
92 mm. pressure and 12,000 c.c. volume condensation begins and goes on
at constant pressure till all is water. This change is represented by the
line b'c. The point c will represent about 1*01 c.c., and is, therefore,
very near the line cd. Subsequent increase of pressure corresponds to
the line c'd', this again only slightly leaning towards OP. Starting anew
with the steam at 100° and at 1 mm. pressure, the initial volume is
about 1,300,000. The subsequent changes are represented by a" if" c" d",
b" corresponding to 760 mm. pressure and 1700 c.c. volume, c" represent-
ing about 1'04 c.c. volume. The successive curves for each temperature
are termed isothermal s.

We may represent on the diagram any change of volume and
pressure occurring in a quantity of water substance. For instance, if 1
gramme of water at 0°, and, say, 92 mm., be gradually heated at
constant pressure in a closed extensible vessel, it will be represented
by a horizontal line starting from the point on cd level with c, and
cutting all the water-isothermals till the temperature reaches 50°, when
it will have reached c'. It now normally begins to turn into steam, and
its course is represented by the line c'b', all at 50°. After b' is reached
the temperature rises again, and successive steam-isothermals are cut
along b'f. But if suitable precautions as to vessel and freedom from air
bubbles are taken, the water may be heated above 50°, still remaining
water. This implies that the water-isothermals do not end in their down-
ward course at the points cc'c", but may be prolonged, as represented in
Fig. 109 to ee. Or, again, we have seen that water may easily be heated
in a very clean glass vessel at the atmospheric pressure to 106° without
boiling. This would be represented by the point g, where the 106°
water-isothermal produced cuts the line of 760 mm. pressure. Dufour's
experiment shows that even the 178° isothermal may be prolonged down
thus far.

Another case is afforded by a phenomenon sometimes observed in
barometer-tubes. If the tube is very clean and the mercury free from
air, it is possible after filling and inversion, to raise the top of the tube
far above 30 inches without detachment of the mercury from it. This

CHANGE OF STATE— LIQUID— VAPOUR.

187

may be done even when some water is above the mercury. The water is
then at a negative pressure, showing that its isothermal may be prolonged
even below the line OV. Worthington has succeeded in subjecting
alcohol to a negative pressure or pull of seventeen atmospheres, and has
measured its expansion under this pull (Properties of Matter, p. 123).

But not only may the water-lines be prolonged below the normal turn-
ing point, but, as described on page 168, if air containing water vapour
is filtered through cotton-wool into a clean flask, so that it is quite free
from dust, then it may be either compressed beyond, or cooled below,

938

- 760

\\tf

FlG. 109. — Indicator Diagram showing Prolongations of Water and of Steam
beyond Normal Turning Points.

the normal condensing-point of the vapour present. The curves ah, a'b't
may, therefore, be prolonged as represented in Fig. 109 by bf, b'f. Just
as the change from water to steam appears to require some gas nucleus,
so does the change from gas to water seem to require some liquid or
solid nucleus. Professor James Thomson suggested that the cui'ves ce,
bf ultimately turn round and join, as represented in Fig. 110.

If this is the case we can hardly hope to experiment on the state re-
presented by gJi, as it would represent that of a substance increasing
in volume with increase of pressure, evidently an unstable condition.
But if we could imagine it possible to take the substance round the
cycle ckhgbhc in a " reversible " manner (chap, xvii.) then the total work
done is zero since the temperature is uniform and the area ckh must
therefore equal the area hgb.

188

HEAT.

Going back to Fig. 108 it will be seen that the length of the horizontal
portion of the isothermal decreases as the temperature rises, so that we
should expect from this diagram alone that above some limiting isother-
mal the flat parts of the curve will cease to exist — that is, that the two
states, liquid and vapour, will gradually approach each other, as the
temperature rises until all distinction ceases.

Critical Point. — This merging of the two states into one, above a
certain temperature, was first suggested by some experiments carried
out by Cagniard de la Tour in 1822. He showed that if a proper quantity

of alcohol was heated
in a confined space, in
the presence of its own
vapour only, up to a
certain temperature the
contents of the vessel
were partly liquid,
partly gaseous, but that
at this temperature —
about 225° C. — the
surface of separation
between the two dis-
appeared, and above it
the contents of the
vessel were entirely
homogeneous, the two
states appearing to
meet. Before the dis-
appearance of the sur-
face of separation the
surface - tension gradu-
ally diminished, tend-
ing to show that the
distinction was disap-
pearing. He observed
the same phenomenon
with other liquids, and

e\

1

\

FIG.

110. — Continuous Form of Isothermal
suggested by James Thomson.

Faraday continued the
researches, making the

important suggestion that for such gases as oxygen, hydrogen, and
nitrogen, this limiting temperature is far below ordinary temperatures,
so that they cannot be liquefied by pressure alone (Researches in Chemistry
and Physics, p. 99.)

But our definite knowledge on the subject was greatly increased by
the researches of Andrews, who made a careful map of the isothermals
of carbon dioxide from actual experiment, verifying the existence of the
limiting or critical temperature, and showing the shape of the isothermals
both above and below this critical temperature.

His apparatus consisted of a long glass tube (Fig. Ill), about 2*5 mm.
in diameter from c to b, and with a capillary bore from b to a, the whole
being carefully calibrated. A current of dry carbonic acid gas was
passed through the tube for some hours, until the proportion of air in the

CHANGE OF STATE— LIQUID— VAPOUR.

189

gas issuing from the tube was small and constant. It was found always
to be not less than y^n^j- of the whole, and the proportion had to be
determined and allowed for. The end a was then sealed, the other end
closed and af terwards opened under mercury. By suitable manipulation,

c b a.

FIG. 111. — Andrews' Tube for Experiments on Carbon Dioxide.

partly by heating, partly by reduction of external pressure, about one-
fourth of the gas was expelled, and a short column of mercury intro-
duced into the tube cb to act as stopper and index. The tube was then
placed in position with the open end c in the compression chamber
(Fig. 112), which was filled with water. By means of the screw S, any
desired pressure could be applied, the amount being
measured by a second exactly similar tube and com-
pression-chamber, containing air instead of carbonic acid
gas, and placed by the side, a cross tube connecting the
two chambers. Each tube was surrounded by a water-
bath with plate glass sides, not shown in the figure, that
round the air tube being kept at a constant temperature,
while that round the carbonic acid gas was raised to any
desired temperature.

The diagram (Fig. 113) taken from Maxwell's Theory
of Heat (5th ed., p. 120) shows Andrews' results. The
13'1° and 21'5° isothermals exhibit the normal change
from the all-gas curve on the right to a horizontal line,
indicating condensation at constant pressure and a
coexistence of the two states, followed by a steep rise of
the all-liquid curve. In the actual results, through a
small trace of air in the tube, the corners were rounded
off and the flat part of the isothermals sloped slightly
upwards, showing that the last part of the condensation
required a greater pressure than the first. The higher
isothermals slope upwards throughout, showing no con-
densation at all. During the compression the substance
remains homogeneous, and the only relic of the co- LgJ

existence of two states is in the diminished slope at one
part of the isothermal. Even this has disappeared at
48*1°. By several experiments the lowest temperature
at which condensation does not occur was fixed at 30'92°. FIG. 112. — Dia-
This therefore is the critical temperature for carbon grammatic Re-
dioxide. Anlrews^Com!

If we draw a curve (dotted in Fig. 113), the " border pression Appa-
curve," through the points where the isothermals below ratus.
this temperature change into and from the horizontal
straight line, its area includes all the conditions of coexistence of gas and
liquid at one temperature. To the right the substance is a gas, but one
for which the volume decreases more than in accordance with Boyle's
law, that is pv decreases as p increases. To the left, the substance is a
liquid, and pv increases as p increases.

190

HEAT.

The critical isothermal just touches the vertex of this dotted curve,
and, being there nearly horizontal, a slight change in pressure corre-
sponds to a great change in volume. Andrews noticed that about this
point any change in pressure was accompanied by flickering movements
in the tube, showing considerable local alterations in density.

FIG. 113.

The Critical Constants, — The vertex of this dotted curve may be
termed the critical point. It therefore represents a temperature of
30 '9 2°, and a pressure, termed tha critical pressure, of about 73 or 75
atmospheres. The pressure could not be determined quite accurately,
as the air in the attached manometer did not change volume exactly in

CHANGE OF STATE— LIQUID— VAPOUR.

191

accordance with Boyle's law. The volume of unit mass, termed the
critical volume, was about T^ of the volume at the same temperature
and 1 atmosphere.

Above the critical isothermal, the only survival of the vaporous con-
dition is found in the fact that on the right-hand part of each curve,
pv decreases as p increases to a certain point. After this it increases
as p increases, and this may be regarded as a survival of the liquid
condition. The point of minimum value of pv may be considered to
represent what in the lower isothermals is a change of state.

Regnault (whose work has been verified by Amagat) found that with
pressures up to 20 atmospheres, pa for air and nitrogen decreased as p
increased, while it increased for
hydrogen. (See Properties of
Matter, p. 124.)

Andrews found that it was
quite possible to take a quantity
of carbon dioxide gas round from
the undoubtedly gaseous to the
undoubtedly liquid condition
without any condensation. The
mode in which this was effected
may be understood from Fig 114.

Starting with a quantity
of gas, say at 13°, represented
by A, let it be heated at con-
stant volume till it reaches,
say, the 40° isothermal at B.
Let it then be compressed at
40° until the volume is less
than that occupied by the
liquid when just entirely con-
densed at 13°, this being re-
presented by 0. Let it then
be cooled at constant volume to
D on the 13° isothermal. It is
certainly gaseous at A, certainly
liquid at D, for, on increasing
the pressure at the former point, condensation occurs, and on removing
the pressure at the latter point, ebullition occurs. But throughout the
passage from A to D the substance remains homogeneous.

Andrews made the suggestion that the term gas should be restricted
to the condition of a substance above the critical point, that the sub-
stance below that point and to the right of the border curve should be
termed vapour, while to the left of the border curve it is liquid.

Determinations of the critical constants of temperature and pressure
and, in some cases, volume have been made for a large number of sub-
stances by Andrews and others experimenters. Some have used
Andrews' method more or less modified, but various other methods have
been employed. For example, Cailletet and Collardeau (Ann. de Chimie
et de Physique, 6, xxv. p. 519), experimenting on water, heated the sub-
stance in a steel tube, and observed the point at which the vapour

FIG. 114.— A, B, C, D, path from gas to liquid
without discontinuous change of state.

192 HEAT.

pressure ceased to be one-valued and became dependent on the amount
of water present. Up to the critical point the pressure of vapour in con-
tact with the water was definite at a definite temperature, but as soon
as that point was passed the contents of the vessel were homogeneous,
and the pressure depended on the amount of water dealt with. They
found a critical temperature of 365° C., with a critical pressure of 200-5
atmospheres. Traube and Teichner (Ann. der Physik, xiii., 1904, p. 620),
found the critical temperature to be 374" 0. They used a quartz tube
filled to J with distilled air-free water and heated by mercury boiling
under pressure. The meniscus disappeared at 374° and reappeared
when the temperature was again brought down to that point.

The refractive index has also been used to find the critical point.
In the researches of Prinz Galitzine and Wiliss (Congres International
de Physique, 1900, i. p. 668), the refractive index of a substance near
the critical point was measured either by using the substance as a
cylindrical lens and observing the refraction, or by inserting a small
angled prism in a tube containing the substance and finding the refrac-
tion due to the prism. The refractive indices of the liquid and vapour
approach each other as the critical point is approached, and at that point
are equal. For ether the meeting point of the refractive indices was
found to be p»l-12 at 1937° C.

In these experiments as in others it was found that near the critical
point very great variations in density occurred in the successive layers,
variations only eliminated by stirring.

The isothermal is very nearly horizontal at the critical point and
very minute changes of pressure may bring about large volume changes,
and with the volume changes temperature changes may ensue, so that
it might be expected that there would be difficulty in obtaining homo-
geneity of the contents of the vessel even just above the critical point.
But the variations are so large that many observers doubt whether the
liquid and the vapour do actually meet in all their physical qualities at
the critical point. They regard the critical point as that at which the
densities coincide. They suppose that slightly above the point the
liquid is still liquid, and may have a different density from the gas and
a different molecular grouping ; but the evidence is hardly conclusive.

Ramsay and Young, and later Young (Phil. Mag., xxxiii., 1892,
p. 153, and in other papers) have measured a number of critical con-
stants. The pressures and temperatures were obtained by the direct
method of raising the temperature and observing the vapour pressure
at the critical point, when the liquid meniscus disappeared. The
critical volume was found by heating the substance above the critical
temperature, and then cooling it by sudden expansion till there was
a temporary separation between liquid and vapour. The volume
at which this just took place on slight expansion was taken as the
critical volume. MM. Cailletet and Mathias (C.R., cii. p. 1202 ;
civ. p. 1563) have found that in many cases the critical volume can
be calculated on the assumption that the mean of the liquid density
and saturated vapour density changes uniformly with the temperature,
so that on a density temperature curve it would be a straight line.
If these densities for a lower range of temperature and the critical tem-
perature are known, the critical density is easily determined.

CHANGE OF STATE— LIQUID— VAPOUR.

193

We give below a few critical constants, but it must be remembered
that the results obtained by different observers are not coincident : —

Temperature.

Pressure in
mm.

Volume of
1 gramme.

Molecular
Volume.

Benzene

288-5

36395

3-293

2563

Ether .

194-4

27060

3-801

280-7

Ethyl alcohol
Water

243-1
365

47850
152380

3-636

166-9

Van der Waals' Equation — The equation of Van der Waals' (p. 152)

is of great interest in connection with the critical constants of pressure
volume and temperature, in that above a certain value of 6 it is repre-
sented by curves having the general form found by Andrews, while
below that value the curves are of the form suggested by J. Thomson,
and represented in Fig. 110.

The equation may be arranged as a cubic in v, viz.,

p

p

and this has either one or three real roots for given values of p and 6.

Taking a given isothermal, say the lower curve in Fig. 115, it will
be cut only once by an equal pressure line if the pressure is less than
that at B, and only once again if the pressure is greater than that at D.
Between these values it will be cut three times as in A, 0, E.

In the first case the substance is a vapour; in the second, a liquid ;
while in the third case it may be either a liquid, at A, or a vapour at E.
The volume 0 is only realisable experimentally if ACE represents the
vapour-pressure at the temperature chosen. It is then a definite mixture
of vapour and liquid. But the equation hardly means this. It rather
indicates that it is conceivable that the substance might be brought quite
continuously all in one state round the curve EDCBA. We shall suppose
that if ACE is at the height of the vapour-pressure then the areas
ABO, CDE are equal (p. 187). Now plotting all the isothermals: the
equation agrees with observation in showing that ABODE gets less and
less, and that ACE draw nearer and nearer together as the critical
temperature is reached. At that temperature they coincide. Hence,
if we insert in the equation the critical values pe and de the equation has
three equal roots, each equal to vc, the critical volume.

Comparing then

a
Vs

/, R#A 9 , av ab A
- [b + — c I v2 + --- = 0

\PcJ PC PC

with

Vs -

194

we get at once

HEAT.

whence

PC

ab

P!
a

ve=3b

e=27 6R'
Now, if the equation is a correct representation of the behaviour of

FIG. 115.

gases and vapours, a can be determined from the change of pressure with
temperature of the gas when at constant volume, while b can be deter-
mined from the change in pv with change of pressure at constant tempera-
ture. R may then be found by inserting any three values of p, v, and 6 in
the equation. For details of the calculations we refer to Van der Waals'
paper (Physical Memoirs, vol. i., p. 390). He finds for carbonic acid
the equation

/ + 0-OQ874V _ 0.0Q23) = j .OQ646 ^ + ^

where the unit of pressure is 1 atmosphere, the unit of volume the
volume of 1 gramme at 0° and 1 atmosphere, and t is the temperature
centigrade.

CHANGE OF STATE— LIQUID— VAPOUR. 195

From this the critical pressure is 977,2 = ^1 atmospheres,

while the critical temperature is — — = 32*5°,

27011

which are values not very far distant from those found by Andrews.

The values of a and b, however, are not determined very exactly, and
the best test of the equation is probably to be found in the comparison
with experiment of certain deductions from it which we shall now
discuss.

Reduced Isothermals— Corresponding Temperatures.— If we

express the pressure, volume, and temperature for a given condition of a
substance as fractions of the critical values, viz. : —

ea

v «= nv = Srib

and substitute these values in Van der Waals' equation, it at once
reduces to

from which a and b, which refer to a particular gas, have entirely dis-
appeared.

If the same value of m is taken for different gases which have critical
points Qc, #„', &c., the temperatures indicated by mOc, mOc', &c,, are " cor-
responding temperatures." Thus the absolute critical temperatures for

ether and water are 470° and 638° about. If we put m = -, then a temper-
ature of 235° absolute, or - 38° C. for ether corresponds to 319° absolute,
or + 46° 0. for water. The above equation shows that if m is constant,
i.e. if different substances are taken at corresponding temperatures, then
the relation between e and n is definite, and the same for all substances.
We have in fact a cubic in n, a Van der Waals' equation, in which

1 8

a = 3, &= o>R= o> an(^ giving to m a succession of values we obtain a
o o

family of curves representing the relation between e and n, of the same
general form as the isothermal curves given by the original equation.
These general isothermals are called Reduced Isothermals. To apply
any reduced isothermal to a particular gas we must multiply the m for it
by Bc to get the temperature, and must multiply the values of e and n at
any point by pe and vc respectively to get p and v.

The line on a reduced isothermal corresponding to the horizontal
vapour-pressure line on a pv diagram is the same for all substances. We
may prove this as follows : — The horizontal vapour-pressure line cuts the
continuous pv curve in three points, and, as we have seen, makes equal

196 HEAT.

areas above and below, as bgh and Me in Fig. 110. This is expressed by

V3

pdv

where pv relate to the continuous curve, P is the vapour pressure, Vl
and V3 the liquid and vapour volumes.

If we now substitute in terms of the critical values, putting p = epey
v = nva P = E£>C, Yx = Wjz;c, V3 = nsve, pcvc divides out and we have

nl

an expression which shows that the areas cut off by the E line on the en
diagram are equal above and below. Substituting from equation (1) put
in the form

_ 8m _ 3
3n-l n*

we see at once that we can integrate. Then using the equations which
state that Ex, nv E1? nz are on the curve, we get two more equations which
might enable us to find E, n^ and n3 if this were needed. But it is enough
to note that for the same value of m, E, n^ and ns are the same for all
substances. In other words, at corresponding temperatures the saturated
vapour pressures are the same fraction of their critical pressures for all
substances, and the liquid volumes and the vapour volumes are the same
fractions of the critical volume for all substances.

Corresponding Pressures and Corresponding Volumes.— If

we take corresponding temperatures for different substances, the vapour
pressures at those temperatures are corresponding pressures. The
liquid volumes just before evaporation begins and the vapour volumes
just when it is completed are termed corresponding liquid and corre-
sponding vapour volumes.

This constancy of E, n^ and n2 for given m gives us a result which
can be more easily compared with observation than the original equation
between p, v, and 6. Thus Van der Waals used it to calculate the
boiling point of carbon dioxide from its known constants and those of
ether. The critical pressure of carbon dioxide is 72 atmos. and its

critical temperature is 303*9° absolute. For 1 atmosphere, then, e = —

72.

Now the critical pressure of ether is 36 '9 atmos. and the critical tempera-
ture is 463°. The corresponding pressure is 36-9 x 760 x e = 384 mm.,
which is the vapour-pressure of ether as obtained by direct experiment

9SQ-Q

at 16-9° 0. or 289'9° absolute. Then m=- -1=0-625, and the corre-

463

spending temperature for carbonic acid is 303'9 x 0'625 = 190° absolute,
which is very nearly in agreement with observation of the boiling-point.
Young (Phil. Mag., xxxiii., 1892, p. 153, and in later papers) has examined
the behaviour of a number of liquids and vapours with regard to their
corresponding quantities. He finds that Van der Waals' result is not

CHANGE OF STATE— LIQUID— VAPOUR. 19?

far from true for the halogen derivatives of benzene, but that for many
other substances it does not hold. The equation and its deductions
are to be regarded not so much as expressing the actual relations for
substances in general as an approximation for some substances sufficiently
near the truth to suggest lines of experiment, while its divergence from
the truth in other cases found by these experiments may ultimately lead
to some more correct and more general equation. The modifications
hitherto suggested, while more difficult to deal with, are hardly more in
accordance with experiment.*

Liquefaction Of Gases. — When Andrews made his celebrated
researches, several gases, as oxygen, nitrogen, carbon monoxide, and
hydrogen, had never been liquefied, and such gases were often termed
permanent gases. But the results which Andrews obtained with carbon
dioxide gave weight to the suggestion of Faraday that these gases were
far above their critical points at ordinary temperatures, and that they
could only be liquefied by cold as well as pressure. Acting on this idea,
Cailletet and Pictet, working independently, both succeeded at nearly
the same time in 1877 in liquefying oxygen and other gases. The
general principle of their experiments, though carried out with very
different details, consisted in compressing a quantity of gas under 300 or
500 atmospheres, cooling it by surrounding the containing vessel with
liquid sulphurous acid or liquid carbonic acid evaporating freely, and
then making a sudden expansion of the gas thus compressed and cooled.
The work done by the sudden expansion implied a further transfer of
energy from the gas, which fell below its boiling point and became
liquid. In Cailletet's early experiments the volume of the containing
vessel was suddenly increased, and the liquid appeared as a mist. In
Pictet's experiments the gas was allowed to issue through a nozzle, the
gas behind doing work on that in front, and cooling so far that ulti-
mately it issued as a liquid jet, and sometimes even gave signs of solidifi-
cation. There were doubtful indications of the liquefaction of hydrogen. f

Much valuable work on the liquefaction of gases was done by
Wroblewski and Olzewski, at first working together and later inde-
pendently. They used the evaporation of liquid ethylene as a cooler,
and to find the temperature reached they used either a hydrogen ther-
mometer or a thermo-electric couple. Olzewski ultimately used a
platinum resistance thermometer. In some of his experiments he
placed a graduated test tube in the tube containing the compressed
gas, but separated from it by a narrow space. On the expansion, liquid
collected in and round the test tube. That inside the test tube was
protected by an evaporating liquid jacket, and could be preserved for
some time. The boiling point could then be found. Collecting the
vapour or gas given off from a measured volume of liquid, the liquid
density could be determined.

Dewar was also one of the early workers in the field, and has made
very valuable researches, on a much larger scale than any other worker.

* Further information as to Van der Waals' equation will be found in Nernst's
Theoretical Chemistry.

t A full account of the various methods of liquefying gases, with a valuable
history of the subject, will be found in Harden's Liquefaction of Gases and Travers'
Study of Gases. We only mentioii a few workers to illustrate methods.

198

HEAT

He greatly aided the study of the liquids ana tneir use as cooling agents
by the introduction of a special containing vessel, to catch the condensed
gas issuing from the expansion nozzle and to keep it for a long time
liquid at the atmospheric pressure. The vessel is a double-walled glass
globe with a vacuum between the walls, as in
Fig. 116, the surfaces in the evacuated space
being silvered or, later, covered with a film of
mercury, secured merely by condensation of
mercury vapour rising from a few drops of
mercury left in the space. The evaporation from
liquid air or oxygen in such a globe is only a
small fraction of what it is in a single-walled
globe.

About 1894-95 a new step was taken in the
introduction of the "regenerative" method. There
has been much discussion as to priority of dis-
covery, but it is enough here to say that it was
used very nearly at the same time by Dewar,
Hampson, Kammerlingh-Onnes, Linde, and Tripler.

FIG 11 0. Dewar's ^e Principle of the method may be understood

Vacuum Vessel (Sec- from Fig. 117, which is merely a diagrammatic

tion). sketch. The gas is forced by a pump P at very high

pressure through a cooler C, which removes the

heat developed by the compression, and the gas flows on at the ordinary
temperature to the regenerator R, where it passes down a spiral pipe to
a nozzle N. From this it issues with a sudden fall of pressure from
that of the pump to that of the atmosphere. There is considerable
cooling of the issuing gas, which in turn cools the spiral pipe by which
succeeding portions are
coming to the nozzle.
Thus the next issuing
gas falls still lower in
temperature, cools the
spiral still more, and
so on, till ultimately
the gas issues as liquid
and falls into the col-
lecting vessel V. No
cooler below the ordi-
nary temperature is
required, the cooler 0
merely reducing the
gas to the temperature
of the room.

It is to be noted
that the expansion at
the nozzle is not adiabatic, i.e. not one in which the gas has to use up
its own energy in pushing out its surroundings. The pump behind is
continually doing work and supplying very nearly all the energy which
the gas requires to push its way onwards, when it conies out of the
nozzle. The conditions are much more nearly those of the porous

Fia. 117. — Diagram of Regenerator Process for
Liquefying Gases.

CHANGE OF STATE— LIQUID— V A POUfc.

199

plug experiment of Thomson and Joule (chap, xix.), and the cool-
ing is chiefly due to the conversion of molecular kinetic energy
into potential energy in the work done against the internal forces.
At the ordinary density of a gas so few of the molecules are at any
one instant near each other, that we may consider that all the
molecules have the potential energy of complete separation from their
fellows. But when the gas is at several hundred atmospheres, and
still more when it is cooled far on its way to the absolute zero, a very
large proportion of the molecxales are close together and are exerting
cohesive forces. If now expansion occurs, work has to be done in
separating them, or at any rate in separating a very large proportion of
them, and cooling results.

In Joule and Thomson's researches hydrogen alone showed no certain
evidence of such cooling, but even hydrogen at the pressures used in the
regenerative method appears to cool with the rest when it is at a
sufficiently low temperature initially.

Though hydrogen was not liquefied in the earlier experiments,
Olzewski was able to reduce it to a very low temperature, and to calcu-
late its boiling-point from its behaviour, and in 1898 Dewar succeeded
in obtaining and keeping it liquid by cooling it first to - 205° 0. at a
pressure of 180 atmospheres, and then allowing it to issue from a nozzle
into a vacuum vessel.* In 1899 he succeeded in obtaining solid
hydrogen. A small vacuum vessel containing liquid hydrogen was
suspended in a larger one also containing the liquid, and on allowing
evaporation to take place below 60 mm., a slight air leak caused the
hydrogen to solidify as a white froth-like foam in the inner vessel
(Nature, lx., 1899, p. 514).

The temperatures of the boiling points and melting points of these
ordinary gaseous substances are not found with perfect consistency
with different instruments, but the various experimenters obtain results
not very far apart.

The following are some of these (Travers, Study of Gases, p. 247) : —

Critical
Tempera-

Critical
Pressure

Boiling-
Point.

Melting-
Point.

Absolute.

atmos.

Absolute.

Absolute.

Argon . . .

155-6

53

86-9

Carbon Dioxide .

304

72-4

193

208f

Carbon Monoxide

137

33-4

83

66

Hydrogen .

35

15-3

20

16

Nitrogen . . .

124

27-6

75

60

Oxygen .

154 .

58

90-5

below 50

* Science Abstracts, i. 1898, p. 562. An account of the liquefaction of hydrogen
is given by Travers in the Phil. Mag., 1901, i. p. 411.
t Andrews.

CHAPTER XIL
CHANGE OF STATE. SOLID— LIQUID.

Melting of Ice and Melting of Wax — Melting of Ice at a Definite Point and on the
Surface only — Latent Heat— Supercooling — Regelation— Effect of Pressure on
Melting Point — Melting Points of Solids— Explanation of Melting on the Kinetic
Theory— Resemblance of Solution to Fusion — Evaporation from Solids.

Melting Of Ice and Melting Of Wax. —The change from the solid
to the liquid state may be illustrated by the familiar examples of the
melting of ice and the melting of fatty or waxy substances. The latter
are probably mixtures, and the process may be complicated by the melt-
ing of one constituent before another. But, however this may be, we at
once observe a difference between the two cases. A piece of sealing-
wax, as its temperature rises, gradually softens from a solid into a
viscous liquid without any discontinuity, and on cooling it gradually
hardens again into a solid.

Melting of Ice at a Definite Point and on the Surface only.

— A piece of ice, however, remains solid to the moment of melting — it
is never really soft, and in the converse process of freezing, the water
does not gradually get thick, but crystals of quite solid ice form within
it. If we take a quantity of ice below the melting point, and gradually
supply heat, keeping the ice at a uniform temperature throughout, the
ice rises up to 0°, still continuing solid. The surface of the ice then begins
to melt, and if care is taken to keep the temperature still uniform through-
out, no further rise takes place till the whole of the ice is melted, the
temperature remaining at a perfectly definite point. Indeed, on the
constancy of this melting point depends our system of graduating ther-
mometers. The ice melts at its surface alone, and there is no softening
into a plastic condition, as in the case of wax.

Latent Heat. — The heat taken up by each gramme ot ice, in
turning from ice at 0° to water at 0°, is constant in amount, and is
termed the "latent heat" of water.

Freezing. — Taking the converse process, if a quantity of water be
cooled down gradually, the temperature being kept uniform meanwhile
by stirring, the water remains liquid without any sudden or remarkable
increase in its viscosity. At 0°, crystals of solid ice begin to appear, and
grow at the expense of the water. The temperature is arrested at 0°
throughout the mixture, if the stirring is kept on while the ice is grow-
ing. The same amount of heat per gramme is given up to the surround-
ings as was absorbed in the change from ice to water.

Supercooling. — If we begin again and, instead of stirring the water,
we keep it as quiet as possible during the cooling, then, in general, the
temperature falls continuously past 0° to perhaps - 3° or - 4°. Next,

CHANGE OF STATE— SOLID— LIQUID. 201

freezing sets in, and the temperature immediately leaps up to 0°, and
remains there, during the rest of the process of solidification, the latent
heat yielded by the ice first formed being sufficient to raise the tempera-
ture of tho mixture to 0°. This frequently occurs in jugs of water left
standing quietly in cold weather. There is no ice in the water while it
is quiet, but the moment the jug is lifted up and its contents poured out,
a number of minute crystals of ice form in the water, showing that the
latter must have been below 0°, for the latent heat of the water given
up on freezing can only be accounted for by supposing that it has raised
the temperature of the water. If the water be covered with a layer of
oil, it is still easier to produce this phenomenon, and without any trouble
the temperature may be lowered to - 6° or - 7° without freezing. In
this state the water is said to be superfused. If a crystal of ice be
dropped into the liquid, freezing at once commences on this as a nucleus,
and the superf usion ceases. Or a sudden shock will often put an end to it.

There is a sudden change of volume on the change from water to ice.
As we have already seen, water is most dense at + 4°, and it expands
slowly as the temperature falls. If the water be kept superfused, the
expansion is continuous. But if the water changes to ice at 0°, there is
a sudden increase of volume, 1 c.c. of water becoming 1-09 c.c. of ice.
The ice contracts on cooling below 0°

Similar phenomena are noticed in the melting and freezing of other
substances. For instance, phosphorus melts definitely at 44 '2 ; if kept
under water it may be cooled in the superfused condition far below this ;
but a sudden shock, or the introduction of a glass rod previously rubbed
with phosphorus, is followed by solidification. With sulphur, melting
at 115°, the superfusion is still more marked. These substances also
have definite latent heats on melting, and show a sudden increase of
volume — -1 c.c. of solid phosphorus melting to 1'034 of liquid and 1 c.c.
of solid sulphur melting to 1-05 of liquid.

Regulation. — Faraday first directed attention to a remarkable
phenomenon, now known as " Regelation." If a number of blocks of ice
at 0° be pressed together they adhere at the points of contact. This
effect, as remarked by Faraday, is very familiar in another form. Snow
near the melting-point will easily bind into snowballs ; but if below 0°
it is powdery, and will not form a cohesive mass. Similarly the blocks
of ice will not adhere if below 0°.

Tyndall showed that ice at 0° could be moulded by great pressure,
the ice breaking up under the pressure, and then freezing together at
the points of contact. By continued fracture and regelation, the
crevices are filled up, and the ice forms a transparent continuous mass
which has the shape of the mould.

The motion of glaciers has been ascribed to regelation. It has been
observed that the whole mass of a glacier flows like a stream of water
down the valley in which it lies, but with exceedingly small velocity, the
centre of the glacier flowing more rapidly than the edges. Forbes found,
for example, that the Aletsch glacier flowed in " mid-stream " at one point
about 1 4 inches in 24 hours, while near the edge it flowed only 3 inches in
the same time. In winter, the speed is considerably less than in summer.
Now the ice of a glacier is pretty nearly at 0° through the penetration
into all the crevices of the water melted at its surface. The enormous

202 HEAT.

strains to which the ice is subjected continually fracture it, and the
fracture allows slight yielding to the strain. Owing to the pressure,
regelation immediately occurs and the pieces join together again. This,
no doubt, is a partial explanation, but from experiments made by
M'Connel and Kidd (Proc. R.S., xliv., 1888, p. 331) it appears that ice
when consisting of an irregular aggregation of crystals behaves like a
viscous solid. They found that bars of ice of regular crystalline struc-
ture showed only exceedingly minute changes in extension and com-
pression under continued pulls and pressures, while bars of irregular
structure, formed of aggregations of crystals with their axes in all
directions, went on extending or lessening in length under continued
forces. The changes may be due to slipping along the cleavage planes.
The possession of a sensible vapour-pressure by ice shows a certain
mobility or power of escape of the molecules near the surface, and if we
suppose this mobility to exist within the mass we may suppose that there
will be considerable variations of strain from point to point in the irre-
gular aggregation. The mobile molecules will tend to move from points
of greater to points of less strain, and will perhaps fill up the gaps which
the slipping tends to form. Thus the mass keeps continuous. The
crystalline structure of glacier ice is of this irregular type.

The viscosity of metals is probably due to a similar irregular
crystalline structure, and though we have not such clear indication of
internal molecular mobility it doubtless exists.*

Regelation has been explained by supposing that the interior of the
blocks in contact is slightly below 0°. The points in contact at 0° are by
the contact surrounded by portions of ice below 0°, and are thus cooled,
and the water on the surface is frozen so as to form a solid bridge.
Doubtless there is sometimes an effect of this kind. But it would take
place with very great slowness, as the latent heat of the newly formed
ice would have to be transmitted through the badly conducting surround-
ing ice to the colder ice inside the blocks.

In order to remove this difficulty, Forbes and others have supposed
that ice passes gradually into water, so that on the surface of a melting
block there is a layer which may be regarded as either " plastic ice "
or "viscid water." When two such blocks are brought together,
any water between them need not give up its whole latent heat, but just
enough to become either "plastic" or "viscid," and the connecting-
bridge is formed. But there is no independent evidence of this plastic
condition. In freezing water, the ice always forms at definite points,
crystallising out from the water — the water showing no tendency to
become viscid. Indeed, as we have seen, water may be cooled easily
below 0°. A mixture of ice and water kept at 0°, and neither gaining
nor losing heat from its surroundings, remains ice and water, whereas
if the plastic condition existed the whole of it should, as pointed out by
Helmholtz, gradually assume the viscid condition at a uniform tempera-
ture. The use of ice calorimeters, too, is founded on the supposition
that there is no such gradual transition but that the heat given up is
used to convert true ice to true water. If there is any intermediate
condition, we must make the improbable supposition that the contraction
in passing through any part of that condition is proportional to the heat
* See Properties of Matter, p. 59.

CHANGE OF STATE— SOLID— LIQUID. 203

received. We must, therefore, reject the supposition of a continuous
change from liquid to solid, and regard the change from ice to water as
discontinuous, like the change from water to steam.

Effect of Pressure on the Melting Point. — The true explanation
of regelation is probably given by a fact first predicted from theory by
James Thomson, and since verified by his brother, Lord Kelvin, viz. : —
that the melting point of ice is lowered by pressure. The following
general explanation of this effect is not to be regarded as at all strict,
but is given merely as enabling us to think of the main features of the
process. A more complete proof will be found in chapter xix.

Let us take a mixture of ice and water at 0°. Any further melting
will decrease the volume ; any further freezing will increase it. Now,
subject the mixture to great pressure, and let us assume that the mixture
will yield to this pressure in the most effective way. This will take
place if some of the ice melts. In melting, latent heat will be taken up,
and the mixture will consequently cool until there is once more equilibrium
between the ice and the water at a lower temperature — in other words,
the melting point is lowered. If the pressure is still further increased
more ice melts to yield to it, more latent heat is taken up, and the
mixture will cool still further till a new melting point is reached.

Similarly, with substances in which there is an expansion on melting,
pressure will raise the melting point. For, on subjecting a mixture of
solid and liquid to pressure, the mixture will yield most effectively
by solidification, latent heat is given up and the temperature is raised.

To explain regelation by the lowering of melting point by pressure, let
us suppose that two blocks of ice at 0° are pressed together. The pressure
lowers the melting point below 0°. Some of the ice at and near the
surface is subjected to this pressure, and is, therefore, above the new
melting point. It melts, and taking up latent heat it cools the liquid
and the surroundings to the new melting point. The cooled water is
squeezed out to the space between the two blocks where there is no
pressure, so that the water is now below its freezing point. Therefore,
being in contact with ice, so that there is no superfusion, it freezes again
and forms a bridge connecting the two blocks. In fact, the effect of the
pressure is to transfer the ice from points of greater to points of less
pressure, the ice, however, liquefying in order to effect the passage. On
the viscous theory of ice, it also passes from points of greater to points
of less pressure, but without liquefying.

The lowering of the melting point of ice by pressure was verified by
Lord Kelvin* as follows: He enclosed a mixture of ice and water
in an Oersted's piezometer, consisting of a stout glass cylinder provided
with a screw. A thermometer — with ether in place of mercury, to secure
greater sensitiveness — was enclosed in a protecting glass tube, and placed
in the mixture, the part of the scale to be used being kept free from ice
by a lead ring, so that it could be seen. The pressure was indicated by a
glass tube closed at one end, and just full of air at the atmospheric
pressure. This was put into the piezometer with the open end down-
wards.

When a pressure of several atmospheres was put on, the temperature
fell and remained steady, the fall being about 0-0075° per atmosphere,
* Mathematical and Physical Papers, vol. i. p. 166.

204

HEAT.

FIG. 118.— Bunsen's

Raising of

this fall agreeing very closely with that predicted by Prof. James

Thomson. Dewar (Proc. R.S., xxx., 1880), using apparatus like that of
Cailletet for the liquefaction of gases, found that up
to 700 atmospheres the reduction was proportional to
the pressure and at the rate of 0-0072° per atmosphere.
Experiments have also been made by Bunsen and
Hopkins. Bunsen used a bent tube (Fig. 118), con-
taining a small quantity of the substance to be
tested at the end of the shorter limb OD, A^hich was
closed after its introduction. Mercury extended from
D round the bend, filling the wider part E of the
tube, and rising some distance in the capillary tube
AB. The end A was closed when the quantity of
mercury was suitably adjusted. The mercury, on
heating, expanded and compressed the air which
served both as a manometer and as a spring to exert
pressure. This pressure and the melting point when
it was exerted could be observed. Bunsen found that
100 atmospheres raised the melting point of sperma-
ceti about 3'2°, and that of paraffin about 3'6°.

Hopkins, using an iron cylinder in which the
5£ melting was indicated by the fall of a ball of iron
the within the cylinder, was able to put on greater pres-
Melting Point by sures, raising the melting point of spermaceti, for
Pressure. instance, from 51° to 80'2° by a pressure of 792

atmospheres — a result nearly agreeing with Bunsen's.
Melting Points Of Solids. — Various methods have been used to find

the melting points of solids. The simplest is to employ an arrangement

similar to that used for fixing the zero point of thermometers. Heat

is very gradually supplied to a mixture

of the liquid and solid, and the steady

point observed. The melting point of

a substance like paraffin may be de-

termined by filling the capillary part

of a bent tube (Fig. 119), with the

substance when liquid, freezing it, and

then immersing the tube in a bath,

and noting the temperature at which

melting occurs. Carnelly determined

the melting points of many salts which

melt at a high temperature, by heating

a small quantity in a platinum crucible

of mass large compared with that of

the salt, and then, at the moment of

melting, dropping the crucible into

a calorimeter. The quantity of heat

given up allowed the temperature at FIG. 119.— Melting-Point Apparatus.

melting to be calculated from the

known specific heat of platinum — the heat given up by the salt being

negligible.

Latent Heat. — The latent heat of many solids may be found by

CHANGE OF STATE— SOLID— LIQUID. 205

an experiment resembling the determination of the specific heat by the
method of mixtures. If the specific heat of a substance in the liquid
state is o-, and in the solid or', while its latent heat is L, then on cooling
w grammes of it from the liquid state, 0 above the melting point, to the
solid state, & below the melting point, the total heat given up is

10(0- 0 + L + o-' ff),

and knowing the capacity of the calorimeter in which the cooling takes
place, and the rise of temperature of the calorimeter, this heat may be
measured, o- and a-' may be found beforehand and then one observation
will give L. If not, separate experiments in which 6 and & are varied
will enable us to determine all three quantities.

This method was employed by Person, who warmed the substance in
a small copper vessel, and then immersed the vessel and its contents in
the calorimeter.

For the latent heat of ice, Bunsen made use of his calorimeter, de-
termining the quantity of ice melted by pouring into the calorimeter a
known quantity of water at a known temperature.

The following are a few values of latent heat : —

Person. Bunsen. Smith.

Water, . . . 79'2 80'2 79*818

Phosphorus, . . 5'0

Lead, . . . 5'36

The Explanation of Melting on the Kinetic Theory.

We have already pointed out that, in cases of fusion resembling that
of ice, there is no gradual change from solid to liquid — no softening
throughout the mass — and that the melting always occurs at the surface.

Again, in the converse process, ice is not spontaneously formed within
the body of the water till at least many degrees below zero, even if then,
and, in general, solidification requires the presence of ice or some dis-
turbance to start it.

We cannot, therefore, regard the two states as in any way continuous
one with the other. At melting, there is an abrupt change. If we have
a piece of ice in water at 0°, on supplying heat the water grows at the
expense of the surface ice, and on withdrawing heat the ice grows at its
surface at the expense of the water. This closely resembles the change
from liquid to gas by surface evaporation, which we explained by suppos-
ing interchanges of molecules between the gas and liquid, and the
resemblance suggests an explanation of a similar kind for the change
between solid and liquid, in which this also is regarded as a matter of
exchange of molecules.

Perhaps we may make this explanation clearer by considering an
ideal experiment.

Let A and B (Fig. 120) be two bulbs containing ice and water
respectively, and connected by a tube C, the space above the ice and
water in A and B and in the tube C containing water vapour only. Let
the bulbs be surrounded by a constant temperature bath, so that both
are at exactly the same temperature.

If this temperature is 0° C., the vapour-pressures of ice and water

206

HEAT.

are each equal to 4*6 mm., and each will be in equilibrium with the vapour,
neither growing nor diminishing. If the temperature is - 1° 0., the
pressure of water vapour exceeds that of ice vapour by nearly •£, mm.
Hence, when the water vapour has saturated the space, the pressure in
A will exceed the maximum for the ice, and the ice will grow. Distilla-
tion will go on into the bulb A till all the water has gone from B.

If the temperature of the bath rises ever so little above 0°, the
vapour-pressure of ice exceeds that of water, and the vapour from the
ice tends to supersaturate the space above the water, and condensation
goes on in B. But the condensation is not now confined to B ; it occurs
on the surface of the ice, and melting takes place there.

Suppose now that we have a mixture of ice and water at 0° ; imagine
each block of ice to be separated from the surrounding water by an
indefinitely thin vacuous layer ; evaporation will take place into this till
the common vapour-pressure is reached, and then a steady state of equal
interchanges is arrived at. If the temperature falls below 0°, the ice

. 120,

receives more than it gives up ; if the temperature rises above O8, the ice
gives up more than it receives.

Now let us do away with this vacuous space, and replace the evapora-
tion by the " mobility " of the molecules, using this term to describe the
tendency of the molecules to escape from their position — in this case
from the surface. This mobility is to some extent proved to exist by the
evaporation. Then, at 0° the mobilities are equal. Above 0° the
mobility of the ice molecules is the greater. Below 0° the mobility of the
water molecules is greater. We may remark that probably the passage of
molecules across the bounding surface of ice and water is much greater
than the passage across the surface of ice or of water into a space
only containing gas. For in the one case, we have the pull of the
solid or liquid molecules in front assisting the escape, while in the other
we have only the feeble pull of the gas in the space.

There is one fact which remains to be explained on this theory, that
is, the melting of ice at 0°. No attempt to raise ice above 0° has as
yet succeeded — melting always takes place at the surface.

The effect of pressure in lowering the melting point is explained by the
greater mobility due to the pressure. We know that in general pressure
increases mobility. Thus, in viscous solids there is a transfer from
points of greater to points of less pressure, or there is more mobility
at points of greater pressure ; and we know, too, that the vapour-pressure

-2°

CHANGE OF STATE— SOLID— LIQUID. 207

of a liquid or of a solid is slightly greater at greater pressures (see
chap, xix), which again implies that pressure increases the mobility. It
can be shown that the effect on the vapour-pressure of a given increase
of pressure is greater for ice than for water. Hence, we may suppose
the mobility of ice increased more than that of water by a given pressure,
and by putting a sufficient pressure on to a mixture of ice arid water
below 0°, the mobilities can again be made equal.

