.
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 V 2 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 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 C 309
174. Triple Point on Temperature Pressure Diagram 310
175. Difference of Vapour Pressures of Ice and Water below 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 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 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
_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 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 and 100 will
agree, intermediate indications may differ by, perhaps, more than 0'1C.
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. 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 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 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
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 Q l be put in at the higher temperature which
we will denote by O l and let a quantity Q 2 be put out at the lower
temperature 2 . Then we fix the ratio of these temperatures by
putting
If we keep l and Q l constant, Q 2 is less the lower # 2 . If all the
heat Qj is turned into work, none remains to be put out at 2 . In this
case Q 2 is zero and therefore 2 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 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 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. 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 T J^th of the increase of
pressure between 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 C to 100 C. Let E, be the resistance at any temperature, R
and R 100 the resistances at 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. 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 ...
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 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 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 contain- so that the bars
ing cast-iron bar b'V with uprights w/ fixed near ends , : nvar : n ui ft
carrying microscope eyepieces, that on v' being provided *
with a micrometer m, length. Near the
ends of the bar, bb y
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 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 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 for each rise of 1. This fraction is termed the co-
efficient of expansion, and is usually denoted by k. Hence, if 1 l t are
the lengths of a bar at 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 :
FlO. 14. Mirror Method of Reading
Deflections.
The length at 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
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 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 Une 1 ual 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 becomes I (1 +M) at t. The
volume, therefore, increases from Z 3 to Z 3 (l + Jct) & = Z 3 (l + 3kt + 3/^ 2 + k s t s ).
Now Jet is, for moderate temperatures, so small that we may with more
or less exactness neglect Jc 2 t 2 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 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 H are the two heights, hot and cold, and p and p are the
respective densities,
But if a volume Y at expands to V (1 + A t ) 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 for a rise of t can be
found from H and H .
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 is equal to that at 0', both being equal to the
air-pressure in the reservoir M.
Then, pressure at - 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 p is the density at 0, the volume of the same mass being
inversely as the density, we have
P 1 P' 1
Po 1+A T
Then
H
1 + A T
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 A T . 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
A T ,).
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 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
A T = 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 g l 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 O a t is the mean co-efficient of expansion
between C. and t C., then
10 10 x O a t = 1805553 + 12444^/100 + 2539^/10000.
For approximate work we may put
10 8 x O a,= 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 p is the specific gravity at and if p t is that at t"
and A is the expansion of unit volume at when raised to t, we have
Po = Pt (l+ A).
The specific gravity of mercury at 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. to
100 C. We first fill the bottle, when surrounded with ice at 0, with
mercury, and find the weight W of this mercury. Then we fill it when
surrounded with steam at 100 and we find the new weight W 100 of the
mercury.
Let V be the internal volume of the bottle at 0. and
Y 100 100 C.
Let p a be the density of mercury at and
Pm " 100
Let G be the expansion of 1 cc. of the internal volume of the bottle
from to 100
7 mercury.
Then V 10Q = V 8 (l + G) and Po = Ploo ( 1 + y).
From the weighings
V = W / Po : V 100 = W 100 //>ioo-
Then Y (l +G) = W m / 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 and 100. Let 1 cc. of the liquid ut expand to 1 + A
at 100, and let <r , cr 100 be its densities at and 100.
Then o- = o- 100 (l + A).
Filling the bottle first at and then at 100 let a> , w 100 be the weights
of the liquid.
We have o- = <o /V , and tr m = w 100 /V 100 .
Then 1 + A = = -^ ^P.
"ioo w ioo v o
- -"(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 W be the weight of mercury, density p , filling the bulb at from
e up to the zero of the scale. Then the volume V = W //o .
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 AV . A, will be a
very small fraction.
Now start with the bulb filled with mercury at up to the zero, and
raise the temperature to 100. Suppose that the mercury rises N
divisions.
Let V expand to V 100 = V (l + G).
Then the total volume of the mercury is
V :00 + NAV 100 - V (l + G)(l + NX).
:00 100
But if y is the expansion of mercury we have
V (l+y) = V (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 to 100. Let 1 cc.
expand to 1 + A cc. Then the total volume at 100 is V 100 (l +n\)
But this is also V (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< 2 9. 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
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
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, = V (l + at + W + ct 3 + &c.).
Usually at is the most important term, but in the case of water, as we
have seen, bt 2 is very important. The terms after ct 3 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 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 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 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
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 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 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 ...... = -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
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. In each there are 100 degrees between C. and 100 C.
In the first case, if a is |y, f the expansion between 0. and
100 C. of a volume which is 1 at 0., the temperature t, measured from
C. is given by
EXPANSION OF GASES. 47
where v is the volume of the gas dealt with at C, v its volume at the
temperature to be measured.
Since 1 v m - V
a -foo x ~^T
we may put
t= JLJ^.. x iQQ (i)
If we choose a new zero - below 0., we may term this the gas
zero, and if we put
e=t+-
a
~t +
then
or putting 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., the temperature t, measured
from 0. is given by
when p is the pressure at O 8 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., on which 0. is - = 9 we
have 6=- P xlQQ
PlOO - Po
and 60
_
~
. .
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. on the work scale,
t v the temperature defined by (1) for air, t p 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
t v -t
t p -t
+
100
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. 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 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
0C.
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 Y x (Fig. 38) represents the air bulb. M is the
measuring tube initially filled with mercury. ^ is the connecting tube.
V 2 is a bulb of volume equal to that of Y I} and with equal " dummy "
connecting tube v 2 , following as nearly as possible the course of v l 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 6 , 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 V