If we seek to represent the behaviour of ice and water on an indicator
diagram, we get curves whose general course is shown in Fig. 121.
There is no attempt here to draw to scale, the diagram merely represent-
ing the characteristic features. It is probable that the horizontal
portions would diminish gradually after a time as the expansion of water
for fall of temperature increases, and ultimately there may be a critical
point at which the volumes of ice and water become the same. Below
this temperature there would,
perhaps, be a gradual passage
from ice to water.*

In mixtures, softening
may take place gradually
with rise of temperature
through the melting of one Waler

constituent, which then, per- 260- •
haps, dissolves the others.
If this occurs on reversing '30 • •

othe process, supersaturation \\\ .

i r e • ,*, - \ • • Mixture

and superrusion may come in

to lower the solidifying tem-
perature. This is observed
with fats. If the melting is
a true continuous change
from solid to liquid beyond Fl(J' 121.-Ice and Water laothermals.

the critical point, then, on

reversing the order of change of temperature, the change of state should
be exactly reversed. Perhaps this kind of melting may be found to
occur in metals which soften as the temperature rises.

Resemblance of Solution to Fusion. — In several respects solution
and precipitation from solution resemble fusion and solidification. If a
salt is placed in excess in a solvent, a state of equilibrium is reached,
when the liquid is said to be saturated. If the mixture of salt and
solution is subjected to pressure, more salt is dissolved if the solution
occupies a volume less than that previously occupied by the constituents.
If the volume of the solution is greater than that of the constituents, the
pressure is accompanied by deposition from solution.

Again, in most cases, rise of temperature increases the amount of
salt taken into solution. If the warm saturated solution is pcured off
the salt into a clean vessel, it is easy to cool the liquid without de-
position of the salt, though the amount dissolved is now greater than
that corresponding to the saturated condition. The liquid is then said to

* Tammann (Ann. der Physik, 1900, II. 1, 424) finds that at -22° and under 2130
atmos. ice tends to pass into two other crystalline forms. Possibly ordinary ice
could not be subjected to much greater pressure than 2130 atmos.

208 HEAT.

be supersaturated. On adding to the liquid a small quantity of the salt,
deposition at once occurs.

This may be illustrated by a solution of sodium sulphate in water.
At 33°, water dissolves its maximum amount of this salt. Filtering the
saturated solution into a clean vessel at this temperature, the liquid may
be cooled down without crystallisation. But on the inti'oduction of a
crystal of sodium sulphate, the excess of salt at once crystallises out,
forming a mass of crystals throughout the liquid.

Solution resembles fusion also in requiring, in general, a supply of
heat to effect the change of state, though in some cases this " heat of
solution " is negative.

From these general resemblances, it appears probable that solution is
an exchange phenomenon, a saturated solution in contact with its salt
taking up and depositing equal quantities of the salt.

We shall return to the subject of solution in chap, xix., where we
shall discuss it by the aid of Thermodynamics.

Evaporation from Solids. — Many solids evaporate sensibly.
Camphor, for example, gives out a characteristic odour through evapora-
tion at ordinary temperatures, the solid turning at once into gas without
passing through the liquid stage. Ice also evaporates slowly, as may be
shown by the slow disappearance of snow and ice in dry winds too cold
to allow of melting.

There is a maximum vapour-pressure for solids, just as for liquids,
definite for each temperature. Hence, we may have a process corre-
sponding to distillation. If a piece of camphor is heated in a test-tube,
the upper part of the tube being kept cool, the vapour-pressure at the
high temperature of the lower part of the tube is above the maximum
for the temperature in the upper part of the tube, and some of the
vapour rising there is condensed to the solid form on the side of the
tube.

This process is termed " Sublimation." It is used extensively in the
preparation of sulphur, the product of sublimation being known as
flowers of sulphur.

If a solid is supplied with heat in such a manner that evaporation can
take place freely, it appears that there is a definite subliming-point, at
which there is an arrest of temperature corresponding to the boiling-point,
the temperature remaining at the point at which the pressure of the
vapour equals the external pressure (Ramsay and Young, Phil. Trans.,
Parti., 1884, p. 37).

CHAPTER XIII.
WATER IN THE ATMOSPHERE.

Hygrometry — Relative Humidity — Dew-Point and its Determination — Regnault's
Researches on the Density of Water- Vapour — Cloud — Convective Equi-
librium— Halos and Parhelia — Coronas — Rate of Fall of Cloud Drops — Hail
— Fog — Dew.

Hygrometry, or the Measurement of the Amount of Water-Vapour
present in the Atmosphere. — The atmosphere always consists in part of
water- vapour, which is present in varying quantity, the mass in any
given mass of air depending on its preceding history. If it has been
moving over the sea or over damp ground, meanwhile being at a high
temperature, it will contain a large quantity. If it has lately been cooled
down, so that much of its vapour has been condensed, and has fallen
as snow or rain, and if it has since been moving in dry regions or been
maintained at a low temperature, it will contain a small quantity.

Relative Humidity. — It is of great importance in meteorology to
measure the quantity present at any given time, for should this approach
the quantity required to saturate the air, a slight lowering of temperature
may result in condensation. It is usual to determine the amount present
in terms of the pressure which it exerts, and the ratio which this bears
to the maximum vapour-pressure at the temperature is frequently termed
the "relative humidity."

Dew-Point and its Determination.— The simplest mode of

determining the pressure of the water-vapour present in the air is by
means of some form of "dew-point" apparatus in which a surface is
gradually cooled down until dew is deposited on it. At the temperature
at which deposition begins, the " dew-point," the air is just saturated,
and from the tables of the pressure of water-vapour, the pressure for the
dew-point may be obtained. Now in merely cooling the air the pressure
of the water-vapour in it does not change, since both dry air and vapour
contract equally. Hence the pressure of the vapour in the air before
cooling is the maximum pressure at the dew-point.

A very good and easily used form of dew-point apparatus was devised
by Regnault. It consists essentially of a tube AB (Fig. 122) like a test-
tube, the lower part AB being of very thin highly polished silver. The
upper end is closed by a cork through which passes a thermometer T and
a narrow glass tube G reaching nearly to the bottom of the silver tube.
A side tube DM is connected through the stand MN and by the tube
NO with an aspirator. Some ether is placed in the tube up to PQ, and
when the aspirator is set working it draws air down through the tube G
and up through the ether. The free evaporation into the air-bubbles
lowevs the temperature of the ether and of the silver surface, and when
the dew-point is passed the bright surface clouds over. The instant this

210

HEAT.

clouding is seen the thermometer is read and the aspirator is closed.

But the observed temperature will be a little too low, as the very first

deposition of dew will be too small
to be visible. Communication
with the aspirator being cut off,
the ether gradually rises in tem-
perature, and when the dew-point
is exceeded the dew will evaporate
again. When the surface is once
more clear the thermometer is
again read, but the temperature
now observed will be slightly above
the dew-point, as the evaporation
will not be instantaneous. After
a few trials the two tempera-
tures, one above and the other
below the true dew-point, are
made to close in upon it until the
mean of the two can only differ
from it by a very small quantity.
The tube A'B', similar to AB, is
not connected with the apparatus.
It only contains a thermometer T',
which gives the temperature of
the air, while the silver surface
always remains bright and serves
as a standard with which to com-
pare the surface on which the dew
is deposited.

The form of the Kegnault
hygrometer has been modified in

FlG. 122. — Regnault's Dew-point
Hygrometer.

various ways so as to make the deposition of dew on the silver surface as
evident as possible, and with practice an observer determines the dew-
point speedily and accurately.

Dines's Hygrometer. — A somewhat simple instrument is that devised

FlO. 123.— Dines's Hygrometer.

by Dines. V is a vessel containing water in which ice is placed to cool
it. T is a tap which allows the cold water to trickle out of the vessel

WATER IN THE ATMOSPHERE.

211

into a chamber C of which the top is closed by a black glass or silver
plate P. Immediately under the plate is the bulb of a thermometer th
round which the water flows. The water trickles out at O. As it passes
the plate P it cools the bulb and plate approximately at the same rate,
and at the dew-point P is clouded over. The tap T is then turned off
till the dew disappears. The deposition and disappearance are observed
several times, just as with the Regnault hygrometer,
until the two temperatures are sufficiently near together,
when the mean is taken as the dew-point.

Wet and Dry Bulb Hygrometer.— This consists of

two vertical thermometers arranged side by side on a
frame with the bulbs projecting below the frame (Fig.
124). One bulb d is freely exposed to the air, while the
other iv is covered with muslin or wick which dips down
into a cistern of water just below the bulb. The water
rising up the threads keeps the bulb always moist. The
hygrometer is hung up so that the air can always flow
past it. If the air is saturated it takes up no water from
the wet bulb, and the wet a,nd dry bulbs show the same
temperature. If the air is not saturated then evapora-
tion goes on from w, and the latent heat of evaporation
is abstracted from the air, which cools and so keeps the
temperature of the wet bulb below that of the dry bulb.
We can see, as follows, without attempting an exact
investigation, how the instrument will work. Let t be
the temperature of the dry bulb thermometer, i.e. the
temperature of the air arriving, let t' be the temperature
of the wet bulb, i.e. the temperature to which the air
is cooled by the latent heat taken up. Let F be the
pressure of the water- vapour present in the air, / the
saturation pressure at the temperature of the wet
bulb. If the air passing the wet bulb becomes saturated, FlGfi 124.— Wet
the pressure of the vapour is raised from F to / and and Dry Bulb
the latent heat taken is proportional to this rise, or is Hygrometer,
equal say to A (/-F). But the air is cooled from
t to t' giving up heat proportional to the fall or equal say to B (t — t')
Equating the two quantities of heat

or

F = f-^(t-

when the wet bulb is

Apjohn gave for — - the value

A u\j

above 32°F. and the value -01042^-^ when it is below 32°F.,p being the

barometric height in inches and F and/ being in inches. Various
modifications of the formula have been proposed with the aim of making
it more in accordance with facts. But at the best the instrument is not
very exact, its indications varying with its situation and its conse-
quent exposure to wind, and it is probably best to use a simple formula,

212 HEAT.

and be content with an approximation to the truth. In Hazen's Tables *
the formula given is

using a constant barometer reading of 29 '4 inches. A variation of two
or three inches in the barometer does not seriously affect the result.

If the pressures are in millimetres and the temperatures are centi-
grade the formula may be replaced by

F=/- -00068(^-0750

As an example of the first formula suppose £ = 65°F. and £' = 50°F.f
so that t-t' = 15. At 50°F. /= '3598 in., whence

F = -3598 - -165 = -1948 inch.

which is the vapour- pressure at 34°, and this is the dew-point.

The relative humidity is obtained by dividing the actual pressure
•1948 by the vapour-pressure at 65° which is -6163. This gives the
f&lue '31.

To save time and arithmetic, tables are constructed in which the dew-
point and relative humidity are given for every depression read to half a
degree below the temperature of the dry bulb. Thus opposite 65° in the
table 34° and -31 are entered under t-t' '= 15.

The Chemical Method. — In this method a measured quantity of
air is drawn through drying tubes which take from it all the water
vapour, and weighing the tubes before and after the passage of the air
through them, the gain in weight gives the amount of vapour which was
present in the air. For details of the method we refer the reader to
Glazebrook and Shaw's Practical Physics, p. 233.

In Fig. 125 A is the aspirator from which the water sipnons out;
CO are the drying tubes ; B an intercepting drying bottle to prevent
any vapour passing back from the aspirator to CO. The volume of
water drawn through from the beginning to the time when the aspirator
is empty is measured. Suppose it to be V. Let the mean temperature
of the air entering at a be t and let the final temperature of the air in
the aspirator be t'. Let the barometric height be H. Then we have a
volume V of air at if saturated with vapour at t'. If the saturation
pressure at t' is P' the pressure of the dry air is H — P' and the weight
of the dry air drawn through the tubes

W_.VH-FX -001293
760 l+at'

If the gain in the weight of the tubes is w, the weights of water-
vapour and dry air present in any volume of air are in the ratio w : W.

Since, according to the results given below, a given volume of water-
vapour weighs 0'622 as much as an equal volume of dry air at the same
temperature and pressure, the pressures exercised by these are in the

an

ratio — --- : W, their sum making up the barometric pressure H.

"D £tit

If then P is the actual pressure of water-vapour

* Handbook of Meteorological Tables (Washington).

WATER IN THE ATMOSPHERE.

213

or

w
•"622

II

w
:622

•001293 x -622
l + at'

This method has the disadvantage that it takes a long time to
execute.

Regnault's Researches on the Density of Water- Vapour.—

Regnault investigated the density of water-vapour at pressures below
the maximum, both at ordinary temperatures, and near the normal
boiling-point. He also determined the density of the vapour present in
saturated air at ordinary temperatures, expressing the result in all cases
as the ratio of density of water-vapour to that of dry air at the same

FIG. 125. — Chemical Method for Determining Dew-Point.

temperature and pressure, in order to test the applicability of the laws
of Boyle and Gay-Lussac.

If we calculate the density of water-vapour from the densities of its
constituents, assuming that two volumes of hydrogen combine with one
of oxygen to form two of water-vapour at the same temperature and
pressure, the density compared with that of air is

•0896x2 + 1-43

2x1-293

= -622

Regnault's results, as will be seen below, show that under the various
circumstances of the experiments, the density is very nearly this, or we
may, to a close approximation, apply the ordinary rules connecting
temperature, pressure and volume.

At Ordinary Temperatures. — Regnault introduced a small glass
bulb containing a known weight of water into a large glass globe, which
was then exhausted, the small quantity of residual air being carefully
dried. A manometer attached to the globe gave the pressure within it.
The bulb was then broken and the temperature of the globe was raised
to a point above that at which all the water evaporated. The vapour-
pressure was, therefore, below its maximum. The pressure indicated

214 HEAT.

by the manometer was then observed, and subtracting that due to the
residual air, the difference gave that of the water-vapour.

He found that if the pressure did not exceed '8 of the vapour-
pressure for the temperature of the experiment, the relative density
of the water- vapour was about '621.

When near saturation the density appeared much greater, but this
must have been due to some error in the method, inasmuch as the
density of saturated vapour in air agrees very nearly with the value '621.
Probably the vapour condensed on the glass before saturation.

At Temperatures near the Boiling Point, and at Pressures

far below 760 mm. — A large globe containing some water was kept
for a long time in steam until all the air was expelled, its place being
taken by the water-vapour. Then the globe was connected with a con-
denser outside, some of the vapour being thus withdrawn. All the
water in the globe having evaporated, the vapour-pressure was reduced
below 760, its value being determined by a manometer. The globe was
then closed, allowed to cool, and weighed. It was once more heated in
steam and some more of the vapour withdrawn by external condensation,
the pressure being still further diminished. The new value was observed
and the globe was then closed. After cooling it was weighed again.
Assuming that the laws of Boyle and Gay-Lussac could be applied, the
loss in weight would be the weight of the vapour at the steam tem-
perature filling the globe at a pressure equal to the difference in the
pressures before the two weighings. Of course allowance was made for
any variations in the steam temperature in the two parts of the
experiment.

As long as the pressure did not approach 760 mm. the results agreed
in all cases with a relative density of about -620, but near 760 mm. the
density became decidedly greater. Calculations founded on thermo-
dynamic principles also show that the relative density of saturated steam
is decidedly above *623.

Using the method of Dumas, Cahours has shown that at the at-
mospheric pressure the relative density falls again to about the normal
value when the temperature is raised considerably above 100°.

At Ordinary Temperatures in Saturated Air.— Air was drawn
by an aspirator through a balloon filled with wet sponge into a box in
which hung wet linen cloths, so that it was thoroughly saturated at the
temperature of the box. It was then drawn on through drying tubes,
where all the water-vapour was left behind, into the aspirator, where
it became thoroughly saturated once more. Let us suppose the
volume of the aspirator to be Y, and for simplicity let its temperature
be taken as equal to that of the box, both being t° ; let the gain in
weight of the drying tubes while the aspirator is once filled with air,
be w. Let P be the maximum pressure of water-vapour at t", and let
us suppose that the relative density of the water-vapour is '622, then

.
760 1 + at

The observed values very nearly agreed with this calculated value
though in all cases they were slightly lower — in general by about 1 per
cent. — a difference which may be due either to a real difference in

WATER IN THE ATMOSPHERE.

215

density or some error in experiment. Still the agreement is sufficiently
close to justify the statement that the relative density of the water-
vapour present in the atmosphere is practically the same as if the
vapour alone were present at the same pressure.

Cloud. — When a mass of air is cooled below the saturation point —
the dew-point — the excess of water- vapour is deposited as cloud, which
consists of minute drops of water forming in general on " dust-particles "
as nuclei (see p. 168). The cooling may sometimes result from a
mixture of warm and cold air, and it is quite possible that a mixture
of two quantities of air at different temperatures, neither saturated,
may form a supersaturated mass. This may easily be seen from
Fig. 126. Let PRQ represent the vapour-pressure curve, PM, QN being
the saturation pres-
sures at ^ and t2.
Let equal masses at ^
and t2 contain vapour
at pressures AM and
BN respectively.
When mixed, the
temperature will be

1 2, and the vapour
a

will be sufficient to
produce pressure OL.
But this is in excess
of the saturation
pressure RL, and
OR represents the
amount which can be
deposited as cloud.
Clouds appear to be
formed in this way
on changes of the
wind. When a south-
west wind comes on in

the upper regions over a north-east wind in the lower regions of the air,
clouds are formed where the two streams mingle.

But a much more obvious mode of cloud formation is that in which a
large mass of air expands and, in doing work, cools below its saturation
point, depositing the excess of vapour as cloud. Such formation may be
often watched on mountain sides and summits. A stream of nearly
saturated air may be blowing against a mountain side, and being
deflected upwards it comes into regions of less pressure, where it expands
and cools, and the hitherto transparent vapour is converted into visible
cloud. Sometimes beyond the mountain top the air descends again, and
the clouds trailing off are gradually dissolved in the air which is now
becoming warmer.

Expansion may also result from ascent of masses of air due to
destruction of vertical equilibrium of the atmosphere. The limiting
condition of equilibrium of the atmosphere, if it is uniform in composi-
tion, is that known as a condition of convective equilibrium.

• lemperaturt

FlG. 126. — Mixture of Equal Masses of Unsaturated
Air at Different Temperatures.

216 HEAT.

Convective Equilibrium. — In this condition the temperature falls
as we rise at such a rate that any mass carried upwards will cool by
expansion so that it will always be at the same temperature, and of the
same density, as its new surroundings. This means that the fall of
temperature with decreasing pressure is that given by the adiabatic
relation, and near the earth's surface it is easily shown that the rate is
about 1° F. for each rise of 183 feet, or 1° C. for about 100 metres. If
the lower layers of air are cooled so that the rate of fall of temperature
upwards is less than this, or if the temperature rises as we ascend, the
equilibrium is stable. But if the lower layers are heated so that the fall
is more rapid, equilibrium is destroyed and circulation occurs. The lower
air rises, and the upper air falls to take its place. This circulation
frequently occurs on a hot summer day, when the surface of the earth,
heated by the sun, in turn heats the air in contact with it, and there are
small streams of ascending and descending air. These streams are
rendered evident by the quivering of objects seen through the surface
layers, the light being refracted by passage through a medium of density
varying from point to point. The quivering is often seen with the naked
eye, but is still more evident when a distant outline is looked at through
a telescope. Some of the haziness of outline is no doubt due to this
quivering.

If a large mass becomes simultaneously heated, it may rise as a whole
and be thrust up through the upper layers. In rising it expands and cools,
and if it rises far enough and cools down enough, it may become super-
saturated at a certain height and begin to form cloud. Often such
formation may be watched in progress, a big piled-up mass of " cumulus
cloud " somewhat as in Fig. 127 being formed, the outline of the cumulus
marking the upper boundary of the rising column, the horizontal base
marking the cloud level or the point at which the air is cooled to
saturation point. When thunderstorms are gathering in hot weather,
the formation may be seen on a gigantic scale, the columns pushing far
up into the higher regions. The " strato-cumulus " clouds, which are
those with rather irregular globular or fleecy outlines, seen, say, in a
south-west wind with a broken sky, were probably formed originally as
cumulus clouds at the top of ascending columns, but they have drifted
far from their birthplace, as their form shows. The upper part is always
in advance of the lower, the upper wind travelling faster than the lower.

The so-called " Ripple Clouds," almost like ripple marks in sand, have
been explained by Helmholtz &s due to alternate expansion and contrac-
tion of the air. When one stream of air moves over another, there will
be undulations in the surface of separation, and a mass of air at this
surface will alternately rise and fall as it travels on. If it is just at the
saturation point, when it rises to the top of an undulation it expands,
cools, and deposits cloud, which it dissolves again in falling into the
succeeding trough, where it contracts and heats. The parallel lines of
cloud then mark the summits of the undulations, and the air may be
travelling on at quite a different rate from the clouds.

HaloS and Parhelia. — If the temperature of the air is below the
freezing point, any excess of water-vapour is deposited as ice in the form
of minute crystals. The presence of these crystals, when forming only
a thin veil of cloud, is shown by halos round the sun and moon and by

WATER IN THE ATMOSPHERE. 217

parhelia to the right and left of the sun when it is near the horizon.
The halo at 22° from the sun or moon is the most frequent and con-
spicuous effect, and has value as a weather sign. Very often after clear
weather, the arrival of a cyclone and rain is preceded by a damp wind in
the upper air, so cold that ice crystals are formed in it. These are in
the form of regular hexagonal prisms with angles of 120°. Alternate
faces therefore make angles of 60° with each other, and through prisms
of 60° the minimum deviation for ice with refractive index 1*31 is 22°.
At this minimum deviation there is a concentration of the rays, and so
the rays from the sun striking a crowd of such 60° prisms will be some-
what concentrated in a direction making 22° with the original direction.
When an observer looks through the crowd towards the sun and then
outwards from it, he will receive no light refracted through the GO"
prisms till he has reached a distance of 22°, when there will be a concen-
tration of refracted light. Of course the radius of the bright halo will
differ slightly for different colours, but partly through the appreciable
diameter of the sun and the consequent overlapping of halos due to
different parts of his disc, and partly
due to the scattering of other light in
all directions, the colours are hardly
evident, the inner edge of the halo at
the best only showing a red tinge.

The ends of the ice prisms are
planes perpendicular to the faces,
and so form a second series of prisms
with angle 90° and minimum deviation
46°. These form a second halo, which,
though rarely observed in our latitude, FIG. 127.— Cumulus on Horizontal
is sometimes seen farther north. Base of Cloud.

When the sun is near the horizon,

and the air is in such a condition that a large proportion of the hexagonal
prisms have their axes in a vertical position, the refraction is much
greater in the horizontal direction through the sun to right and left, and
sometimes quite brilliant patches of light are seen with well-developed
colour, at 22° on each side of the sun just above the horizon. If at the
same time a large proportion of the prisms have their axes horizontal,
a patch is also seen above the sun at the same distance, though this is
much rarer. These patches are termed parhelia. Sometimes the parhelia
lie on a circular halo. The phenomena are more frequent and much
more marked in northern latitudes. Other circles, not all yet explained,
have been occasionally observed passing through the parhelia.*

Coronas. — The coronas of colour seen within a few degrees of the
moon when shining through the edge of fleecy clouds, and round the sun
(seen best when reflected in a pool so that the excessive light is dimin-
ished), are due to diffraction by equal sized water drops. The explanation
is given in the volume on Light.

Rate Of Fall Of Cloud Drops. — The drops of water in a cloud are
subject, of course, to gravity, and they therefore begin to fall directly they
are formed. But the viscosity of the air through which they pass
introduces a resisting force which increases with their velocity, and is

* Tait, Light, p. 132.

218 HEAT.

very soon equal to their weight, and then the velocity of fall is uniform.
Stokes (Carrib. Phil. Trans., vol. ix. p. 94) showed that this limiting
velocity is proportional to the square of the radius of the drop, and his
formula, with the now accepted coefficient of viscosity of air, gives the
velocity for drops '001 inch radius as about 0-8 inch per second. From
the observed radii of coronas round the sun and moon, we know that
the drops in clouds are often much less than this, so that the velocity
of fall is very much less than 1 inch per second.

The change of the cloud-drops and crystals into rain-drops and snow-
flakes has not been made out in full detail. It appears probable that
rain-drops are formed by coalescence of cloud-drops. Defant (Sitz d.
Wiener Akad., May 1905 ; Sci. Abst., 1905, p. 634) found drops of various
sizes in the same rainfall, but those most frequent had weights in the ratios
1:2:4:8, apparently formed by the coalescence of drops of equal size.

In thunderstorms the clouds are usually of very great depth. A
drop therefore tends to grow, perhaps by coalescence, through a very
considerable range of fall, and this may account for the largeness of the
drops as they reach the ground.

Hail. — The genesis of hail is much more complicated than that of
the ordinary raindrop. The fall of hail accompanies very great atmos-
pheric disturbances, in which there is ascent of large masses of damp air
to very great heights. The hailstones usually show a structure of several
coatings of ice and snow round a core of snow, and it is supposed that
they may begin as snow in the upper regions, then descend some distance
where the air is above the freezing-point. Here they condense on them-
selves a coat of ice. Then they may be carried up again into the snow
region and acquire a coat of snow. Descending again they will receive
another coat of ice, and the number of coatings is supposed to show the
number of alternations.*

Fog is cloud formed on the surface of the earth. The condition for
its formation appears to be that the upper air is warmer than the surface
layer and nearly saturated. The lower layers cooling below the satura-
tion point, the excess is deposited as fog. As the number of dust
particles, especially in town air, is very great near the ground, the drops
may be very minute and very numerous, and the fog may be much more
opaque than ordinary cloud. When the upper air is warmer than the
lower air, the smoke rising from chimneys does not tend to rise so high,
as its excess of temperature above its surroundings is less. Hence in
a town fog the smoke hangs near the ground and mixes with the fog
instead of rising away from the surface as when the upper air is colder.
In confirmation of this account of fog, it may be noticed that often a fog
comes on just before the break up of a frost. A warm damp wind has
already come on overhead, and the vapour diffusing from this into the
still, cold surface layers, supersaturates them and produces deposition of
the excess of vapour as fog.

Dew. — The formation of dew was first successfully explained by
Wells in a celebrated " Essay on Dew," in which he pointed out that
when the surface of the earth radiates out its heat at night so far as to

* An account of the atmosphere and of the mode of determining the condition
of the upper regions by means of kites is given in Rotch's Sounding the Ocean
of Air.

WATER IN THE ATMOSPHERE 219

cool the surface layers of air below the saturation, or dew-point as it is
now termed, the excess of water vapour is deposited as dew, and he
made out the chief conditions for a plentiful deposit. When the sky ia
clear the radiation is greater, and when the surface layers are at rest
they are more rapidly cooled below the dew-point. Hence the conditions
favourable are, air with considerable relative humidity, at rest and under
a clear sky to allow free radiation. The nature of the ground has, of
course, some effect. If it is a good radiator and a bad conductor, its
surface will become the more rapidly chilled.

Aitken (Proc. R.S. Edin., 1886) has supplemented Wells' theory of
dew by showing that there are really two kinds, the dew which comes
from the air and the dew which is the extruded water or the condensed
vapour exhaling from the surface. The surfaces of leaves are always
tending to give off vapour, and if the air is not saturated it takes this
vapour up. But if the air is saturated, then the exhalation forms drops
of dew on the surface.

CHAPTER XIV.
GENERAL ACCOUNT OF RADIATION.

Radiant Energj — Radiometers — Radiant Energy and Light resemble each other —
Light is Radiant Energy to which the eye is sensitive — Radiant Energy has a
much greater range of Wave-Length than Light — Radiometers only measure
Energy Slreams and do not indicate Quality — Comparison of Emissive Powers —
Radiation of different Wave Lengths — Comparison of Absorptive Powers —
Comparison of Reflecting Powers — Diffusion — General results — Radiating and
Absorbing Powers vary together — Illustrations — Transparency and Opacity-
Radiation and Absorption by Gases and Vapours.

Radiant Energy. — When heat is received from a hot fire through
an intervening space of air, the air is not permanently affected by the
passage of the energy through it ; there is no continuous downward slope
of temperature from the fire to the receiver, and frequently, as on a cold
winter's day, the air may be far colder than the receiver. The energy of
the hot body spreads out on all sides from the body as a centre if unim-
peded by obstacles, and it is therefore said to be radiated. The process
is quite distinct from conduction, a good conductor like copper being as
efficient an obstacle to the propagation of the energy as a bad conductor
like wood, while air (as we have seen, an exceedingly poor conductor),
allows the energy to pass freely. Before leaving the radiating body the
energy radiated is evident as heat, and on reaching the receiver it is
again evident as heat ; but since in transit it does not make itself
evident by warming the medium, we must ascribe to it a form distinct
from heat which we term radiant energy.

In this chapter we shall investigate the nature of radiant energy,
and discuss its emission from sources and its absorption and reflection at
surfaces upon which it falls.

Radiometers. — It is necessary, of course, to have some instrument
to receive, and indicate the rate of reception of, the energy arriving at a
given point. We may term such an instrument a radiometer, though
that name is frequently applied to one particular instrument, Crookes's
Radiometer, described on p. 150.

When the rate at which the energy is arriving is large, we can
detect it directly by the temperature sense, that sense which tells us
when the skin is warmed or cooled. But obviously the temperature
sense can only give us qualitative indications, and is no more suited for
radiation measurements than for measurements of temperature.

Differential Thermometer. — Leslie, a pioneer in research on
radiation, used the differential thermometer, which consists of a U tube
with a bulb at each end of the U. A small quantity of coloured liquid
is introduced before the bulbs are sealed off, and the air in the two bulbs

GENERAL ACCOUNT OF RADIATION.

221

Hot

is so adjusted that the liquid extends round the bend of the U and stands
at the same level on each side when the bulbs are at the same tempera-
ture. One of the bulbs is exposed to the radia-
tion to be measured, while the other is protected
from it, and the liquid is depressed by the
expansion of the air in the exposed bulb.

Thermopile. — Melloni, another pioneer,
used a thermopile, which is much more sensitive
than the differential thermometer. It consists
of a number of bars of bismuth and antimony
connected at the ends as represented in Fig. 128,
the bars being insulated from each other along
their sides. The beginning of the first bar and
the end of the last bar are connected to binding
screws, which in turn are connected to a galvano-
meter. If one set of alternate junctions is heated
by exposure to radiation, a current flows from
bismuth to antimony across the hot junctions,
proportional to the difference of temperature of
the hot and cold junctions, and the current is
indicated by the galvanometer.

Thermopiles have been made containing
many hundred pairs of bars, the E.M.F. being
proportional to the number of junctions. But
the resistance increases with the number of

bars and with a properly designed low -resistance galvanometer very
little is gained by this multiplication of bars.

Bismuth \\ Antimony |
FIG; 128.— Thermopile.

Radio-Micrometer. — This point is well

brought out in the radio-micrometer, an
instrument first invented by D'Arsonval
and subsequently reinvented and made into
an instrument of extraordinary delicacy by
Boys.* It may be regarded as a combina-
tion of a thermopile of two bars and a
galvanometer, all in one. Fig. 129 shows
the principle of the instrument. The
lower ends of two short bars, respectively
of bismuth and antimony, are soldered to a
small copper disc Cu in the figure. To the
upper ends are soldered the ends of a copper
wire, completing the circuit and making it a
very narrow rectangle. This circuit depends
from a glass fibre ff, on which is a mirror m,
and this in turn from a quartz fibre qq.
The rectangle hangs between the two poles
of a strong permanent magnet. The whole
is enclosed in a protecting case of wood,
but there is a window for a beam of light to

fall on m, and a tube is directed on to the disc Cu. If radiation passing

through this tube warms the disc, it in turn warms the junctions with the

* Phil. Trams., 1889, A., p. 158.

S6

FIG. 129. — Kadio-micrometer.

222

HEAT.

bismuth and antimony bars, and the same E.M.F. is produced as if the
bars were directly joined and raised to the same temperature. A current
flows counter-clockwise round the circuit as represented, and it tends to
turn at right angles to the plane of the paper, and the torsion of the fibre,
being the only resistance to turning, the instrument may be made ex-
ceedingly sensitive.

Bolometer. — In 1881 Langley * described an instrument which he
named the bolometer, which is probably of the same order of sensitive-
ness as the radio-micrometer, but has the advantage of portability and
capacity for receiving radiation in any direction. The bolometer is a
Wheatstone Bridge and may be represented diagrammatically by Fig.
130. AB is a thin flat strip of metal — in the original instrument a strip

of iron 7 mm. long, -177 mm. broad,
and '004 mm. thick. It had a re-
sistance of 0'9 ohm. CD is a wire oi'
equal resistance. EF and GH are the
ratio arms, here of course nearly equal,
and capable of adjustment to give a
balance in the galvanometer. The
bridge is enclosed in a protecting case,
but opposite AB is a shutter which
can be opened to admit radiation.
When AB receives radiation which
warms it, its resistance rises and the
balance in the bridge is destroyed.
The current through the galvanometer
is proportional to the rise in tem-
perature of AB. The instrument is
variously modified according to the pur-
pose for which it is used, but it still re-
tains its simplicity of principle. By its
aid Langley has greatly extended our
knowledge of radiation, and it is gener-
ally used by other workers, having
largely displaced the thermopile in investigations on radiant energy.

Radiant Energy and Light resemble each Other.— Quite com-
mon experience tells us that radiant energy has many properties in
common with light, with which it is so frequently associated, as with
sunlight, firelight, and so on. It is propagated in straight lines, and
forms shadows — as we know when we cross from the shady side of a
street on a winter's day, or when we use a fire-screen to keep off the heat
as well as the light of a fire. It is also reflected and refracted with
light, as we know by the charring of a piece of paper or wood placed in
the focus of the sun's light formed by either a mirror or a " burning "
lens. We may also reflect the dark radiation from a heated ball by a
mirror, just as we reflect light. It further resembles light in travelling
with great rapidity, for the obstruction of the light of the sun by the
most distant cloud is accompanied by a diminution of the warmth
received from him ; and in eclipses by the moon there is a marked fall of
temperature as well as a loss of light.

* Nature, vol. xxv., 1881, p. 14. For the construction of bolometers we may refer to
Kurlbaum, Wied. Ann., xlvi. p. 204, 1892; or Langley, Am. Journ. Sci., v. p. 241, 1898.

FIG. 130.— Bolometer.

GENERAL ACCOUNT OF RADIATION.

223

Then again, radiant energy and light have the same laws of propaga-
tion. The amount radiated varies with the slope of surface, just as in
the case of light, and the quantity of radiant energy falling on a given
area, from a given source in a given time, varies inversely as the square
of the distance — the law of light-propagation.

This may be proved very simply by placing a thermopile in front of
a blackened tin vessel full of hot water, and between the two a screen
with a hole in it (Fig. 131). So long as the hot vessel entirely fills up the
field of view of the thermopile through the hole in the screen, the
deflection of the galvanometer remains the same, whatever the distance
between the screen and the hot vessel, provided that the distance of the
thermopile from the hole in the screen remains unchanged. That this
proves the law may be seen from a simple case. Let the hot vessel be
placed successively at two distances 1 and 2 respectively from the pile,
the screen remaining fixed. Then the areas seen from the pile, and
from which it is receiving radiation, are as 1:4. But the total energy
received is the same in each
case. Then the energy received
from the single area at the
double distance is J of that
received from it at the single
distance. Further, we may alter
the slope of the hot vessel
without affecting the heat re-
ceived by the thermopile, just
as we may alter the slope of a
light-giving surface. Common
observations and simple experi-
ments, therefore, show us that
radiant energy resembles light
in its most evident properties.
We may supplement these

observations and experiments by various other experiments, and they
all confirm the conclusion.

Light is Radiant Energy to which the Eye is Sensitive.—

Formerly, it was thought that this resemblance was all that we could
assert and that corresponding to our two distinct sense-perceptions, one
by the retina and the other by the warmth-sense in the skin, there were
two distinct external agents travelling together. This was quite natural
before physicists had begun to think of phenomena in terms of energy
and its transformations. But now we clearly realise that light itself is
a form of energy sent out by a heated source, which, in parting with the
light, loses an equivalent of heat, and that this light energy, falling on
an absorbing surface, is transformed into heat again. In this respect
it is undistinguishable from radiant energy. We have now no
reason whatever to suppose that there is the double agency. When the
radiant energy falls on the retina — if it is of suitable quality — it gives
rise to the sensation of light. The same energy falling on an absorbing
substance, say, a blackened thermometer, will be transformed into heat.
The difference, then, between light as received by the eye and radiant
energy as studied by the thermometer, thermopile, or other radiometer is

FIG. 131.— Proof of the Law of
Inverse Squares.

224 HEAT.

not in the external disturbance, but in the effect upon which we
concentrate our attention.

Radiant Energy has a much greater Range of Wave-length

than Light. — But though we now believe that all light is radiant energy,
the converse is obviously not true, for we may receive heat by radiation
from an absolutely dark body in a dark room. To illustrate the distinc-
tion let us consider the formation of a spectrum. If a beam of sunlight,
or the light from an electric lamp falls on a slit S (Fig. 132), and passing
thence is received on a lens L, it will.be brought to a focus at a point
F, the focus conjugate to S, and an image of the slit is formed on a screen
placed at F. If a prism P be interposed at the angle of minimum devi-
ation, then the light is bent round, each colour at a different angle, and
forming its own image, so that a spectrum consisting of an infinite num-
ber of overlapping slits of different colours is formed on a screen rb,
placed at the same distance from the prism as F, the red being the least,
the violet the most refracted. In various experiments on interference
of light we obtain a similar spectrum, arranged in the same order from
red to violet. Now, whatever view we hold as to the nature of the dis-

FIG. 132. — Radiation spread out into Spectrum.

turbance constituting light, there can no longer be any doubt that it is a
wave-form of energy, this supposition affording the only conceivable
explanation of interference phenomena. And it can be shown that
difference in position in the interference-spectrum must correspond to
difference in the length of wave constituting the light. We learn thence
that red light has the longest wave, the length diminishing through the
spectrum to the violet. The total range of wave-length of the visible
spectrum appears to lie between ^ ^^ and ¥Tr A^ of an inch, or between
•000075 cm. and '00004 cm.

We may turn the light into heat by placing a thermopile at various
parts of the spectrum. At the red end it is sensibly affected, but less
and less as it moves from that end, the heating being quite insensible to
ordinary experimental arrangements in the blue. It was formerly
supposed that the visible spectrum was accompanied by a heat spectrum
especially strong in the red.

Now, were there any difference in kind between heat radiation and
light, we might expect by some process to filter out the one from the
other. But we cannot separate red radiation into two parts, one warm-
ing and the other lighting. When we diminish the light, we equally
diminish the heat, as Jamin has shown.* We conclude, then, that there
is a radiation of one kind only for each wave-length. The eye trans-
lates the radiation of the different wave-lengths into different colours ;

* Cours de Phytique : Etude des Radiation, p. 62.

GENERAL ACCOUNT OF RADIATION. 225

but when they are absorbed by lampblack, say, they all become merely
heat, of the same quality after absorption, whether absorbed red or
absorbed yellow. The absence of heating effect at the blue end is merely
due to want of sensitiveness of the heat detector, the eye being far more
easily excited than the thermopile.

But this does not tell us anything about the radiation received from
dark bodies. We may easily, however, explain the nature of this dark
radiation in answering a question which naturally arises. Does this
narrow range in the wave-length of the visible spectrum correspond to
limits existing in the external disturbance, or merely to limits in our
light-sense perception?

Suppose that we replace the eye, as the recording instrument, by a
thermopile deflecting a galvanometer-needle. Placing this in various
parts of the visible spectrum, we get, as we have seen, deflections hardly
visible in the blue and green, but rising towards the red, showing that
the light is converted into heat in the thermopile. Now, taking the
thermopile out beyond the red, it still gives indications which show
clearly that longer wave-radiations are falling on the screen there, and
these are termed "infra "-red radiations But just as a coloured glass
absorbs light of a certain wave-length in the visible spectrum, while
transmitting that of other lengths, so what we call transparent glass
absorbs some of the radiation of long wave-lengths, though transmitting
that of the visible spectrum. The effect beyond the red is, therefore, dimin-
ished by absorption in the glass lens and prism. If we use rock-salt
instead of glass for prism and lenses the absorption is much more limited,
and the effect on the thermopile is greater On moving the thermopile
to the ultra-violet, we find no sensible effect, and we must use a more
delicate test for the presence of the energy. If we paint a sheet of
paper with a solution of sulphate of quinine in dilute acid, and expose it
in the ultra-violet region, there is an action on the solution, and it gives
out visible radiation. Or, there are ultra-violet short wave-length radi-
ations, which are absorbed by sulphate of quinine, and given out again
by it as radiations of wave-length long enough to excite the retina. Or
still more marked is the effect on an interposed screen of the kind used
for the detection of X-rays, brilliant bands of green light appearing in
the region beyond the violet. These ultra-violet radiations are especially
active in exciting the chemical changes employed in ordinary photography.
That these extreme radiations are also of wave form, has been shown
from the fact that they exhibit the phenomena of interference and of
polarisation, which we can explain only by the wave theory.

We conclude then, that radiant energy is, in form, a wave dis-
turbance, the lengths of the waves extending over a wide range, but to
one particular part of the range the retina is sensitive, and we call the
energy within this range light. This is probably the part of the energy
which is present in greatest quantity in sunlight. As we see most things
by reflected sunlight, we see them more easily by being especially sensitive
to that kind of radiation which is present in greatest quantity.

Radiometers only Measure Energy Streams and do not
Indicate Quality. — The various radiometers, unlike the eye, take no
account of the wave-length of the stream of radiation which they mea-
sure. Their indications are proportional to the rate at which they are

P

226 HEAT.

absorbing energy. For when they are in a steady state they are giving
out as much as they receive. Now, with the small excess of temperature
produced by the radiation, the amount given out is proportional to the
excess, and therefore the excess is proportional to the amount received.
It is just the same whether the energy was blue light, yellow light, or
ultra-red radiation before it was absorbed and converted into heat. If the
total energy absorbed is the same the indication of the instrument is the
same. When we wish to study wave-length as well as quantity we must
first sort the radiation out by a prism or grating. In much of the earlier
work this was not attempted, and it was really by the invention of the
bolometer by Professor Langley that the detailed study of quality as well
as quantity of radiation was rendered practicable. But a great deal of
valuable knowledge was obtained by the earlier workers who merely
investigated radiation totals, and we shall give here a brief account of
their results.

Comparison of Emissive Powers. — Leslie* compared emissive
powers by an arrangement represented in Fig. 133. C was a tin cube

FIG. 133. — Leslie's Comparison of Emissive Powers.

filled with boiling water. M was a mirror of tinplate beaten into parabolic
form and about 12 inches aperture, th was the thermometer bulb placed
at the focus conjugate to C. The cube had its vertical face either plain
or coated with the substance to be tested. He obtained the following
results : —

Emissive Power of Surfaces at 100°. Lampblack = 100.

Lampblack . . . . . . . 100

Writing paper 98

Indian ink ....... 88

Bed lead . 80

Tarnished lead 45

Clean lead .19

Polished iron ....... 15

Tinplate, gold,'! 12
silver, copper /

Melloni obtained results confirming those of Leslie, but De la Provostaye
and Desains showed that through faulty arrangements both the earlier
observers overestimated the emissive power of metals. They obtained

* An Experimental Inquiry into the Nature and Propagation of Heat. 180$.

GENERAL ACCOUNT OF RADIATION. 227

the following results, the temperature of the radiating body being
120°—

Lampblack . . . . . . 100

Indian ink
Platinum
Copper
Silver

, 88

, 11

5

2 to 3

They also found that the relative powers changed with the temperature.
For this purpose, they used a platinum plate coated on one side with
lampblack, and on the other side with the substance to be tested, and
heated it to various temperatures with an electric current. The radia-
tions on the two sides were compared by means of two thermopiles, one
placed on each side of the plate. Now, had the relative emissive powers
been independent of the temperature, the indications of the two thermo-
piles would have always been in the same ratio. But the ratio was found
to change. Thus the radiation of borate of lead was equal to that of
lampblack up to 100°, but after that it fell in comparison, being only ^ at
550°. This is exactly what we should expect, since the radiation is a
mixture of different wave-lengths or " colours," and we have no reason
to expect that the rise of temperature would produce the same increase in
emission of all the different wave-lengths.

Radiation of Different Wave-Lengths.— The difference of wave-
length in invisible radiation was shown conclusively by Forbes, who
succeeded in showing that as the temperature of a source rose, the
refractive index of the most energetic radiations emitted, as determined
by a rock-salt prism, rose also.* Langley,f using his bolometer, showed
that, while the refractive index of rock-salt for light ranges from about
T58 to 1'53, he was able to obtain radiation from a Leslie cube at 178° C.,
which passed through a rock-salt prism with deviation corresponding to
a refractive index not greater than 1'45, and a wave-length probably
many times that of the D line. For the details of the work the reader
should consult the original papers, but a sketch of Langley's method may
serve to show how the quality of radiation may be investigated.

When a beam of light from a slit falls on a diffraction grating, and
the rays are brought to a focus, a central bright band is formed, and a
series of spectra is arranged on each side, the distance of a given colour
in any spectrum from the centre being proportional to its wave-length.
The blue rays are therefore the nearest, and the red furthest, from the
centre in each.

Considering the first spectrum on one side of the centre C, Fig. 134,
the various rays appear at points distant from the central bright band C
proportional to their own wave-length. If then Sj (Fig. 134) is the
position of the D line, and we erect a height Sx Dx equal to the wave-
length of the D line, 589/x/x,, and join ODa producing it onwards, the
height of this line above any point of CS4 will represent the wave-length
of the ray of the first spectrum diffracted to that point. But even in the
solar spectrum there is no radiation perceptible below about 290/x/u,.
Hence, the spectrum only begins at Ax where AXBX equals 290. But each

* For an account of Forbes's researches on radiation ; see Balfour Stewart's Heat.
t Phil, Mag., xxi. 1886, p. 394 ; xxii. 1886, p. 149.

228

HEAT.

ray is also diffracted into the other spectra ; in the second, at double the
distance from C ; in the third at treble the distance from 0, and so on.
Hence if we make CS2 = 2 CSj ; CS3 = 3 CS15 &c., the D line of the second
spectrum is at S2, of the third at S3, &c. Making S2D2 = S3D3 = SjDp and
joining CD2, CD3, &c., the heights of these lines above any point give us
the wave-lengths of the various spectra diffracted to that point. Drawing
through Bj a parallel to CjSj, we need not consider any line below that,
as the corresponding radiation is insensible.

It is evident that, except near the centre, the spectra overlap. At
the point S4, for instance, we have

A =589 of the 4th

= | x 589 of the 3rd

= 2 x 589 of the 2nd

= 4 x 589 of the 1st.

The first of these is, of course, the yellow line easily seen, the second is
in the far red, the others dark rays.

If, now, the slit of a spectroscope be placed at S4, and a rock-salt train

FIG. 134.

of lenses and prisms be used to minimise absorption, since the different
rays have different refractive indices, they are sorted out into different
positions in the field of the spectroscope. The bolometer is made to
travel along, first being placed in the yellow line, which is visible. As
it passes each of the others in succession, the effect is apparent, and
though the ultra-red rays are invisible the bolometer feels them. By
this method, Langley was able to detect radiation received from bodies
at quite low temperatures, and in positions in the spectrum correspond-
ing to very great wave-lengths.

Comparison of Absorptive Powers. — The absorptive powers of

various substances were first compared by Leslie, who covered his
thermometer bulb with a layer of lampblack, and placed it in the focus
of the mirror, noting the rise, and then compared this with the rise
when the bulb was covered with other substances. This method is not
satisfactory, as the thermometer is steady only when the absorption is
equal to the emission, and the emission as well as the absorption varies
with the substance. Melloni put plates of thin copper between the

GENERAL ACCOUNT OF RADIATION.

229

source and the thermopile, the side towards the source being covered
with the substance and that towards the pile with lampblack.*

If the quantity of heat emitted in one second by one side, per degree-rise
of temperature of the plate above the surroundings, is E for lampblack, and
Et for the substance, and if the excess of temperature is tv the emission is

If the quantity absorbed by the surface towards the source is Ax, then in
the steady state

If both surfaces are lampblacked, and the excess of temperature is t, then if
A is the quantity absorbed by the lampblacked surface towards the source,

Then

2E~

E

Now the indications of the galvanometer in circuit with the pile are

/ ~F

proportional to ^ and t, so that we know -i. Knowing also — J from the

t -El

J^

previous determinations of emissive power, we can find — -i,

A.

Just as the light absorbed by various surfaces depends on the nature
of the incident light, red paper, for instance, absorbing more of blue light
than it does of red, so the absorptive power for radiation in general
depends on the source. For example, among other results Melloni
obtained the following absorptions from the sources given at the head
of the columns, and taking in each case the absorption of lampblack
as 100:—

Absorbing Surface.

Locatelli
Lamp.

Incandescent
Platinum.

Copper at
400°.

Copper at
100°.

Lampblack
Indian ink .

100
96

100
97

100

87

100
85

White lead

53

56

89

100

Metal

14

13-5

13

13

De la Provostaye and Desains worked in another way, coating a

* A simple method of illustrating the difference of absorptive power of metal and
lampblack consists in placing two equal metal vessels in front of some source of
heat — equidistant, say, from a gas-flame, the surface of one being lampblacked on
the side turned towards the source. The rise in temperature of the lampblacked
vessel is much more rapid than that of the other. A good way of arranging the
experiment is to make each of the vessels into an air thermometer on the principle
of Fig. 32, chap. iv.

230

HEAT.

thermometer with the substance to be tested, and finding its rate of cool-
ing for different excesses of temperature above the surroundings. It
was then exposed to radiation from a source, and the steady state was
attained. The heat lost per second was proportional to the rate of cool-
ing previously found for the temperature now observed, and the loss was
equal to the heat gained by absorption. The results obtained agreed in
their general character with those of Melloni.

Comparison of Reflecting Powers. — The earliest work on this

subject was done by Leslie. He employed an arrangement represented
in Fig. 135. 0 was a Leslie cube filled with boiling water and with a
blackened face turned towards a tin spherical mirror M. The focus
conjugate to C was /, but the rays were intercepted on their way
thither by a plate of the substance to be tested placed at ab. The rays
came to a focus at /', and at that point was one bulb of a differential
thermometer. The rise in temperature given by the thermometer was

FIG. 135. — Leslie's Comparison of Reflecting Powers.

proportional to the reflecting power. The following are some of his
results : —

Reflecting power of surfaces for radiation from a black cube at 100° C.
in terms of brass taken as 100 —

Brass .

Silver

Tin .

Steel

Glass

Lampblack

100
90
80
70
10
0

Lampblack practically reflected none of the radiation falling on it.

Melloni obtained results generally confirming those of Leslie. But
we know that the reflecting power for light varies with the quality of the
incident light and also with the particular specimen of the substance
tested. And there is no doubt that there is a like variation in the case
of invisible radiation. We should expect the results of different observers
to differ, and they do differ.

As illustrating this we give the following results of De la Provostaye
and Desains obtained from a Locatelli lamp as source. It will be seen
that the order is slightly different from that found by Leslie : —

GENERAL ACCOUNT OF RADIATION. £31

Reflection of radiation from a Locatelli lamp —

Silver . . 97 per cent, of incident light

Gold 95 „ „

Brass 93 „ „

Speculum metal 86 „ „

Tin . 85 „ „

Steel 83 „ „

Diffusion. — When radiation falls on a surface it may be in part
diffused or irregularly reflected. The diffusion depends on the polish of
the surface.

A method of studying diffusion will be understood by the aid of
Fig. 136, which represents the apparatus used by Melloni to determine
the variation of diffusion of radiation received from different sources.
The incident beam falls on a thin plate of copper a b, coated with lamp-
black. Two thermopiles c, d are arranged symmetrically in front and

Incident Beam

VlQ. 136.— Diffusion.

behind the plate. The front one c registers emission plus diffusion,
while d registers emission alone. The heat received by the former was
about 1 '18 of that received by the latter. On coating the front of a & with
white lead the difference was very much greater, the diffusion by the
white lead being much larger.

Knoblauch found that lampblack diffused about 10 per cent, of the
incident radiation whatever the source from which it was received.

General Results. — Even though no reliance is to be placed on the
exactness of the numbers obtained for the different qualities of absorp-
tion, emission, and reflection, a comparison of their magnitudes at once
leads to an important conclusion. For instance, lampblack has no regular
reflection, low diffusive power, and the highest emissive and absorptive
power. Indian ink has emissive power about 88 per cent, of that of
lampblack, and absorptive power 85 per cent. Metals reflect over 75
percent, of the incident radiation. Their emissive and absorptive powers
are less than 20 per cent, of those of lampblack. These may serve as
typical cases, and we conclude that high reflectors, which must therefore
have low absorbing powers, have also low emitting powers, while low reflec-
tors, which, if opaque, have high absorbing powers, have also high emitting
powers. In other words, Radiating and absorbing powers vary together.
High radiators are high absorbers. Low radiators are low absorbers.

232

HEAT.

This statement must not be taken too generally. It requires limita-
tion on account of the variation of the absorbing power with the nature
of the radiation falling on it, and we shall see later that the limitation is
that the law applies to each kind, or wave-length, of radiation emitted. So
that a high or low absorber of radiation of a given wave-length is a high
or low emitter of that same radiation. If, as in the case of lampblack,
nearly all wave-lengths are absorbed to a very great degree, the surface
is a high radiator of nearly every kind of radiation.

The way in which absorbing and radiating powers go together may be
illustrated in a very striking way by the experiment represented by
Fig. 137. A Leslie cube AB is placed midway between two tin plates
A' B'. These have small thermoelectric junctions at the back, and the
circuit is completed, as shown in the figure, through a galvanometer G.
If the plate B' is at a different temperature from the plate A' a current
goes through the galvanometer. If A and A' are lampblacked surfaces,
while B and B' are bright tin, no current is generated. A radiates far

more than B, but B'
absorbs [a far less propor-
tion of the radiation fall-
ing on it than does A',
and if E, E' are the emit-
ting powers of A and B,
while a a' are the absorb-
ing powers of A' and B',

E E'

A further illustration
of this equality of emitting
and absorbing powers is
given by a thermometer
placed in an enclosure at
a different temperature.

If the thermometer is 6" above the enclosure, it will fall towards the
temperature of the enclosure at the same rate as it will rise towards it if
it is 0° below the enclosure.

Illustrations of Emission and Absorption of Radiation.—

The low radiating and absorbing power of metals is frequently used in
physical experiments. Thus the external surface of a calorimeter is
brightly polished, and is usually surrounded in turn by a second metal
vessel brightly polished. Most of the small quantity of heat lost by
radiation from the calorimeter is reflected by the outer vessel, and the
low radiating quality of the outer vessel protects the inner vessel from
external changes. It is probably advantageous to make the two vessels
of different metals, since, if of the same metal, the outer vessel will
absorb more of the rays emitted by the inner one than if it is of a
different metal. It should as far as possible reflect them back

A familiar example of the use of low radiating power of bright metal
is afforded by a metal teapot.

FIG. 137.

GENERAL ACCOUNT OF RADIATION. 233

If we wish to keep ice from melting, it is advisable to enclose it in a
box of low conducting power, lined outside with bright metal. The outer
surface will ultimately take a temperature below that of the room, and such
that the temperature slope from the outside to the inside of the box will
just conduct to the ice all the heat supplied by the surroundings. The
lower the absorbing power of the metal surface, the less will be the slope
in the non-conducting case needed to convey the heat to the ice, and the
less the heat so conveyed. This may be seen more exactly as follows : —

If A is the absorption of heat per second by the metal surface for each
degree that it is below the temperature of the room, and C the quantity
of heat conducted in through the box per second per 1° difference of
temperature between inside and outside, and if t° is the temperature of
the room, 0° that of the outer surface of the box, 0° that of the inside,
the steady state is given by

A(t-8) = C6

whence 0=

-; - ~«

A + C

The quantity of heat received by the ice is, therefore,

AC*

+

which diminishes as A and C diminish.

Deposition of Dew on Different Surfaces. — A very interesting

illustration of the effect of variation in radiating power is afforded by
the phenomena of dew formation. We owe the explanation of the
formation of dew, now generally accepted, chiefly to the experiments and
observations of Dr. Wells, made about 1812. He found that dew is
most freely deposited on calm, clear nights, on substances close to the
surface of the earth and not shielded from the sky. These observations
led him to suppose that the deposition is due to the cooling of the
surface by radiation, and the consequent cooling of the air in contact
with the surface below the saturation-point of the vapour contained
in it. He verified this cooling by radiation by showing that the
surface cooled down far below the air a little above it, and he showed
also that the cooling was greater for high radiators than for low:
He also found that any protection from the sky, by hindering the
radiation outwards, lessened the cooling. The lesser deposit on surfaces
raised above the surface of the earth he attributed to the falling down
from them of the cooled and therefore heavier air. There was, there-
fore, a constant renewal of the air in contact with the elevated surface,
and this air had not time to cool sufficiently to deposit much of its
vapour. Wells's theory has since been supplemented by Aitken, who
has shown that much of the dew found on the surface of vegetation
arises from the vegetation. When the air is not saturated, the water
evaporates from the surface ; but when the air is cooled to saturation-
point, it cannot take up this water, which therefore remains as dew.

Formation Of Ice in India. — Ice has been formed in India on
nights when the air does not fall to freezing-point by forming square
pits (according to Dr. Wells) about 2 feet deep and 30 inches wide, and
filling them about 8 inches or 1 foot deep with straw. On this rows of

234

HEAT.

small unglazed earthen pans were placed, about l£ inch creep, filled with
boiled soft water. Evaporation and radiation, acting together, cooled the
water below freezing, and ice was formed. Dr. Wells pointed out that
as the formation was most successful on calm clear nights, the effect was
chiefly due to radiation. Probably evaporation would aid the first cool-
ing until the dew-point was reached, any subsequent cooling being due
to radiation.

Transparency and Opacity. — Substances differ as markedly in
their transparency for dark radiations as for light, and all the phenomena
of selective absorption are repeated with these longer waves.

Many bodies which are transparent to light are opaque to long wave
radiations, but the opacity differs with the source, that is, with the
quality, or wave-length, of the radiation received. Thus, Langley finds
that glass is more or less transparent to all the radiations reaching us
from the sun. But Melloni found that a plate of glass about 2 mm.

from Gasholder through
Drying Tubes'

Tube Containing Vapour closed
by Rocksalt Plates

FlO. 138. — TyndalTs Experiment on Emission and Absorption of Gases.

thick would absorb half the radiation from an argand burner, while in
front of a Leslie cube at 100° it was as efficient a screen as a plate of
metal.

Again, it was found by Melloni that rock-salt transmits a very large
proportion of dark radiations, and it was formerly supposed to be trans-
parent to all kinds. But Balfour Stewart showed that a plate of cold
rock-salt was exceedingly opaque to the radiations from hot rock-salt.

Lampblack, which is opaque to nearly all radiations, has been found
by Langley (Phil. Mag., 5th series, xxi. p. 403) to transmit some of very
great wave-length. Tyndall has found that a solution of iodine in
bisulphide of carbon, though quite opaque to light, is transparent to
many long wave radiations. A very thin plate of ebonite is also trans-
parent to long wave radiations, and if placed in front of the condenser of
a lantern, allows enough radiation to pass through to heat the hand if
held at the focus.

A thin ebonite prism spreads the invisible long-wave radiation into a
spectrum, which is quite sensibly detected by the radio-micrometer.

Radiation and Absorption by Gases and Vapours.— Tyndall

GENERAL ACCOUNT OF RADIATION.

235

made a great number of researches on the transparency and opacity of
gases and vapours, his principal mode of experiment consisting in passing
a beam of radiation through a tube about 4 feet long and 2 or 3 inches
in diameter, with its two ends closed by rock-salt plates, from a source
which, in many of the experiments, was a Leslie cube filled with hot
water, as represented in Fig. 138. The beam fell on one face of a thermo-
pile, and the heating effect was compensated for by the radiation from
a second cube falling on the other face of the pile. A chamber which
could be exhausted intervened between the cube and the nearer rock-
salt plate, so that no absorption could occur by the air before the radia-
tion reached the gas. The air could be pumped out of the tube, and the
dry gas to be experimented on could be introduced from a gasholder.
When vapours were experimented on, a flask containing some of the liquid
to be vaporised replaced the gasholder. By these experiments Tyndall
showed that air, oxygen, nitrogen, and hydrogen exercised only a very
slight effect, but that compound gases showed decided absorption. When
the source was a copper plate heated by a Bunsen burner, Tyndall found
the comparative results given below. It will be seen that at a low
pressure the differences are still more striking than at the pressure of
the atmosphere. In the latter case, probably, the absorption was in
some instances so complete that practically all the radiation which could
be absorbed was sifted out of the beam before it had traversed the entire
thickness, so that the whole length of the tube was not effective for
the gases lower on the list.

Amount absorbed
at Atmospheric
Pressure.
Absorption by
Air=l.

Amount absorbed
at 1 inch Mercury
Pressure.
Absorption by
Air = l.

Air .....

1

1

Oxygen ....

1

1

Nitrogen ....

1

1

Hydrogen ....

1

1

Chlorine . .

39

60

Hydrochloric acid . .

62

—

Carbon monoxide . . .

90

750

,, dioxide . . .

90

972

Nitrous oxide . . .

355

1860

Sulphuretted hydrogen .

390

2100

Marsh gas ....

403

—

Sulphurous acid . .

710

6480

Olefiant gas ....

970

6030

Ammonia ....

1195

5460

The effect of vapours was, in general, very great — in some cases
enormously greater than that of the simple gases, and he found also
that they followed the same general order as the liquids from which
they evaporated.

Tyndall also investigated the emissive powers of gases and vapours,

236 HEAT.

either by directly heating the gas and measuring the radiation from it,
or by a very ingenious method suggested by an apparent anomaly
noticed in the course of his work. On admitting air into a tube pre-
viously containing vapour, he noticed that, apparently, the vapour became
much more transparent, as the thermopile indicated a larger amount
of heat falling on to it. But he found that this was really due to the
heating of the vapour by compression. This suggested that he might do
away with the external source of heat, and allow the gas in his tube to
be heated by compression. He was able to measure its radiation when so
heated. He thus found that radiating and absorbing powers were in
the same order. One result obtained by Tyndall is that water-vapour
has a very considerable absorption for dark radiations, a result which
was contradicted by other experimenters. The various methods, how-
ever, by which Tyndall obtained evidence of the absorption, and the
latter experiments of Paschen and others, leave little doubt that his
experiments warranted his conclusion. It is confirmed by the observa-
tions of meteorologists on the greater or slower rate of cooling of the
earth at night, when the air contains less or greater amounts of water-
vapour. If we accept this conclusion, we can also account for the great
variation in the absorption of the solar spectrum noticed by Langley *
and others as occurring with changes of weather.

We may notice one very interesting result obtained by Tyndall, that
water-vapour was especially opaque to the radiation from a hydrogen
flame, while a thickness of about a quarter of an inch of liquid water
quite cut off this radiation.

The foregoing account of the results obtained by measurement of
quantity without regard to exa,ct quality of radiation, amply confirms the
general conclusion that radiating and absorbing powers go together, both
being either great or small together. But they do not tell us anything
as to the relation to each other of the wave-lengths of radiations
emitted and absorbed by the same body ; and for the long wave-length
radiations the experimental difficulties in the way of such an investiga-
tion have only recently been overcome. But certain general conclusions
have been arrived at, chiefly by the aid of the Theory of Exchanges, of
which we shall now give some account. These conditions have been
directly tested and found to hold good with regard to light radiations
and, in some few cases of dark radiations, and we cannot doubt that
they are true with regard to all radiations.

* Rfiearch.cs on Solar Heat, p. 184.

CHAPTER XV,
THEORY OF EXCHANGES.

Theory of Exchanges— Uniform Temperature Enclosures— Full Radiation — Pro-
positions regarding Uniform Temperature Enclosures — Bodies exchanging
Kadiation at different Temperatures — Bodies in the same Physical State con-
tinue to absorb the same kind of Rays independently of Change of Temperature
— Eadiation of every kind emitted by a Body increases as the Temperature
rises — Application to Special Cases.

Theory Of Exchanges. — If an enclosure is maintained at a constant
temperature, and we put another body — say, a thermometer — inside it,
the thermometer, if at a lower temperature, will receive energy from
the surrounding walls by radiation, while, if at a higher temperature, it
will radiate out energy, till, in either case, it arrives at the temperature
of the enclosure, after which everything remains steady. But according
to the kinetic theory of matter, the molecules of the enclosure and the
body are still in vibration, still in a condition to send out radiation. We
are led, therefore, to suppose that the radiation is going on just as before,
and that the apparent cessation of transfer, when equilibrium of tem-
perature is reached, is really due to a balance of exchange, the ther-
mometer radiating out just as much as it absorbs. The space between
the two bodies is not to be regarded as no longer affected, but as the
medium of two equal and opposite streams of radiation. Before the
balance is arrived at, the thermometer, if hotter, is sending out more
than it receives ; if colder, it is receiving more than it sends out.
We may suppose, in fact, that there is, in a given transparent medium,
(we say " given " since, as we shall see later, the medium has an effect)
a total radiation from a body depending on its own temperature alone,
and not on the surrounding bodies. When a body ceases to lose or gain
energy, it is receiving an amount equal to its own radiation. We
imagine a process more complex than that for which we have direct
experimental evidence.

Perhaps we may illustrate the nature of the theory by considering
the money payments between two commercial firms, each supplying the
other with goods. Leaving out of consideration altogether the goods
passing, at each settling-day a certain balance would generally have to
be paid over by one of the firms to the other. But, examining the
books, we should find each transaction entered separately, and the effect
is, therefore, the same as if each purchase were paid for separately —
as if each firm paid for the whole of the goods it purchased, and as if it
received the whole sum due on the goods it supplied. The book-keeping
takes account of the separate items, but the money passing only concerns
the balance. Now, in the case of radiation, experimentally we only

2S8 HEAT.

detect the balance paid over, but the theory of exchanges supposes the
payments all gone through on each side ; it is, as it were, a complete
system of book-keeping. Our justification for its use is, that it tells
us how the balance to be paid over is made up, and under what con-
ditions the balance will exist. How far the double process is gone
through in nature is, perhaps, open to question. In the closely related
theory of secondary waves in light, we have a similar " book-keeping "
account of the disturbance at any point to which a wave travels. We
consider that this is made up of disturbances sent by all the points in
the wave-front in some previous position, and we find the actual distur-
bance to be the resultant or balance of these. But we do not suppose
that each secondary wave exiats, in the sense that each is sending energy
to the point. The supposition of their existence is to keep an account
of the process of wave-propagation, and whenever on reckoning up we
find a balance of uncompensated disturbance at a point, observation
shows us that the disturbance actually exists, and that the calculated
amount of energy has been supplied for it.

The theory of exchanges consists of two parts : —

In the first we consider the transactions when bodies are entirely,
surrounded by bodies at their own temperature, that is, are in uniform
temperature enclosures.

In the second we consider the transactions when bodies are exchang-
ing radiations with bodies at a temperature different from their own.

1. Uniform Temperature Enclosures.

We shall not at present consider the effect of the medium on the
radiation, but suppose that we have the same medium in all cases and
that its refractive index for all radiations is unity.

By experiment we find that a body placed in a uniform temperature
enclosure finally comes to the temperature of the enclosure, whatever
may be the shape or nature of the walls and whatever the position of the
body with respect to them. On the theory of exchanges, therefore, the
body is receiving everywhere just as much as it radiates, that is, it is
receiving the same total quantity at all points. Not only is it receiving
the same quantity, but also the same quality. For, suppose the body
able to absorb more of one kind of radiation than of another — to fix our
ideas, say more of red than of yellow — and suppose that at A the pro-
portion of red to yellow is greater than at B. Placing the body at A, it
will be able to absorb more energy than when placed at B, but it radiates
equal amounts at the two points. Hence, if the temperature remains
constant when the body is at B, it must rise when the body is at A,
which is contrary to experience. The proportion of the two kinds
must, therefore, be the same throughout. The radiation must also be
the same in quantity and quality in all enclosures of the same tempera-
ture, for, removing the body from one enclosure to another, the preced-
ing argument applies.

It is, therefore, independent of the nature of the surface of the
enclosure. Suppose part of the surface lampblacked, part covered with
polished silver, and part with a plate of rock-salt. The lampblack absorbs
nearly all that falls on it, but makes up for this by its great radiation ; the

THEORY OF EXCHANGES. 239

silver radiates very little, but then it reflects nearly all that falls on it.
The rock-salt neither reflects nor radiates much, but transmits freely most
of the radiation from the surface against which it is placed.

There is, therefore, a stream of radiation going in all directions in a
constant-temperature enclosure, the amount and quality of the stream depend-
ing only on the temperature. Since a lampblacked surface absorbs nearly all
the radiation falling on it, it must send out a stream nearly equal to this
total radiation, and hence the radiation in a constant - temperature
enclosure is sometimes termed the lampblack radiation. But we know
that even lampblack does not entirely absorb the radiation falling on it,
diffusing and transmitting small fractions. Hence it does not radiate
out quite as much radiation as falls on it in the enclosure. It is better,
therefore, to term the stream in a constant, uniform temperature en-
closure, the full radiation for the given temperature.

From the foregoing we may immediately deduce the following
results : —

(a) If a body absorbs any kind of radiation it must also emit the same
kind of radiation at the same temperature, and if it is placed in a uniform-
temperature enclosure the emission equals the absorption.

This follows at once from the fact that, in order to make the stream
of issuing radiation full, it must restore to the incident stream just what
it took from it.

(b) A body at a given temperature cannot emit more of a given kind of
radiation than exists in the full radiation for that temperature.

For, if it could, and if it were placed in a constant-temperature en-
closure, the issuing stream would contain more of the radiation than the
incident stream — or, the stream would not be uniform in character.*

(c) A body opaque to any kind of radiation must emit that kind of
radiation.

For, construct a uniform-temperature enclosxire entirely lined with
the substance, and let the temperature be such that the given radiation
is a constituent of the full radiation at that temperature. The given
radiation now existing in the enclosure must have come from the lining
substance, for it could not get through from behind. Hence, opacity is
always accompanied by the power of emission, and, therefore, of
absorption.

To enable us to think a little more definitely of such an enclosure,
let us suppose a hollow iron ball, coated inside partly with lampblack,
partly with silver, partly with rock-salt. Let it then be heated red hot
and kept at a constant temperature. If we can imagine ourselves placed
within, but protected in some way from the high temperature without
affecting the full radiation, then, on looking round, everything will
appear of the same colour. The lampblack radiates out most but it
reflects little, the silver radiates little but reflects much. The rock-salt
is nearly transparent to the radiation from the iron behind it, and the
uncovered iron partly radiates and partly reflects enough to make up the
full radiation. We shall, therefore, have one uniform glare, and shall be
quite unable to see the boundary lines between the different substances.

* This does not necessarily hold if the body is phosphorescent, or if it is under-
going chemical or molecular change. In these cases molecular energy may be
transformed into radiant energy without a corresponding rise of temperature.

240 HEAT.

If any bodies are suspended in the enclosure, they too will be indis-
tinguishable. A piece of coloured glass will absorb special rays, but will
also emit an equal amount of those rays. A plate of tourmaline will
polarise the light transmitted through it, but it will emit light polarised
in a perpendicular plane, making up the issuing stream to the full radia-
tion without polarisation.

A difficulty in accepting these statements arises, perhaps, from our
want of experience of true constant-temperature enclosures. A room
without a fire in daylight may possibly be at a nearly constant and
uniform temperature, yet our experience shows that it is far from
answering to the above description, since we can see the various objects
in it quite plainly. But it is to be marked that we see them not by
their own radiation, but by reflected daylight which is originally sun-
light. In fact, the room is not an enclosure. The window being tran-
sparent, the sun forms part of the boundary from which radiations are
received, and it is at an enormously higher temperature than the rest of
the room, while the sky or clouds probably form another part of the
boundary far colder than the room. In order, then, to have a uniform-
temperature enclosure, the radiations received at any point within it
must all proceed from bodies at the same temperature. Hence it would
be impossible to make glass or rock-salt enclosures, unless, indeed, the
thickness were so great that the impurities and the natural absorption
rendered the walls opaque to external radiation.

A room darkened by thick shutters, or a cellar without windows,
gives a very near approach to a uniform-temperature enclosure, but our
eyes are not sensitive to the radiations given out at ordinary tempera-
tures. A very hot coal-fire, however, frequently has cavities forming
nearly complete enclosures, into which we may look, and so verify some
of the properties stated above. The boundaries between the different
pieces of coal are very indistinct. Pieces of glass of different colours
put into the cavity soon become nearly indistinguishable, each restoring
to the stream of radiation what it takes from it. Similar observations
may be made on heating an ordinary clay crucible, or even a clay
tobacco-pipe, to red heat and putting small pieces of glass or of any
other non-combustible material at the bottom. On looking in, the same
indistinctness of outline will be marked.

Effect of the Medium on Radiation. — We have seen that in all
enclosures at the same temperature, containing the same medium, the
stream of radiation is the same, our proof depending on the assumption
that the radiation from an enclosed body is independent of the nature
of the radiating enclosure.

But we have no ground for supposing it to be independent of the
surrounding medium. A body surrounded by rock-salt may be able to
send out more or less energy in a given time than the same body
surrounded by air.

Let us now investigate the effect of a change of refractive index on
the stream of radiation. The following proof is due to Balfour Stewart
(Treatise on Heat} : —

Let AC A' (Fig. 139) represent a spherical constant-temperature
enclosure, the lower half being filled with a substance of refractive
index //., for some given radiation. Let B be a very small area on the

THEORY OF EXCHANGES.

24-1

surface of the substance at the centre of the sphere. Let CBD represent
a cone of rays of very small angle, with ABA', the normal to B, as axis,
and draw the cone BC'D' such that

sin ABD = /zsin A'BC'
or, since the angles are small, ABD = p, A'BC'

If B is sufficiently small the refracted part of all the rays reaching B
through the cone CBD must pass through the cone C'BD'.

We have an exchange, then, of radiations through the area B, and
the total radiation, measured by the energy received per second, sent
downwards must equal the total radiation returned upwards, and this
will be true for the two corresponding elementary cones OBD, C'BD'.

Now, the quantities of radiation incident on B through the cones will
be proportional to the solid angles of the cones, which, since they are
small, are proportional to the squares of the
angles ABC, A'BC', that is, are as /u,2 : 1.

If, then, R, R' represent the amount of
the given radiation in the full radiation in
each medium, then the radiation incident
on B through OBD is to the radiation
incident on B through C'BD' as ju,2R is
to R'.

But by optical theory and experiment
the fraction of the incident light trans-
mitted is the same, whether the radiation
falls on the surface from above or below.

Hence, the radiations transmitted into
the two cones are also in the ratio ;u,2R : R'.

But these are equal, or

p, _ 2T} FIG. 139. — Effect of the Medium

H — fj, xl. on Radiation.

To illustrate the meaning of this result,

suppose that a body is suspended in an enclosure colder than itself, the
space between the body and walls being a vacuum — it will radiate out
more than it receives, and tend to fall in temperature. If, now, the body
is surrounded by a layer of rock-salt of which the refractive index is
about 1*5, the stream of radiation from the body and the stream to it
will each be multiplied by ju,2. The difference is, therefore, increased in
the same proportion, and, therefore, the body will cool more than twice
as rapidly, even neglecting the effect of conduction.

We may summarise the foregoing by the statement that in a uniform
constant-temperature enclosure there is a definite stream of radiation in
all directions depending only on the temperature and the medium, and
not on the nature of the radiating surfaces. Further, that any body
placed within the enclosure and at the same temperature absorbs and
radiates the same kind or kinds of radiation and to the same amount,
leaving the radiation issuing from it the same as the full radiation
incident upon it.

The principle that bodies absorb radiations of the kind which they
emit is perhaps a special case of the general principle of resonance, of
which we have examples in sound and mechanics, according to which

242 HEAT.

principle bodies naturally vibrating in a given period are set vibrating
by receiving waves of that period from different sources.

2. Bodies exchanging Radiations at different Temperatures.

We can apply what we have learnt from uniform-temperature
enclosures to cases of unequal temperature by the aid of two general
principles : —

(a) Bodies in the same physical state continue to absorb the same kind
of rays independently of change of temperature.

The experimental evidence in favour of this is that bodies keep the
same colour through wide ranges of temperature, that is, they absorb the
same constituents of the radiation falling on them. If they change
colour, we usually find some evidence of chemical action or of change of
molecular aggregation, as when yellow phosphorus changes to red phos-
phorus, a change which might almost be considered chemical.

(6) The radiation of every kind emitted from a body increases as the
temperature rises, as long as the physical state remains the same.

We may verify this for light rays by heating an opaque body till it
becomes incandescent. When first visible, it only gives out the longest red
rays in sufficient quantity to be seen and these only in small quantity ; but
as the temperature rises, the spectrum gradually extends towards the blue
till the body becomes white-hot, when it gives out a full spectrum. But
as fresh rays are added to the spectrum, those previously existing become
stronger — the red, for instance, being much more intense than it was
when the body first began to glow. The same holds good for the dark
long wave radiations, as has been shown for blackened copper by Langley.
He investigated the wave-lengths emitted at various temperatures from
40° C. up to about 800° 0., and found that, as the temperature rose, the
spectrum became extended towards the shorter wave-lengths, and that
the previously existing radiations also became more intense.

In the case of solids and liquids, the spectrum is always more or less
continuous, and we may lay down the general rule that, at low tempera-
tures, only the very long wave-lengths are sensible, but that, as the
temperature rises, the spectrum gradually extends to the shorter wave-
lengths, and ultimately comes into the visible range.

Gases, however, only emit, at least in the visible spectrum, waves of
definite length or in definite groups, and their spectra consist of bands or
lines. As the temperature rises, these bands or lines become brighter.
We may easily show this by comparing the sodium flame in a spirit-lamp
with the hotter sodium flame in a Bunsen burner.

Application of the foregoing Principles to Special Cases. — We

know that a black surface such as lampblack absorbs nearly all the radia-
tions which fall upon it. If we raise its temperature, it will still continue
to absorb. It must, therefore, also emit rays of nearly all kinds, and at a
given temperature it will emit more radiation than other bodies, or when
incandescent will shine more brightly. In illustration of this, if a piece
of platinum-foil with inkmarks on it, which will withstand heat, is heated
to incandescence, the inkmarks shine more brightly than the rest of the
foil. If a piece of white porcelain has a dark pattern on it, the white
porcelain reflects a great part of the light-radiations falling on it, and the

THEORY OF EXCHANGES. 24S

dark pattern absorbs most of them. On heating the porcelain to incan-
descence, the pattern is still the best absorber, and it is, therefore, also
the best radiator, and the pattern stands out as bright on a less bright
ground.

Blue glass absorbs red more freely than blue rays. If heated to
incandescence, it radiates red more freely than blue, and therefore it
appears a very bright red. Transparent glass absorbs little, if any, of
the light rays, and therefore it radiates poorly, and a piece of colourless
glass-tubing heated white-hot only gives out a faint white light as com-
pared with a wire heated to the same temperature.

We have a converse case with sodium vapour in a Bunsen flame,
which emits yellow light, its spectrum consisting of the two well-known
D lines. It, therefore, also absorbs yellow light of the same wave-lengths.
If, then, we place a cooler sodium flame — that of a spirit lamp — in front
of a hotter sodium flame given by a Bunsen burner, the cooler flame will
absorb from the hotter flame behind it, but will not restore to the out-
going radiation as much as it absorbs. Hence it appears smoky, especially
round the edges, where the absorbing vapour is coldest.

This is applied to explain the existence of the dark lines in the
spectrum of the sun. The body of the sun is intensely hot, and so dense
that it sends out radiations of all kinds, forming a continuous spectrum.
But round it is an atmosphere consisting of a mixture of gases and
vapours cooler than the interior mass, though they are still exceedingly
hot, sufficiently so to send out their own radiations. But they absorb
from the stream passing outwards more than they restore, each selecting
the kind of radiation it emits. Thus, sodium vapour absorbs the two D
wave-lengths, while the light on each side passes on unabsorbed. On
examining the solar spectrum, then, these two D lines appear dark, not
from absence of sodium light, but because the sodium light is weakened
relatively to the neighbouring radiation. We may quote as another
example the cases already mentioned of absorption by cold rock-salt of
the radiation from hot rock-salt, and of the absorption by water of the
radiation from a hydrogen flame.

CHAPTER XVI.
RADIATION AND TEMPERATURE.

Variation of Rate of Radiation with Temperature — Newton's Law of Cooling—
Dulong and Petit's Law- — Rosetti's Law — Stefan's Law — Constants of Radia-
tion— Radiation from Surfaces which Absorb Selectively — Rate of Solar Radia-
tion— Solar Constant — Pouillet's Pyrheliometer — Violle's Actinometer — Lang-
ley's Researches — Crova's Researches — Effective Temperature of the Sun —
Source of Solar Energy.

Variation of Rate of Radiation with Temperature-— When a

body is in a constant temperature enclosure, at an excess e above the
temperature 0 of the enclosure, it is radiating out at a rate depending on

e e*e

*• Absolute Temperature

FIG. 140a.

its own temperature 0 + e, but if its absorbing power is the same as when
it was at 0 it is absorbing as much as it would radiate out at 6. For at
that temperature it would radiate out as much as it receives. Hence if
we represent its total radiation per second at any temperature t, by R, it
is losing energy by radiation at a rate

If we could surround the body by an enclosure at the absolute zero
the enclosure would probably cease to radiate and we might put R, = 0.
The rate of cooling would then represent the total radiation per second
from the body at the temperature e.

244

RADIATION AND TEMPERATURE.

245

Various attempts have been made to find how the quantity R0 repre-
senting the radiation from a body at absolute temperature e, in an
enclosure at the absolute zero, depends upon the temperature. Since we
cannot have a zero enclosure it is obvious that we must deduce E, from
observations of the difference

where 6 is the temperature of the enclosure or surroundings and e the
excess of the radiating body above that temperature. Let us suppose that
we have succeeded in determining the form of the curve representing the
radiation of a body at different temperatures in a zero enclosure, and that
it is OPQ in Fig. 140a. Let us draw the ordinate PM at 0 and QN at 0 + e
and PR parallel to the temperature axis. Then PQ, with axis PR,

FIG. 1406.— Curves of Cooling for Temperatures e, 0 + e,
6 + 2e of the Surroundings.

will represent the cooling curve in an enclosure at temperature 6, for
QR = Rs+e - R«. Suppose that we obtain successive cooling curves, say
first between Q for the enclosure and 0 + e for the body, then for 6 + e for
the enclosure and 6 + 2e for the body, and so on. Let these curves be re-
presented on a single diagram, as on Fig. 1406, each curve being marked
with the temperature of the enclosure ; by putting these curves end to
end we can build them up into the radiation curve of Fig. 140a.

Newton's Law Of Cooling. — .Newton * made the first experiments
on rate of cooling, and found that the rate was proportional to the excess
of temperature above the enclosure. This may easily be verified by raising
a thermometer some 20° or 30° above the temperature of the room and
observing its fall. Plotting the logarithm of excess of temperature, against
the time, the result will be found to be a straight line. If e be the excess
at time t, and if E be the initial excess, the observations show that

e
lo= -at

* Phil. Trans., 1701, p. 828. Newton really investigated the convection effect
in a current of air. See Russell, Phil. Mag. (Q), xx. p. 599, 1910.

246

Differentiating

or

HEAT.

dt

-Tt
dt

that is, the rate of fall is proportional to the excess.

If this law held good for any value of the excess the radiation curve
would be a straight line. But it is only true for small excesses, so small

19
18

drawn, through, observed points

^^-'--^^

0 20 40 60 80 100 120 140 160 180 200 220 240
Excels of temperature of bulb over temperature of enclosure

FIG. 141. — Dulong and Petit'e Cooling in Vacuo

that we may regard the radiation curve as straight, and consider that we
are moving along the tangent. It therefore gives us no information as to
the shape of the radiation curve.

Dulong and Petit's Law Of Radiation. — Very extensive researches
were made on the cooling of thermometers in enclosures by Dulong and
Petit about 1817.* Their chief work was carried out by suspending a
thermometer raised to any temperature up to 300° 0. in a copper globe
surrounded by water-. The globe was then exhausted to a pressure of 2
or 3 mm. and the rate of cooling of the thermometer was observed.
Corrections were applied for the effect of the residual air and for other
disturbances, and the results were taken to give the cooling due to radia-
tion. They are represented in Fig. 141 by the curves I. to V., the tern-

v

* Ann. de Chemie, vii., 1817, or Thomson's Annals of Philosophy, xiii. , 1819.

RADIATION AND TEMPERATURE.

247

perature of the enclosure being marked on each curve. Now the heights
of these curves are in a constant ratio to each other, each being 1*16
times as high as the preceding. Then if we have obtained the rate of
cooling for a given temperature of the enclosure, when we raise that
temperature by 20° we must multiply the ordinate of the curve by 1'16,
when we raise it 40°, by 1-162, when we raise it 60", by T163, and so
on. Then for a difference of 1° we should multiply by 1'ld*, or by
1-0077, and for a difference of n by 1-0077". Taking curve I., where the
enclosure is at 0° 0, if we multiply the ordinates. by 1'0077~278='123
we shall obtain the curve of cooling when the enclosure is at 0° absolute,
and this is the radiation curve on the assumption that the results
obtained between 0° C. and 300° C. allow us to extrapolate.

We have already shown that the cooling curves are successive pieces
of the radiation curve, and if we break the radiation curve up into suc-
cessive 1° steps, beginning at the absolute zero, as in Fig. 142, the height

0°abs

I' 2" 3'

FlG. 142. — Eadiation Curve broken up into Cooling Curves.

of each step above the last is, by Dulong and Petit's result, 1 '0077 times"
the preceding height, or B6= 1-0077 Aa ; Gc = 1-0077 B6, and so on.

But these 1° steps up are each - - for the radiation curve so that

dv

^~ is multiplied by the factor 1'0077 for each step of 1°, and at 6 we

dB

shall have

^=1-0077" Aa
dv

where Aa is the height of the first step. Integrating we get

R = 7»(l-0077'-l)

where m is a constant for the radiating body.

For a long time this result was supposed to represent the radiation
curve, at least within the range of the experiments, but there is no

248 HEAT.

doubt that the residual gas played some part not eliminated, and that
the radiation is not truly represented by the formula.

Rosetti's Law. — Rosetti* found that with a Leslie cube or an iron
vessel filled with mercury, but not surrounded by a lampblack enclosure,
the cooling between 300° C. and 0° 0. was much more nearly represented

by

when T is the absolute temperature of the body, 6 that of the enclosure,
and a and b are constants for the body.

Stefan's Law. — In 1879 Stefan f gave an entirely new turn to the
researches on radiation by the suggestion that the rate of radiation is
proportional to the fourth power of the absolute temperature. He was led
to this by observing that a result of Tyndall's accorded with it. Tyndall
found that the radiation from a platinum wire at 1200° 0. was 11*7

/1200 4- 273\*
times its radiation at 525° 0. Now ( -=^= — ===- ) is very nearly 1 1 -7,

\ t)ZO + 2tlO J

or the two radiations were almost exactly in proportion to the fourth
powers of the absolute temperatures. Re-examining the work of
Dulong and Petit and their successors, Stefan found that on taking con-
duction by the surrounding gas into account, the radiation was more in
accordance with the fourth-power law than with the formulae previously
used. In 1875 Bartoli applied thermodynamic reasoning to radiation,
and in 1884 Boltzmann followed up Bartoli's work. Starting with the
supposition, now proved, J that radiation presses against any surface on
which it falls, Boltzmann treated the radiation in an enclosure as the
working substance in a Carnot cycle, and showed that the full radiation
at any temperature must, if the method is justifiable, be proportional to
the fourth power of the absolute temperature. We shall give an account
of this thermodynamic treatment in chap, xx Here it is sufficient to
say that the subject lias been developed by Wien§ and others, and that
Wien has shown that if a body is emitting full radiation then the wave-
length of maximum radiation is given by

A^fl = constant,

where 6 is the absolute temperature of the surface, and that the energy
radiated through a small range dX at this maximum is

ETOc? A. = constant x 65.

Wien on certain assumptions deduced a formula to express the
energy radiated for every wave-length. But the results of experiment
did not confirm Wien's formula so well as a modification obtained by
Planck.||

* Phil. Mag., viii., 1879.

t Wien. Alcad. Ber., 1879, Ixxix, pp. 391-428.
$ Lebedew, Congres International de Physique, ii. p. 133.

§ Wien, Congrts International de Physique, ii. p. 23 ; Larmor, " Radiation,"
Encyclopedia Britannica, xxxiii., 10th ed.

|| Deutsch. Phys. Gesell. Verh., 2, xiii. p. 202; 2, xvii., p. 237, 1900.

RADIATION AND TEMPERATURE. 24-9

Wien's formula for radiation in the small range dX is —

x ~ X» 6
Planck's modification is

x = -f—
1 ew-l

where Cl and 02 are constants. These last formulae must for the present
be regarded as of less weight than those for ATO and Em, as they are
obtained by the aid of special molecular hypotheses.

Since the enunciation of Stefan's law and the foundation of the
thermodynamics of radiation, much work has been done to test the
formulae obtained by the theory.* Lummer and Pringsheim used as
radiating source a constant-temperature enclosure with a small hole in
the side, the enclosure being of different materials according to the
temperature. The radiation emerging from the hole was practically the
full radiation for the temperature of the enclosure. For imagine that a
stopper at the temperature of the enclosure is put into the hole, the full
radiation strikes on the stopper ; but it only exceeds that passing out
through the hole by the amount emitted by the stopper and reflected by
the sides of the enclosure back to the stopper again, and this is
negligible if the hole is small enough.

The issuing radiation was measured by a bolometer, and the fourth-
power law was very exactly verified. Using a fluorspar prism, the
radiation was dispersed, and a bolometer travelling along the spectrum
gave the position of maximum energy and the amount of that energy.
The formulas XmO = constant and Em = constant x #5 were verified.

In subsequent work f Planck's formula has been shown to accord
very closely with observations on the energy in different parts of the
spectrum — more closely than that of Wien. But it should be noted that
C2 is in Wien's formula 5Xmd, where AOT is the wave-length for which E
is a maximum and in Planck's very nearly the same. Hence we have

&\rn

practically e~*~ in the denominator, and this is great compared with 1,
so long as A. is less than Xm. The two formulas nearly agree, therefore, on
the shorter wave-length side of the maximum.

The confirmation of these formulae justifies their use to determine
the temperature of bodies emitting full radiation. Thus Lummer and
Pringsheim J have determined the temperature of a uniform-temperature
enclosure (1) by measuring the total radiation emitted, (2) by measuring
the wave-length of maximum energy, (3) by comparing the brightness of
a given part of the spectrum with the brightness of the same part of a
spectrum emitted by a surface at a lower known temperature and using
Planck's formula. All three methods gave the temperature as 2325°
absolute within 20°. The constants in the formula had been previously
determined.

* An account of the subject will be found in the CongrJs International d<
Physique, ii. p. 41, by Lummer, who has been one of the chief workers.
+ Paschen, Ann. d. Physik, 4-2, Feb. 1901, pp. 277-298.
t Berichtc der Deut. Phys. Oes., 1903, p. 3.

250

HEAT.

Constants Of Radiation. — These constants may be taken as having
nearly the following values* : —

If R, is the energy emitted per second per square cm. from a full
radiator at temperature 6 on the absolute scale,

where o- = 5'32 x 10~5 ergs per second per sq. cm., or 5*32 x 10~12 watts
per sq. cm.

In Am0 = A

where Xm is the wave-length of maximum energy at 6 expressed in terms
of //,= 10" 6 metre as unit A— 2Q4-0

In Wien's formula

0,9*

= l «

02=5A=14700.f
Cj is of the order 1000 for \ = I/A.

The Radiation from Surfaces which Absorb Selectively.— The

foregoing results do not apply
to surfaces which absorb dif-
ferent proportions of different
wave - lengths. Even if a
surface always absorbed the
same proportion of the same
wave-length it need not absorb
the same fraction of the full
radiation at different tempera-
tures. Suppose, for instance,
that I., Fig. 143 represents the
full radiation for one tempera-
ture, while II. represents it for
another. Suppose that at L a
certain surface has an absorp-
tion band, L being the wave-
length of maximum intensity
of II., and suppose this the
only absorption band, then in
II. a certain amount at the
absorbed, while

* wave length

FIG. 143. — Radiation Curves for Spectrum
at Two Different Temperatures.

maxmum s

in I. it is not at the maximum and is therefore not such a large fraction
of the whole.

But we cannot assume that a surface does always absorb the same
proportion of the same wave-length. It appears possible that the
fraction increases in general as the temperature rises, for exceedingly hot
surfaces appear to approach fulness in the quality of their radiation even
though at low temperatures they are low radiators and only give a small
fraction of full radiation.

* Congrks International de Physiqtte: Lummer, loc. cit. The valus of <r is due to
Kurlbuum ; Wied. Ann., Ixv., 1898, p. 748. A simple experiment giving an approxi-
mate value of ff is described by Denning: Phil. Mag., x., 1905, p. 270. Uncertainty
arises from the fact that the receiving surface, though black, is not a full absorber.
See Kurlbaum, Wied. Ann., Ixvii. p. 846, 1899.

t Holborn and Valentiner (Ann. d. Physik, xxii., 1, 1906, p. 1) give 14200.

RADIATION AND TEMPERATURE. 251

So far no general formulae have been obtained for such surfaces,
though special cases, such as bright and black platinum, have been studied,
and formulae have been devised to suit the results obtained.

The Rate of Solar Radiation.

The Solar Constant. — One of the most interesting problems in solai
physics consists in the determination of the rate at which the sun is
radiating energy. The rate is expressed in terms of the amount falling
on 1 sq. cm. at the distance of the earth but outside the earth's atmo-
sphere. When this is stated in calories per minute it is called the solar
constant If we multiply by (radius of earth's orbit/radius of sun's

lamp
Blacked

FIG. 144. — Pouillet's Pyrheliometer.

radiating surface)2, i.e. by (92400000/430000)2 = 46000 we get the
amount passing out from each square centimetre of the sun.

Pouillefs Pyrheliometer. — The first successful research on the solar
constant was made by Pouillet by means of the pyrheliometer. This
consists of a flat, thin, cylindrical vessel A, Fig. 144, of which the upper
face is lampblacked and the rest is silvered. This contains water and
serves as a calorimeter. It is mounted at one end of an axis D round
which it can be rotated to secure mixing of the contents, and the thermo-
meter stem lies along this axis, so that the temperature of the calorimeter
can be read easily.

The instrument is movable about another axis on the stand, so that
it can be pointed in any direction, and a disc A' of the same diameter as
A, is fixed near the other end of the axis D, and this is exactly covered
by the shadow of A when D is directed towards the sun.

Pouillet's method of observation was as follows. He first directed
A to a part of the sky away from the sun, protected the instrument from
sunlight, and noted its fall of temperature, say 6V during five minutes.
He then directed it towards the sun for five minutes, and noted the rise,
6. Finally he again directed it away from the sun, and noted the fall #2
during five minutes. He assumed that the mean cooling during the time

252 HEAT.

n . s\

of exposure to the sun was * 2> so that, had all the heat received from

m

f) 4- 0

the sun been retained, the rise would have been 6 + l ~ -

Knowing the capacity of the calorimeter, the total heat received per
unit of surface per unit of time could be measured. This, of course, only
gave the radiation reaching the earth after much had been absorbed by
the atmosphere. That this absorption is very considerable is sufficiently
shown by the fact that we receive sensibly more heat from the sun as he
rises higher, and thus diminishes the thickness of air through which his
rays must pass.

To determine the absorption by the air, Pouillet made observations
when the sun was at different heights, and found that as long as the air
was exceedingly clear, and the observations were made on the same day,
the quantity of heat received per centimetre per second might be
represented by Aa**, where d is the thickness of air passed through on
the supposition that the air is a limited ocean of uniform density.
A is, therefore, the heat per centimetre per second which would be
received outside the atmosphere. By aid of this formula and the ex-
perimental results, A and a could be determined. Suppose, for instance,
that observations are made with a vertical sun, and again when it is 60°
from the zenith. The thicknesses of air may be taken as 1 : 2.

Hence, Q! = Aa" Q2 = Aa**,

Therefore, Qi2 _ .

or

It was found that a changed from day to day, and with the clearness
and hygrometric state of the atmosphere, but the mean absorption of the
radiation from a vertical sun was, according to Pouillet, about 20 per
cent.

He obtained 0*02939 calories per second per sq. cm., for the value of
A, or 1'7634 calories per minute per sq. cm.

Multiplying by 46000, the radiation from a square cm. of the sun
would be about 1350 calories per second, or 1350 x 4-2 = 5650 watts = 7'6
horse-power. This gives about 7000 horse-power per square foot.
Later observers have obtained values somewhat higher than that of
Pouillet. His calorimeter was probably somewhat too slow in responding
to the heating effect of the sun. The outer surface being at a higher
temperature than the water as registered by the thermometer, the
radiation outwards was most likely greater than that allowed for.

Again, he assumed that the air absorbed all rays in the same pro-
portion, whereas it absorbs or diffuses some much more than others.
For instance, the blue colour of the sky shows that there is greater diffusion
of the shorter wave-lengths, and the spectroscope gives us clear evidence
of marked " selective absorption." As Langley has shown, Pouillet's
supposition that the radiation may be represented by Aad would always
lead to an underestimate of the solar constant.* This may be seen by
considering a simple case. Imagine that the radiation outside the atmo-
sphere consists of two kinds, respectively A and B in quantity. Let A

* American Jownal of Science, September 1884 ; Phil. Mag. xviii., 1884.

RADIATION AND TEMPERATURE. 253

pass through unchanged, but let B be reduced by one-half in passing
vertically through the atmosphere. Then with the sun in the zenith we

T>

observe A -t- — • If we observe the radiation when the sun is 60° from

a

•p

the zenith we obtain A + — • Pouillet's rule would give the outside

radiation as

. B

B

which is necessarily less than A + B.

Again, if any ray were totally absorbed by the atmosphere even with
vertical sun, Pouillet's method would take no account of it.

Violle's Actinometer. — Violle used an instrument which lessens the
fault of Pouillet's Pyrheliometer
by diminishing the lag behind the
true temperature. A thermometer
.t, Fig. 145, is placed in the middle
of a double-walled enclosure AB,
the space CO between the walls
being filled with melting ice or with
water at a constant temperature.

An opening provided with a shutter V \ / / corslant

,i , f n ,1 \ X. A/ / temperature

allows the sun's rays to fall on the \ ^ ---- ~ / baifi

thermometer bulb. The shutter

being closed the thermometer comes

to the temperature of the enclosures FIG. 145. — Violle's Actinometer.

which is noted. The shutter is then

opened and in about fifteen minutes a stationary temperature is reached

when the heat from the sun balances that lost to the enclosure. To

find the latter, the shutter is closed and the initial rate of cooling of the

thermometer is noted. The water equivalent of the thermometer being

known, we have the heat gained from the sun. Violle found for A the

value '04233 calories per second per sq. cm. or 2'5398 calories per minute.*

Langley's Researches with the Bolometer. — Langley was the first to take
selective absorption by the atmosphere thoroughly into account. In his
researches on Mount Whitney f he sought to measure the absorption of
each part of the spectrum separately, using a diffraction grating to
separate out the various wave-lengths, and a bolometer to compare the
heating effect with different thicknesses of atmosphere passed through.
His results for wave-lengths between '4/i and I 'Op are given in Fig. 146,
ju, being one-millionth of a metre.

Curve I. is for a given position of the sun at noon, termed " high
sun," II. for a position — "low sun" — in which twice the thickness of
atmosphere is passed through. Curve III. is obtained by drawing
ordinates bearing the same ratio to those of I. that the ordinates of I.

* Violle's researches are described in various papers in the Comptcs Rcndus.
t Researches on Solar Heat (Washington 1884) : Phil. Mag., xv., 1883, p. 153.

254

HEAT.

bear to II., and therefore III. is the curve he would have obtained had
he been outside the atmosphere altogether. The general absorption
steadily increases as the wave-length diminishes, though there is marked
absorption of particular rays in the infra-red. This can easily be seen by
inspection of the ratios of the ordinates of I. and II. at H with those at A.

Curve III. , therefore,
contains a larger proportion
of the blue rays than I. or
II., and Langley concludes
that outside the atmosphere
the sun would appear of a
bluish tinge. Continuing
the curves beyond 1-0 as
far as he was able, and com-
paring the areas of curves

1200

llOO

'000

900

\

800

700

600

500

400

300

200

100

I. and II., he found
Area high sun _ .. r_
Area low sun

Hence also area of curve
III. = 1'57 area of curve I.
Now by using Violle's actino-
meter, he obtained as the
total radiation received per
square centimetre per
minute at high sun on
clear days, 1'81 calories.

Then outside the atmo-
sphere this would be in-
creased by the fraction 1-57,
giving 2 '84 calories.

Langley believed that
even 2 '84 was an under-
estimate and put the solar
constant as at least 3.

Crova's Researches. —
Crova,* who devised and
carried on a series of re-
searches with an actino-
meter> put the value of the
constant still higher, be-
lieving that it is probably
as high as 4 calories per minute per square centimetre.

Recent work tends to show that these values are too large. Abbott
and Fowle,f using the Bolometer, found almost the same value at Wash-
ington and at Mount Wilson, 1800 metres higher, just below 2'1 calories
per minute per square centimetre. Millochau,J working with the Fdry
pyrometer on the summit of Mont Blanc and at different levels below,

* Cong. Int. de Phys., hi. p. 453, contains a discussion of the value of the solar
constant by M. Crova.

f Terrestrial Magnetism, xiii. p. 79, 1908 ; Ast. Phys. Journ,, xxix. p. 281, 1909.
j Journ. de Physique, viii. p. 347, 1909.

.3

.40

FIG. 146. — Langley's Curves of Solar Radiation.

I. High sun, through one atmosphere.
II. Low sun, i.e. through two atmospheres.
III. Estimated curve outside atmosphere.

RADIATION AND TEMPERATURE. 255

found a value about 2-4, and Bellia * found from his own and previous
work a value 2*1.

There are indications that the so-called constant is not constant, but
varies with the condition of the sun. It would be better if some such title
as " Solar Radiation Stream " were adopted to replace " Solar Constant."

Taking 2*1 as the value, it is equivalent to 0*147 x 107 ergs per
second or 0*147 watts.

Multiplying by 46000, the radiation from a square centimetre at the
surface of the sun is 6800 watts, about 9 horse-power.

The Effective Temperature of the Radiating Surface of the Sun. — We
receive radiation, no doubt, from different layers of the sun, and these
are at different temperatures. We cannot, therefore, accurately say that
the radiating surface has one definite temperature. But we may find
the one definite temperature of a fully radiating surface which is giving
off energy at the same rate as the sun, and this is defined to be the
" effective temperature" of the sun.

Before there was any approximation to a correct law of radiation, a
temperature was assigned to the sun which we now know to be immensely
exaggerated. Thus Watterston, using Newton's law, according to
which the temperature is proportional to the radiation, found a value
7,000,000° 0.

Violle, on the other hand, found far too low a temperature by
using the law of Dulong and Petit. He obtained an effective tem-
perature of 1500°, and considered that the actual temperature might
be 3000°.

The first determination on what we may describe as modern methods
was made by Rosettit He measured the heat received by a thermo-
pile from a lamp-black surface at various temperatures up to 300° C.,
then from a copper ball suspended in a flame, the temperature being
determined by a specific heat experiment. By a subsidiary experiment
he determined the ratio of emission of bare copper to that of lampblacked
copper, and ultimately arrived at the law of radiation already quoted
(p. 248). He then observed the radiation received from the sun and,
correcting for the absorption by the atmosphere, he obtained for the sun's
effective temperature a value about 10,000°.

Another important determination was made by Wilson and Gray.*
They compared the radiation from the sun with the radiation received
from a platinum strip which they could raise by an electric current to
any desired temperature, the radiation being measured by Boys's radio-
micrometer. That of the sun was reflected into the instrument by a
mirror of known reflecting power. They first found that the radiation
from the platinum increased according to Stefan's fourth-power law, up
to the highest temperature which they could measure. Then they used
a result found by Rosetti, that at high temperatures the emission from
bright platinum is 1/2*9 of that from lampblacked platinum, and so they
calculated the temperature of lampblacked platinum at which it would
give off the same energy per sq. cm. as the sun, and they found for
this a value varying from 6200° 0. to 7400° C., according to the value
assigned to the absorption by the earth's atmosphere.

In a later paper Wilson § described a modification of the experiment,

Science Abst., xiii., No 660. f Ph*1- Mag., viii., 1879.

Phil. Trans., A., 1894, p. 361. § Proc. R.S., Ixix., 1901-2, p. 312.

256 HEAT.

in which the radiation from the sun was compared with that from a hole
in the wall of a constant-temperature enclosure. The enclosure was an
iron and porcelain tube heated in a gas furnace. Its temperature was
given by a Callendar platinum thermometer. One end of the tube was
open, and in front of the opening was a rectangular area of variable
width, and this was arranged so that the radiation from the sun was
equal to that from the enclosure.

If p is the fraction of the sun's rays passing through the atmosphere,
and if q is the fraction reflected by the mirror, we may put

pqO4 x solid angle subtended by the sun at the receiving surface
= (T4 - T04) x solid angle subtended by the rectangular aperture.

where 0 is the effective temperature of the sun,

T the temperature of the enclosure,
and T0 that of the receiving surface,

T04 was negligible in comparison with T4, since T was of the order of
1000°. Then we may put

.34 _ T4 solid angle of aperture
pq solid angle of sun

The value of 0 depends on the value assigned to p. If, with Rosetti,
it is put at 0'71, the effective temperature of the sun comes out as
5768° absolute. If, with Langley, it is put at 0-59, the effective tem-
perature comes out as 6085° absolute.

It is interesting to compare with this the value given by the solar
constant and the constant of radiation determined by Kurlbaum (p. 250).
With constant 2*1, the radiation from the sun is 6800 watts per sq. cm.
The radiation constant, according to Kurlbaum, is 532 x 10~14 watts.

If the effective temperature of the sun is 6, then

5x3210-1404 = 6800
whence 0 = 5980° nearly.

If we adopt Langley's value of the solar constant, which is 3, we get
0 = 6500° about,

slightly above the value obtained by Wilson.

Another value may be obtained from the equation (p. 250)

Am0 = constant =2940
in conjunction with Langley's value for A^, viz.,

A..-O-5/fc
This gives 0 = 5880° absolute.

But this should hardly be taken as another mode of determining
the sun's temperature, since the equation A^fl = 2940 applies only to a
surface emitting full radiation. The close agreement of the result it
gives should rather be taken as indicating that the sun is nearly a full
radiator, when we consider its radiation as a whole.

Whatever the temperature of the solar surface, the interior on the
average is doubtless much hotter. If, as we are entitled to suppose from
the high temperature, the interior of the sun is in a mobile condition,
the tendency will be towards a condition of "convective equilibrium,"

RADIATION AND TEMPERATURE. 257

the fall of temperature outwards from the centre being such that a mass
moving outwards arid expanding as it moves will cool, owing to dimin-
ished pressure through expansion, at such a rate that it is in equilibrium
with its surroundings.

If the temperature falls as we go outwards less rapidly than this, a
mass raised up from the inside will be colder than its surroundings, and
therefore denser, and will tend to sink, or the equilibrium is stable.
But if the temperature falls more rapidly, a mass raised up from the
inside will be warmer than its surroundings, and will be less dense, and
will, therefore, continue to rise, or if equilibrium exists it will be un-
stable. Hence, as the cooling goes on from the outside, there is a limit
to the slope of temperature according to this convection law. As soon
as the slope is greater than this limit, the arrangement is unstable, and
circulation takes place, cooler material from the outside sinking down
and being replaced by hotter material from the inside, until the slope is
equal to or less than that of convective equilibrium. We probably see
such disturbances taking place in sunspots, which appear to be due to
the increased absorption of the colder layers thickened by the disturbance.

Source Of Solar Energy. — The enormous rate of loss of energy by
the sun — a radiation of probably 10,000 to 15,000 h.p. per square foot —
requires some explanation. We cannot suppose that the supply of energy
is merely yielded by the cooling of the body of the sun, for at such a
rate the sun would have cooled very appreciably within historic times,
and of this there is no evidence. There must be, therefore, some other
form of energy continually being converted into heat to keep the tem-
perature up. This cannot be chemical energy, for not only is the sun
above the temperature at which we believe chemical combination to be
possible, but even had it been composed originally of separate elements
giving out the greatest known amount of energy by chemical com-
bination, the energy would last but a few thousand years. One very
probable source was pointed out by Helmholtz, who suggested that the
sun's heat is the equivalent of the potential energy of the matter of the
sun gradually converted into kinetic energy, and then into heat, as the
body of the sun contracts.

As the sun cools slightly, the matter comes nearer together. The
potential energy thus disappearing is converted into heat, which not
only keeps the temperature up nearly to its original value, but yields an
enormous surplus which is radiated out.

If the sun were of uniform density throughout, and if as it con-
tracted every part were of the same density, a contraction of the radius
of the order of 200 feet or 6000 cm. per annum would yield potential
energy mechanically equivalent to that radiated out. This would not
amount to more than one second of arc in 6000 years, so that we can
hardly expect as yet to obtain direct evidence for the contraction.

It appears possible, too, that the solar energy is in part maintained
by the disintegration of radio-active bodies present in the sun. This was
suggested by Rutherford and Soddy,* and Rutherford states that 2'5
parts of radium per million by weight in the sun would account for its
present rate of emission of energy.

* Rutherford, Radio-activity, chap. x.

OHAPTBK XVIL

THERMODYNAMICS.

The Second Law of Thermodynamics — The Indicator Diagram — Isothermals —
Adiabatics or Isentropics — Heat Engines — Carnot's Reversible Heat Engine —
Carnot's Cycle — Conditions for Reversible Working — Examples of Reversible
Processes — Of Irreversible Processes — Efficiency of an Engine — Absolute or
Work Scale of Temperature — Efficiency expressed on the Absolute Scale —
Comparison of the Absolute with the Air Scale — Water-wheel Analogue
— Reversible Cycles in general — Entropy — Entropy-Temperature Diagram —
Quantities Analogous to Entropy — Entropy tends to increase — Dissipation of
Energy — Intrinsic Energy — Available Energy — Possible efficiency of a Steam
Engine.

WE shall now investigate the conditions under which the transformation
from heat into other forms of energy occurs, and the extent of the trans-
formation under given conditions.
We shall make use of a general
principle, which we may con-
veniently state here, known as —
The Second Law of Thermo-
dynamics.— We cannot transform
heat into work merely by cooling a
body already at the lowest available
temperature.

The significance of this state-
ment will be appreciated better
when we apply the law to re-
versible heat engines. Being a
negative statement, it is hardly
founded on direct experiment, but
rather on the general observation
that when we seek to transform
heat to work we usually derive the
heat from a hotter body, we trans-
form some of it, and are obliged to
give some of it to a colder body,

«

X

Q

"\

3

H

t

«

N

Volume
FIG. 147. — Indicator Diagram.

actually heating that colder body.
Any apparent exception will, on examination, be found to involve some-
thing beyond mere cooling of the already coldest body. For instance, if
a quantity of air be under great pressure it may be allowed to do a con-
siderable amount of work in expanding, and its temperature will fall
very appreciably, perhaps far below that of the surroundings. But it is
not merely cooled. It has not given up its heat by conduction. It has
changed its volume while doing the work. If the air could be put

THERMODYNAMICS.

259

through any process by which it returned finally to the same volume,
but was appreciably colder, and if it had meanwhile produced a balance
of work, then the second law would be untrue.*

The application of this principle will enable us to find the maximum
fraction of heat which under given conditions can be transformed to
work, and will further give us the temperature changes occurring when
bodies undergo known strains or are subjected to known stresses.
Incidentally, it will give us a scale of temperature which is definite
and quite independent of any particular substance — Lord Kelvin's
work scale.

As our investigations will be carried out chiefly by the aid of the
Indicator Diagram, we shall first give some account of that diagram as a
mode of setting forth the conditions of a body and the relations between
its temperature, pressure, and volume.

The Indicator Diagram. — In this diagram two axes at right angles
are chosen as volume and pressure axes respectively. A point on the
diagram represents by its ordinate the pressure of the body, supposed to
be uniform throughout, and by its abscissa
the volume. Thus in Fig. 147 A represents
the condition of a body which has volume OM
and pressure AM.

If any change occurs in the body the
successive conditions may be represented as
regards pressure and volume by a series of
points which will together form a line either
straight or curved, joining the points repre-
senting the initial and final conditions, as
APQB in Fig. 147.

We may represent the work done by or on
a body during a change of volume by an area
on the diagram. Let S (Fig. 148) represent
the surface of the body when in the condition represented by P on the
diagram, and let S' represent the surface when the body is in the con-
dition Q very near P, so that S' is only very slightly larger than S. Let
a be a small area on S, which has moved against the pressure through a
distance d.

If p and p' are the initial and final pressures, the change is so small
that we may suppose the pressure to vary uniformly from p to p'. Then
the mean pressure is ^(p +p), and the work done is ^(p +p')a.d. Summing
up all over the surface, the total work

FIG. 148.— Work done by
a Body in Expansion.

P+P'

"2, ad.

= ™ • x volume between S and S'.
2

p+p' . . ,

mt-*—~£- x increase in volume.

* A statement less general but sufficient for our purpose in Thermodynamics
would be : We cannot by a cyclical engine derive work continuously by conducting
heat into the engine from the coldest part of the surroundings.

260

HEAT.

But

p +p PH + QK .

in Fig. 147,

and increase in volume
Therefore, total work

2

= HK
PH + QK

xHK

= area of slip PK.

The total work from A to B is evidently the sum of all such slips, or the

area AMNB.

This work represents the energy given out by the body merely

through expansion against the outside pressure. If the body contracts,

moving, say, from B to A, the same area will represent the work done

011 the body, or the energy given into
it through contraction under outside
pressure. If the point representing
the condition of the body moves from
A to B and back again along the
same course, evidently, on the whole,
no external work is done, the two
quantities being equal and opposite.
But if the return course is not
the same, we may have a balance
left over.

Suppose that the body goes
along ACB (Fig. 149), returning
along BDA. Marking work done
by by shading thus ///, and

volume
FIG. 149.— Work done in a Cycle.

work done on by shading thus
\\\, where there is a cross
shading the two neutralise each
other, and the balance of work done
by the body is represented by the

area included by the curve ACBDA. Had the change of condition been
represented by a counter-clockwise motion round the curve, the same
area would have represented the work done on the body. A clockwise
motion round the area then represents the giving out of so much energy.
Isothermals. — If the temperature of a body is kept constant, for
each pressure there is in general a single definite value for the volume,
and the relation between volume and pressure will be represented by
a curve. For instance, in the case of a gas p x v is nearly constant for
a given temperature, and the curve will be nearly an equilateral hyper-
bola. Such a curve is termed an isothermal. We may draw a series of
isothermals, each corresponding to a different temperature. The reader
can easily plot the isothermals for a gas at 0° 0., 100° C., and 200° 0 on
the assumption that pv = RT when T is the temperature on the gas scale
and K is a constant for a given gas.

The isothermals for liquids and solids are in general nearly parallel
to the pressure axis. Thus for water an isothermal as it rises will only
approach the pressure axis by --- of the volume for an increase of 1

THERMODYNAMICS. 261

atmosphere, so that it is impossible to represent it to scale on an ordinary
diagram.

AdiabatiCS or IsentrOpiCS. — If a body is not allowed to gain or lose
heat by conduction, and if the volume and pressure alter, but in such a
way that no kinetic energy is acquired by the body or its parts, the
change is said to be adiabatic. When, for example, a mass of gas is
enclosed in a non-conducting cylinder under a loaded piston, a very
gradual change in the load will produce an adiabatic change in the gas.
But a sudden change, such as a sudden finite decrease of load, will result
in rapid motion of the piston and rapid motion of the gas. Some of the
energy of this internal motion of the gas will be converted into heat by
viscosity, and the body will thus receive heat, though from itself. In the
celebrated experiment of Gay-Lussac and Joule on the expansion of a gas
when no external work is done (p. 120), no heat is given to or taken from
the outside, but the gas acquires considerable kinetic energy which is
ultimately converted into heat, each element receiving heat from the
surrounding elements through the viscosity. Hence the change is not
adiabatic. As we shall show later, the temperature would have fallen
about 70° had the change been truly adiabatic, the gas losing energy
through external work. This cooling with adiabatic expansion shows
that the adiabatics of a gas on the indicator diagram are steeper than the
isothermals, for the adiabatic through a given point must move down to
a lower isothermal. We shall show later that the adiabatics are steeper
than the isothermals in all cases. For a reason to be given later the
adiabatics are also termed isentropics.

Isopiestics and Isometrics. — Lines parallel to the horizontal axis
indicate a change of volume at constant pressure and are termed Iso-
piestics. The lines of equal volume parallel to the vertical axis are
termed Isometrics,

Heat Engines. — Any arrangement for the transformation of heat
into mechanical energy is termed a heat engine.

By considering the most familiar instance, the steam engine, we may
see what are the essentials of a heat engine.

There is a working substance, the water and steam. This is made to
expand on conversion into steam by the communication of heat from the
sides of the boiler — the source of heat. But only a part of the heat
communicated by the source is turned into work. In order to make the
transformation continuous, the steam is ejected from the cylinder and is
either allowed to expand into the cooler air, carrying off with it much of
the heat originally given, or it is passed into a cold condenser, where it
yields up heat to the condenser and is turned into cold water. The
general nature of the process then consists in the communication to the
working substance of heat from a hotter body — the source — the trans-
formation of some of this heat to work by the expansion of the working
substance and the communication of heat not transformed to a colder
body, either the outer air or a condenser. In thermodynamics the colder
body is termed the Refrigerator.

It is of the utmost practical importance to find what is the maximum
fraction of the heat leaving the source which can be transformed to work
when the temperatures of the source and refrigerator are known. For
a comparison of the actual fraction transformed in n given engine with

262

HEAT.

the maximum enables the engineer to determine how far the engine is
working economically.

The steam engine is a complicated engine unsuitable for the investiga-
tion of the general theory of heat transformation. We shall use, instead,
a much simpler ideal machine, first imagined by Carnot, the founder
of this part of thermodynamics, and described in a celebrated essay,
Reflexions sur la Puissance Motrice du Feu, published in 1824. This
engine is outside the range of practical experience, but inasmuch as we
can see how it would work if the practical details of construction and
manipulation were surmounted, the abstract conception is perfectly
legitimate.

Carnot's Reversible Heat Engine. — Let us suppose that we have a
cylinder (Fig. 150) with absolutely non-conducting walls, and containing a

Conducting

Source
kept at f

Variable
Load

Working Substam V,

A/o/7 Conducting
Table

Conducting

Refrigerate'

kept el t'"

FlG. 150. — Carnot's Engine. Shaded parts non-conducting.

non-conducting piston perfectly fitting yet without friction. Let the cylinder
contain air or other working substance under pressure applied by the
piston, which we may suppose loaded to any desired extent. Let the
bottom of the cylinder be perfectly conducting. As source and refrigera-
tor let us have two bodies arranged as stands, as in Fig. 150, with a non-
conducting table between them, their temperatures being maintained
constant somehow at t" and t'° respectively. We shall take the working sub-
stance through a series of changes such that it ends in its initial condition.
This series, known as Carnot's Cycle, is represented on the indicator
diagram by the four-sided figure, ABCD (Fig. 151), bounded by BC and
AD, the t and t' isothermals, and two adiabatics BA and CD. We
suppose the substance in the cylinder initially in the condition represented
by the point A, Fig. 151, at the temperature £'. Placing it on the central
table (Fig. 150), we gradually increase the load, decreasing the volume
adiabatically until we arrive at the point B, when the temperature has
risen to /. Now, sliding the cylinder along the table on to the source we

THERMODYNAMICS.

263

gradually decrease the load, allowing slow expansion. In general a
substance tends to cool on expansion, but the substance is here in contact
with a conductor maintained at tf, from which it will take in heat, and if
the change is slow enough, the temperature of the working substance
will never fall sensibly below t. When some arbitrary point C is reached
we slide the cylinder back on to the non-conducting table, and then
gradually unload the piston, the substance expanding adiabatically till
the temperature has fallen again to t', the state being now represented by
D. Finally we slide the cylinder on to the refrigerator and increase the
load. In general, compression tends to raise the temperature of a
substance, but we must here effect it so gradually that all excess of heat
is conducted away to the refrigerator and the temperature never rises
sensibly above t'. When the pressure arrives at its original value the

FlG. 151. — Carnot's Cycle.

volume is alro at its original value since the temperature is the same, or
the state of the body is again represented by A.

The balance of external work done by the working substance is
represented by the area ABOD, and this is the mechanical equivalent of
the excess of the heat taken from the source over that given to the re-
frigerator, for the substance in the cylinder has exactly the initial
amount of energy and the only changes are in the source and refrigerator.

The process is reversible, i.e. we might work counter-clockwise round
the cycle ABCI), first sliding the cylinder on to the refrigerator and
taking in heat from it during a sensibly isothermal expansion, this heat
being equal in amount to that given up in the direct working, then
sliding it on to the table and compressing adiabatically till the tempera-
ture is t, then compressing isothermally till B is reached, giving up to
the source heat equal to that taken in in direct working, and finally ex-
panding adiabatically till we have the initial temperature, volume, and
pressure. In each part of the cycle the work done is equal and opposite
to the work done in the same part during the direct working. Then by

264 HEAT.

doing work on the substance represented by ABOD, a certain quantity
of heat has been taken from the refrigerator and a quantity exceeding
this by the heat equivalent of ABOD has been given to the source.

Conditions for Reversible Working1. — In order that the same

figure may represent the direct and reverse working, that is, in order
that indefinitely small changes in the external conditions shall reverse
the order of change, certain conditions are necessary. These conditions
are —

1. That the working substance shall never differ sensibly in tempera-
ture from the bodies to which it is giving or from which it is receiving
heat ; for suppose that while the working substance goes along BC in the
diagram, the source is at a higher temperature T, then it is impossible to
make the working substance return along CB merely by a gradual in-
crease of load. The source being T° - 1° hotter, the substance cannot
yield heat up to it.

This condition requires that the isothermal parts of the cycle BO and
DA should be traversed exceedingly slowly.

2. That the pressure exerted by the working substance on the piston
shall be sensibly equal to the load, for then, and then only, will in-
finitesimal changes of load suffice to reverse the direction of motion.
This condition requires that the motion shall always be exceedingly
slow.

Some processes, in which kinetic energy is generated by a difference
between internal pressure and external load, may be reversible at the
initial and final points though not at intermediate points. For example,
if a gas is contained in a vertical non-conducting cylinder under a
frictionless piston, and if the load is suddenly decreased by a finite
amount, the piston will spring up and move out to a certain point, the
same, if we can neglect the effect of viscous friction, as if the load had
been decreased gradually to that corresponding to the final volume, and
the whole change had been adiabatic. At this final point the piston will
itself reverse the motion, and will in fact continue to move harmonically.
The work done between the extreme points is equal to load x distance
moved out. But at intermediate points, the motion evidently is not
reversible by an infinitesimal change of load.

3. That the machinery moves without friction. If there is friction
the load will have to be altered by a finite amount to reverse the motion.
The heat generated by the friction is also " irreversible," for it will be
given up by the working substance to the surroundings, whether the
motion is direct or reverse.

Examples Of Reversible Processes.— It will be realised from the
foregoing that exactly reversible processes are ideal, in that exact reversi-
bility requires exact equilibrium with surroundings, that is, requires .a
stationary condition, while a process is necessarily a changing condition.
But we can approximate as closely as we like to the conditions of
reversibility, by making the conditions as nearly as we like those re-
quired, and lengthening out the time of change.

As an illustration, imagine a quantity of water at 100° 0. under a
piston loaded to 1 atmosphere. Imagine the cylinder to be kept exactly
at 100° 0. by, say, a current of steam round its outside. There will be
equilibrium. But if we decrease the load ever so little, a minute bubble

THERMODYNAMICS. 265

of steam will suffice to start evaporation, which will continue until all
the water has become steam at 100°. At any point we may restore
equilibrium by holding the piston while we restore the infinitesimal load
removed ; and at any point we may reverse the motion and condense the
steam by adding an extra infinitesimal load so that the pressure is just
over 1 atmosphere. As another illustration, consider a quantity of ice
and water under atmospheric pressure at 0° 0, Keep the surrounding
temperature 0° and the proportion of the two remains unchanged. But
lower the temperature ever so little below 0° and the ice increases ; raise
it ever so little above 0° and the water increases. The change from
water to ice or ice to water may be reversed by indefinitely small
temperature changes.

Examples of Irreversible Processes. — We may contrast with
these the case in which water at 100° is under a piston with a load of,
say, half an atmosphere. Evaporation with expansion will ensue, and
the piston will ultimately rise about twice as far as in the previous case,
and will then be in equilibrium. But an indefinitely small increase of
load will now only lead to an indefinitely small compression, and not to
a reversal of the whole process. Clearly, too, the process is very far
from reversible at intermediate points. Or suppose that a lump of ice is
thrown into water at 10°. The ice will melt, but no known process will
suffice to make the water at 10° change back to ice.

"We conclude, then, that in Carnot's reversible cycle the machinery
must be free from friction, the working substance must never differ
appreciably in temperature and pressure from its surroundings, and that
the process must be indefinitely slow.

Efficiency Of an Engine. — The fraction of the heat received from
the source which an engine converts into work is termed the efficiency of
the engine.

It is convenient to denote any quantity of heat in calories by H, and
its work equivalent by Q = JH when J is the mechanical equivalent of
1 calory.

If an engine receives H units of heat from the source and produces
W units of work its efficiency is

We shall now prove that

All reversible engines working- between two given tempera-
tures and taking- in equal quantities of heat from the source
are equally efficient.

If possible, suppose that one reversible engine A is more efficient
between a given source and refrigerator than another reversible engine
B. Sending B through a cycle, let it take S of heat from the source and
give up R to the refrigerator producing W of work. Or, in reverse
working, when W of work is done on it per cycle, it will take R from
the refrigerator and give S to the source. Now, by hypothesis, for every
quantity of heat S which A receives from the source, it produces more
than W of work, say W'. Allow A to work forwards, and out of every
quantity W of work which it yields, take W to work B backwards
through a complete cycle. Then on the Avhole the source will neither

266

HEAT.

FIG. 152.

lose nor gain heat, for B restores what A takes. But after each cycle
which B goes through, a balance of work W - W remains over. This
can only come from the refrigerator, which must give up to B more heat
than it receives from A. Allowing the process to continue indefinitely
we should obtain any quantity of work by abstracting heat from the
refrigerator. Now the refrigerator may be arranged to be the coldest

body in the system, so that we should
be obtaining work by merely extracting
heat from the already coldest body.
For the working substances are at the
end of each cycle in their initial con-
dition in every respect. This is con-
trary to the experience embodied in
the Second Law of Thermodynamics.

Hence we conclude that A cannot
be more efficient than B, or that the
two reversible engines are equally
efficient.

We may further prove that the
efficiency of a reversible engine working
between given temperatures is the same
whatever quantity of heat is put in at

the higher temperature. That is, if on an indicator diagram (Fig. 152)
ABCD represents a Carnot cycle for a reversible engine, the area ABCD
is proportional to the heat taken in along BO, so long as BO and AD
are given isothermals. For suppose that in a second case double the
heat is taken in, the working substance at the higher temperature
moving to E along the BO isothermal, and suppose that BEFA now
represents the cycle. We may imagine a second engine exactly like the
first working round the cycle ODFE. Then from the preceding proposition
it has the same efficiency
for the same quantity of
heat taken in. But the
heat given along CE equals
the heat given along BO
by supposition. Hence the
area CDFE equals the area
BADC. Or, if the heat
given along BE is double
the heat given along BO,
the work BF is double the

work BD. That is, the efficiency is the same whatever quantity of heat
is taken in.

Absolute or Work Scale of Temperature. — We may express the
equal efficiency of all reversible engines between given temperatures,
and its independence of the quantity of heat put in, by saying that the
efficiency between given temperatures depends only on those tempera-
tures, and not on the nature or conditions of the particular substance
used. This independence of the working substance suggested to Lord
Kelvin that the efficiency might be used to indicate the temperature on
an absolute scale — i.e. one in which the given intervals would have

~P

py

rs

A

B

c

D

1'iG. 153. — Equal Temperature Intervals
on the Work Scale.

THERMODYNAMICS.

267

the same proportion whatever the substance used. Let ABCD (Fig.
153) be a number of heat reservoirs, arranged in descending order of
temperature, and let reversible engines «3, /3y, yd ... be set working
between the successive pairs. Let «/3 go through a cycle, taking QA
from A and yielding QB to B. Let 167 go through a cycle, taking QB
from B and yielding Qc to 0. Let yb go through a cycle, taking Qc from
C and yielding QD to D and so on, each engine taking from its source
what its predecessor had yielded to it. The quantities of work yielded
will be QA - QB, QB - Qc, Qc - QD, &c., the heat being expressed in
mechanical measures.

Now, adjust the temperatures so that the quantities of work given
up by the successive engines are equal, or so that

Then we define the intervals of temperature as all equal. Or we may

say that if a quantity of

heat is sent down a succes-

sion of temperature steps,

the quantity lessening as

it goes by reversible trans-

formation into work, the

steps are equal when the

amount transformed in

each step is the same.

Though this defines equality

of temperature interval, it

still leaves it open to us to

fix on any desired interval

as 1°, and to start from

any desired zero point.

We may represent this
process on the indicator
diagram. Drawing an iso-
thermal AB (Fig. 154) and
the adiabatics ACE, BDF,
the isothermals drawn so as to make the areas AD, CF, &c., equal, are
at equal intervals of temperature.

We may also note that if BG, GL represent distances along which
the same amount of heat is taken in as along AB, the adiabatics through
G and L must form a series of quadrilaterals equal in area to AD. This,
of course, is merely equivalent to saying that the efficiency between given
limits is independent of the quantity of heat taken in.

Efficiency expressed on the Absolute Scale. — Any number of

the engines a/3, $7, &c., may be joined together so as to work in concert,
and they will take heat from the source of the first, yielding heat to the
refrigerator of the last, transforming the difference to work. On the
whole, the intermediate reservoirs will be unaffected. Evidently the
process may be exactly reversed on putting in the work at the bottom,
so that the arrangement forms a compound reversible engine.

If we keep to a given temperature of source indicated by Os, on a
scale at present arbitrary as to length of degree and zero, and if QR be the

FIG. 154.

268 HEAT.

temperature of the final refrigerator, the efficiency is evidently propor-
tional to the number of engines or steps down, and we may write—

Efficiency = A(0S - 0K)

where A does not depend on 0R, but is a function of Os alone.

Let us now suppose that it is possible to continue the steps down so
far that all the heat put in is converted to work, and that none remains
over to put into the lowest refrigerator. The temperature of this
refrigerator is the lowest conceivable consistent with the First Law of
Thermodynamics, for a still lower temperature would enable us to get
more work out of a given quantity of heat than its mechanical equiva-
lent. We choose this temperature, then, as the zero of the new scale,
and shall term it the absolute zero.

If the refrigerator is at the absolute zero 0R = 0, and the efficiency is
1, or all the heat put in at Os is converted to work. Then we have

A=

With any other temperature of refrigerator the efficiency is

This efficiency has been obtained from a peculiar compound engine ; but
since that engine is reversible, its efficiency equals that of any other
reversible engine working between the same temperatures and the
result is therefore general.

If Qs is the heat taken in at the source and QR that given out to the
refrigerator by a reversible engine, then, equating the temperature ex-
pression to the work expression for the efficiency, we have —

Efficiency = %^R=^B
Vis PS

whence 5 = 5

-

PS #B

We may put this result into the following form. If any quantity of
heat is allowed to go down a temperature slope, some of it being inter-
cepted and transformed, then if the transformations are reversible, so
that by exact reversal the original quantity of heat could be returned

to the source, the quantity -^ remains constant down the slope. We

may here mention that this quantity, ^, is termed the Entropy put in GO
the system. We shall return to the subject of Entropy later.

THERMODYNAMICS.

269

Comparison of the Absolute with the Air Thermometer Scale.

—To make the formula just found of practical importance, it is necessary
to show what relation the absolute bears to known scales. Fortunately
it can easily be shown that it nearly coincides with the air thermometer
scale. Since all gases give nearly the same temperature indications if
far above their condensing points, and the gas scale is therefore nearly
independent of the particular gas used, we might perhaps be led to
expect this.

Let us work a mass of air in a reversible engine between two tem-
peratures expressed on the air scale. Our knowledge of the properties
of air enables us to determine the efficiency of the engine in terms of

FIG. 155. — Air Scale and Absolute Scale.

the air temperatures, and we may compare the efficiency so found with
its expression in terms of the absolute temperatures.

Let AD, BO (Fig. 155) be two neighbouring isothermals of the air,
the temperatures measured from - 273° C. being t and t' respectively.
Let AB, DC be adiabatics, and let AD and t' — t be so small that ABCD
is sensibly a parallelogram. Working round the cycle reversibly, heat
is taken in along BO and given out along DA, the difference being
converted into work represented by the area ABCD.

The Efficiency =

Area ABOD

Mechanical Equivalent of Heat taken in along BC.

But by Joule's experiment on the expansion of gases, a gas expanding
and doing no external work remains very nearly at the same temperature
without any supply of heat. If it does work and still remains at the
same temperature, the heat supplied must be equivalent to the work
done. But this is the condition of the air along BC, and the work done
is represented by BCNM, which is therefore the mechanical equivalent of
the heat taken in along BC.

270 HEAT.

Then the efficiency —

_ABGD.

" BONM.

KBGL

~BONM

BK + OL

xMN

BM + CN
= x

2

BK + GL
"BM+CN

Now BM and KM are the pressures at constant volume correspond-
ing to t' and t, and are therefore proportional to t' and t, this being really
the definition of t' and t with the air thermometer.

. KM t

BK t'-t

and SM— r

CL t'-t

Similarly

.". efficiency =

CN~ t

BK + CL t-i

BM+CN" H

If 6 & are the temperatures on the absolute scale corresponding to
t and t' —

ff-e t'-t

& t

0=1

Or the temperatures on the one scale are approximately proportional
to those on the other, and we may conveniently choose the degrees so
that 0°C. shall be - 273 on either scale, or, more exactly, that the interval
between 0° C. and 100° C. shall be 100 degrees on either scale.

This is only an approximate result, for it assumes that in Joule's
experiment (p. 120) there was no change of temperature, an assumption
not quite exact, as we know by later experiments. It also assumes that
the air scale for the volume OM is proportional to that for ON, an
assumption exceedingly near the truth, but still probably slightly inexact.

Non-Reversible Cyclical Engine.— The foregoing propositions
relate to reversible engines only, but if we have an engine not reversible,
say one in which the working substance is at a lower temperature than
the source when it takes in heat and is at a higher temperature than the
refrigerator when it gives out heat, and yet goes through a cycle so that
it recurs to its initial condition, it is easy to show that

No cyclical non-reversible engine working between given temperatures is
more efficient tJian a reversible engine. For let the non-reversible engine

THERMODYNAMICS. 271

work forward, driving a reversible engine backward. If the first is more
efficient it can get more work from a given quantity of heat. Put it so
that it will be able to drive the reversible one at such rate that the revers-
ible restores to the source the heat taken out and there is a balance of work
over, which can only come from the refrigerator. If this is arranged to
be the coldest body of the system the result is contrary to the Second Law.

Water-Wheel Analogue of the Reversible Engine. — The reader may be
aided in understanding and remembering the foregoing theory of the
reversible heat engine by considering an analogous case of energy tran-
formation.

Let us suppose that we have two reservoirs of water at different
levels, and that we wish to transform the potential energy of the water in
the higher by means of a water-wheel, allowing the water to fall in the
wheel from one reservoir to the other. Evidently we cannot get all the
potential energy transformed by the wheel. We put in so much at the
top and take out a smaller quantity at the bottom. If the wheel works
very slowly, receiving the water at the level of the surface of the higher
reservoir, and parting with it at the level of the surface of the lower one,
and if there is no leakage and no friction, then the whole of the potential
energy lost will be usefully transformed by the wheel, and no wheel
between the same levels could be more efficient. We see at once that
the conditions of maximum efficiency and reversibility coincide. It is
obvious that it is impossible to get mechanical effect out of the lower
reservoir if that is already in the lowest available position — the analogue
of the Second Law of Thermodynamics. The total potential energy of a
mass of water in the upper reservoir is the work which would be done in
lifting it from the absolute zero of level — the centre of the earth.

If then we define

Efficiency = Work got out
Energy put in

the denominator is very nearly the same for the same mass of water at
different places, and the efficiency is very nearly proportional to the
difference in height, in feet, between the two reservoirs. But the height
in feet does not give an absolute scale of efficiency, for, as gravity varies,
the work for a fall of one foot depends to some extent on the situation of
the wheel. This is analogous to the gas scale of temperature. We may
get an absolute scale by discarding the length measure of difference of
level and substituting for it a work measure. Dividing up the distance
between the centre of the earth and the level of the higher reservoir into
steps, so adjusted that a reversible wheel in each step will give the same
amount of work from the passage of the same quantity of water, we may
define these steps as having equal differences of level. Now the efficiency
of any wheel becomes

No. of steps occupied by wheel.

Total No. of steps from centre of earth to higher level,
or if n = No. of steps from centre of earth to higher level

and n' = the No. from centre of earth to lower level.

-ncc. • n — ri
Lmciency =

272

HEAT.

and the new scale of level is analogous to the absolute scale of tempera-
ture.

Since the total quantity of energy is proportional to the total number
of steps, and the quantity transformed is proportional to the number-
descended, the quantity still remaining in the water is proportional to
the number of steps remaining.

If then a quantity of water is started downhill, the energy being
transformed as it goes by reversible wheels, the quantity

Energy still possessed
level of water possessing it
is constant.

This is really equivalent to saying that the quantity of water remains

FIG. 156. — Any Reversible Cycle.

constant.

Q

Hence, quantity of water is analogous to the quantity

~, which we have termed entropy.
6

Reversible Cycles in General ; with Varying Temperatures
of Source and Refrigerator, — We have found that if Qs of heat is

taken in by a reversible engine at a definite temperature 6S, and QK is
given out at a definite temperature 0R, then ~? — ^? = 0.

But in many cases the heat is not all taken in at one definite tem-
perature and given out at another. We have, therefore, to generalise
this result.

Let the working substance go through a cycle represented on the
indicator diagram by ABCKc&aA (Fig. 156).

Take a succession of points ABC . . . very near together, and through
them draw adiabatics Aa, B&, Cc across the area. Draw the isothermals

THERMODYNAMICS. 273

AD, BE . . . ad, be . . . &c. Instead of sending the substance round the
curve, let it go through the series of steps ADBEC . . . cebda . . .

In the limit the heat received along each piece AD is the same as
that along each piece AB, for, sending the substance round the cycle
ABDA, the area ABD is the mechanical equivalent of the difference
between the heat received along AB and that given out along DA, since
BD is an adiabatic ; and this area is an indefinitely small fraction of the
area ADda, which is itself only a fraction of the heat received along AD.

We may, therefore, as far as concerns heat taken in or given out,
replace the curve ABO ... by the steps ADBEC . . .

Let QAB QBC QCD ... be the heats received along AB, BO, CD . . .
respectively, or, as we have just shown, along AD, BE, OF . . .

Let Qc6 Q6a ... be the heats yielded along cb, ba . . . which are
sensibly equal to those along eb, da . . . If now we imagine the sub-
stance to be sent round the cycle ADda, we see that this is the kind of
cycle already considered.

where 0A and Oa are the temperatures at A and a.
Sending it round the cycle BEei

QBC Qrf>

0B and 06 being the temperatures at B and b and so on.

If we regard the negative sign as denoting that the substance receives
negative heat we may add up all the equations and denote the result by

where each element of heat received is to be divided by the temperature
of the substance when receiving it. Using the notation of the Integral
Calculus the result may be written,

fdQ

)~e

round a reversible cycle.

This deduction is often quoted as the Second Law of Thermodynamics
but it appears preferable to consider the fundamental experience, given
at the beginning of the chapter, as constituting the law. In reversible
cycles then we have

by the first law I dQ = Work done,

by the second law I ^ = 0.

J v

the former telling us the relation between the heat disappearing and
the work done, and the latter giving a relation between the heat and the
temperature.

8

274

HEAT.

Entropy.

Definition of Entropy. — If a substance at a temperature 6 receives

Q

heat Q, it is said to receive entropy -g .

Similarly if it parts with heat Q, when at a temperature 6, it is said
to lose entropy *? . If the gain or loss of heat affects the temperature,

we may divide the gain or loss into elements each at practically constant
temperature, and then the amount of entropy gained or lost is

The Entropy gained by a body in passing from one state
to another depends solely on the initial and final states. Let

FIG 157. — Entropy. Change in passing from A to B
the same by all paths.

A in the indicator diagram (Fig. 157) represent some standard state of
the body, B some other state. Then we shall show that, however the
body passes from A to B, the entropy received is the same, or the
entropy at B depends solely on the position of B and not on the mode in
which the state changes from A to B.

Let ACB, ADB, be two paths from A to B. Let the substance be
carried round the cycle AOBDA by a reversible process,* then

2~ from A through C to B + 'sS- from B through D to A = 0.

* Inasmuch as the conditions of reversibility are external conditions, any series
of changes of temperature, pressure, and volume in a body can be imagined to be
carried out reversibly by suitably adjusting the external surroundings.

THERMODYNAMICS.

275

Now if BDA is described in the reverse direction ADB, heat taken in
becomes heat given out and we must change the sign of Q in the second
term of Ahe left side of the equation. Changing the sign and transposing

2^ from A through 0 to B — 2~ from A through D to B

V 0

or the entropy received by either path is the same.

The Adiabatics are Isentropics. — If a body is compressed or expanded
adiabatically, i.e. so that, whatever energy may be transferred as work,
none is conducted as heat, the entropy remains constant and the adia-
batics on the indicator diagram are lines of equal entropy, or isentropics.
The different adiabatics may be
denoted by the value of the entropy
along them, reckoned from some
standard condition, just as the dif-
ferent isothermals are denoted by
the value of the temperature along
them. The entropy of a body is
usually denoted by <f>.

The Entropy Temperature

Diagram is a diagram in which
the ordinate represents the tempera-
ture, and the abscissa the entropy
reckoned from some standard con-
dition.

If AB (Fig. 158) represents a
small change of condition of a body,
on an entropy temperature diagram, FIG. 158. — Entropy-Temperature
the temperature changing from AM Diagram,

to BN and the gain of entropy being

MN, and if Q be the quantity of heat received in the change from A
to B

Q

H

entropy

M N

Q = MN.

= area ABNM.

Similarly for every element in a finite change. Then the heat received
in a change from P to Q is equal to the area PQKH.

Now if we take a body through any cycle, returning to the original
state the entropy and temperature both have their original values, and
a cycle is therefore represented by a closed curve on the diagram. The
area included by the curve is the difference between the heat taken in
and the heat given out, and so represents the amount of heat transformed.
We shall make use of the diagram later, but without further discussion
now, the reader will see that a Carnot's cycle is represented by a rect-
angle with sides parallel to the axes, and he may easily prove that for a

276 HEAT.

gas the lines of equal pressure and of equal volume are represented by
curves of the type

where, for the one, c is the specific heat at constant pressure, and for the
other the specific heat at constant volume.

The total Entropy of a system is unchanged by the perform-
ance of reversible cycles between its parts. When a body is

worked between two given temperatures we have seen that — , the

Os

entropy taken from the source, equals ^K, the entropy given to the re-

0R
frigerator ; the total quantity is therefore unchanged.

But if the cycle is a perfectly general reversible one we still have

Y— — 0

2,0-0

or the total entropy received by the working substance is zero. But in
order that the cycle should be reversible, all passages of heat must take
place between bodies differing infinitely little in temperature, so that Q
and 0, being the same for giver and receiver, the entropy gained by the
working substance in any part of the system is equal to that lost by the
source whence the heat comes. Hence the total entropy lost by the
surroundings of the working substance is also zero.

Quantities Analogous to Entropy. — The conception of entropy is some-
what difficult in that, as it does not directly affect the senses, there is
nothing physical to represent it. It may assist the reader if we point
out some already familiar quantities which are analogous to it. Since

Entropy = heat energy -f temperature,
. •. Heat energy = entropy x temperature.

Also in the Carnot reversible cycle,

Energy transformed = Qs x

0 — f)

-

= Entropy x fall of temperature.
Comparing this with gravitation energy —

Gravitation energy = Mass x level above zero,

the level being measured by work done on unit mass raised from zero level,
and Energy transformed = mass x fall in level.

Then entropy is analogous to mass and temperature to work measure
of level.

Or taking the case of electrical energy,

Electrical energy = charge x potential.

THERMODYNAMICS. 277

If the charge is allowed to pass by an insulated conductor to a lower
potential,

Energy transformed = charge x fall in level.

Then charge corresponds to entropy and potential to temperature.

Entropy Tends to Increase. — These analogies are not to be pursued
too far, for while mass and electrical charge (taking into account the whole
field) are constants, entropy is only constant in the very special case of
reversible transformation, when the maximum amount of heat is converted
into other forms of energy. In other cases it tends to increase. When-
ever there is any conduction between bodies at a different temperature,
there is a gain of entropy, for the hotter one loses Q at 6V and the cooler

gains Q at 02; the one loses entropy ^, while the other gains entropy

i

~, which is greater. If, then, the source is sensibly hotter than the work-

2

ing substance, or if the refrigerator is sensibly cooler, there is an increase
of entropy. Or if friction occurs, some of the mechanical work goes to
produce heat and the body receiving it gains entropy, which is not lost
by any other body, but which starts into existence through the friction.

Now in actual engines there is always conduction going on through
finite differences of temperature and there is always friction, so that
there is always gain of entropy.

And more generally in any change of condition of a body, there is no
loss and usually a gain in the sum total of entropy. If the body expands
or contracts along an adiabatic, then the entropy is unchanged. But if
it changes in any other way, heat must be taken in or given out by con-
duction. The entropy would only remain unchanged in amount (though
passed on from one body to another) if the conduction went on between
bodies at sensibly the same temperature. In practice there is always a
finite temperature slope, and always therefore a gain in the entropy of
the heat conducted.

On the whole, then, entropy tends to increase. This principle was
first laid down by Clausius.

If we compare transformations from heat with other transformations,
we at once notice a marked contrast. In heat transformation there is a
constant tendency on the part of the transformed energy to revert to the
original heat form, and there is a tendency for it to pass from one body
to another by conduction without transformation. But with gravitation
energy, for instance, we may allow a mass to fall from one level to
another, transforming all the available potential energy to other kinds to
the utmost possible extent, though not necessarily to useful forms, and
there may be no tendency of the transformed energy to revert to the
potential form. Or with electrical energy we may allow a charge to
pass from one conductor to another and obtain all the available electrical
energy as work or heat. There is nothing in either case corresponding
to conduction of heat without transformation. Whenever a rearrange-
ment of mass or of electricity occurs without communication /from the
outside, there is a loss of potential energy or of electrical energy.

If we could imagine a frictionless machine in which a mass descending
to a lower level lifted up another mass from the absolute zero level to

278

HEAT.

meet it, the mass being such that the total potential energy remained
unchanged, or if we could imagine an electrical system in which the work
gained by passage to a lower potential turned some machine, increasing
the charges at the lower potential so that the total electrical energy
remained unchanged, we should have cases analogous to the conduction
of heat.

The difference really turns on the fact that entropy tends to increase,
while its analogues, mass and quantity of charge, are constant.

Dissipation or Degradation of Energy. — Through conduction

and radiation there is a continual tendency to reduce differences of
temperature in the universe, and therefore to reduce the amount of heat
available for transformation. Though not diminishing, but on the
contrary increasing in amount, the heat is continually approximating
more to a dead level, being " degraded " or " dissipated." The quantity
of other forms of energy available for transformation is also diminishing,

FlG. 159.— Intriusic Energy.

for in every transformation which occurs some heat is generated, and this
can never be wholly retransformed.

There is, as it were, a tax levied on every transformation, the proceeds
being put into an untransformable fund. The amount of possible trans-
formation by any method which we can at present conceive is therefore
gradually diminishing, a fact which is expressed by saying that the
energy is undergoing dissipation or degradation. More and more,
energy is assuming the form of heat, which may be regarded as the
stable form, and a dead level of temperature is the state which we are
gradually approaching.

Intrinsic Energy. — The energy which a substance possesses in
itself is termed its intrinsic or internal energy. We have no means of
measuring the total quantity of a body's intrinsic energy, but we can
find the amount added in changing from one condition to another.

It will obviously be the amount of heat added less the external work
done. We may give some attempt at representation on the indicator
diagram. Let AB (Fig. 159) be the curve representing the change from
the condition A to the condition B Draw the adiabatics Aa, B/3, down to

THERMODYNAMICS.

279

8

the absolute zero isothermal a/?y. Of course we have no experimental
justification for drawing this isothermal, but for diagrammatic repre-
sentation we may suppose it to exist.

Taking the substance round the cycle AB/3aA, the whole of the heat
taken in along AB will be converted into work since no heat will be
given out along (3a, its temperature being zero. Then the area AB/3a
represents the heat taken in along AB. The external work done along
AB is ABNM. Then the gain of intrinsic energy along AB is

AB£aA - ABNM
= PB/3aP-APNM

or if y be a point at some distance along a/3 and yQ the ordinate, adding
PayQNP to each, the gain of intrinsic energy is

B/3yQNB - APa^yQMA.

The change in the intrinsic energy is represented then by the change in
the area AMNQy/3aA.

Available Energy. — The amount of energy in a body or system
which can be transformed to useful
work depends entirely on the condi-
tions under which the work is to be
done.

If we have, for example, a sub-
stance in an envelope under pressure,
and we can allow it to expand to a
given lower pressure, the work done
will depend on the nature of the
envelope. If it allows heat to pass we
shall get more work by an isothermal
than by an isentropic or adiabatic
expansion. For the adiabatics on the
indicator diagram, as we shall see
later, are always steeper than the
isothermals, so that, if in Fig. 160, AB
is the isothermal, AC the adiabatic through A, the work ABNM is
greater than the work AOPM. Of course the body in the first case
may have taken in energy through the envelope, and may have con-
verted some of this to work.

If the envelope is non-conducting, then the most work will be done if
the external pressure differs only by an infinitely small amount from
the internal pressure, so that the expansion will be adiabatic. For if
there is a finite difference, kinetic energy will be generated in the sub-
stance, which will ultimately tend to become heat, and some of the
energy will thus be lost for transformation to work.

Thus, in the gas engine, in which a mixture of coal-gas and air is
ignited with transformation of chemical energy to heat, the temperature
and pressure of the gas are raised. Obviously the cylinder should be
non-conducting, otherwise heat will pass out of the gas which might be
transformed. And the expansion should be adiabatic till the pressure
is as near as possible to the air pressure, the lowest available.

The Available Energy of an Isolated System. — If we have a system of

M P
FIG. 160.

280

HEAT.

bodies under such circumstances that they can neither part with heat
to, nor receive it from the outside, the entropy of the system cannot
diminish. In order to obtain the maximum work from the system, it
must evidently be brought to a condition of uniform temperature and
of uniform pressure, for if these are not uniform still more work may
be obtained by coming to uniformity. There will be some limiting
condition given as to pressure or volume. In coming to this, the best
course will be by adiabatic transformations, so that the final equals the
initial entropy, for if there is any gain, some of the energy which might
have been transformed has gone to increase the entropy of the system.
Now, if in the final state the entropy and some other quantity, say the
pressure, is given, the state of the system is determinate, and therefore
its intrinsic energy is determinate. Hence we may calculate the work

done in any convenient method of passing from the initial to the final
state, in which we obey the condition of constant entropy. For instance,
we may first allow all the bodies to come to the standard pressure adia-
batically. We may then use a reversible engine to bring them to the
same temperature, first say working between the two coldest as source
and refrigerator ; then when they are at one temperature, working
between the next above as source, and these two as refrigerator, and
so on.

We may represent the work obtained in the latter part of the process
on the entropy temperature diagram. Let us take the case of two
bodies only, at the same pressure but different temperatures. Let HAK
(Fig. ]61) represent the final pressure line for the hotter substance, AN
its temperature ; LBM the line for the same pressure of the colder sub-
stance, BN its temperature. If we turn the curve LBM round AB as
axis till BM is in the plane of the paper on the other side, lying ulong BH
and cutting HAK in H, evidently as the hotter body passes along AH,
giving up heat to a reversible engine with the colder as source, the latter

THERMODYNAMICS.

281

8

passes along BH, the two meeting at the common temperature H. For
this secures that the entropy remains constant. The heat converted is
given by the area AHB. In other words, if HCM is the horizontal line
from one curve to the other, bisected in 0 by AB, ON is the common
temperature and

HAG + BOM is the heat converted.

The Possible Efficiency of a Steam-Engine.— We shall conclude
this account of reversible engines, and of the idea of entropy arising
therefrom, by a calculation of the possible efficiency of a steam-engine
supposing that there is no waste of heat and no friction.

Let us for definiteness take an engine in which the boiler is at
140° C., say 413° absolute, and the condenser at 30° 0., say 303°
absolute. We shall suppose that the steam is admitted into the cylinder
at the temperature and pressure of the boiler till the moment of cut off,
that then there is adiabatic expansion till the temperature is reduced to
that of the condenser, and that
then the contents of the cylinder,
which will be partly saturated
steam at that temperature and
partly water, are let into the
condenser and there condensed
to water. We shall further
suppose, in order that we may
take the working substance
round a cycle, that the con-
denser water feeds the boiler.

The path may be simply
represented on an entropy tem-
perature diagram, Fig. 162.
Using absolute or work-scale
temperatures, A represents the
water returned to the boiler from the condenser at 303°, AB the water
as it rises in temperature to 413°, BC the conversion into steam at 413°,
CD the adiabatic expansion till the temperature is 303° again. In this
expansion water will at once be formed, and the result will be a mixture
of saturated steam and water, the proportion of water gradually increas-
ing as the temperature falls.* When 303° is reached, the further
condensation into water is represented by DKA.

The heat given during the cycle is represented by the area LABCNL,
and that transformed by ABCDA.

FlG. 162. — Entropy Temperature Diagiam
for Steam-Engine.

Then the efficiency =

ABCDA ABK + BCDK

LABCNL ABML + BCNM'

In calculating the value of this, we shall make very slight error in

* The reader may calculate the proportion of water at 303° — very nearly one-fifth
— by assuming that the entropy is the same at the beginning and at the end. We
may refer to Perry's Steam-Engine, § 200, for the mode of calculating the entropy of
wafer and of steam, or to p. 320 of the same work for tables giving the entropy at
different temperatures.

282 HEAT.

taking AB as a straight line, and the area LABML as 110 per gramme
of water. We have the values

BONM = latent heat at 413°, or 140° C.

= 606-5 - -695 x 140 = 508 (Regnault)

AL = 303; BM = 413.
To find ABK

ABK = IAK-BK = ^AK(BM - AL),

2 2

also ABML=!AK(BM+AL)=HO,

2

A -ry-n- BM - AL A T.T..T

whence ABK= - , ABML

110
-jj- x508

110x110 110 ~a

Then the efficiency = — -— ; -- 1- — — x 508

716 413 _

618
= 0-246 about.

If the cycle were a Carnot cycle, with heat put in only at the higher
and taken out only at the lower temperature, the efficiency would be

This is only a particular cycle, and the efficiency might be brought still
nearer the Carnot cycle value by allowing, say, isothermal expansion of
the steam at 413°, until the pressure was such that, on further adiabatic
expansion till the temperature fell to 303°, condensation was then just
beginning. But such a cycle would be less like that in the actual steam-
engine than the one considered.

The engine we have imagined is a perfect one for the particular
cycle, for we have supposed no loss by conduction or friction. Let us
now see how much coal would be required per horse-power per hour in
such an engine.

We may estimate that one gramme of coal gives a supply of 8000
calories in the source, or 8000 x 4'2 x 107 ergs.

Of this only 0-246 is transformed by our engine, or 1 gramme of coal
yields 8-27 x 1010 ergs as work.

THERMODYNAMICS. 283

Now, 1 H.-P. is 76 kgm. metres/sec.,

say 746 x 107 ergs/sec.,
and 1 H.-P. for 1 hour is 746 x 3600 x 107,

= 2680xl010 ergs say.
This requires 2680 -f-8'27 = 324 grammes of coal,

say 0'71 Ib. of coal.

Actual engines have by no means reached this efficiency over such a
temperature range. Probably 2| to 3 Ibs. of coal per horse-power hour
would be a good result at present.

It may be noted that if the whole of the heat-yield of the coal were
transformed, then we should have

1 H.-P. for 1 hour requiring (2680 x 1010) -r (8000 x 42 x 107),

= 80 grammes nearly,
or = 0-181b.

CHAPTER XVIII.

THERMODYNAMICS OF ISOTHERMAL AND ADIABATIC
CHANGES.

Heat taken in when a Body expands Isothermally — Heat to a neighbouring Adiabatic
the same by all paths — Change in Temperature when a Body undergoes a small
Adiabatic Change — Adiabatics steeper than Isothermals — Specific Heats at
Constant Pressure and Constant Volume — Their Ratio y equal to the Ratio of the
Isentropic and Isothermal Elasticities — Experimental Determinations of y for
Gases — Adiabatic Gas Equation— Fall of Temperature Upwards with Convective
Equilibrium — Internal Energy taken up by a Gas in Expanding — Comparison
of Air Scale with Absolute Scale— Generalisation of Indicator Diagram for
any Stress and corresponding Strain.

FROM the two laws of Thermodynamics, which, for reversible cycles,
may be stated in the forms

3=W and (2) l^S = 0

we may deduce the temperature changes or the heat developments in

many alterations of physical con-
dition, in terms of the known
properties of the bodies affected.

Some of the results may be
obtained by elementary methods,
using the indicator and entropy-
temperature diagrams. We pro-
ceed to obtain several of great
importance.

The Heat taken in when a
Body Expands Isothermally.—

Let the body expand at tempera-
ture 6 by a small amount dv, its
path being represented on the
indicator diagram by the element
AB of an isothermal in Fig. 163.
Then dv is represented by MN.

Let us make AB part of a Carnot cycle, the lower isothermal DC being
at temperature 0 - dd. Produce CD to cut AM in H. If all changes
are small, ABCD may be regarded as a parallelogram. We shall equate
its area expressed by aid of the Second Law of Thermodynamics to that
expressed in terms of the constants of the body.

Let Q be the work measure of the heat along AB.

Then ^ = ABCD = AH x MN.

0

FIG. 163.

ISOTHERMAL AND ADIABATIC CHANGES.

285

If we denote the change of pressure at constant voluma per degree rise
in temperature by u, then AH = (W0, and putting

. QcW

= <i>dd'dv

' e

and Q = <*>0dv. (1)

The qtuintity u> is only observed in gases, for which it is approximately

i-t but we can express it in all cases in terms of eg, the isothermal
v

elasticity, and a, the coefficient of expansion at constant pressure.
Let AB be an isothermal at 0 (Fig. 164), and let AC, CB be
drawn parallel to the axes, meeting at
C, of v. Inch the temperature is 9 — dO.
The delinition of e<> is

_ v x small increase of pressure
consequent decrease of volume

where the changes are at constant
temperature

AC

But

60=-.

(2)

FIG. 164.

Substituting in (1) from (2) we get
Q = ae<,ddv.

If we put eedv = vdp, we get the heat in terms of the pressure
change, viz.,

Q = avddp.

The heat taken in or given out by a body in passing from
a given condition to a neighbouring adiabatic is the same along

all paths. Let H in Fig. 165 indicate the initial condition of the body,
and let BA be a neighbouring adiabatic. Let HA, HB be any two
different paths to the adiabatic. If the body is worked reversibly round

HAB, then ^ = 0. Or if Qj be the heat taken in along HB, Q2 that

along HA, Ol 02 the average temperatures at which it is taken in along
the two respectively, then since no heat is taken in or given out along AB

a

In the limit when H approaches AB, -1 = 1, and therefore Q1 = Q2.

^2

The reader will easily see how this proposition comes out at once by
the use of the entropy-temperature diagram.

The change in temperature when a body undergoes a small
adiabatic change of volume by a given change of pressure.— Let

286 HEAT.

AB represent, in Fig. 165, the small adiabatic change due to an increase
of pressure HB, a unit mass of the body being dealt with. Let the
temperature at A be 0, and that at B be 0 + d6. Let BO be the 6 + dd
isothermal, and HAG the line of equal pressure through A. Then by
the last proposition the heat along AC equals the heat along BC.
But heat along BC = a.v6dpt

while heat along AC = Kpd6,

where Kp is the specific heat at constant pressure, as expressed in work
measure. Equating

or

avOdp

= -

If 64, is the adiabatic elasticity, i.e. the elasticity along AB defined by

HB

Though the area of ABC is of the
second order and vanishes in com-

^^^^^_^ parison with the heat received

/i c along either AC or BC, it may be

of interest to show how it may be
used to obtain the above result
when equated to the corresponding
area on the entropy-temperature

- '• - diagram, the two areas represent-
FIG. 165. *nS *he work done, and the heat

transformed, respectively.

The area ABC = ^HB -AC

z

= -dp-avd6
On the entropy-temperature diagram, Fig. 166, the corresponding area is

abc=-ab~bc

2

1 db

= - - — • area omiic

2 bm

But in the limit bmnc = amnc

= heat along ac
= K,d0

ab d0

and -

ISOTHERMAL AND ADIABATIC CHANGES.

287

Equating the two areas, we get as before

,/j avOdp

7

The adiabatics are steeper than the isothermals on the

indicator diagram. If «, is positive, i.e. if a body expands with rise
of temperature at constant pressure, then dO is positive if p is increased.
The adiabatic, as AB, Fig. 165, rises then from a lower to a higher
temperature, and must slope up more rapidly than the isothermal.

If a is negative so is dO. In this case the isothermals for lower
temperatures are above those for higher temperatures, on the diagram,
and as the adiabatic AB rises to a lower temperature, again it must slope
more rapidly.

The formula dQ—avOdpfK^ has been verified in various cases, first
by Joule for water and sperm oil (Scie?itific Papers, vol. i. p. 474), the
liquid being suddenly compressed in a
copper vessel by the addition of weights
on to a piston in a small connected
cylinder. The change of temperature was
determined by a thermo-electric circuit, of
which one junction was in the centre of
the copper vessel, the other in a constant
temperature water bath outside. To
justify the use of the thermoelectric
circuit it was necessary to prove that the
pressure alone did not affect the E.M.F.
This Joule did by having the difference of
temperature between the junctions in one
case small and in another great, and show-
ing that pressure produced the same effect
in each case. Had the E.M.F been altered
by the pressure the alteration would have been greater for a greater
temperature difference.

We may take as a specimen an experiment on water at 30° C.
The pressure put on was 53,634 Ibs. per sq. ft., equivalent to 2-57 x 107
dynes per sq. cm.

At 30° a =-00032
0 =303
v =1
Ep = 4'2 x 107, the mechanical equivalent of 1 calory.

Then by the formula we should have

J0 -00032 x 303 x 2-57 x 107

FiG. 166. — Entropy-Temperature
diagram.

Joule found by experiment

4-2 x 107
= -059°

According to the formula dd should change sign with a and, other things
being the same, should be proportional to it. This conclusion was verified

288 HEAT.

by Joule for water. He showed that on compression there was a very
slight cooling at 12° C., which changed to a very slight warming at
5° C., and the higher the temperature above 4° C. the greater was the
warming.

The Specific Heats at Constant Pressure and at Constant

Volume. — The specific heat of a substance at a given temperature is not
an invariable quantity but depends to some extent on the conditions
under which the heat is communicated, this dependence being most marked
in the case of a gas. It is usual to consider especially two kinds of specific
heat, that when the pressure is maintained constant, and that when the
volume is maintained constant. For gases (as we have already seen in
chap, vi.) these two specific heats differ very nearly by the heat equivalent
of the external work done in the expansion, and the difference is a large
fraction of either. For solids and liquids the difference, though appreci-
able, is not nearly so important. We shall denote the specific heats at
constant pressure and constant volume, respectively, by Cp and Cu when
in heat units, and by Kp and K,, when in work units. The elasticity of a
substance, i.e. the ratio of a small change of pressure to the consequent
change of volume per unit volume, is also a variable quantity depending
on the circumstances of the change. There are two especially important
cases, the elasticity when the change is adiabatic — a condition \vhich
holds if the change is very sudden ; and the elasticity when the change
is isothermal — a condition which holds if the change is suificiently slow.
The two are denoted respectively by e$ and eg. We shall now prove a
very important relation connecting the specific heats and the elasticities.
The ratio of the two specific heats y is equal to the ratio of the two
elasticities,

K» e*

or ^r = -

K,, et

In Fig. 165 let BA and BC represent adiabatic and isothermal
elements through B, cutting a line of equal pressure in AC, and let BH
be perpendicular to AC

change of pressure along BA
change of volume along BA

BH
XHA

Similarly

BH

Then

Now HO _ heat received along HC (since both are at constant
HA " heat received along HA pressure)

heat along HC

•»,— (since AB is an adiabatic)

heat along HB

ISOTHERMAL AND ADIABATIC CHANGES. 289

T&jdO (where dd is the temperature difference between
= K^ Hand BO)

Then

This relation only involves the Second Law of Thermodynamics in
the assumption of the smallness of HBA, and it was obtained by Laplace
before the law was known.

The ratio is usually denoted by y.

The Difference between the Two Specific Heats. — Taking unit mass of a
substance round HBO, and using the value for the heat along BO found
on p. 285, we have ^ ^

Kvd6 + a6e*dv - Kpd6 = area of HBO

avdOdp
~2~

= 0 in the limit,

, -rr " TT vftuiu

whence Kp - K0 = — - ^—

Putting dv = avdO we have

Kp-K.

Dividing by Kp we get at once

From this we see that y is constant when a?veel'Kp is constant.
The Numerical Value of 7, the Ratio of the Two Specific Heats. — The
last formula at once gives us y, when the quantities on the right are
known.

Thus let us take : — For air at 0° and 760 mm.,
a =-00367
v =773-4 '
6 "l/«

ea =p = 1 atmosphere = 1013600 dynes per cm2
K, » -2389 x 4-19x10*
whence y = 1*41 nearly.
If we take Swann's value (p. 8G) as 0'2414 we get y = 1'396.

For water at 30°.

a =-00032

v =1

0 =303

ft = 1-014 xlO«x 22000

(since the decrease in volume is 1 in 22000

for 1 atmo.)
Kp = 4-2xl07
whence =1-017

290 HEAT.

For iron at 30° we may take the quantities as approximately : —

a =-000033

• -

6 =303
e. =1x10^
K;; = -l 12x4-2x10*
whence y = 1-009

Experimental Determinations of y for Gases. — This ratio has been the
subject of much research since Laplace first showed that the formula
for the velocity of sound, found by Newton, must be multiplied by the
square root of y, and it has been determined from that velocity, from
comparisons of the two elasticities, and from measurements of the two
specific heats. We shall give short accounts of some of the methods used.

1. The velocity of sound.

The velocity of propagation of waves of longitudinal disturbance in
air was shown by Newton to be U = ^elasticity / density. He supposed
the elasticity to be es.

Laplace pointed out that the elasticity used in the propagation of
sound waves in air is the adiabatic elasticity et, for the alternations of
compression and rarefaction are so rapid and the conductivity of air is so
low that practically no conduction of heat takes place out of the com-
pressions or into the rarefactions. This supposition is confirmed by the
consideration (Lord Rayleigh's Sound, vol. ii. p. 26) that if heat were par-
tially diffused out of and into the waves by conduction, the entropy would,
on the whole, increase, and the operations would then not be reversible.
When, therefore, a rarefaction followed a compression, the work done in
expansion outwards would be less than that done in compression inwards,
and with the dissipation of energy the sound would die away. Lord
Rayleigh has shown that if the elasticity were only slightly below the
adiabatic elasticity, the rate of dying away would be very rapid. This
consideration only applies when there is partial conduction. With con-
densations and rarefactions taking place so slowly that the temperature
remained constant, all heat transfers would take place at the same tem-
perature and the entropy would remain constant. The operations would
therefore be reversible, and the waves would be propagated without
dissipation of energy. As the waves of sound are propagated with very
little dissipation of energy, the elasticity is either isothermal or adiabatic,
and not between the two.

The former supposition, that it is isothermal, gives

U= \/pressure —• density
«= s/1013600 x 773-4
«e 28,000 cm. per sec.

But experiment gives between 33,100 and 33,300, say 33,200, as the
value at 0° and 760 mm. Then the latter supposition, that the elasticity
is adiabatic, is alone admissible.

ISOTHERMAL AND ADIABATIC CHANGES.

291

But e+ — yea = yp for air

.'. 33200 = \/y x pressure -h density

= Jy • 28000

whence 7=1-406

2. By comparison of the two elasticities on change of volume of the gas.

This method was devised by Clement and Desormes. They used a
large glass reservoir A (Fig. 167), furnished with a tap B and a water
manometer act' to indicate the pressure of the contained air.

To begin with, A was partially exhausted, and when its contents were
at the temperature of the surroundings, the pressure indicated by the
level a of the manometer was
read. The tap B was now turned
on for a moment to establish
equilibrium with the external
pressure, and then turned off
again. The temperature having
risen by this compression, which
was taken as adiabatic, it was
allowed to fall again to its original
value, and the final level a' of the
manometer was read.

The experiment may be re-
garded as one to determine the
ratio of the elasticities. The air
originally filling A was regarded
as crushed down to a fraction of
the volume of A, the same at FIG. 167.

both readings of the manometer.

But if we make the same change of volume, first suddenly and adia-
batically, and then slowly and isothermally,

et sudden increase of pressure

"y = — = - - - .

' e9 slow increase of pressure

Now, in this experiment the volume of the gas is altered by a definite
amount suddenly, and the pressure increase noted. The volume being
kept the same, the temperature falls to that of the surroundings, and
then the increase of pressure above its original value is the same as if
the same change of volume had been made slowly, so that

ab

Taking one of their results as a specimen —

Atmospheric pressure = 766-51

7Ko 7 / 1 ULf = itj °

l762'-9 )rta' = 10'2

Initial
Final

, Q.0

=li

138
~= 1-353.

292 HEAT.

This assumes that the pressure changes are exceedingly small, which is
obviously not exact.

Taking p0, P, and p as the initial, atmospheric, and final pressures,
and V0, V, and V as the corresponding volumes, the first change being
adiabatic, we have (p. 295)

j,0V/ = PV (1)

The second change being isothermal,

P.V.-PV (2)

Then 4g

-(£°)y from (2)

logf?
whence y =

This gives y = 1'348 in the above example.

The method, as originally devised, has a serious fault in the assump-
tion that the newly introduced air will have the same temperature the
moment after introduction as the air already in the receiver, and that
all will cool down equally. This fault was avoided in a modification
due to Gay Lussac and Welter, in which, to begin with, A contained
air at a pressure p0 higher than the atmospheric pressure P. On turning
the tap the pressure fell to P, while the air which had previously occu-
pied a part V0 of the volume expanded to fill the whole volume V.

Finally the pressure rose to p.

Then

and p0~V0 =pV

whence as above y =

log%

p

In one of the experiments (Baynes's Thermodynamics, p. 136), the pressures
on an arbitrary scale were —

P= 1-0096, Po= 1-0314,^= 1-0155.
whence y = 1-3745.

There are still sources of error in the method. The change of
volume is not accurately adiabatic, for momentum is produced in the
issuing gas, and if the tap is turned off at the first moment of equalisa-
tion of pressure with that of the atmosphere, the air being still in
motion, its kinetic energy will be gradually converted to heat, and so the
entropy will be increased. If, however, the tap be left on for a longer
time the outrush will cease and be followed by an inrush, and there will

ISOTHERMAL AND ADIABATIC CHANGES. 293

be a series of oscillations, the energy of the oscillations being gradually
converted to heat, and the entropy being thereby increased. M. Cazin
showed the existence of these oscillations, and was able to turn off the tap
at the end of the first outrush. The expansion up to that point might
be considered as equivalent to an adiabatic one, for nearly all the work
done by the gas has gone out of it, the amount retained through viscosity
being so far negligible. M. Cazin's determinations were much nearer
those given by the velocity of sound than the earlier results.

Rontgen used as a pressure indicator a corrugated plate like that in
an aneroid barometer, fixed over an opening in the reservoir. Any
movement of the plate was communicated to a mirror in which was
viewed the reflection of a scale.

He obtained the following values for y : —

Air ...... 1-4053

Carbon dioxide . . . 1'3052

Hydrogen . ... 1-3852

The value for hydrogen is no doubt too small. The gas so rapidly
regains the temperature of the surroundings that the adiabatic change
of pressure cannot be observed accurately.*

3. By direct comparison of the two specific heats.

If a quantity of heat H is given to a unit mass of gas at constant
pressure it raises the temperature by dd where

If V is the initial volume and dv the increase, a the co-efficient of
expansion,

dv = aVdO,

If the same quantity H is given to the unit mass with constant volume
it raises the temperature by dd' where

If P is the initial pressure, dp its increase, /3 the co-efficient of pressure
increase

••H-7sr

Equating the two values of H, we get

Cp<fo =

°P
OTpPcto

* Among later researches we may refer to a paper by Capstick, " On the Ratio of
the Specific Heats of the Paraffins and their Monohalogen Derivatives," Phil. Trans.
A. 1894, Ft. I., p. 1.

29* HEAT.

Janiin and Richard carried out a method based upon this result, in which
a definite quantity of heat H was communicated to the gas contained in
a large vessel by means of a wire heated by an electric current.

In one case the gas was allowed to expand at constant pressure, and
dv was measured, and in the other the volume was kept constant and dp
was observed. No absolute measures could be made owing to the loss of
heat from the wire by radiation, which was unknown. But it was the
same in both parts of the experiment, and so the quantity H remaining
in the gas was the same.

They obtained as values of 7 —

Air ..... . 1-41

Hydrogen ..... 1-41

Carbon dioxide . . . . 1'29

Joly has succeeded in making direct measurements of the specific
heat at constant volume by the steam calorimeter, as already described
in chap. vi. His result for air at 0° and 760 mm. is

Ov = 0-17154.
If we take Wiedemann's value for that at constant pressure, Cp = 0*2389,

C
we get 7 = ^=1-393.

With Swann's value taken as 0-2414 at 0° 0. we get 7= 1-407.

The Determination of the Absolute Temperature at 0° C. from the Value
of y. — If in the equation for the difference of specific heats

Kp-K,, = a2ve,0

•pr

we substitute for K, its value —2

7

we get KD = — J— o?veeO

y-l

Rankine, taking the value of y given by the velocity of sound, used
this equation to determine Kp and Cp, but if we use the value of Kp
found by direct experiment, it may also be regarded as an equation
giving 6, the absolute temperature at 0° 0., the length of the degrees of
the absolute and air scales so nearly coinciding at 0°, that a may be
taken as the same for each.

Putting the equation in the form

y a.*ve

if we take Kp= -2389 x 4-2 x 10T

a = -00367
v = 773-4
e» = 1-0136 x 106 (Paris atmosphere),

and use the value of y given by the velocity of sound as 1 -406 we get

(9=274-4
But if we use the value of 7 obtained by Joly, viz. 1-393, then we get

0 = 268-1

ISOTHERMAL AND ADIABATIC CHANGES. 295

Inasmuch as a difference of '002 in the value of y makes a difference
of about 1° in 0, and as the values of y obtained by different methods
differ by far more than this, the value of 0 thus obtained can only be
regarded as an approximation justifying the use of the gas scale for the
absolute scale when no great exactness is required. We shall see later
how a much closer approximation may be found.

The Relations between Volume, Pressure, and Temperature
of a Gas expanding Adiabatically. — We may use the approximate
equation —

pv=p0v0(l+at)

where R =p0v0a and T is the gas temperature measured from — below

a

0° 0.

Now the elasticity of a substance is, by definition,

dp

~V^T
dv

If T is constant

vdp +pdv — 0

vdp
or e» = -=£ =p

dv

But e+ = ye* = yp

Hence if any small adiabatic changes of p and v occur

vdp

- -j = yp

dv

dp dv

or — = -y —

p v

whence log p = log Cv~

or pv = C where 0 is a constant. If we eliminate v by means of the
equation pv = HT we have

Or if we eliminate p we have

= R ' f

As an illustration let us find the fall of temperature if a mass of air
at 15° 0. expands adiabatically to double its volume, the value of y being
1-4. If the new temperature is T' we shall have the two equations

«-i-° JL

R ' 288

296 HEAT.

and dividing

T,_288_288
~ '

a fall of 70°.

Decrease of Temperature Upwards with the Atmosphere in Convedivt
Equilibrium. — As already pointed out (p. 216), the limiting condition
of equilibrium for the atmosphere is one of " convective " equilibrium,
i.e. one in which the temperature decreases upwards in such a way that
a mass of air in ascending will expand and cool in doing work so as
always to be at the temperature of its surroundings. Such expansion is
adiabatic. Hence the temperature and pressure should be connected in
this condition in air in which there is no condensation by the equation

Let us apply this to find the decrease per 100 feet rise from the sur-
face, which we will suppose to be at 0° 0.
Taking logarithms we have

(y - 1) log p = log A + y log T
whence -l =

or

7 P
If dp is the decrease due to a rise of 100 feet,

dp= 100
p "26000

since 26,000 feet is the height of the homogeneous atmosphere.

T = 273

•41 100

X

1-41 26000x273
= •3°

Or, when the atmosphere is just in equilibrium, the temperature
decreases by about '3° per 100 feet rise, when the mean is about 0° C.
With a more rapid decrease there will be circulation.

The Internal Energy taken up by a Gas in Expanding.— We

have already mentioned (chap, viii.) Mayer's assumption that a gas in
expanding does no work against internal forces, and have described the
experiment, first made by Gay Lussac and later made by Joule, to test
the truth of the assumption. In Joule's experiment air was compressed
in one vessel to a pressure of 22 atmospheres, and was then allowed to
expand into another equal vessel previously empty, so that no energy was
given to or taken from the air by external work. If in mere expansion
the gas had done work against its own cohesion, it would have drawn on
its own heat for the supply, and the temperature would have fallen.

ISOTHERMAL AND ADIABATIC CHANGES. 297

Joule could not detect any alteration of temperature on the whole. But
the heat capacity of the gas was so small compared with that of the
calorimeter and its contents that a very small change in the temperature
of the gas would not have affected the temperature of the calorimeter
sensibly. The result of the experiment, therefore, only showed that
Mayer's assumption was not far from the truth.

Lord Kelvin afterwards proposed a method of experiment giving a
much more delicate test for the presence or absence of internal work in
gaseous expansion. This method was put into practice by himself and
Dr. Joule.* Their researches showed that in Joule's original experiment
there must have been a slight cooling effect, i.e. that some work must
have been done against internal forces. The method of experiment also
gave a means of comparing the absolute scale of temperature with the
indications of the air thermometer.

In order to understand the principle of the method, let us suppose
that a long pipe AB (Fig. 168) is obstructed in the middle by a narrow
passage 0, and let air be gently driven towards 0 by a piston at A, on
which the pressure is P. On the other side of 0 let the air expand
against a less presssure p, applied by a piston at B. Let the volume of
unit mass be V on the A side and v on the B side. Then the work done

A C B

:L_ ±

FIG. 168.

on the gas on the A side is PV per unit mass, and the work done by the
gas on the B side is pv for the same mass. Let the walls of the enclosure
be either non-conducting or everywhere at the temperature of the gas in
contact with them.

It may appear at first sight that this is an adiabatic transaction, since
no heat is communicated from the outside as heat ; but if, as we suppose,
the passage at C is very small, there will be a " rapid " there, the air
passing through it acquiring a considerable amount of kinetic energy.
But this kinetic energy is quickly lost on the B side through the viscosity
of the surrounding air, i.e. some of the work done on the gas gives rise
to kinetic energy and this kinetic energy is transformed to heat, so that
there is, on the whole, a gain of entropy.

Let us first suppose that the temperature is found to be the same on
the two sides of 0 at a sufficient distance away from the " rapid." Let
us also suppose that Boyle's law is true.. Then

or the work done on the gas on the A side is equal to that done by the
gas on the B side. On the whole, then, no energy is given to the gas
from the outside, and since the temperature is unaltered, we conclude
that, while the volume has increased from V to v, the gas has not trans-
formed any of its own heat into energy of molecular separation. In other

* Joule's Scientific Papers, vol. ii. p. 217, or Thomson's (Lord Kelvin's) Papers,
vol. i. p. 333.

298

HEAT.

words, the internal or intrinsic energy is the same for the same temper-
ature whatever the volume.

In such a case, therefore, Mayer's assumption would be justified.
But if, as was actually the case in Thomson's and Joule's experiments,
there is an alteration of temperature on the passage through 0, this
alteration may indicate either that Boyle's law is untrue or that Mayer's
assumption is untrue, or that both are untrue. We know already that
Boyle's law is only an approximation, and it remains therefore to de-
termine whether the alteration of temperature indicates that Mayer's
assumption is also only an approximation.

In order to measure the change of temperature
accurately, Thomson and Joule forced the gas through a
porous plug consisting of a boxwood cylinder bb (Fig. 169),
1'5 in. internal diameter, containing two perforated brass
plates A, B, 2*72 in. apart, the space between these plates
being filled with cotton or silk, more or less compressed.
The gas was driven up to A through a long pipe immersed
in a constant-temperature bath, and flowed out from B
round a thermometer, being allowed there to come to the
atmospheric pressure. When a steady state of tempera-
ture was everywhere attained, there was with air a sensible
cooling, with carbon dioxide a much larger cooling, and
with hydrogen a very slight heating.

Let us first investigate the effect which we might
expect if Boyle's law does not exactly express the relation
between volume and pressure, neglecting meanwhile the
effect of change in intrinsic energy.

According to Amagat's researches, in the case of air
the product pv decreases by about 12 in 1000 for an
increase of pressure from 1 to 31 atmospheres. We are
perhaps not far wrong in assuming that the decrease is
nearly uniformly at the rate of 1/2500 per atmosphere
within these limits. With this assumption, if the air
is kept at the same temperature on the two sides of the
plug, and if ib expands to the atmospheric pressure p
on the further side, then

B

P Y

FIG. 169.

PV

/, P-p

=pv { 1 ±-

\ p

' 2500/

and

or

P - p t pv
~~^T 2500

pv exceeds PV by--~? . ~

And on the whole external work

--
2500

is done by the air, and in

order to keep the temperature constant this amount of energy must be
supplied from without.

ISOTHERMAL AND ADIABAT1C CHANGES. 299

If it be not supplied, the air will cool by an amount nearly given by

P— p pv 1
p ' 2500 ' K^

where K^ is the mechanical value of the specific heat at constant
pressure. In fact the cooling will be rather less, because with lowering
of temperature the expansion against the atmospheric pressure is rather
less, pv is diminished, and therefore pv — PV is less.

In some of the experiments P was about 130 inches of mercury and
p was 30 inches.

pv is about 780 x 106 in C.G.S. units, while Kp is '2375 x 4'2 x 107.

Substituting these values in the expression for the cooling, we find
that it should be about 0'1° C. But the actually observed cooling was
about 0-9° 0. Then some of the energy was also taken up in the
separation of the molecules, or there was an increase of intrinsic energy
on expansion.

In a similar manner it may be shown that the cooling in the case of
carbon dioxide was chiefly due to change in intrinsic energy.

In the case of hydrogen the product pv increases with the pressure.
Then, if the intrinsic energy is independent of the volume, energy must
be subtracted from the gas to keep the temperature the same in the
expanded condition. If it be not subtracted, the temperature will rise.
The actual effect observed was a slight heating, and of the order due to
the increase of pv. Hence it was concluded that the change in intrinsic
•energy due to change in volume is very small.

Use of the Porous Plug Experiment for the Comparison of the Air Scale
with the Absolute Scale. — We may express the change in intrinsic energy
of the gas in two ways ; (1) in terms of the energy communicated from the
outside and (2) in terms of the observed changes of physical condition,
temperature, pressure, &c., and equating these two expressions we obtain
a. value for the absolute temperature.

In considering the energy received from the outside, we remark that
on the whole there is very little change of temperature in passing through
the plug, though no doubt there are very great local variations in the
" rapids " — the fine streams issuing from the pores. The gas is therefore
very nearly in temperature equilibrium with the surrounding walls and
the temperature slopes in these walls are very gradual. This is still
more nearly true since the gas was allowed to flow for so long a time
before observations were made that affairs had arrived at a steady state
and the wall at each point was at practically the same temperature as
the gas passing it. The walls were bad conductors so that the heat con-
ducted into or out of the gas might be neglected. The kinetic energy in
the gas was also small when it had settled down to steady motion beyond
the " rapids." Then the gain of intrinsic energy is practically due to the
difference between the works done on and by the gas on the two sides of
the plug or if E is the initial, and e the final, intrinsic energy,

e - E = PY -pv.

In finding the value of e — E in terms of the observed changes in
physical condition, we may suppose the change made in any convenient

300 HEAT.

way, for the intrinsic energy corresponding to a given state is quite
independent of the manner in which that state is reached. Let A
(Fig. 170) represent the state of unit mass of the gas on the entrance
side, and B that of the same mass on the exit side of the plug.

We shall suppose the change from A to B to take place along the
isothermal AC, and the equal pressure line CB, and to avoid consideration
of the variation of the different quantities, such as coefficient of expansion
and specific heat, we shall suppose A and B so near together that these
quantities are practically constant.

The heat put in along AC is

a0V(P -p) or atfVdP.
The heat put in along CB is Kp dO

where dO is the rise in temperature. The external work done is ACBNL.
Then e-E

e - E = P V — pv

= AHOL-BKON

aOVdP + Y.pdB = AHOL - BKON + ACBNL
= ACKH

A l K»dd m

e=a-aV'dP <*>

If dO = 0 then 9 = - so that if a substance had the same temperature on

a

the two sides of the plug, equal expansions of that substance would
indicate equal intervals of temperature on the absolute scale. Hydrogen

approaches3 most nearly to this con-
dition, hence its use in the standard
gas thermometer.

In their experiments Thomson
and Joule found that dd was propor-
tional to dP for a given temperature,
throughout the range of pressures
employed, and inversely as the
square of the absolute temperature,
so that we may put

d6 A 02

L *» N dP~ H tf2

FIG. 170. where 00 is the absolute tempera-

ture of 0° 0., II the atmospheric

pressure, and A a constant for each gas, being the cooling per atmosphere
difference of pressure at 0° C.

The observed values for A were

Air + '275

Carbon dioxide .... +1*388
Hydrogen - 0'03 about

ISOTHERMAL AND ADIABATIC CHANGES. 301

fjf)
Though — was obtained with finite differences of pressure, it is evident

Ct-lr

that, since it is independent of the pressure, we may use it in equation (1),
where the difference of pressure is supposed to be very small. We obtain

(2)
W

-

a aVII02

We may at once obtain from (2) the absolute temperature at 0° C.,
assuming that the degrees have the same value as those of the air scale
about that temperature. Thus, for air, since at 0° 0. 60 = 6,

-2389 x 4-18 x!07x -275

-00367 x 773 x 1013600
•95

or

Or the absolute temperature does not differ from the air temperature by
as much as 1° CJ.

We refer the reader to Thomson's article on " Heat," Ency. £rit.,
9th ed., § 60-2, for the mode of comparing the absolute with the gas scale
not assuming that the degrees have the same length about 0° C., but
taking the interval from 0° to 100° 0. as the same on both.

It will be observed that equation (2) implies that a, the coefficient of
expansion, is not strictly independent of the density of the gas, for the
second term on the right is not constant, but when A is positive it
increases as V decreases or as the density increases.

Then - must decrease and a increase with the density for gases such
a

as air and carbon dioxide. For hydrogen, on the contrary, a decreases as
the density increases.

Starting from equation (2) Thomson (loc. cit. § 63) has calculated the
expansion of air and other gases at different temperatures and obtains
results agreeing very closely with those found experimentally by
Regnault.*

The change in internal energy of a gas on expansion has been brought
into greater prominence by the " regenerative " method of liquefying
gases, in which a gas is forced from a pipe, where it is under great pressure,
out through a fine nozzle into the air, the process corresponding very
nearly to that of the " porous plug" experiment. At low temperatures

* Maxwell (Heat, 5th ed., p. 214) gives an equation obtained by integrating
equation (1), which is equivalent to

In the integration it is virtually assumed that a is constant, an assumption which is

shown to be unjustifiable by the consideration that, if a were constant, dO would be

p
proportional not to P -p but to log — . The proper treatment of the equation after

P
its reduction to the form (2) will be found in the Article " Heat," Ency. Brit. loc. cit.

302 HEAT.

and expansion from high densities there appears to be great cooling in
all cases, even in that of hydrogen. And this we might expect, for with
the high densities the molecules are, in much larger proportion, within
each other's sphere of action, and work is needed to separate them.*
Generalisation of the Indicator Diagram for any Stresses and

the Corresponding Strains. — If we take abscissae to represent strains
and ordinates stresses, so measured that areas represent work done on or
by the quantity of matter considered, we may use the indicator diagram
for any corresponding stresses and strains and draw isothermals and
adiabatics just as with pressure and volume.

As an illustration let us suppose we have a wire under an end pull
and let the relation between stretch ds per length 1 and pull P per
square centimetre section be given by

where Y is Young's modulus. This implies that we are dealing with unit
cube of the material.

If Y is constant for a given temperature, the isothermal for, say, 0° 0.
may be represented by a straight line going through the origin as OH
(Fig. 171), the tangent of the slope representing Y. Now as the tempera-
ture rises the length under zero pull in general increases. Let a be the
ordinary coefficient of length expansion. Drawing OP = a, PK will
represent the 1° isothermal, and since Y in general decreases with rise of
temperature it is at a slightly less slope than OH. Similarly QL will
represent the 2° isothermal.

It may be noted that these isothermals, if straight lines, will meet
beyond O, at a point in the lower quadrant to the left, say at T. Draw
TM to represent the pressure. If Young's modulus at any temperature

is given by Y = Y0( 1 - A*)

then Y0

TM TM

OM + OP~OM + o

Y0 OM + a •

ana -ff- = A -t- A = x-^-..- = i +

OM " OM
whence OM = y.

11 2

With iron a is of the order -y^-6, while A is of the order . 4> so that

T = OAA» a compression which is never approached. But if it could be

A Z(JO

reached without disintegration, and if the isothermals were straight lines
as supposed, the interpretation would be that at this compression the
increase in yielding due to rise of temperature would just neutralise the
expansion due to the same rise of temperature.

* For a full discussion see Callendar, "On the ThermodTnamical Correction of
the Gas Thermometer." Phil. Mag. v., 1903, p. 48.

ISOTHERMAL AND ADIABATIC CHANGES.

303

Let us now see how the wire can be taken round a Carnot cycle.
Let AB, DC represent neighbouring isothermals at 0 and 0 — dO. Let
AD, BC represent adiabatics. We draw these steeper for we may expect
the change in tension for given stretch to be greater if no heat is
allowed to pass, i.e. if we introduce a constraint and the results obtained
confirm the supposition. We may take ABCD to represent a Carnot
cycle. But we must note that since work is done by the body in con-
tracting we must go round counter-clockwise to get a balance of work
done by the body in returning to the starting-point. Heat is taken in

Fia 171.

along AB and given out along CD, and an adiabatic stretch along AD
cools the wire.

Just as in the case of the pressure- volume diagram, it can be shown
that the heat along AB is

where ds is the change in length along AB, and J3 is the change in length
under constant load per 1° rise in temperature. If a is the ordinary co-
efficient of expansion, X the temperature coefficient of Young's modulus,
and s the total stretch up to A, we may put (3 = a + Xs.

Similarly the change in temperature under adiabatic change of pull

when p is the density, KP the specific heat under constant pull — probably
nearly the same as the specific heat under no pull — and Y# is the adiabatic

304 HEAT.

Young's modulus. We must give the negative sign, since the lower
isothermals are higher in the diagram.

In the case of india-rubber f3 is negative, for if a weight be hung on
to a rubber cord, the weight rises if the cord be heated. Hence dd is
positive for a sudden increase of stretch. This may be verified by
suddenly pulling out an india-rubber band and applying it to the upper
lip — a sensitive thermoscope — when we can easily detect the warming.
Keeping the band extended, it soon falls to the temperature of the air.
Now allowing it to contract suddenly, it is very appreciably cooled. The
ratio of the specific heats under constant stress and constant strain is
easily seen to be equal to the ratio of the adiabatic and isothermal
moduli, and we can obtain as the formula corresponding to that on
p. 289—

For small stresses /3 = a + \s may be generally assumed equal to a, the
coefficient of expansion. If we take as the values of the constants for
steel

P = ^ : Y, = 2 x 1012 : PKV = 4. 2 x 107 x 7.8 x 0. 1 12
we get 7 = 1.002

or the adiabatic elasticity is greater than the isothermal by about 1
in 500.

The term Xs would only seriously affect this result when of the order
10~6. Since A. is of the order 10~4, this requires a stretch of the order
10~2, which cannot be given without permanent set, when our assump-
tions are all inadmissible.

But though not seriously affecting y the existence of As implies that
the adiabatic elasticity increases with the strain. It will follow that
when longitudinal waves travel along a wire, the elasticity at the points
of greatest tension will be greatest, and if we represent the waves by the
curve of tension, the crests in the curve will move most rapidly and will
therefore gain on the hollows.

Similar formulae can at once be worked out for the case of shear
strains and stresses involving the rigidity modulus. If abscissae repre-
sent angles of shear and ordinates tangential stresses per unit area, areas
on the diagram will represent work done with unit cube of the sub-
stance. We shall not now have anything corresponding to the ordinary
coefficient of expansion a, but if we put for the modulus of rigidity

n = n0(l - kt)

a shear e at constant stress will increase by Xe for a rise of 1°, so that
we must use A.e instead of ft in the last case.

The adiabatic change of temperature for change de in shear is

Xen+Ode

that is, there is a cooling for increase of shear. But in general it is of
the second order, since both A and e are usually small.

ISOTHERMAL AND ADIABATIC CHANGES. 305

The consideration of the ratio y is especially important in this case
since the torsional resistance of wires is so frequently used to measure
small torques, the torque being deduced from the angle of twist and the
time of vibration of the system acted on. Now the angle of twist may
be produced slowly and the condition of the wire may be isothermal ; the
vibrations, on the other hand, may be rapid and the conditions may be
somewhere between isothermal and adiabatic conditions. If the adiabatic
and isothermal elasticities differed seriously, error might arise in such
experiments as that of Cavendish on the constant of attraction.

The error would be the greatest if the conditions in vibration were
truly adiabatic and those in steady deflection by the torque truly
isothermal.

We may show that

7 = T

2

For steel A. is not far from — 3. Suppose a wire 1 mm. radius and

104

1 metre long is turned through 1 radian at its lower end, then the

extreme value of e is — -; n is about 1012.
ICr

Substituting the numerical values we get

r = T^lcF«

and the ratio will be still more nearly 1 for less shear. This implies
that the two rigidities are for all practical purposes identical for steel
wire.

If we have any other type of stress and strain we have the results

( + if the effects of rise of temperature and increase of stress are opposite)

1
and y 1-/32E,0

where ft is the change of strain under constant stress for 1° rise in
temperature, E* is the isothermal modulus of elasticity, KP is the specific
heat under constant strain, and p is the mass in the volume so strained.
Thus in a spiral spring we may deal with unit length measured along the
axis of the spiral, when p will be the mass of the spring per unit length.
The reader will find that for such a spring y is practically 1, as might
be expected when we remember that the strain is chiefly torsional.

CHAPTER XIX.

THERMODYNAMICS OF CHANGE OF STATE AND
OF SOLUTIONS.

First Latent Heat Equation — Volume of Saturated Steam — Triple Point and Dif-
ference of Vapour- Pressures of Ice and Water below 0° C. — Second Latent Heat
Equation — Alteration of Vapour-Pressure with Curvature of Liquid Surface —
Connection with Change in Melting-Point by Pressure — Solutions — Vapour-
Pressure less than that of the Solvent — Osmotic Pressure — Raising of Boiling
Point — Lowering of Melting-Point — Semi-Permeable Membranes— Van'!. Hoff's
Application of Thermodynamics — Molecular Theory of Osmotic Pressure.

The First Latent Heat Equation. — If a substance is at such a
temperature, that with suitable pressure and volume it can exist in two

B! M

FIG. 172.

states at once, the two being in equilibrium with each other, then the
isothermals on the indicator diagram will have a horizontal position.
For example steam and water at 100° C. may coexist and the isothermal
for 1 gramme at 100° will be represented by a curve, of which the
general course only (it is quite out of scale) is represented by ABOD,
Fig. 172. AB is water, BO is steam and water at 100° and 1 atmos-
phere, and CD is steam. If the volume is fixed, say at QM, then the
proportions of steam and water will be fixed, and they will be in

306

THERMODYNAMICS OF CHANGE OF STATE, ETC. 807

equilibrium with each other at 1 atmosphere, as long as the temperature
is 100°. If the volume increases, then more steam is formed till at C
all is steam with a volume about 1640 c.c. On the other hand, if the
volume diminishes more water is formed till at B all is water with a
volume about 1'04 c.c. The process is reversible at every point.

The general course of a substance at the melting-point may also be
represented by a curve like ABCD. For ice and water at 0° DC will
represent ice. At C the ice begins to melt, having then a volume about
1 '09 c.c. At B it is all water having a volume about 1 c.c., and BA
represents water.

If the temperature is not that of the'normal boiling-point or the normal
melting-point the curve will have the same general character, but the
horizontal portion will be at a different level. For steam and water at
99° C., for example, it is at 733*21 mm. instead of 760 mm. For ice
and water at — 1° C. it is, as we shall see below, raised up to about 134
atmospheres. We do not, in ordinary experience, meet with ice and
water in equilibrium except at or very near the normal melting-point.
The possibility of such equilibrium at other temperatures was indeed first
discovered as a result of thermodynamical reasoning by Professor James
Thomson (Trans. R.S. Edin., xvi., 1849, p. 575), and was first experi-
mentally demonstrated by his brother shortly afterwards (see chap. xii.).

The following investigation applies to either change of state. Let
ABCD, A'B'C'D', Fig. 172, be isothermals at 0 and 6-dO respectively
for 1 gramme of the substance. Let B represent volume vl and let C
represent volume vz. Let L be the latent heat taken in along BC in
changing state. Through B and C draw adiabatics BM, CN, where
M and N are on the 6 - dO isothermal. Then BCNMB may be taken
as a Carnot cycle, and we have

— £- = external work = area MBCN.
u

If the isothermals are sufficiently near to each other, any want of
parallelism of BM and CN may be neglected, and the area of MBCN
may be taken as

Hence

7--* >

If then v2 is greater than vv as in the case of steam and water, -¥

du
is positive, or the boiling-point rises with the pressure.

If, on the other hand, i>2 is less than vv as in the case of water and

fjff\

ice, -~ is negative, or the pressure at which equilibrium is possible

increases with fall of temperature. That is, pressure lowers the melting-
point.

In change from solid to liquid, in which there is an increase of volume,
a rise in pressure corresponds to a rise in temperature, or pressure raises
the melting-point.

308 HEAT.

The Volume of Saturated Steam. — The latent heat equation at once
gives us the volume of a saturated vapour at any temperature at which
we know the latent heat, the liquid volume and the rate of change of
the vapour-pressure. Thus, for water and steam at 100° we have

L = 537 x 4-19 xlO7
6 = 373
vl =1-043

Pim-5— 77 '371 cm. of mercury
= 74-650

whence -^ = 2'721 cm. of mercury

(W

= 2-721 x 13-596 x 981 dynes per sq. cm.
Substituting in equation (1), we have

= 1 -043 4- 537 x 4-19 x 107

373x~2T21 x 13-596 x 981
= 1663 c.c.

Fairbairn and Tate (Phil. Tra?is., cl., 1860) found by direct experi-
ment that v2 =161 6. While there is some uncertainty about the
thermodynamic result depending on the uncertainty in the values of the
constants, it is easily seen that we cannot accept so low a result as that
obtained by Fairbairn and Tate, which was probably vitiated by the
adherence of steam to the sides of the containing vessel. The thermo-
dynamic method is preferable here to direct methods.*

It is interesting to compare the value «2=1G63 with that which
steam would have if it followed the same law of expansion as air. We
know from Regnault's researches t that the density of steam at low tem-
peratures and pressures is 0*623 of the density of air — almost exactly
the value obtained on the supposition that two volumes of hydrogen
unite with one volume of oxygen to form one volume of steam. If it
maintained the same relative density at 100° 0. and 760 mm., its volume
would be

GIG i rtf\n

x ^- =1696

0-001293x0-623 273

which is certainly greater than the actual volume. Hence steam does
not expand according to the laws of Boyle and Charles at higher tem-
peratures.

The Change of Melting-Point under Pressure. — Taking the latent heat
equation we have, for water-ice —

L = 80x4-19xl07
0=273
0.-1-09
4-1

whence |= - 8-?2**™*™! - 136500000 dynes/cm.

* llamsay and Young, " The Properties of Water and of Steam," Phil. Trans.
A., 1892, p. 107.

+ Ann. de Chim. et de Phys., 3rd series, t. xv. p. 141.

Ice waler

vapoui

vapour

THERMODYNAMICS OF CHANGE OF STATE, ETC. 309

Taking 1 atmosphere as 1014000 dynes/cm., this gives 134 atmo-
spheres as the pressure required to produce a lowering of 1° C., or the
lowering per atmosphere is -0074° 0.

This result, due to James Thomson, was confirmed by Lord Kelvin
(Phil. Mag, [3] xxxvii., 1850, p. 123). He used a strong glass piezometer,
in which was a mixture of ice and water. A manometer and a sensitive
thermometer enclosed in an outer tube were inserted in the piezometer,
and when the pressure was increased a mean lowering of -00735° per
atmosphere was obtained.

Other experiments have already been described in chap. xii.

The Triple Point and the Difference of Vapour- Pressures of Ice and
Water below 0° 0. — When ice and water are at the melting-point under
the pressure of their vapour alone, that point is above 0° C. by about
0'0074°, since the vapour-pressure is less than 5 mm. or nearly an atmo-
sphere less than 1 atmosphere. At this point it is easily shown that the
vapour-pressures of ice and water are exactly equal, by taking the
substance through a " one temperature " reversible cycle, or a cycle in
which the temperature is throughout the same. Since the cycle is

reversible, \-fr — Qi and if 6 is constant lcZQ = 0, or the total heat

given is zero. Then the total external work done is zero. Now
imagine some such apparatus as that
in Fig. 173, where ice and water
at this melting-point, + 0-0074° C.,
are in contact through the pipe re-
presented in the lower part of the
figure, and let all be maintained at one
temperature. Suppose, if possible,
that ice-vapour has a greater pressure
than water-vapour. Then, in allowing FIG. 173.

ice -vapour to expand into water-
vapour, it will do work. Allow this expansion to take place isothermally
and reversibly through an engine in the pipe represented in the upper
part of the figure. Then the ice will continually give off vapour which
can be passed through the engine, where its pressure is reduced to that
of water-vapour. It will then condense on the water surface, and the
excess of water can freeze on to the lower surface of the ice. Here
is a reversible cycle which can continue endlessly, giving off work
through the engine. But this is contrary to the principle that the total
work must be zero. Then our supposition is wrong, and the vapour-
pressure of the ice cannot be different from that of the water.

At this point, -0074° 0., ice, water, and their vapour are all in
equilibrium with each other at the same pressure. The three states or
the three " phases" can co-exist without changing from one to another.
The point is therefore called the triple point.

If we use a temperature -pressure diagram, as in Fig. 174, T will
represent the triple point, TA the vapour-pressure of water above the
temperature of the triple point, TB the vapour-pressure of ice below that
temperature, TO the line giving the relation between melting-point and
pressure — a line going up very nearly vertically, sloping in fact only
•0074° per atmosphere.

S10

HEAT.

The three included regions may then be marked vapour, water, ice
respectively. If we have all three states or phases present, T is the only
point of equilibrium, and the system is said to be non-variant.

If, however, we have two phases only, say water and vapour, we may
have equilibrium anywhere along TA. We can vary the temperature,
but given any one value of the temperature the pressure is fixed. The
system is therefore said to be monovariant.

If we have only one phase present, as water, we can vary both tem-
perature and pressure so long as they give a point within the region
OTA, and the system is said to be divariant.

When Regnault made his celebrated researches on the vapour-
pressure of ice and water, he supposed that the line AT was continuous

Lc.

vapour

temperature

FIG. 174.

through T. But Kirchhoff (Pogg. Ann., ciii. p. 206) showed that the slope
of AT continued was different from that of TB, a conclusion which
follows at once from the latent heat equation. Putting that equation in
the form

dp _ L

dB~ '0(v2-va)

p may represent the saturation pressure of the vapour of either ice or
of water, while v2 — v^ is the change in volume from the denser to the
rarer state. Let o> represent the pressure of water- vapour, &/ that of ice-
vapour. Let L be the latent heat from water to steam at 0°, L' the
latent heat of fusion of ice. Then the latent heat from ice to steam is
L + L'. We may evidently neglect vv the volume in the denser state,

since it is exceedingly small compared with v2> and if we put — = tr, the

*2

vapour density

THERMODYNAMICS OF CHANGE OF STATE, ETC. 311

dd) _ L _ Lor ,. ,

39"3^~T

%-&+*% (2)

^-fe-^ <»>

or dividing by (2)

(dot _ d<a\ _._ do/ _ L' x . v
d0~&) i5ff~T+i?

These equations obviously apply to any substance at its triple point. In
the case of water at 0°

L = 606-5 from Regnault's formula (p. 180)

L' = 80 ; —Q = 4'60 - 4-26 = 0'34 mm. from Regnault's researches on

vapour-pressure,

cZw d<a 80 x-34 n

---='

That is, the vapour-pressure of ice below 0° falls down more rapidly than
Lhat of water by about 0*04 mm.
per degree.

Then the line AT, if con-

tinned beyond the triple point, ["""

lies above TB; see the dotted

line TB' in Fig. 174. FIG. 175.

The difference in vapour-
pressures of ice and water just below 0° is too small to measure accu-
rately, though Regnault's work on re-examination indicates its existence.
But if, instead of taking the difference of vapour-pressures for the same
temperature, we take the difference of temperatures for the same vapour-
pressure, we get a quantity large enough to be measured. For let T (Fig.
175) be the triple point on a pressure-temperature diagram. Let TA,
TA' represent the ice- vapour and water-vapour lines. Let A' AM be a
line of equal pressure. Then AA' is the difference of temperature of ice
and water in equilibrium with their vapours at that pressure.

AA' A'N , . A, A'JST. AM

But ~ and AA = =-=

AM TM TM

o> -a/ _ -040
= ~d^' ~ ~^4

dd

(where 0 is the temperature of the water below the triple point)

= 0-116x0
or 0'116° per degree below the triple point.

312

HEAT.

This has been verified by Ramsay and Young (" On the Influence
of Change from the Liquid to the Solid State on Vapour-Pressure,"
Phtt. Trans., Part II., 1884, p. 461). They evaporated water and
ice in separate vessels at the same pressure below that of the triple
point, and found that the water when in equilibrium was colder than
the ice.

While it is possible to supercool water in contact with its vapour, i.e.
to have it to the left of the line TO, Fig. 174, so that TB' has an actual
existence, no method of superheating ice has yet been devised. The pro-
longation of BT beyond T, therefore, represents nothing physical. But
ice can be obtained in a condition represented by points slightly to the
right of TO. For instance, if a block of ice at the triple point is subjected
to pressure, it is raised in temperature slightly (from the formula of p. 286
about 0'002° C. per atmosphere if we take its coefficient of expansion as
Q'00015 and its specific heat as 0'5), while it only melts at its surfaces.

N N

enlropy
FlG. 176. — Equal Pressure Lines on the Entropy-Temperature Diagram.

Thus it will be represented on Fig. 1 74 by a line from T nearly vertical
but sloping slightly outwards as it rises. And no doubt it would be pos-
sible to obtain ice under tension in a condition represented by the pro-
longation of this line downwards into the vapour region. In the case of
water substance there are only three phases and one triple point. But
there are substances, such as sulphur, with four phases, i.e. two solid forms
as well as the liquid and gaseous, and for these substances there are three
triple points. For an account of these triple points and for the general
phase rule of Willard Gibbs, giving the condition of equilibrium of
phases, we refer the Dreader to Whetham's Theory of Solution, chap, ii.,
or to Findlay's The Phase Rule.

The Second Latent Heat Equation. — Another equation connect-
ing latent heat and its change with the specific heats of the two phases
may be readily obtained from the entropy temperature diagram. Let
ABOD, A'B'C'D' (Fig. 176) be equal pressure lines, the horizontal
portions representing the mixture undergoing change of state with the
receipt of latent heat at 0 and Q—dO respectively.

Taking the substance round the cycle BCC'B'B, the heat given along
BC is L. That given along CO' is - Gzd6 where C2 is the specific heat

THERMODYNAMICS OF CHANGE OF STATE, ETC. 313

of the body in the second state, always just on the point of changing
into the first state. That given along C'B' is

and that given along B'B is + 0^0, where Cj is the specific heat of the
body in the first state, always just on the point of changing into the
second. Thus the total heat is the sum of these,

or T7r + 01-0aki0.

\dO l V

But this is equal to the area BCO'B', which is practically equal to the
rectangle with base BO and height dd. Now BC x BM = total heat along

0
Then the area

.

6

Equating the two values of the heat given we have

<£L n n L

W+QI G* r

In the case of water and steam at 100° 0.,

_^= -0'695 from Regnault's equation (p. 180),

C1=l L = 537 0 = 373
whence C2= -1-135,

or the specific heat of steam when kept saturated is negative.

This means that if we have a quantity of saturated steam at 100°, i.e.
atapressureof 760 mm., and we increase
the pressure adiabatically to 787 mm.,
which is the saturation pressure at
101°, the work done will raise the tem-
perature above 101°, and heat must be
abstracted to keep it down to that

TTjr1 1 77

temperature. Or if we decrease the

pressure adiabatically to 733 mm., the

saturation pressure corresponding to 99°, the temperature will fall below

99°, and heat must be given to restore it to that point.

In this latter case of adiabatic expansion, if dust nuclei are present,
condensation will occur, for as the temperature is reduced the pressure
will always be above the saturation pressure. For example, when the
temperature has reached 99° the pressure will not yet have fallen to 733
mm., and the excess of vapour will condense on the dust nuclei. This
agrees with the common observation that the adiabatic expansion of
saturated water vapour produces cloud. Let A'B', AB, A"B" (Fig. 177)
represent the three isothermals at 101°, 100°, and 99° on an indicator
diagram, and let MBN be the adiabatic through B ; then the above result
implies that BM cuts the upper isothermal in the vapour part to the

314

HEAT.

right of B', while BN cuts the lower isothermal in the mixture part
to the left of B".

Usually for saturated vapours the specific heat is negative, but for
ether vapour (Clausius's Mechanical Theory of Heat, p. 136)
it is positive at ordinary temperatures. Hence an adiabatic
expansion of ether vapour does not produce condensation.
Alteration of Vapour-Pressure with Curvature of

Liquid Surface. — Lord Kelvin (Proc. R.S.E., February 7,
1870) pointed out that the vapour pressure of a liquid with
a curved surface must be different from that of the same
liquid with a plane surface. Suppose that a capillary tube
(Fig. 178) dips in a liquid contained in a closed vessel, and
that no gas other than the vapour of the liquid is present
above the free surface. Further, suppose that the liquid
rises in the tube, and is in equilibrium at the level A.
Then there must be equilibrium between liquid and vapour
at A, as well as at the plane surface B ; that is, the concave
curved surface is in equilibrium with the vapour at a
pressure less than that at the plane surface B by the
weight of unit column of vapour of height AB. For if not,
suppose the vapour rising from A can have a pressure
greater than the pressure of the vapour at the same level
FIG. 178. outside the tube. The space above A will not be saturated,
and circulation will take place through continuous evapora-
tion from A and condensation at B, and work can be obtained. We may

make the idea more definite by imagining

a small engine above the capillary surface,

as in Fig. 179, the whole being maintained

at one constant temperature. If the pressure

of the vapour from A exceeds the pressure

of the surrounding vapour, let evaporation

take place from A at the maximum pressure,

and let the vapour go through the valve V

and push against the piston P with this

pressure. When some quantity of vapour

has passed through V, let V be closed and

let P move forward with isothermal expan-
sion of the vapour between it and V till

there is the same pressure on both sides of

the piston. The excess of vapour pushed in

front of P will produce condensation at B.

Then let a valve be opened in P, and let

P move back to V. It is now ready for

another stroke. This process can be carried

on endlessly, and the piston does work while

the substance is carried through a cycle which

can be made perfectly reversible by merely

reversing the action of the valves. But in

such a cycle no balance of work can remain over, so that our supposition

of difference of pressure between the vapour from A and its surroundings

must be false.

FIG. 179.

THERMODYNAMICS OF CHANGE OF STATE, ETC. 315

If o) be the pressure of vapour at B and <o' be its value at A, and if
h, the intervening height, is so small that we may regard the density
of the vapour <r as uniform, we have

We may express h in terms of the surface tension T and the radius of curva-
ture r of the surface. The pressure in the liquid just under the surface A is

, 2T

CO - — ,

r

therefore the pressure in the liquid at the level of the plane surface is

where p is the density of the liquid. But this is equal to the pressure
of the vapour outside at the same level, viz.,

to = a/ + gcr h

equating we get

2T
~=9(p-<r)h

, 2T <r

whence w-to = -- .

r p — °"

If the surface is convex, so that the liquid is depressed in the tube, we
have

2T <r

(0—0) =

r p-a-

The expression is simpler if we introduce the difference of hydrostatic
pressures P under the curved and under the plane surfaces. This is

or

P<r

whence w — w = — .

P

If the height Ji is great we cannot take <r as uniform. We must
then integrate the change in pressure with the varying density as we
descend from A to B.

Let dli be a small step down, and let c?co be the change in pressure in
the step, cr being the density.

Then d(a

and a) = Ko- (Boyle's law)

whence d« = gdh

316 HEAT.

and integrating from o>' to w, and h = 0 to h = h

, w qh
log -, = 4

= Ptr

(up

where P is the difference of hydrostatic pressures under the two
surfaces.

We may apply this result to the case of a small spherical drop,
radius r surface tension T. The hydrostatic pressure just within the
drop will be greater than that under a plane surface at the same level

2T
by P= — . Hence the vapour-pressure from the drop will exceed that

2T cr
from the plane surface by — • — if P is not very great or r is not very

T cop

small. If we cannot make these limitations, then we must use the
logarithmic formula, and put

co 2T o-

log^'=T ' Jo>

Hence a space over a plane liquid surface and saturated for that
surface, is not yet saturated for a small drop, and such a drop if formed
in any way will tend to evaporate and disappear. We can see, then,
how in a dust-free space vapour can exist without condensation, though
supersaturated as far as a plane surface is concerned.

If, however, dust is present, a particle of it may have such small
curvature that if any condensation occurs on it the vapour pressure of
the liquid is practically that of a plane surface. The liquid may spread
round the particles, and from the beginning form drops of such size that
the space is saturated for them and they continue to grow.

It is to be noted that the alteration of vapour-pressure is very small
until the radius of curvature is exceedingly minute.

Thus, taking water vapour at 0°, we have its density - density of air.

o

Now, for air at 0° and 760 mm.

co 101 4,000
cr ~ -001296

whence for water vapour

5-£8?-««»'

THERMODYNAMICS OF CHANGE OF STATE, ETC. 317

Putting p=l and T = 80

eo_ 160 1
gcto'~l-25xl09 ' r

1-28x10-7

to 1-28x10-7
logl» ^ 2-30 xr

•56 x 10-7

If r=10-5cm. "=1-013

CO

r=10-«cm. -,= 1-138

CO

r= 10-7 cm. = 3-631

CO

or it is only when the drop is of the order of a millionth of a centi-
metre that its vapour-pressure is sensibly greater than that from a plane
surface, and if the dust particles are greater than this, a very small
excess above normal saturation may make drops form on them.

If no dust particles are present, we may perhaps still see how drops
are formed if the pressure is great enough (chap. x.). For there may
be groups of the vapour molecules formed here and there by collisions
which are virtually small liquid masses. Now the radius of the sphere
of action of a molecule is of the order 10 ~8 (chap, ix.), so that if a group
of size 10~7 is formed it may find the space saturated if the pressure is
three or four times its normal value and it may continue to grow.

The Connection between Alteration of Vapour-Pressure by
Pressure on the Liquid and the Change in Melting-Point by

Pressure. — An alteration in the hydrostatic pressure to which a liquid
or a solid is subjected, affects the vapour-pressure, whether the altera-
tion is produced by capillarity, as in the rise or fall of a liquid in a
tube, or by increasing or decreasing the atmospheric pressure on the
surface, as when ice or water is contained in a vessel into which more or
less air may be pumped.

If we have a mixture of ice and water in such a vessel with their
vapour only above the mixture, then their equilibrium point is the
triple point. But if air be pumped in till the pressure is P, the equili-
brium temperature of the ice and water falls by

where vl is the volume of 1 gm. of ice, v2 that of 1 gm. of water, and L'
is the latent heat of fusion.

318

HEAT.

^^

/Ur

and

Ice

Vapour

V

c

-_ 1^

8

8=3

FIG. 180.

At this temperature we may show that the two vapour-pressures of
solid and liquid are so changed by the pressure P that they are again
equal, or we have, as it were, another triple point. This may be proved
by imagining an arrangement to take the substance
through a one temperature reversible cycle. That used
on p. 309 for the ordinary triple point is no longer appli-
cable. Suppose that the two vapour-pressures at the
new equilibrium pressure are different, that of water say
being the greater, then if there be water in a vessel
(Fig. 180) containing air at pressure P, at the level of
the water surface the vapour in the air would be super-
saturated for ice. But if the vessel be sufficiently lofty,
then the pressure both of air and of water vapour will de-
crease from below upwards, and at some height A, ice will
be in equilibrium with the water vapour present. Then
arrange a side tube of this height containing a cylinder
of ice and supported a little above its base by upward
pulls T. If now the cylinder is slowly pulled up by
forces applied at the level C, evaporation will take place
at A, condensation will go on at the water surface, and
solidification at the base of the ice. Or if the cylinder
is slowly let down, doing work against T, everything is
reversed, and we have a one-temperature reversible
cycle from which work may be obtained. Our supposi-
tion of different vapour pressures must therefore be false.
It may be shown that the alteration given by the formula of p. 315
just accounts for this new equality of vapour-pressures. For if TA TA'
(Fig. 181) represent the vapour-
pressures of ice and water, re-
spectively, below the triple point
T, a pressure P on the surface
alters the ice vapour pressure by

TI = — = Po-y1} where v^ is the

Pi
specific volume of ice. That is,

the new vapour-pressure line is

IT7, P<rw, above TA. Similarly,

the new water vapour pressure

line is represented by WT',

P<rv2 above TA' where v2 is the

specific volume of water. W is

lower than I, since u2 is less

than vv and IW = PO^VJ - f2)- If

WT, IT' meet at T', and if r

be dd below the triple point,

evidently IW is equal to the

difference of height of TA, TA' for the same temperature.

p. 311 (3) this is

LW0
W — CD = — g —

Temperature
FIG. 181.

But by

THERMODYNAMICS OF CHANGE OF STATE, ETC. SJ9

T, /

whence * ~ v

and

Lt

or the two vapour-pressures are equal at the new melting-point.

The point T' is really above T, or as the melting-point falls by
pressure the vapour-pressure rises. For if we take the water vapour
pressure dd below T when the water is under pressure P, it is

. d(» T/1 T-,

w = a) - 7adu + Per • vz
do

where to is the vapour- pressure at the triple-point

d<a Ixr

But ~

and

<rdO/ L'v8 T\
then u> «= a> + — TP ( - • — Li ]

e ^-v^ j

, Lcr^fl/ L't>2 - L(vt - v2) \
1F\ Lto-*,) /

L' 80 , v, - v9 Aft

but T- = c7^-R and J - '~^

L 606'5 v2

then the second term on the right is positive, and o>' is greater than o>.

The results just obtained enable us to give a molecular account of
the phenomena of melting under pressure. The vapour-pressure of a
substance depends on the number of molecules escaping from the surface
per second, or upon the " mobility " of the molecules, and we may take
the vapour-pressure as measuring this mobility. At the triple point the
three states are apparently in equilibrium ; but this is only a " mobile
equilibrium " due to the equality of the number of molecules passing in
each direction across each separating surface. In other words, the
mobility of each state is the same. Below the triple point the vapour-
pressures differ, and the mobilities differ also. Thus the liquid gives to
the solid more molecules than it receives in return and the solid grows.
The liquid evaporates more than the solid, and would distil over on to
the solid in the space above their surface. But if pressure is put on ,the
mobilities are increased, though not to the same extent, the denser state
being less increased than the less dense. Hence at the triple point
equilibrium is destroyed, the ice having a greater mobility than the
water, and the new point of equal mobility is below the triple point.

We have applied the foregoing equations to the case of ice and
water ; but it is evident that they apply to any case of a substance
capable of existing in the three states or phases at the same time.

320

HEAT.

Solutions.
Vapour-Pressure of a Solution always Less than that of the

Solvent. — It has long been known that when a stilt is dissolved in a
liquid the vapour-pressure of the solution is less than that of the pure
solvent, the reduction of pressure increasing with the proportion of salt
dissolved.

It follows that if we have two vessels in an enclosure, as in Fig. 182,
one containing pure solvent and the other containing a solution, the rest
of the space being filled with vapour of the solvent — the salt being
supposed to have no appreciable vapour — then the solvent will distil
over into the solution, and the level of the latter will tend to rise till its
surface is so high that the pressure of the
vapour actually present at the new level is
equal to the vapour-pressure of the solution.

Let h be the difference in levels of the
surfaces when equilibrium is attained, let o>
be the vapour-pressure of the pure solvent,
o>' that of the solution, and cr the density of
the vapour. We suppose that the solution is
very dilute so that h will not be too great to
suppose a- uniform in the column intervening
between the levels of the two surfaces. Then
we have

lutio i

vapour

I 7

= o> + gph—

where p is the density of the solution, and if
FIG. 182. this is very dilute it is equal practically to that

of the solvent. Let us put P = ypli, then P is

the excess of hydrostatic pressure in the solution over that in the solvent
at the level of the surface of the latter.

Then

Po-

(1)

But (p. 315) if we put pressure P on to the solution its vapour- pressure

j u Po-

is increased by — , or since

P

, Po-
tt) = CO -\ ,

p

it is increased to the vapour-pressure of the solvent. Hence the solution
will be in equilibrium with the vapour at such a height above the surface
of the solvent that the hydrostatic pressure due to that height of liquid
will make the vapour-pressure of the solution equal to that of the pure
solvent.

We can see this also by imagining a capillary tube, which the
solution does not wet, to be fixed in the side of the solution vessel, as in
Fig. 182, and of such narrow bore that the solution is depressed in it to
the level of the surface of the solvent. Evidently the vapour-pressure

THERMODYNAMICS OF CHANGE OF STATE, ETC. 32J

at this surface must be equal to that of the solvent, otherwise a one-
temperature reversible cycle giving work could be arranged. Or,
again, the vapour-pressure of the solution is made equal to that of
the solvent by putting pressure on it, ~P = yph, where h is the height
at which the solution is in equilibrium with the vapour above the
solvent.

The pressure P which must be put on to the solution to equalise the
vapour-pressures is, for a reason to be given later, called the OSHLOtic

pressure.

If the difference of vapour-pressures is too large to allow us to assume
the vapour-density constant, we must use the logarithmic formula, whence

to Po-
loge— , = —
to top

or

This may be obtained, of course, exactly as on p. 316.

Raising Of the Boiling-Point. — One effect of the lowering of the
vapour-pressure by solution is a rise in
the boiling-point. For, if the solvent
vapour has a pressure of 1 atmo. at a
given temperature, the solution has a
pressure less than 1 atmo. at that tem-
perature, and it must therefore be raised
to a higher temperature before it will
boil.

We can easily calculate the rise of
boiling-point in terms of the osmotic
pressure.

For if AO, BO, Fig. 183, represent the j>
pressures of solvent and solution vapours |_
at the normal boiling-point of the solvent,
and AD, B(J, the vapour-pressure lines as
the temperature rises, if we draw AC
parallel to the temperature axis to meet
BC in C, AO is the rise in boiling-point,
say dO. As we suppose the solution to be dilute, AD and BC will be near
together and may be regarded as parallel straight lines. This follows
also from Raoult's researches (see below) confirming earlier work, which

~Df)

showed that for a given solution -— is independent of the temperature.

AO

AB

Then AD is practically parallel to BC if — ^ is very small.

o

Temperature

FIG. 183.

Then

CD CD
AC

322

HEAT.

by p. 320 and p. 311

(2)

dO
Por Ixr

-7+-zr
E?

"Lp

where L is the latent heat of the solvent, and 6 is the absolute tempera-
ture of the normal boiling-point.

Lowering of the Melting-Point. — Another effect of salt in solution
which has long been known is a lowering of the melting-point. When
the solution does begin to freeze, the solid which crystallises out is pure
solvent. At the new melting-point, when solution and solid solvent

are in equilibrium with each
other, their vapour - pressures
must be equal. For if not,
T if, for example, the vapour of
S the solution were the greater,
a reversible one - temperature
cycle could be arranged giving
work.

Let T, Fig. 184, represent
the triple point of the pure
solvent, TA the liquid solvent
vapour- pressure line, TB the
solid solvent vapour -pressure
line, SB the vapour-pressure
line of the solution, lower than

TA by — where P is the osmotic
P

pressure of the solution ; o- the

density of the vapour, and p that of the liquid, and let TB, SB meet in
B. B is the melting-point in the solution. Let it be dQ below T.

We have AB = TS = —

P

But AB is equal to the difference of pure solid and pure liquid solvent
vapour- pressures dO below the melting-point, whence, as on p. 311,

Temperalure

FIG. 184.

where L is the latent heat from solid to liquid. Equating the two values
of AB

P<r

or

dd =

P

P6
Lp

(3)

The quantity P, the osmotic pressure, or as we are now regarding it, the

THERMODYNAMICS OF CHANGE OF STATE, ETC. 323

hydrostatic pressure .which must be put on the solution to restore its
vapour-pressure to equality with that of the solvent, can be found from
either of these effects, lowering of melting-point or raising of boiling-
point, or going back to equation (1) it may be found from the difference
of vapour-pressures.

Though much work had been done by earlier observers * the effect of
solution on vapour-pressure, and on melting-points was first put into a
satisfactory form by Raoult, who worked not only with salts in water,
but with solutes in other solvents. Herein lay the success he attained in
formulating laws. For aqueous solutions are generally electrolytic and for
these the osmotic pressure is irregular, probably for a reason which we
shall give below. But with other solvents it is easy to make great
numbers of non-conducting solutions, and these give more regular values
leading to definite laws. Raoult showed f that

1. or the proportionate lowering of the vapour-pressure, is

a)

independent of the temperature for a particular solution, ether being the
solvent.

2. — is proportional to the amount of salt or solute dissolved.

„ T» grammes dissolved in 100 of solvent

molecular weight of solute

= "gramme molecules " of salt per 100 grammes of solvent
then in the case of ether as solvent

0) — (I)'

-T- n is constant,

0)

or the lowering per gramme molecule is the same for different substances.
Later he showed J that if n is the number of molecules of solute in N of
the solvent, then for very dilute solutions

a) — to n i

= — very nearly.

0)

Still later § he found that

would hold for much stronger solutions if c were 0*9, and that as the
dilution increased c tended towards 1.

For electrolytic solutions, c may be very much more than 1, rising
towards 2 for such salts as NaCl or KC1 in water, and towards 3 for such
salts as Ca C12 or Ba C12, in water.

. The law — = — implies that the lowering of the vapour-pressure

* See Ostwald's Solution, chap, vii., for a history of the subject.

t C.R., ciii. p. 1125, 1886.

J C.R., civ. p. 976, 1887.

§ Zeit.fiir Phys. Chem., ii. p. 353, 1888

324 HEAT.

is proportional to the number of molecules of solute present, whatever
their nature. When we have to multiply by c, where c is nearly 2, it is
taken to imply that the molecules of salt are for the most part dissociated
each into 2. When we have to multiply by a number nearly 3, it is
taken to imply that the dissociation is from one into three molecules and
so on.

For very dilute non-electrolytic solutions n, the number of salt
molecules, is very small compared with N, the number of solvent molecules,
and we may put

— a n

(4)

Substituting in (1), p. 320, we get for the osmotic pressure

n /K\

N

If we put the relation between the pressure o> and the density ir of
hydrogen as

(T

we may put that for any rarefied vapour of molecular weight M as

°=2-ne

<r M

whence P = T'£ <6>

Further, if s is the density of the salt in the solution or the number

of grammes per c.c., and if S is its molecular weight, then _ is the number

S

of gramme molecules per c.c. Similarly ~ is the number of gramme
molecules of the solvent per c.c. if the solution is sufficiently dilute.

5-5*4

and substituting in (6) we get

Comparing this with - = ^R# we see that the osmotic pressure P is the

pressure which the salt would exert if it were gaseous, and of the density
at which it actually exists in the solution. This most remarkable result
was first obtained in a very different way by Van t'Hoff, and is known
as Van t'Hoff's law of osmotic pressure. We shall give some account of
his theory later.

THERMODYNAMICS OF CHANGE OF STATE, ETC. 325

It will be useful to collect the results which we have obtained. We
have for very dilute solutions

Osmotic pressure P = (w-a>')£ (1)

Raising of boiling-point or lowering of freezing-point of liquid density p

d9 = ~p (2) and (3)

Raoult's experimental value of the difference of vapour-pressure,

w — to' en en

= _ nearly

ft> N + n

where c is nearly 1 for non- electrolytes. For such non-electrolytes by
substituting in (1)

,.».- (5)

<}

Whence Van t'Hoff's result P-^r • &# 00

S

For the purpose of illustration let us calculate the actual values for
1 gramme molecule in solution in water. It must be remembered that
as a rule aqueous solutions are electrolytic, and it is only for non-
electrolytic solutions, such as one of cane sugar, that the results hold.

We may determine the value of R from the density of hydrogen
•0000896 at 0° 0. and 1 atmo. or 1014000 dynes/sq. cm.

we get R = 4- 15 x 10T when o> is in dynes/sq. cm.

or — 40'9 when to is in atmos.

8

When the solution is 1 gramme molecule per litre of water, ~- which is
the number of gramme molecules per c.c., is j^rry Then (7) becomes

P = 8-3 x 104 x 0 dynes/sq. cm.
or '08180 atmos.,

and if the temperature is 0° C., putting 0 = 273, this becomes

P = 22-3 atmos.
If there are n gramme molecules per litre

P = 22-3xraatO°
or =22-3n(l+a*)at«cO.

It must be noted that this only holds for dilute solutions, so that in
general n must be small, indeed a small fraction.

326 HEAT.

The lowering of the vapour-pressure per 1 gramme molecule per litre is

co -a/ _ n _ 1 18 1
a> ~N~ 1000 = 1000 ^SJHT
~18~

since N, the number of gramme molecules of water in a litre, is 1000/18.
Thus at the boiling-point the lowering per gramme molecule in mm. of
mercury is

The raising of the boiling-point is obtained from (3) and (7) by putting
0=373; L = 537 x 4-19 xlO7; p = 1/1-043

jfi - 8'3 x 104 x 3732 x 1 -043
537 x 4- 19 x 107

= 0-54°.

The lowering of the melting-point is obtained from the same formulae by
putting 0 = 273; L = 80 x 4-19 x 107 ; p=l

,„ 8-3x 10*x2732
80x4-19xl07

= 1-85°.

Thus for cane sugar, C12 H22 On, S = 342, and a solution containing 342
grammes per litre of water will boil at 1 00*54° C., will freeze at - 1-85° C.,
and will have a vapour-pressure less than that of pure water by 1 in 55.
These results agree very closely with direct observation.* Observations
have been made with a large number of salts and solvents, and when
the solutions are non-electrolytic the agreement is equally close.

If the lowering of the freezing-point or the raising of the boiling-point
is greater than the values obtained from Van t'Hoff's formula — if, in
fact, c in Kaoult's formula is greater than 1 we can reconcile the observa-
tions with the theory that each gramme molecule has the same effect by
supposing that dissociation of the salt molecules has taken place, and that
the dissociated ions are acting independently, each producing its own
effect. Thus, if we have such a salt as KC1, and if a fraction A of it is
dissociated, then of the total n molecules originally present only n (1 — A)
remain single, while nX. have split each into two, so that we have
n (1 - A) + 2nA = n(l + A) molecules actually in solution.

Hence the results just obtained must be multiplied by 1 + A, where
1 + A may rise to 2 if dissociation is complete.

Thus with KOI in water a solution of 1 gm. molecule per 10 litres of
water gives a depression not of 0'185°, but about 1'86 times as much;
whence we may conclude that 86 per cent, of the salt is dissociated. As
the dilution increases so does the molecular depression of the freezing-
point, whence it is concluded that the dissociation also increases. This
supposition of dissociation is confirmed by observations on the electric

* Wbetham, Solution, chap. vL

THERMODYNAMICS OF CHANGE OF STATE, ETC. 327

conductivity of the solution, where the percentage dissociated comes out
to nearly the same value as that given by the depression of the freezing-
point.

The theory of osmotic pressure which we have just given appears to
be the most satisfactory mode of treating the subject, in that it approaches
it through the phenomenon of vapour-pressure and the effects of change of
vapour-pressure, which are most easily subjected to direct measurement.
But historically the idea of osmotic pressure was arrived at through the
study of diffusion through membranes, and the difference of vapour-
pressure was regarded as a secondary phenomenon. We shall give a
brief account of the subject from this point of view, as it gives some
excellent examples of the application of thermodynamic reasoning.

Semi-Permeable Membranes. — It has long been known that if a
mixture or solution of spirit and water be separated by bladder from pure
water then the water passes into the solution more rapidly than the
spirit passes out, and will maintain an excess of pressure on the spirit
side. Dutrochet, who first carefully investigated this passage through
membranes, gave to it the name osmosis. Graham in 1854 {Phil. Trans.,
1854, p. 177) showed that osmosis took place rapidly into alkaline solu-
tions ; and in 1861 he published his celebrated memoir on Dialysis (Phil.
Traits., 1861, p. 1183), in which he divided substances into two classes :
the crystalloids, which in solution will pass through membranes, and
which include bodies susceptible of crystallisation; and colloids, or glue-like
substances, such as gum, gelatine, albumen, which will not pass, or only
with great slowness.

A new turn was given to the subject by Pfeffer's discovery in 1877
that membranes could be made through which a salt could not pass,
while water could pass freely. One such membrane he made by filling a
porous pot with a solution of potassium f errocyanide and immersing it in
a solution of copper-sulphate. Where the two solutions met in the pores
of the clay an insoluble gelatinous precipitate was formed, supported by
the clay and constituting the membrane. This membrane will allow
water to pass through it easily, but is impermeable to sugar. It is there-
fore termed a semi-permeable membrane.

i Pfeffer working with sugar solution inside a semi-permeable mem-
brane and water without, found that the water passed in more rapidly
than it passed out, until a certain definite excess of pressure existed
within the vessel over that without. This excess of pressure, the osmotic
pressure — we shall see later that it is identical with osmotic pressure as
defined above — is found to be proportional to the amount of sugar in
solution and to rise with the temperature. Thus the osmotic pressure
was 53'1 cm. of mercury for 1 per cent, of sugar dissolved at 14'2° C., and
it rose from 50 '5 cm. at 6'8°C. to 54 '8 cm. at 22°. Pfeffer also worked with
some other solutions. In 1884 De Vries (Pringsheim's Jahrbucher, xiv.
p. 427, 1884) found that when certain vegetable cells, those of the epi-
dermis on the under side of the midrib of the leaves of Tradescantia
discolor being the best, are placed in concentrated salt solutions the
protoplasmic contents contract and shrink away from the cell walls, the
protoplasm being apparently covered with a semi-permeable membrane,
which allows the water to pass more freely from the dilute solution in the
protoplnsm than in the opposite direction. If the external solution is

328

HEAT.

gradually diluted a point is at last reached at which the protoplasm tends
to fill the cell again, or the osmotic pressure of the contents of the cell
equals that of the solution. De Vries called the solutions at this stage
isotonic. The same cells could be used for solutions of different salts, and
he found that equimolecular solutions of similar salts are isotonic.

Van t'Hoff s Application of Thermodynamics.—

In 1885 Van t'Hoff (Phil. Mag., xxvi., 1888, p. 81)
pointed out that the osmotic pressure of sugar solution
as obtained by Pfeffer was the same as would be exerted
by the sugar in the gaseous form at the same density.

Thus at 14° a 1 per cent, solution of sugar, or -^-r-^, °^ a

°

top-in

solvent.

J/L

solution

FIG. 185.

gramme molecule should produce a pressure

D 22-3/ 14 \ na
F= ( l + -— ) x 76 cm. ot mercury.
o4 2t\ 2iioJ

= 52-2 cm.

almost exactly that found by Pfeffer.

Van t'Hoff also showed that in the variations of the pressure with
temperature as found by Pfeffer the pressure is proportional to the
absolute temperature. He then applied thermodynamic reasoning to
show that in any case the osmotic pressure is proportional to the absolute
temperature if a certain assumption is made which will appear in the
course of the proof. He used the conception of a semi-permeable mem-
brane which will allow the
solvent to pass, but which
is impermeable to the solute,
even though no such mem-
brane may have been found /
in practice. dp

The following is equiva-
lent to Van t'Hoff's work.
Suppose that we have a
cylinder containing a quantity
of solution with osmotic pres-
sure p, separated from the
pure solvent by a semi-per-
meable piston, A, as in Fig.
185. If there is a pressure p
on the piston A so that the
solution is under excess pres-
sure equal to the osmotic

pressure, there will be equilibrium between solvent and solution through
the membrane, and the piston can be moved either way reversibly.

Let us take the piston through a reversible cycle represented on the
diagram, Fig. 186, by ABCD, where abscissae represent volumes, ordinates
osmotic pressures, i.e. pressures by the piston on the solution. Beginning
at temperature 6 at A, allow MN = dv of solvent to flow into the solution,
the temperature being maintained at 0 by the addition of heat H. Then
allow a f urther flow of solvent, with no further addition of heat, represented

volumeM

FIG. 186.

THERMODYNAMICS OF CHANGE OF STATE, ETC. 329

by the adiabatic BC, the temperature falling to 6 - dd. Then let the
piston be pushed back at 6 - d6 along CD, and finally back along the
adinbatic DA to its initial position.

Then the work done is ABCD = AEFB,

and by the second law this is

or

-$»

dp

dO

dv =

H

Now, if the dilution is sufficiently great, further dilution neither gives
out nor absorbs heat. Whatever in the action of
solution does produce heat changes, is all finished by
the time the solution is sufficiently diluted. We may
picture the action perhaps as a separation of the salt
molecules, and as a surrounding of each salt molecule
by solvent molecules. When this is once done, it does
not matter how many more solvent molecules we add,
for we shall not alter the condition of the salt mole-
cules. Hence H is, as with a gas isothermally expand-
ing, equivalent to the external work done, and is 0 if
no such work is done. •

We put
whence

II

pdv,

dp _p
dO~~B

only

gas
insolulioi
Os m
Press
P

liquid
only

tp.m

pe.rm-.lo
qas only

or p = cd.

When the dilution has not proceeded so far that no A r \permito

heat is evolved or absorbed on further dilution, we
cannot assume that H =pdv, and we cannot draw the
conclusion that p = c6.

Having thus shown that osmotic pressure follows FIG. 187.

one of the gas laws, Van t'Hoff then showed that
when the solute is a gas, it follows from Henry's law that the
osmotic pressure is actually equal to the pressure the dissolved gas
would exert if it occupied the same volume in the absence of the
solvent. Henry's law states that a liquid at a given temperature
always dissolves the same multiple of its own volume of a given
gas whatever the pressure of the gas. Or, put in another way — if a
space contains a liquid and a gas, the liquid dissolving some of the gas,
say n volumes at a given pressure, then if the volume and pressure
be varied the relation between them will be given by Boyle's law, on the
supposition that the one volume of the liquid counts for n volumes. To
apply Henry's law to osmotic pressure, imagine a cylinder containing a
quantity of liquid with gas dissolved in it (Fig. 187). Let A represent
a movable piston permeable to the liquid only. Let B represent a fixed

330 HEAT.

partition permeable to the gas only, and 0 a solid piston against which
any desired pressure can be exerted. Let us begin with A at the
bottom and C close against B, and let the pressure of the gas
necessary to keep the solution at its actual strength be P ; let the
osmotic pressure be p. Then A and C are in equilibrium if the pressures
on them are p and P. Let V be the volume of the liquid, and let the
dissolved gas have volume nV at pressure P. Now move the piston A
upward through the solution, pure liquid being left behind it, and at
the same time allow the gas to pass through B, and C to move forward
so that the solution is always of the same strength between A and B.
Then C must move n times as far as A, and ultimately when A has
reached B the volume of gas between B and C is nV. The work done
is ^;Y — PnV. Now, allow the gas between B and C to
expand isothermally to some new volume riV + V and
pressure P'. Let the osmotic pressure, of the gas in
6 solution under pressure P' =p' . Now let the pressure on
A be made equal to p, and let 0 be moved down under
pressure P' while A is moved down under pressure p. C
moves through riV while A moves back through V.
Then we have nV volumes of gas dissolved under pres-
sure P' and V volumes still outside. If we now force
the piston C down till all the gas is dissolved, at every
pressure n volumes at that pressure will be in the liquid,
so that the gas compresses into the liquid just as if we

vapour

solvent
Pi

•solution

started with wV + V volumes at P, and ended with «V
FIG. 188. at P. Hence the work done on the gas in this last com-

pression is just that done by the gas in the expansion from
nV to riV + V. We have now gone through a one-temperature reversible
cycle, and have total work zero, or

pV - PwV - p'V - P'riV + equal and opposite
works in change from nV to nV + V = 0.

or p — nP = constant.

But evidently if P = 0, p = 0, so that the constant = 0
and p = riP.

No method has yet been found of extending this proof to dilute
solutions of solids and liquids, and in the absence of anything correspond-
ing to Henry's law we cannot expect such extension. But we might
perhaps expect the result to hold, inasmuch as in dilute solutions of
solids and liquids the molecular distances of the solutes are of the order
of gaseous molecular distances, even if we did not know it from the
direct investigation of osmotic pressures by Pfeffer.

It now only remains to show that Raoult's value of the lowering of
the vapour pressure in solutions follows from Van t'Hoff's law of the gas
value of osmotic pressure.

Let us suppose that we have a solvent of density />, vapour pressure
a), and density of vapour <r. Let it form a solution in which the osmotic
pressure is P while the vapour pressure is a/ and its density cr'. Let the

THERMODYNAMICS OF CHANGE OF STATE, ETC. 33 J

solution be under a semi-permeable piston A (Fig. 188), and let solvent
be above the piston, and let a movable piston B be above the solvent.
Allow volume v mass pv of the solvent to go through a one-temperature
cycle, thus —

(1) Let A be pushed down under pressure P till volume v of the
solvent has passed out of the solute, work Pv being done.

(2) Let volume v, mass pv, of solvent evaporate at pressure <D, pushing

B up, forming volume £- of vapour and doing work t^.
cr cr

(3) Let the vapour expand isothermally from volume — to —, at

pressure a/, doing work — — . pv ( — , — - ) approximately since w and u/

2 \cr cr/

are for dilute solutions not very different.

(4) Let the vapour condense back on to the solution by some suitable
contrivance, the solution now being under pressure CD', the work done

, . (a pv
being — ?y- .
cr

The cycle is now complete, and the total work is zero, or

I ' /I 1 ^

or 2 \

But - = — , by Boyle's law, whence

<T cr'

p _ O) + co'

~2~'

O) + 0>'

or ft) — ft)' = since o> + co' = 2o/ nearly.

P

If s be the density of the solute, and S its molecular weight, \Tan
t'Hoff's gas value of P gives

P*^ . T> ft
^ — _rt(7.

S
while if M be the molecular weight of the solvent .

""M '

Hence, substituting for P and cr

O) — CD' _ 8 _._ p

~«~"g *a

332 HEAT.

where n : N = number of gramme molecules of salt : number of gramme
molecules of solvent.

It is not necessary to show how the lowering of the freezing-point
follows from the value of the osmotic pressure.

Molecular Theory of Osmotic Pressure. — Since a solvent and a

solution are in equilibrium on the two sides of a semi-permeable mem-
brane when the solution is under pressure equal to what we have termed
the gas pressure of the solute, we are naturally led to look upon the
pressure of the solution as consisting of two parts, that of the pure
solvent and that of the solute, the latter being equal to its gas pressure.
Hence the solvent alone has the same pressure on the two sides of the
membrane. If the solute were actually gas, with the gas velocity
assumed in the kinetic theory, and if its molecules moved among those of
the solvent, quite independent of them, it would produce the observed
osmotic pressure by its impacts against the membrane.

But it is difficult to imagine how this independent motion exists, and
it is remarkable that if, instead of supposing that the molecules of solute
move freely among those of the solvent, we go to the other extreme and
suppose that they enter into some sort of combination with them, we can
obtain the same value of the osmotic pressure.*

Let us suppose that there are n molecules of solute to N of solvent,
and let the vapour pressure of the pure solvent be o>. Now, if each of
the n molecules of solute attached itself to a molecule of solvent, and
loaded it so that it had no tendency to evaporate, there would be N — n
evaporating molecules in the solution to N in the solvent. Then the
rate of escape from the liquid surface would be as N — n : N. If the rate
of return were equally hindered, so that with equal density of vapour
over solution and solvent the return of vapour molecules were also in the
ratio N — n : N, then we should have equilibrium with the same vapour
pressure in each case — just as we have equilibrium with the same vapour
pressure over an open surface and over a surface partially covered with
a perforated plate. But if, as is more natural in this case, we suppose
the rate of return the same for the same vapour pressure, that is, if
we suppose a vapour molecule to run the same chance of getting en-
tangled whether it descends into solution or pure solvent, then over the
solution the return will balance the reduced escape when the pressure of
the vapour is less in the ratio N - n : N. Hence

o> : at = N" : N - n

or ft> : W - a/ = N : n

,

,, a> — a) n
ana = ==

a> N

the approximate form of Raoult's law.

* Phil. Mag., xlii., 1896, r- 289.

CHAPTER XX.
THERMODYNAMICS OF RADIATION.

General Principle — The Pressure of Radiation — The Normal Stream of Radia-
tion, the Total Stream, and the Energy Density — The Pressure on a fully
Radiating Surface — The Relation between E and 0 in full Radiation, the Fourth
Power Law — Full Radiation remains full Radiation in any Adiabatic Change —
Relation between Volume and Temperature in an Adiabatic Change— Entropy —
Application of Doppler's Principle — Change of Energy in a given Wave Length
— Change of Energy of each Wave Length in an Adiabatic Expansion of full
Radiation — Maximum Value of Energy for given Range of Wave Length — Form
of the Function expressing the Distribution of Energy in the Spectrum.

General Principle. — The thermodynamic theory of radiation starts
with the idea that a space containing radiation resembles the working
substance in a heat-engine, in that it exerts pressure on its boundary,
that it can be compressed or extended, that it can receive more energy
as radiation from the boundary and can part with energy to the boundary
by absorption, and finally that it can be taken round a cycle. This most
fruitful idea is due to Bartoli, who in 1875 * showed that it was possible
by means of radiation to transfer heat from a colder to a hotter body in
a manner which will be understood from what follows in this chapter.
Though the details of his calculation require amendment, he showed that
the possibility of such transfer implies that radiation shall press against
any surface upon which it falls, f

In 1884 Boltzmann| took up Bartoli's method. He used Maxwell's
electromagnetic theory of radiation,§ according to which a pencil of
light or radiation, incident normally on a surface, exerts a pressure
on that surface equal to the energy per c.c., or equal to the energy
density in the pencil. This pressure has now been shown to exist
by Lebedew, and by Nichols and Hull.|| Taking an enclosure full of
radiant energy from a fully radiating surface, round a Oarnot cycle, Boltz-
mann showed that the density of the energy in a uniform temperature en-
closure containing full radiation is proportional to the fourth power of
the temperature, that is, he deduced Stefan's law. The subsequent
development of the theory is largely due to Wien.U

* Bartoli's most important paper was not fully published at first. It is given in
full in Exner's Repertorium der Physik, xxi., 1885, p. 198.

f Another proof of such pressure is given by Larmor, Radiation, Encyc. Brit.,
10th ed., xxxii. It depends on the energy in a train of waves of given amplitude
being inversely as the square of the wave length.

J Wied. Ann., xxi., 1884, p. 364.

§ Maxwell's Electricity and Magnetism, 1873, vol. ii., contains the first account of
this pressure.

|| Lebedew, Congres Int. dc Phys., ii. p. 133 ; Nichols and Hull, Am. Acad. Arts
and Sciences, xxxviii., 1903, p. 559 ; Pressure of Light, S.P.C.K.

IT Congres Int. de Phys., ii. p. 23.

333

334

HEAT.

The Pressure Of Radiation. — We shall assume Maxwell's value
for the pressure of radiation. He showed that a pencil of electro-
magnetic waves exerts pressure each way in the line of propagation
numerically equal to the energy per c.c. or energy density in the pencil.
On his theory there is no pressure at right angles to the line of propa-

gation. If, then, E is the energy density
of the radiation falling normally on a
receiving surface, it exerts pressure E per
sq. cm. on that surface. If the radiation
is incident at 6 (Fig. 189), let AB be the
trace of 1 sq. cm. of the surface, BC the
trace of area AB cos 0 x 1 perpendicular
to the pencil. The force on BO is E cos 6,
and resolving, this gives a normal pressure
on AB equal to E cos20. There is also a
tangential force on AB equal to E cos 0
sin 0, but this does not concern us here.
If the pencil is totally reflected, the reflected
ray also exerts pressure E cos20. It may be
noted that the tangential force in this
case is equal and opposite to that due to
the incident pencil, but that if there is
absorption the tangential force is not wholly neutralised.

The Normal Stream of Radiation, the Total Stream, and

the Energy Density. — We can describe the stream of energy from a
full radiator by three quantities, between which there are necessary
relations. These are

1. The Normal Stream of Radiation N.

If a sq. cm. is placed parallel to an emitting sq. cm., at a distance
r from it along the normal, then if the energy
from the emitting sq. cm. passing through the

N
other sq. cm. is -^ per second, N is the normal

stream of radiation from the radiator. Or if
cZS is an element of a radiator, d& an element
along the normal parallel to dS and r from it
the energy from ofS passing through d& is

^ per second.

'9

2. The Total Stream of Radiation R.

The total stream of radiation is the energy
emitted from 1 sq. cm. per second round a hemi- FIG. 190.

sphere. Let <2S be an element of the radiating

surface (Fig. 190). Draw a sphere radius r round dS and let ON be the
normal radius. The stream from rfS passing through an element rfS' of

the sphere at N is 5 — per second. If d& is at the end of a radius

r2

making 0 with ON, the stream through it is, by the law of inclined

.,

pencils,

j

T.. .,

Divide the hemisphere above c/S into rings round

THERMODYNAMICS OF RADIATION. 335

ON as axis. A ring between 6 and 8 + dO will have area ZTrrsind.rdO,
and the radiation passing through it will be

ydSco8g.2irrBiiig.nZg = 27rNdSsin0cosfcZ0

r2

The total radiation RcZS is obtained by integrating this over the
hemisphere, i.e. from 6 = 0 to 6 = ^, and we get

whence R = fl-N

3. The Energy Density in a fully Radiating Enclosure. — The energy
density E can be expressed in terms of R or N. For let the whole
sphere (Fig. 190) be sending out radiation ; that passing through O comes
normally from every element of the surface. If U is the velocity of
travel of the radiation, the energy density at O, due to an element c/S,

.„ , NrfS . NrfS , , , , ..

will be --JJ since — ^- passes per second through a sq. cm. at O, and it

travels U cm. in the second. Then the energy per c.c. is - or

u u
The Pressure on a Fully Radiating Surface.— Put at 0

(F.g. 190), a sq. cm. of fully radiating, fully absorbing surface at the
temperature of the enclosure. The energy density of a pencil falling on

it from an element dS> of the sphere is — - — , and if c£S is at the end of

UH

a radius making 6 with ON this produces pressure -

Ur2

If we divide the hemisphere into rings each of area 2irrzsin0d0, then

..2

total pressure due to incident radiation = / 27ZT2Ncos2#sin#e?0

7.

27TN

3U

But the radiation issning from the area at 0 will be equal in every
direction to that incident, so that this will be doubled, and we shall have

-w-ro-i

Relation between E and 6 in full Radiation. The Fourth

Power Law. — Let us imagine a sphere of which the inside surface
can be made, at will, fully radiating or totally reflecting. Let us suppose

336

HEAT.

that it can be surrounded at will by a perfectly conducting substance
maintained at any desired temperature. Let us further suppose, and
this is the essential feature of the method of working, that the sphere
can be compressed against or extended by the pressure of the radiation

E

P = — without doing any work on or receiving any energy from the

o

matter or ether or whatever we suppose to be the medium in which the
radiant energy is localised. Let us suppose, in fact, that the surface
of the sphere is a sort of semi-permeable membrane, permeable to the

P

BJP

\

volume

FIG. 191.

radiation medium, but impermeable, when we so choose, to the radiation
and its energy.

Now let us take the radiation-filled space through the following
cycle : —

1. Start at temperature 0 with the inside surface fully radiating,

E

and allow the surface to be pushed out by the pressure P = ^ from

o

radius r to radius dr, increasing the volume by 4irr2dr. The surface is
surrounded meanwhile by a conductor maintained at 0, so that not only
is work P x fan&dr done, but radiant energy E x \-nr\lr is supplied to fill
the extra volume. Hence the total supply of energy at Q is

4:7rrzdr x

2. Now make the inside surface of the sphere non-conducting and
totally reflecting, and allow the radiation within to push out the
surface a little further, no energy being received from outside. This is
an adiabatic expansion. The energy density decreases, both through the

THERMODYNAMICS OF RADIATION. 337

increased volume and the work done on the boundary, and the new
energy density will correspond to a lower temperature 0 — </0. The
pressure will fall say to P - d~P. We shall assume that the radiation is
so altered in quality that at every stage of the adiabatic expansion it
remains full radiation for the temperature to which it is reduced. We
shall seek to justify this assumption below.

3. When the adiabatic expansion is completed, and the temperature
is 0-dO replace the reflecting surface by one fully radiating and sur-
rounded by a conductor maintained at 0 — dO. Now compress the sphere
at 0 - dO to such an amount that

4. On replacing the fully radiating surface by a totally reflecting
one, an adiabatic compression will restore the system to the original
volume and temperature.

This cycle may be represented by Fig. 191, and evidently the net
work done is dP'iirr-dr.

Every part of the process is reversible, and indeed we have a Carnot
cycle. Hence

(j
substitute for dQ =

and for dP = ^-

a

<7E

and we have —^- -=-

" Jii

Integrating log 0* = log E + constant

then E = a#4

v 4R

and since -Hi = -=-

4"

Comparing this with the value of the radiation, page 250,

R = «r0*

Ua

we see that — = cr.

4

Full Radiation remains full Radiation in any Adiabatic

Change.— Let us now examine this assumption, on which the rever-
sibility of the cycle depends. Imagine an enclosure containing full
radiation at 6, and let it expand adiabatically till the energy density
corresponds to that in full radiation at 6 — dd. But let us suppose that
the energy, though of the same density, is not of the same quality as
the full radiation for 6 — dO. Suppose, for example, that the red is in
excess and the blue in defect. Put into the enclosure substances at
0 — dB, absorbing and radiating respectively red and blue rays. The

Y

338 HEAT.

red body will be heated and the blue will be cooled, and by establishing
a Carnot engine between them a little work can be obtained by equalis-
ing the temperatures and by gradually bringing back the radiation
towards the quality of full radiation. When we have arrived at this
quality the temperature will be a little below 0 - dO. Now remove the
red and blue bodies and compress adiabatically till the original volume
is reached. Since we started on this compression at a temperature
below 6 — dO, when we arrive at the original volume we shall have a
temperature below 0. Again, we shall have, by supposition, a quality
differing from full radiation, and we can again insert different absorbing
bodies, get out a little more work and lower the temperature still further
in reducing to full radiation. Hence we have in the end the original
volume, but colder, and we have obtained work in the process. That is,
we have obtained work in a process the net result of which is merely
the cooling of a body which may be, if we like, the coldest body in the
system. This we must consider as contrary to experience, and so we
reject the supposition on which it is based, that full radiation after an
adiabatic change ceases to be full radiation.

Relation between Volume and Temperature in an Adiabatic

Change Of Volume. — Let a volume V contain full radiation at tem-
perature 6 and with energy density E. Then the total energy is

W = VE

Now let V change adiabatically to V + rfV. If P is the radiation
pressure the work done is

But this is equal to the diminution in internal energy, or

8

or

Integrating EVi = constant
But E = a04

Then 0*V* - constant

or 0V* = constant

If the space maintains the same shape V* is proportional to the linear
dimension r. Then

dr = constant.

Entropy. — If <£ is the entropy per volume 1, then in an adiabatic
change <f>V is constant.

But EV* is constant also and so is ESV. Hence
<£ is proportional to E* or to 6s

Application of Doppler's Principle in an Adiabatic Expansion.

— If a train of waves is reflected at a surface which is itself moving, then by

THERMODYNAMICS OF RADIATION.

339

Doppler's principle the wave-length of the reflected train is altered. Sup-
pose that the reflecting surface B is moving with velocity u away from the
source A (Fig. 192). Consider first a normal pencil. It will appear,
when reflected, to come from a source A', as far behind B as A is in

2u

B

FIG. 192.

front, so that A' is moving back with velocity 2u and the waves, which
in the incident beam fill a length U equal to the velocity of radiation,
fill in the reflected train a length U + 2u. Hence if A, is the wave-length
in the incident train, A.' that in the reflected,

or if A.' = X + d\

U

If the train is inclined at 9 to the normal, then (Fig. 193) the reflected
train may be supposed to come from A', the image of the source A in the

2u A'

Fro. 193.

reflecting surface B, and again A' is moving back with velocity 2u. But
this only increases the length of the reflected p.ith by 2«cos#. Hence
the waves originally in U will be now spread over U + 2wcos0 and

340 HEAT.

Now let us apply this to radiation originally of wave-length A in a
perfectly reflecting sphere, which is expanding with velocity

= dr
U~~dt

If a ray strikes the surface at 0 to the normal, its path between succes-
sive reflections is 2rcos#, and it strikes the surface „ ^ times per

second. At each reflection its wave-length is increased by

dA = A.

U

so that the increase per second is

dA ZucosO U

-77=A . ===^- X

dt U 2rcos0

r

Xdr
or = - -=-

r dt

Then integrating, - is constant as the sphere expands, or the wave-
length is proportional to the radius.

Change of Energy in Radiation of a given Wave-Length in an

Adiabatic Expansion. — Let us suppose that a spherical enclosure only
contains one kind of radiation of wave-length X, of energy density e, and

exerting pressure p = „• If the radius increases from r to r + dr in an

adiabatic expansion, the original energy is equal to the new energy + the
work done ; then

4 4 e

e . TjTrr8 = (e + de) . -=ir(r + dr)3 + ^ . 4*ndr

de .dr a
whence — \- £— = u

e r

and integrating er4 — constant.

Change of Energy of each Wave-Length in Adiabatic Ex-
pansion of full Radiation. — We shall assume that the above result
applies to each wave-length separately when we have full radiation.
If a sphere, then, containing full radiation expands from radius r to
radius r', 6 changing to & where rO = r'& (p. 338), then

or A.0 is constant during the change.

THERMODYNAMICS OF RADIATION. 341

Let the energy density of the radiation in the range of wave-length
from X to X + dX. be denoted by ed\; e may be termed the energy per
unit range about the wave-length A. By the expansion let A., A + t/A
change to A', A' + dM, while edX, changes to e'dX,'.

As shown above

But dX' = -d\

r

then er5 = eV6

r &
or since ~

or, comparing e, the energy per unit range, or in equal ranges rfA, in two
full radiations at different temperatures and taking corresponding wave-
lengths given by \0 = constant we see that e is proportional to the fifth
power of the temperature.

Q

Suppose, then, at any one temperature we plot 25 as ordinate against

A# as abscissa, we shall obtain a curve which will be identical for full
radiation at all temperatures.
Then we may put

or e

where / is a function of the product A/9 only.

Maximum Value of Energy for Given Range of Wave-Length.

a

For a given temperature, when e is a maximum, — is a maximum.

Then writing the maximum value of e as em, ~ is the same for all

temperatures, and therefore XmO is the same for all temperatures, where
Am is the wave-length for maximum energy per unit range.

Form of the Function expressing the Distribution of Energy

in the Spectrum. — The methods used up to this point do not afford us
any information as to the form of the curve expressing the distribution
of the energy in the radiation spectrum, except that it must be
represented by

At present it appears to be necessary to introduce some hypothesis as to
the way in which radiation is produced, in order to find the form of

S42 HEAT.

We shall not discuss this subject, but shall refer the reader to Wien's
paper already cited for a particular hypothesis by which he obtained as
the value of e

e =

The work of Lummer and Pringsheim * and others shows that Wien's
formula does not agree with experiment through a very wide range of
wave-length.

Planck f by another hypothesis obtains a value

e =

which agrees more closely with experiment, f

It may be noted that if X6 is small compared with c2, that is, if the

C2

radiation is well on the ultra-violet side of the maximum, e™ is large and
Planck's formula is practically the same as Wien's. If, on the other

Cl

hand, Xd is large compared with c2, ew - 1 may be put equal to ^ and
Planck's formula becomes

o • —

C2

or the radiation of a given wave-length well on the ultra-red side of the
maximum is proportional to the absolute temperature.

* Conyres Int. de Phys., ii. p. 41.

t Deutsch Phys. GeseU., 1900, p. 202; Science Abstracts, iv., No. 507, p. 230.
J Kubens and Kurlbaum, Preuss. Ak. Wiss. Herlin, Sitz. Ber. xli. 1900, p. 929;
Science Abstracts, iv., No. 371 p. 167.

INDEX

ABSOLUTE scale of temperature, 10, 12,
266 et seq., 294-295, 299-302; effi-
ciency, 267 ; comparison with air-
thermometer scale, 269

Absorptive powers of surfaces, 228

Actinometer, 253

Adiabatic change, temperature change
in, 285 ; expansion of radiation en-
closure, 338-341 ; gas equation, 295

AdiaLatics on indicator diagram, 261 ;
steeper than isothermals, 287

Air, expansion of, by heat, 3; ther-
mometric use of, 10 ; expansion with
rise of temperature under constant
pressure, 42 ; coefficient of pressure
increase with constant volume, 46-
48 ; specific heat of, 83, 84, 85, 118-
120 ; convection of, 91 ; heat con-
ductivity of, 90-92, 106, 107 ; effusion
of, 141; viscosity of, 146, 147;
mean free path of, 147 ; collision
frequency, 147 ; effect of, on evapo-
ration, 162, 163; dust particles in,
168-172 ; isothermals of, 190, 191 ;
transparency to radiations, 235 ;
value of specific heat ratio for, 289-
296 ; intrinsic energy of, 296 et seq.

Air-pressure, relation of, to melting-
point of ice, and boiling-point of
water, 5, 6

Air-thermometer by Regnault, 48-49;
Gallendar, 50, 51,52; Bottomley, 50,
scale, 269, 294-295, 299-302

Aitken, on boiling, 165 ; on dust as a
condensing agent, 168-170 ; dust
counter, 169 ; on dew, 219, 233

Alcohol (ethyl), used for thermometers,
9 ; boiling-point, 12, 175 ; specific
heat of, 80-81, 182; heat conduc-
tivity of, 106 ; vapour tension of,
159 ; expansion of, under a pull,
187 ; critical constants of, 193 ;
osmosis of, 327 ; (methyl), heat con-
ductivity of, 106

Aluminium, specific heat of, 81, 87 ;
atomic weight, 76 ; atomic heat, 87

Amagat, on the volume-pressure relation
in air, 298

Amaury. See Jamin

Ammonia, specific heat of, 85 ; radia-
tions, absorbed by, 235

Andrews, on the isotbermals and critical
point of carbon dioxide, 188-190

Angstrom, method of measuring heat
conductivities, 98 ; result for copper,
99

Aniline, boiling-point, 12 ; specific heat
of, 80

Anticyclone over cyclone, 58

Argon, value of specific heat ratio for,
138 ; motion of molecules in, 139 ;
critical pressure of, 199 ; critical
temperature, 199 ; boiling-point, 199

Atmosphere, water in the, 209-219 ; de-
crease of temperature with height
in the, 296

Atomic heat, 83, 84, 87, 130

Atomic hypothesis, 129-132

Atomic weight, 84, 87

Atoms, 130 et seq.

Avogadro's law, 137

BALANCE wheel of chronometers, 27

Barnes and Callendar, on specific heat of
water, 79-80, 127

Bartoli, on radiation, 248, 333

Bartoli and Stracciati on specific heat
of water, 78-80, 127

Benzene, critical constants of, 193

Berget, heat conductivity of mercury,
99

Berthelot, on latent heat of steam, 179

Beryllium, atomic heat, 87

Bismuth, heat conductivity of, 100 ;
electric conductivity of, 100 ;
velocity of light in, 100

Boilers, heat conductivity of walls of, 90

Boiling, 163-167

Boiling-points, 175, 199 ; of water as a
fixed point on thermometric scales,
4, 5 ; relation of, to pressure, 163-
168 ; determination of, 175 ; rise of,
321-322

Bolometer, 222, 227, 228, 253

Boltzmann, on energy of molecules, 139 ;
on radiation, 248, 253

Borate of lead, radiation of, 227

Boron, specific heat of, 82, 87 ; atomic
heat, 87

Bottomley, constant volume thermo-
meter, 50 ; air thermometer, 50

843

344

INDEX.

Boyle's law, 41, 46, 134, 137, 152, 189,
298, 308, 315, 329

Boys, on the ice calorimeter, 73 ; on radio-
micrometer, 221

Brass, expansion of, by heat, 3, 9, 19, 22,
26 ; specific heat of, 66-68 ; heat con-
ductivity of, 92, 98, 99, 100 ; electric
conductivity of, 100 ; radiation re-
flective power of, 230, 231

Breezes. See Winds

Bridges, expansion of iron by heat in,
17,25

Bromine, atomic weight, 84, 87 ; specific
heat, 84, 87 ; atomic heat, 84, 87

Bubbles, formation of, 163, 165, 166;
stability of, 167

Bumping, 165, 168

Bunsen, the ice calorimeter of, 72-73,
205 ; on the influence of pressure on
melting-point, 204 ; on latent heat,
205

Bureau International des Poids et
Mesures, method of measuring linear
expansion of metals, 19, 20

CAGNIAED de la Tour, on the critical

point of alcohol, 88
Cailletet, on liquefaction of gases, 197
Cailletet and Collardeau, on the critical

constants of water, 191-192
Cailletet and Mathias, on the critical

volume, 192

Calcium carbonate, molecular heat, 87
Callendar, the platinum resistance ther-
mometer of, 11 ; on pressure increase
of air kept at constant volume with
rise of temperature, 48 ; constant
volume air thermometer of, 50 ;
compensated air thermometer, 51-52.
See also Barnes.

Calorimeter, mode of using, 66 ct seq.,
95, 179 ; Bunsen's ice, 72-73 ; Joly's
steam, 73-75 ; Joly's differential
steam, 75-76
Calory, 65

Camphor, sublimation of, 208
Cane sugar, aqueous solution of, 326
Capstick, on specific heat ratio, 293
Carbon, specific heat of, 81, 82, 86 ;

atomic heat, 86

Carbon dioxide, expansion with rise of
temperature under constant pres-
sure, 42 ; molecular weight, 85 ;
specific heat, 85, 86, 293, 294 ;
molecular heat, 85 ; effusion of, 141 ;
isothermals of, 188-192 ; equation
of, 194, 195 ; critical constants of,
196 ; corresponding pressure and
volume of, 196 ; critical temperature
and pressure of, 199 ; boiling and
melting points of, 199 ; radiations
absorbed by, 235 ; intrinsic energy
of, 299

Carbon monoxide, molecular weight, 85 ;
specific heat, 85 ; molecular heat, 85 ;
critical temperature and pressure,
199 ; boiling and melting points,
199 ; radiations absorbed by, 235
Carnelly, on melting-points, 204
Carnot engine as a temperature scale,
10 ; action of, 262-264 ; cycle, 262-
264, 337

Cazin, on specific heats of air, 293
Centigrade scale, 4
Change of state, 83, 157-208
Chemical energy, 114-115, 117, 129, 136
Chimneys, convection in, 59-63
Chloride of potassium, dissociation of,

in solutions, 326

Chlorine, atomic weight, 83 ; specific
heat, 84 ; radiations absorbed by,
235

Chree, on heat conductivity, 104
Chronometers, balance wheel of, 27
Circulation of air, conditions requisite

for, 60-63

Clausius, on entropy, 277
Clay, radiating power of, 240
Cleavage of matter, 129
Clement and Desormes, on specific heat,

291-292

Clouds, 160, 169, 182, 215, 217, 313;

cumulus, 216-217 ; strato-cumulus,

216; ripple, 216; rate of fall of

drops in, 217-218

Coal, radiating power of hot, 240 ;

available heat from, 282, 283
Cohesion, 133, 199
Collardeau. See Cailletet
Collision frequency of gas molecules,147
Compensation measuring bars and pen-
dulums, 26, 27
Compression of matter, 129
Condensation on nuclei, 168 et seq.
Conduction, definition, 88
Conductivity of electricity, 100
Conductivity of heat, 88-107 ; passage
from one body to another, 88 ; in the
three states of matter, 90 et seq.,
definition of, 93 ; difl'usivity, 94 ;
measurements, 94 ; in solids, by
Peclet's method, 95 ; by bar methods,
96 ; Despretz's experiments, 96 ;
Forbes's experiments, 97 ; Neumann
and Angstrom's method, 98 ; Gray's
experiments, 99 ; Berget's experi-
ments, 99 ; experiments by Wiede-
rnann and Franz, 100 ; by Kundt,
100 ; in solids of low conducting
power and crystals, 101 ; experi-
ments by Senarmont, 101 ; and by
Lees, 101-103 ; in liquids, 103 et
seq. ; in gases, 106-107, 147, N,-,
Conservation of energy, 108, 116-118
Convection in water, 38, 90, 91 ; in
liquids, 90 ; in gases, 90, 91

INDEX.

345

Conversion of thermometric scales, 5
Copper, expansion of, by heat, 17, 22 ;
specific heat of, 81, 87; atomic
weight, 87 ; atomic heat, 87 ; heat
conductivity of, 89, 90, 96, 98, 99,
100 ; electric conductivity of, 100 ;
velocity of light in, 100 ; emissive
power of, 226, 227, 255
Cork, heat conductivity of, 103
Coronas, 217
Correlation of different forms of energy,

116

Corresponding pressures, 196
Corresponding temperatures, 195
Corresponding volumes, 196
Cotton, .heat conductivity of, 91, 103
Coulier, as to condensation on nuclei, 168
Critical constants, 190-195
Critical points, 188-195, 199
Critical pressure, 190-192, 199
Critical volume, 191, 192
Crookes, on radiometer, 149, 150
Crova, on solar radiation, 254, 256
Crystals, heat conductivity of, 101
Cubical dilatation. See Volume-Ex-
pansion
Cyclones, method of forecasting, 57-60

DALTON, volume expansion of e:ases
with heat under constant pressure,
42 ; atomic hypothesis used by, 129

Dark lines in solar spectrum, 243

D'Arsonval, radiomicrometer, 221

Davy, Sir H., on heat energy, 111-112

Davy lamp, 92

Day, W. S., on specific heat of water,
77 ; on the mechanical equivalent of
heat, 124, 125

De la Provostaye and Desains, on radia-
tion, 226, 227, 229-231

Defant, on raindrops, 218

Delayed boiling, 165

Desormes. See Clement

Despretz, method of measuring heat
conductivity, 96, 97, 103, 104

De Vries, on osmosis, 327-328

Dew, 218-219 ; formation of, 233

Dewar, on liquefaction of gases, 197-198,
199 ; on melting-point of ice under
pressure, 204

Dew point, measurement of, 209-215, 219

Diamond, specific heat of, 81, 82

Dieterici, on latent heat of steam, 180

Diffusion of matter, 129 ; of gases, 134,
142 ct seq. ; of radiation, 231

Diffusivity of heat, 94

Dilatometer, use of, in measuring ex-
pansion of liquids with heat, 36-37

Dines, hygrometer of, 210-211

Dissipation of energy, 278

Distillation, 161-163

Doppler's principle applied to radiation,
338

Drops, formation of, 165 ; vapour pres-
sure of, 316, 317

Dulong and Petit's experiments on ex-
pansion of mercury with heat, 30-31 ;
law of heat capacity, 86 ; law of
radiation, 246-248

Dumas, method of deter mining vapour
density, 176-177

Dust, deposit of, on relatively cold
bodies, 152 ; in air, 169, 170

Dust-counter, 169

Dust-free region about hot bodies, 152

Dutrochet, on osmosis, 327

EBONITE, heat conductivity of, 103;
transparent to long wave radiations,
234

Effusion of gases, 140 et seq.

Eider-down, heat conductivity of, 103

Elasticity, relation of specific heat to,
288-290

Electric arc, temperature of, 12

Electric currents used to measure tem-
peratures, 11, 12

Electrical energy, 113, 114, 118, 129;
analogies with entropy, 276-277

Ellicott's pendulum, 26

Emissivity of heat, 94, 226 et seq.

Energy, radiant, 88, 220-236 ; conserva-
tion of, 108 et seq. ; various forms of,
108, 109 ; kinetic, 109 ; potential,
110; heat, 111; strain, 112; sound,
112; light, 113; electrical, 113;
magnetic, 114 ; chemical, 114 ;
identity of, 116 ; ratio between
mechanical, and heat, 118 et seq. ;
relation of, to physical state, 157 ;
to temperature, 162 ; balance of, in
an enclosure, 237-240 ; loss of solar,
257 ; dissipation of, 278 ; intrinsic,
278, 279 ; available, 279-281 ; stream
of, from a full radiator, 334, 335 ;
change of, in adiabatic expansion,
340-341 ; distribution of, in spec-
trum, 341-342

Entropy, 268, 272 ; definition of, 274 ;
gain of, 274-275 ; entropy-tempera-
ture diagrams, 275-276, 280 et seq. ;
quantities analogous to, 276-277 ;
tendency of, to increase, 277-278, 338

Ether, vapour tension of, 159, 160 ; boil-
ing-point of, 175 ; specific heat of.
182, 314 ; critical constants of, 193,
195 ; corresponding pressure and
volume of, 196

Evaporation, 157, 158

Exchanges, theory of, 237-243 ; uniform
temperature enclosures, 238-240 ;
effect of the medium on radiation,
240-242 ; bodies exchanging radia-
tions, 24'2 ; application of princi-
ples, 242-243

Expansion of gases by heal, 3, 41-52

346

INDEX.

Expansion of liquids by heat, 2, 29-40
Expansion of metals by heat, 2 ; between
fixed points varies for each sub-
stance, 9 ; methods and results of
measuring, 17-28
Expansion of solids by heat, 2, 17-28

FAHRENHEIT scale, 4

Fairbairn and Tait, on density of steam,

178, 308
Faraday, on conservation of energy, 108-

109 ; on the critical points, 188 ;

on liquefaction of gases, 197 ; on

regelation, 201

Feathers, heat conductivity of, 90
Fir, heat conductivity of, 103
Fire-damp, 92

Fizeau, method of measuring linear ex-
pansion of solids, 23, 24
Fogs, 160, 169, 170, 171, 172, 218
Forbes, experiment on conductivity of

iron, 97, 98, 99 ; on regelation, 201-

202 ; on radiation, 227
Fourier, on heat conductivity, 107
Franz. See Wiedermann
Friction, 130, 131
Friedlander, on dust particles in air,

170

Furs, heat conductivity of, 90
Fusion, resemblance of solution to, 207-

208

GALITZINE and Wiliss, relation of re-
fractive index to critical point, 192

Gannon. See Schuster

Gas, use of term, 158, 159

Gas-engine, 279

Gas thermometry, 46-48

Gases, expansion of, with rise of tem-
perature, 41-52 ; volume expansion
at constant pressure, 42-44 ; increase
of pressure with constant volume,
44-46 ; gas thermometry, 46-48 ;
Regnault's normal air thermometer,
48-49 ; hydrogen thermometer, 49-
50; Callendar's and Bottomley's
constant volume thermometers, 50 ;
Bottomley's air thermometer, 50 ;
constant pressure gas thermometer,
50-51 ; Callendar's compensating
thermometers, 51-52 ; convection of,
55-63, 90, 91 ; specific heat of, 70-
71 ; heat conductivity of 90, 91,
106, 107, 147, 148; kinetic theory
of, 133 et seq. ; liquefaction of, 197-
199 ; transparency and opacity of,
234-236 ; spectra of, 242 ; adiabatics
of, 261, 295-296; value of 7 for,
290-296 ; energy taken up by ex-
panding, 296 et seq.

Gay-Lussac, on volume expansion of
gases under constant pressure, 42 ;
expansion without external work,

119, 120, 261 ; method of determin-
ing vapour density as modified by
Hoffmann, 176, 177 ; and Welter, on
specific heats of air, 292

Germanium, atomic heat, 87

Gibbs, W., on phases of matter, 312

Glaciers, motion of, 201-202

Glass, sluggishness of volume change
with temperature as a source of
error in thermometers, 7 ; should
be of uniform quality for thermo-
meters, 8 ; expansion of, by heat, 19,
22, 25, 27, 28, 29 ; heat conductivity
of, 89, 103 ; absorption of light by,
225 ; radiation reflective power of,
230 ; radiation transparency of, 234 ;
radiating power of, 240, 243

Glycerine, heat conductivity of, 106

Gold, melting-point, 12 ; atomic weight,
87 ; specific heat, 87 ; atomic heat,
87 ; heat conductivity of, 99, 100 ;
electric conductivity of, 100 ; velo-
city of light in, 109 ; radiation,
emissive power of, 226 ; radiation,
reflective power of, 531

Graham, on osmosis, 327

Graphite, specific heat of, 82

Gravitation, analogies with entropy, 276,
277

Gray, J. H., experiments on heat con-
ductivities of metals, 98, 99

Gridiron pendulum, 26, 27

Griffiths, method of determining specific
heat of water, 76, 78. 80, 127 ; on
the mechanical equivalent of heat,
126, 127; on the latent heat of
steam, 180-181

Guillaume, his nickel steel termed
" Invar," 27

HAIL, genesis of, 218

Halos, 216, 217

Hampson, on liquefaction of gases, 198

Harden, on liquefaction of gases, 197

Hazen, dew-point tables, 212

Heat, idea qonveyed by the term, 1 ;
relation of work and, in steam
engine, 9, 258 et seq. ; quantity of,
64 ; unit or calory, 65 ; specific heat,
" 66-87 ; heat energy, 111 et seq. ;
mechanical equivalent of, 116-128;
latent, 157 ; heat engines, 261-264 ;
relation of, to entropy, 276, 277;
dissipation of energy into, 278

Heat engines, ideal, 9, 10, 262 et seq.

Heat sensations, words used for express-
ing, 1 ; relation of, to thermal equi-
librium, 2

Helium, value of 7 for, 138; motion of
molecules in, 139

Helmholtz, on viscid water, 202 ; ripple
clouds, 216 ; sun energy, 257

Henning, latent heat of steam, 181

INDEX.

34-7

Henry's law, 329

Hering, on vapour densities, 178

Hoffmann-Gay-Lussac method of deter-
mining vapour density, 176, 177

Holborn and Valentiuer, radiation con-
stants, 250

Hope, method of determining point of
maximum density of water, 37, 40

Hopkins, on the influence of pressure on
the melting-point, 204

Hydrochloric acid, molecular weight, 85 ;
specific heat, 85 ; molecular heat, 85 ;
radiations absorbed by, 235

Hydrogen, thermometric use of, 10, 11,
300 ; melting and boiling points, 12 ;
expansion of, with rise of tempera-
ture, 42 ; co-efficient of pressure
increase when volume is kept con-
tant under rising temperature, 46 ;
atomic weight, 84 ; specific heat of,
84, 85, 86, 293, 294; atomic heat,
83, 86 ; heat conductivity of, 106,
107 ; as standard scale, 124 ; value
of 7 for, 139; effusion of, 141;
viscosity of, 146, 147 ; mean free
path of, 147, 149 ; collision fre-
quency, 147 ; number of molecules
in 1 cc. 149 ; diameter of molecules,
149; critical point of, 188, 199;
critical constants of, 191 ; liquefac-
tion of, 197, 199 ; critical pressure,
199 ; boiling and melting-points,
119; transparency of, to radiations,
235 ; radiation of, 243 ; intrinsic
energy of, 299

Hydrogen thermometer, 49-50

Hydrometer, U-tube, for measuring ex-
pansion of liquids by heat, 29-36

Hygrometer, Kegnault's, 209-210 ;
Dines's, 210-211 ; wet and dry bulb,
211-212 ; chemical, 212-213

Hygrometry, 209-215

ICE, melting-point of, as zero of ther-
mometric scales, 4, 5, 12 ; use of, in
calorimetry, 72-73 ; specific gravity
of, 72 ; specific heat of, 83 ; mole-
cular heat of, 87 ; Davy's experi-
ment with, 111-112; action of heat
on, 157, 158 ; vapour pressure of,
173, 206, 309 et seq. ; melting of, 200,
206 ; freezing, 200 ; relative volumes
of water and, 201 ; regelation of,
201-203 ; effect of pressure on melt-
ing-point of, 203-207, 317-319;
isothermals of, 207 ; crystalline
forms of, 207 ; evaporation of, 208,
309-312 ; haloes and parhelia due
to crystals of, 216-217 ; preservation
of, 233 ; formation of, 233, 234
Iceland spar, heat conductivity of, 101
Ice-water change of state, 307-312
Indian ink, emissive power of, 226, 227,
231 ; absorptive power of, 229, 231

Indiarubber, strain energy of, 112; in-
dicator diagram for stress and
strain in, 304

Indicator-diagram for water-steam, 185-
188 ; of work done by a body in
expansion, 259-260; for stresses and
strains, 302-305

Ingenhousz, on diffusibility of heat,
94

Iodine, atomic weight, 86 ; specific heat,
86 ; atomic heat, 86

Iron, linear expansion of, as rails,
bridges, pipes, &c., 17, 22, 25, 26;
Kamsden's method of measuring
expansion of cast-iron bars, 18, 19;
method of Bureau International des
Poids et Mesures, 19, 20; expansion
and contraction of, by heat and cold
used for drawing walls together and
fixing hoops, 25 ; specific heat of, 65,
81, 87 ; atomic weight, 87 ; atomic
heat, 87 ; heat conductivity of, 89,
92, 97, 98, 99, 100; electric con-
ductivity of, 100; velocity of light
in, 100 ; emissive power of, 226 ;
radiating power of, 239 ; value of
7 for, 270; indicator diagram for
stresses and strains in, 302-304

Isentropics, 261

Isometrics, 261

Isopiestics, 261

Isothermals, 186 et seq., 260

Isotonic solutions, 328

JAMIN, on vapour densities, 176; on
light and its heat effect, 224

Jamin and Amaury, on specific heat of
water, 77

Jamin and Richard, on specific heats,
294

Jolly, air thermometer devised by, 48-49

Joly, steam calorimeter of, 73-75 ; dif-
ferential steam calorimeter of, 75-
76 ; on specific heat of water, 78 ;
of air and other gases, 86, 294 ; on
the mechanical equivalent of heat,
119

Joule, on specific heat of water, 76 ;
on the mechanical equivalent of
heat, 120-124, 127 ; on the velocity
of molecules, 140-142 ; on hydrogen,
199 ; on adiabatic changes, 287 ; on
energy taken up by an expanding
gas, 296 et seq. See also Thomson

Joule and Playfair, method of ascertain-
ing temperature of maximum density
of water, 38-40

KAMMERLINGH-ONNES, liquefaction of
gases, 198

Kelvin (Lord), work scale of tempera-
ture, 9, 259 ; on heat conductivities
of substances, 103 ; on effect of
pressure on melting-point of ice, 203,

348

INDEX.

204, 309 ; on work done in expand-
ing gases, 297 et seq. ; on vapour
pressure of curved liquid surfaces,
314-317

Kidd. See M'Connel

Kinetic energy, conception of, 1, 109
et seq., 129 et seq. ; relation of, to
heat energy, 118-128, 130 et seq., 199

Kinetic theory of matter, 129-156 ;
atomic hypotheses, 129-133 ; gases,
133 et seq. ; mean value of the square
of the velocity of translation, 134-
137; mixture of gases, 137; Avo-
gadro's law, 137 ; relation between
V and temperature, 137 ; energy of
translation and internal energy,
138 - 140 ; Joule's approximate
method of calculating the velocity
of mean square, 140; effusion or
transpiiation through a small orifice
into a vacuum, 140; thermal tran-
spiration, 141-142 ; the mean free
path, 142-143 ; length of same, 143-
144 ; the mean free path calculated
from the coefficient of viscosity of a
gas, 144-147 ; conduction of heat in
gases, 147-148; the diameter of mole-
cules and number of molecules per
cubic centimetre, 148 - 149 ; forces
acting on unequally heated surfaces
in high vacua, 140-152 ; the gas
equation of Van der Waals, 152-156

Kirchhoff, on vapour-pressure of ice, 173

Knoblauch, on diffusion, 231

Kopp, on molecular and atomic heats, 87

Kundt, experiments on relation of light,
heat, and electric conductivities of
metals, 100

Kundt and Warburg, heat conductivity
of gases, 107

Kurlbaum, on energy of radiation, 250,
256

LAMPBLACK, absorption of heat by, 225,
231, 232; radiation, emissive power
of, 226, 227, 231, 232, 239; radia-
tion absorptive power of, 230, 237 ;
radiation diffusion by, 231 ; trans-
parency to radiations of great wave
length, 234

Langley, on radiation, 222, 226, 227-228,
234, 242, 252, 253-254, 256

Laplace, on velocity of sound, 290

Larmor, on radiation, 333

Latent heat, 157, 178-182, 204-205 ; first
equation, 306-312 ; second equation,
312-314

Lavoisier and Laplace's method of
measuring expansion of bars by
heat, 20-23

Le Chatelier, thermo-electric thermo-
meter of, 12

Lead, melting-point, 21 ; specific heat,

65, 66, 81, 83, 186; atomic weight,
87 ; atomic heat, 87 ; heat conduc-
tivity of, 100 ; electric conductivity
of, 100 ; latent heat of, 205 ; emis-
sive power of, 226

Lebedew, on radiation pressure, 333
Lees, heat conductivity of brass, 98, 99 ;
of crystals, 101 ; of various solids,
102, 103; of liquids, 104-106
Leslie, on radiation, 220-221, 226, 228-

230

Light, action of, on condensation nuclei,
171, 172 ; resembles radiant energy,
222-224

Light energy, 113, 117, 129, 155, 156
Linde, liquefaction of gas, 198
Liquefaction of gases, 197-199
Liquid-solid change of state, 200-208
Liquid-vapour change of state, 157-199
Liquids, volume expansion, 29 ; as
measured by U-tube method, 29-36 ;
by the dilatometer, 36 ; Matthies-
sen's method, 37 ; convection of, 53-
55, 90; specific heat of, 69 et seq.;
heat conductivity of, 90, 103-106 ;
atoms and molecules of, 132 ; vapour
pressures of, 172-175 ; isothermals
for, 260
Lodge, on dust-free space about hot

bodies, 152

Lorberg, heat conductivity of water, 104
Ludin, on specific heat of water, 79, 80
Lummer and Pringsheim, on radiation,

249, 342

Lundquist, heat conductivity of water,
104

M'CONNEL and Kidd, on ice, 202
Magnetic energy, 114, 118, 129
Magnus, on vapour pressure of water,

175

Mahogany, heat conductivity of, 103
Marble, heat conductivity of, 103
Marsh gas, radiation absorbed by, 235
Mathias. See Cailletet.
Matter. See Kinetic theory of Matter.
Matthiessen, hydrostatic method of

measuring expansion of liquids and

metals, 37

Maximum thermometer, 13
Maxwell, heat conductivity of gases, 106,

107 ; on velocity of molecules, 136,

on mean free path, 144 ; on viscosity,

146 ; on number of molecules per cc.

of gas, 149; on radiometer, 150;

on pressure of radiation, 333, 334
Mayer, on mechanical equivalent of heat,

118, 120

Mean free path, 142 ct seq.
Mechanical equivalent of heat, 116-128
Melloni, on radiation, 221, 226, 228, 229,

230, 231, 234
Melting, explanation of, 205-207

INDEX.

349

Melting-point, 199-204 ; influence of
hydrostatic pressure on, 317-319 ;
lowering of, 322 et seq.

Membranes, permeable to some fluids,
not to others, 327

Menai tubular bridge, allowance for ex-
pansion by heat, 23

Mercury, mode of filling thermometer
tubes with, 3 ; freezing-point, 9 ;
boiling-point, 9, 12, 175 ; melting-
point, 12 ; used in the mercurial com-
pensating pendulum, 27 ; expansion
of with heat, 30-37 ; atomic weight,
87 ; specific heat, 85 ; atomic heat,
87 ; heat conductivity of, 99 ; value
of y for vapour of, 138 ; motion of
molecules in vapour of, 139 ; vapour,
tension of, 159

Metals, heat conductivity of, 89, 90, 92,
96 et seq. ; viscosity of, 202 ; ab-
sorptive power of, 229, 232 ; radiant
reflective power of, 231-232

Meyer, 0. E., on the kinetic theory of
gases, 134 ; on viscosity of gases,
146, 147 ; on gas equations, 155

Meyer, V., method of determining vapour
density, 177

Mica, heat conductivity of, 101

Miculescu, on the mechanical equivalent
of heat, 125, 127

Minimum thermometer, 13

Molecular heat, 83, 87

Molecular sphere of action, 142

Molecules, 130 et seq.; collisions of, 133,
134, 143, 147, 152 et seq. ; 161 et seq. ;
velocities of, 134, 135 ; diameter of,
148 ; number of, per c.c. , 148, 149 ;
size of, 1 52 et seq. ; internal energy
of, 155-156

Morby. See Reynolds

Motion, energy of, 110, 131

Miiller, method of measuring linear ex-
pansion of bars by, 21, 22

MUller-Pouillet, on vapour pressure, 175

NACOARI, on specific heat of metals, 81
Nernst, on Van der Waal's equation, 197
Neumann, molecular heat, 87 ; method
of measuring heat conductivities,
98 ; result for copper, 99
Newton's rings as a means of measuring
linear expansion of solids, 23, 24 ;
law of cooling, 243-246 ; formula for
velocity of sound, 290
Nichols, on specific gravity of ice, 72
Nichols and Hull, radiation pressure, 333
Nickel, specific heat of, 81, 87 ; atomic
weight, 87 ; atomic heat, 87 ; velo-
city of light in, 100
Nickel-steel, expansion of, by heat, 27
Nitrogen, atomic weight, 84 ; specific
heat, 84 ; atomic heat, 84 ; velocity
of molecules, 137 ; value of y for,

139 ; effusion of, 141 ; critical points
of, 188, 189 ; critical constants of,
191 ; critical pressure of, 199 ; boil-
ing and melting points, 199; trans-
parency of, to radiations, 235
Nitrous oxide, radiations absorbed by,

235

Non-reversible cyclical engine, 270, 271
Nuclei, condensation on, 168 et scq.

OAK, heat conductivity of, 103
Olefiant gas, radiations absorbed by, 235
Olzewski, on liquefaction of gases, 197,

199

Opacity, 234-235
Osmotic pressure, 321 el seq. ; molecular

theory of, 332

Ostwald, on the ice calorimeter, 73
Oxygen, boiling - point, 12 ; atomic
weight, 84 ; specific heat, 84 ; atomic
heat, 84, 87 ; velocity of molecules,
136 ; value of y for, 139 ; effusion of,
141 ; viscosity of, 146, 147 ; mean
free path of, 147-149 ; collision
frequency, 147 ; number of mole-
cules in 1 c.c. 149 ; diameter of mole-
cule, 149 ; critical point of, 188, 199;
liquefaction of, 197 ; critical pressure
of, 199 ; boiling and melting points,
199 ; transparency of, to radiations,
235

PAPER, heat conductivity of, 89, 103 ;
emissive power of, 226

Paraffin, heat conductivity of, 103 ;
melting-point of, under pressure, 204

Parhelia, 217

Pe"clet, method of measuring heat con-
ductivities, 95

Pendulum, influence of heat on, as time
measurer, 17, 26 ; principle of the
gridiron, 26, 27

Person, on latent heat, 205

Pfeffer, on osmosis, 327

Phosphorus, atomic weight, 87 ; specific
heat, 87 ; atomic heat, 87 ; physical
states of, 157 ; superfusion of, 201 ;
latent heat of, 205 ; radiation of, 242

Pictet, on liquefaction of gases, 197

Pipes, expansion of iron in, for carrying
water, 171

Pitch, flow of, 132

Planck, on radiation, 248, 249, 342

Platinum, thermometric use of its electri-
cal resistance, 11 ; melting-point, 12 ;
expansion of, with heat, 28 ; specific
heat of, 81, 87 ; atomic weight, 87 ;
atomic heat, 87 ; heat conductivity
of, 99, 100 ; electric conductivity of,
100 ; velocity of light in, 100; emis-
sive power of, 227, 248, 255

Platinum-rhodium wire, used in thermo-
metry, 12

350

INDEX.

Playfair. See Jonle
Porcelain, radiation of, 242, 243
Position, energy of, 110, 131
Potential energy, 110, 111 et set)., 129,

130, 199
Pouillet's method of measuring expansion

of bars, 20 ; researches on the sun's

radiation, 251-252
Pressure, influence of, on volume of gas,

41 ; of gases, 134 ; relation of, to

the boiling-point, 163-181
Pringsheim, on spectra of gases, 156.

See also Lummer
Pyrheliometer, 251

QUABTZ, heat conductivity of, 101

RADIANT energy, 113, 129, 131, 221 et seq.

Radiation, denned, 88, 113, 220-236;
radiant energy, 220; radiometers,
220; thermopile, 221 ; radio-micro-
meter, 221-222 ; bolometer, 222 ;
radiant energy similar to light, 222,
224; wave-length range of radiant
energy, 224-225 ; radiometers mea-
sure energy only, 225-226 ; emissive
powers, 226, 227 ; different wave
lengths in, 227-228 ; absorptive
powers, 228-230 ; reflecting powers,
230; diffusion, 231 ; general results,
231-233; deposition of dew, 233;
formation of ice, 233-234 ; by gases
and vapours, 234-236 ; theory of
exchanges of, 237-243; and tem-
perature, 244-257 ; variation of rate
of, with temperature, 244-245 ;
Newton's law of cooling, 245-246 ;
Dulong and Petit's law of, 246-248 ;
Rosetti's law, 248 ; Stefan's law,
248-250 ; constants of, 250 ; radiation
from surfaces which absorb selec-
tively, 250-251 ; the solar constant,
251-257; source of solar energy,
257; thermodynamics of, 333-342;
the pressure of, 334 ; the normal and
the total mean of, 334-335 ; pressure
on a fully radiating surface, 335 ;
relation between energy density
and temperature in full, 335, 337 ;
full radiation unaltered in adiabatic
change, 337-338 ; relation between
volume and temperature, 338 ; en-
tropy, 338 ; application of Doppler's
principle, 338-340 ; change of energy
in, 340-341 ; distribution of energy
in the spectrum, 341-342

Radiometer, 135, 149, 150, 151, 220;
measures radiant energy only, 225-
226

Radiomicrometer, 221-222

Railways, expansion of steel by heat in,
17, 24, 25

Raindrops, rate of fall of, 217-218

Ramsay and Young, on vapour pressure
of ice and water, 173, 312 ; on
critical constants, 192

Ramsden's method of measuring linear
expansion of metals, 18, 19 ; modi-
fied form, 19, 20

Raoult, on boiling-points and vapour
pressures of solutions, 321, 323

Rayleigh, on gas densities, 176

Reaumur scale, 5

Red heat, temperature of, 12

Red lead, emissive power of, 226

Reduced isothermals, 195-196

Reflecting powers of surfaces, 230

Refractive index, relation of, to the
critical point, 192

Refrigerator, 261

Regelation, 201-203

Regenerative method of liquefying gases,
198

Regnault's experiments on expansion of
mercury with heat, 31-35; of gas
with rise of temperature under con-
stant pressure, 42-44, 118-119 ;
increase of pressure with rise of
temperature and constant volume,
44-46, 118, 119 ; normal air thermo-
meter, 48 ; experiments by, on
specific heat, 68-71, 77, 80, 127, 181 ;
on Dulong and Petit's law, 86, 87 ;
on molecular heat, 86 ; on vapour
pressure of water, 172-175, 310 et
seq. ; on vapour densities, 176, 213-
215 ; on latent heat of steam, 179-
182 ; on the critical constants of
gases, 191 ; dew-point apparatus of,
209-210 ; density of water-vapour,
213 ; on vapour pressure of ice, 310
et seq.

Relative humidity, 209 et seq.

Reversible cycles, 272 et seq.

Reversible heat engines, 262 et seq.

Reynolds, on thermal transpiration, 142 ;
on the radio-thermometer, 150

Reynolds and Morby, on the mechanical
equivalent of heat, 126, 127

Richard. See Jam in

Ring vortex theory of atoms, 131

Rock-salt, absorption of radiant energy
by, 225-243 ; refractive index of,
227 ; radiation transparency of, 234

Rontgen, on specific heats of air, 293

Rosetti's law of radiation, 248 ; observa-
tions on solar radiation, 225, 256

Rotch, on exploration of air with kites,
218

Rowland, on specific heat of water, 77-
78, 80, 127 ; on the mechanical
equivalent of heat, 124, 125, 127, 128

Rudberg, method of determining volume
expansion of gas with rise of tem-
perature under constant pressure
42-43

INDEX.

351

SANDSTONE, heat conductivity of, 103
Schuster and Gannon, on the mechanical

equivalent of heat, 127
Searle, beat given off by gas, 70
Senarmont, heat conductivity of crystals,

101

Shellac, heat conductivity of, 103
Siemens, measurement of temperature
in resistance of platinum wire to
electricity, 11

Silicon, specific heat of, 82, 87
Silk, heat conductivity of, 103
Silver, melting-point, 12 ; expansion of,
by heat, 22 ; specific heat of, 81, 87 ;
atomic weight, 87; heat conducti-
vity of, 90-99, 100 ; electric conduc-
tivity of, 100 ; velocity of light in,
100 ; radiation, emissive power of,
226, 227 ; radiation reflective power
of, 230, 231 ; radiating power of, 239
Singing of water, 163, 168
Six's thermometer, 13
Skin, heat conductivity of, 89
Slag-wool, conductivity of, 91, 92
Slate, heat conductivity of, 103
Sodium, atomic weight, 87 ; specific
heat, 87 ; atomic heat, 87 ; spectrum
of vapour of, 156 ; spectrum of, 242,
243

Sodium chloride, molecular heat, 87
Sodium sulphate, supersaturated solution

of, 208

Solar system, energy of, 111
Solid-gas change of state, 208
Solid-liquid change of state, 208
Solids, expansion of, with rise of tem-
perature, 17-28 ; linear expansion
of, 17, 18 ; Ramsden's method, 18-
19 ; method of Bureau International
des Poids et Mesures, 19, 20 ;
Pouillet's method, 20 ; Lavoisier
and Laplace's method, 20-23 ;
Fizeau's method, 23-24 ; illustra-
tions of linear expansions, 24-28 ;
volume expansion of, 28 ; methods
of determining specific heat of, 66
. et seq. ; heat conductivity of, 90 et
seq. ; atoms and molecules of, 131-
132 ; and of viscous, 132 ; melting-
points of, 199-204 ; latent heat of,
204-205 ; evaporation of, 208 ; iso-
thermals for, 260

Solution, 129, 320-332 ; resemblance to
fusion, 207, 208 ; vapour pressure
of, 320; osmotic pressure, 321, 328-
332 ; rise in boiling-point, 321-322 ;
lowering of melting-point, 322-327 ;
semi-permeable membranes, 327-328
Sound energy, 112-113 ; velocity of, 290,

294, 295

Specific gravity bottle method of deter-
mining the expansion of liquids
with heat, 35-36

Specific heat, 65, 66, 288-290; method
of mixtures, 66-71 ; method of cool-
ing, 71-72 ; melting ice method, 72-
73 ; steam method, 73-76 ; electrical
heating method, 76 ; specific heat of
water, 76-82; influence of state on,
82 ; relation to atomic and mole-
cular heats, 83-87 ; of steam, 181-
182 ; relation to elasticity, 288-290;
ratio of, under constant pressure
to that at constant volume, 288-
296

Spectra of gases, liquids and solids, 155,
156, 227, 228

Speculum metal, radiation reflective
power of, 231

Sperm oil, adiabatics of, 287

Spermaceti, melting-point of, under
pressure, 204

Spheroidal state, 182-188

States of matter, 157-208 ; change of
state, 157 ; liquid-gas change, 158 ;
terms "gas" and "vapour," 158-
160 ; distillation, 160-163 ; boiling,
163-167: stability of bubbles, 167-
168 ; condensation on nuclei, 168-
172 ; vapour pressures, 172-175 ;
vapour density, 174-178 ; latent
heat, 178-182 ; spheroidal state,
182-184; water-steam change, 184-
188 ; critical point, 188-190 ; critical
constants, 190-195 ; isothermals,
195-196 ; liquefaction of gases, 197-
199 ; solid-liquid change, 200-208 ;
melting of ice, 200-204 ; of solids.
204-207 ; solution, 207 ; sublimation,
208

Steam, latent heat of, used in calori-
metry, 74 ; specific heat of, 83, 181,
313 ; molecules of, 133 ; latent heat
of, 157, 178-182 ; vapour density of,
178, 214 ; entropy of, 281 ; volume
of saturated, 308

Steam engine acts by conversion of heat
into work, 9, 261 et seq. ; efficiency
of, 265 et seq. ; 281-283

Steam-water change of state, 185-188,
306 et seq.

Steel, expansion of, by heat, 19, 22, 25,
26; magnetic energy of, 114; radia-
tion reflective power of, 230, 231 ;
stress and strain in, 305

Stefan, heat conductivity of air, 106 ;
law of radiation, 248-250, 333

Stewart, B.,on radiation,227,234,240.241

Stewart, R. W., heat conductivity of iron
and copper, 98, 99

Stokes, on fall of drops in clouds, 218

Stracciata. See Bartoli

Strain energy, 112-113, 117-118

Strains and stresses, indicator diagrams
for, 302-305

Sublimation, 208

352

INDEX.

Sugar and semi-permeable membranes,
,827 ; osmotic pressure of, 328

Sulphate of quinine, absorption of radiant
energy by, 225

Sulphur, boiling-point, 12 ; atomic
weight, 87 ; specific heat, 87 ; atomic
heat, 87 ; heat conductivity of, 103 ;
superfusion of, 201 ; sublimation of,
208 ; states of, 312

Sulphuretted hydrogen, molecular
weight, 85 ; specific heat, 85 ; mole-
cular heat, 85 ; radiations absorbed
by, 235

Sulphurous acid, molecular weight, 85 ;
specific heat, 85 ; molecular heat,
85 ; radiations absorbed by, 235

Sun, temperature of, 12, 243 ; spectrum
of, 243 ; rate of radiation power, 251-
257 ; observations by Pouillet, 251-
253 ; Violle, 253-255 ; Langley, 253,
254, 256 ; Crova, 254, 256 ; Watter-
ston, 255 ; Rosetti, 255 ; Wilson and
Gray, 255 ; Wilson, 255, 256

Supercooling of liquids, 201

Superfusion, 201

Snpersaturation, 207-208

TAIT, heat conductivity of iron and other
metals, 98, 99 ; of earth, 107

Tammann, on crystalline forms of ice, 207

Tate. See Fairbairn and Tate

Telepathy, 115

Temperature, 2 ; Centigrade and Fahren-
heit scale of, 4 ; Reaumur's scale of,
5 ; of human body as a fixed point
for thermometric scales, 5 ; work
scale of, 9, 10, 266 et seq. ; absolute
scale of, 10, 266 et seq. ', air scale of,
10; hydrogen scale of, 11; electric
resistance scale of, 11 ; table of, 12 ;
slope or gradient of, 88, 93 ; re-
lation of, to pressure on gases, 138 ;
to loss of energy in evaporation,
162 ; uniform temperature en-
closures, 237-240 ; radiations at
different, 132, 243; radiation and,
244-257; variation of rate of
radiation with, 244-245 ; New-
ton's law of cooling, 245-246 ;
Dulong and Petit's law, 248 ;
Stefan's law, 250 ; temperature of
the sun, 255-257 ; relation of, to
entropy, 276-277, 287 ; to energy,
278 ; decrease of, with height in
the atmosphere, 296

Thermal equilibrium, 2

Thermodynamics, 258-342; first law
of, 128 ; second law, 258 ; the indi-
cator diagram, 259-260 ; isother-
mals, 260-261 ; adiabatics or isen-
tropics, 261 ; isopiestics and iso-
metrics, 261 ; heat engines, 261-266 ;
absolute or work scale of tempera-

ture, 266-268 ; air thermomctei
scale, 269 - 270 ; non-reversible
cyclical engine, 270-271 ; reversible
engine, 271-272; reversible cycles,
272-273 ; entropy, 274-278 ; energy,
278-281 ; possible efficiency, of
a steam engine, 281-283 ; of iso-
thermal and adiabatic changes,
284-305 ; heat taken in by a body
expanding isothermally, 284-285 ;
adiabatic change, 285-305 ; specific
heats, 288-295 ; temperature of
change of state and solutions,
306-331 ; the first latent heat equa-
tion, 306-312 ; the second latent
heat equation, 312-314 ; alteration
of vapour pressure with curvature
of liquid, 314-317; hydrostatic
pressure in relation to vapour pres-
sure and melting-point, 317-320 ;
temperature of solutions, 320 et seq. ;
osmotic pressure, 320-321, 332 ;
raising of the boiling-point, 321-322;
lowering of the melting-point, 322-
327 ; semi-permeable membranes,
327-328; Van t'Hoff's application
of, 328-332 ; temperature of radia-
tion, 333-342 ; stream of energv,
334, 335 ; the fourth power law, 335-

337 ; full radiation, 337-338 ; relation
between volume and temperature,

338 ; entropy, 338 ; application of
Doppler's principle, 338-340 ; change
of energy in adiabatic expansion,
340-341 ; distribution of energy in
the spectrum, 341-342

Thermo-electric thermometer, 11,12
Thermometer, use of, 2 ; contraction of,
3 ; fixing the points of a, 4-6 ; cali-
brating, 7 ; certificated, 7 ; pre-
cautions in using, 7 ; limits of
accuracy of, 8 ; range of the ordinary
mercury, 8, 9 ; platinum resistance,
11; thermo-electric, 11, 12; maxi-
mum and minimum, 13 ; gas, 48 ;
Regnault's normal air, 48-49 ; hy-
drogen, 49 ; Callendar's constant
volume, Bottomley's constant vol-
ume, 50 ; Bottomley's air, 50 ; con-
stant pressure gas, 50-51 ; Callendar's
compensating constant pressure, 51-
52 ; differential, 220-221
Thermopile, 221

Thomson, J. , on the change from pas to
water, 187, 188 ; on influence of pres-
sure on melting-point, 203-309
Thomson and Joule, on intrinsic energy

of air, 298 ft seq.
Thunder cloud formation, 57
Tilden, on specific heat of metals, 81
Tin, atomic weight, 87 ; specific heat,
87 ; atomic heat, 87 ; heat conduc-
tivity of, 100 ; electric conductivitv

INDEX.

353

of, 100 ; radiation emissive power
of, 226 ; radiation reflective power
of, 230, 231

Tobin system of ventilation, 61, 62
Tourmaline, radiating power of, 240
Trachyte, heat conductivity of, 103
Trade winds, 56

Tradescantia discolor, osmosis in, 327
Translation, motion of, 133, 134 ; energy

of, 134 ; velocity of, 134-135
Transparency, 234

Transpiration or effusion. See Effusion.
Trap, heat conductivity of, 103
Travers, on liquefaction of gases, 197,

199

Triple point, 309
Tripler, liquefaction of gas, 198
Tyndall, on regelation, 201 ; on trans-
parency and opaqueness of iodine
in solution to radiation, 234 ; on
radiation and absorption by gases,
234-236 ; on radiation of platinum,
248

UNIT of heat, 65

VAN DER WAALS, gas equation of, 152,
193-195; on critical constants, 196,
197

Van t'Hoff, on osmotic pressure, 324 et
seq.; application of thermodyna-
mics, 328-332

Vapour, use of term, 158, 159; vapour-
liquid change of state, 157-200;
latent heat of, 178-182

Vapour, density of gases, 175, 178

Vapour-pressures, 172-175; of curved
surfaces, 314-317 ; influence of hy-
drostatic pressure on, 317-319

Velocity of molecules, 133 et seq.

Ventilation, 61-63; Tobin system, 61,
62

Vincent, on specific gravity of ice, 72

Violle, on specific heat of platinum, 81 ;
on solar radiation, 253-255

Visible rays, temperature of, 12

Volume expansion of solids, 28

WATER, expansion of, by heat, 3, 37-40 ;
boiling-point of, as a fixed point in
thermometric scales, 4, 5, 12 ; maxi-
mum density temperature of, 37-40 ;
specific heat of, as the standard for
unit quantity of heat, 65, 76-82,
121-128 ; water equivalent for heat,
capacity for heat, 66, 121-128 ; latent
heat of, 73, 157, 200. 203, 205 ; heat
conductivity of, 89, 90, 96, 103-106 ;
convection in, 90 ; molecules of, 133 ;
physical state of, 157 ; distillation
of, 161-163; boiling of, 163-167;
spheroidal state of, 182-184 ; critical
temperature and pressure of, 192,

195; freezing of, 200, 205-207;
supercooling of, 200-201 ; relative
volume of ice and, 201 ; supposed
plasticity of, 202 ; isothermals of,
207 ; in the atmosphere, 209-219 ;
radiation absorption of, 243 ; en-
tropy of, 271, 281; adiabatics of,
287 ; value of 7 for, 289 ; change
from water to vapour. 309-312;
osmosis of, 327 et seq.

Water capacity for heat, 66

Water equivalent for heat, 66

Water-ice change of state, 307-312 ;
influence of hydrostatic pressure on,
317-319

Water-steam change of state, 185-188,
306 et seq.

Watertight joints, 17

Water-vapour in its relation to thunder
clouds, and winds, 57 ; molecular
weight, 85 ; specific heat, 85 ; mole-
cular heat, 85 ; viscosity of, 146,
147 ; mean free path, 147-149 ; col-
lision frequency, 147 ; number of
molecules in 1 cc., 149; mass of
one molecule, 149 ; diameter of
molecule, 149 ; tension of, 158 et seq.;
condensation of, on nuclei, 168 ( t seq.;
pressure of, 172-175, 206, 211-215;
hygrometry, 209; relative humidity,
209; dew point, 209 et seq.; ab-
sorptive power for dark radiations,
236 ; isothermal for, 260-261 ; change
from water to, and from ice to, 309-
312 ; pressure of, in curved surface,
316-317

Water-wheel, efficiency of, 271-272

Watterston, on temperature of the sun,
255

Wax, electricity of, 113 ; melting of, 200

Weather, type of, in cyclones, 58-59

Weather forecasting, 57-60 ; use of
charts in, 58

Weber, H. F., on specific neat of carbon,
boron, and silicon, 81-82 ; heat con-
ductivity of mercury, 99 ; of water,
104

Wells, on dew, 219, 233; on formation
of ice, 233-234

Welter. See Gay-Lussac

White-heat, temperature of, 12

White lead, absorptive power of, 229

Wiedemann, on specific heat of gases,
84 ; of air, 119, 294

Wiedemann and Franz, experiments on
heat and electric conductivities, 100

Wien, on radiation, 248, 249, 342

Wilson, C. T. R., on condensation of
water-vapour in air and gas, 171,
172 ; on solar radiation, 255

Wilson and Gray, on solar radiation,
255

Winds, 56; land and sea breezes, trade

354

INDEX.

winds, 56 ; origin of, 57, 58 ; in

cyclones, 58
Winkelmann, heat conductivity of air,

107 ; on latent heat of steam, 180
Wood, heat conductivity of, 89, 90,

103

Wool, conductivity of, 91, 103
Work, as a measure of temperature, 9 ;

relation of heat to, 9, 258 et teq. ;

work scale, 259 et seq.
Worthington, on expansion of alcohol

under a pull, 187

Wroblewski, on liquefaction of gases,
197

YOUNG, on critical constants, 192.
also Ramsay and Young.

See

ZEBO point of thermometer scales, 4, 7 ;

absolute, of temperature, 10, 12
Zinc, melting-point, 12 ; specific heat of,

81, 87 ; atomic weight, 87 ; atomic

heat, 87 ; heat conductivity of, 100 ;

electric conductivity of, 100

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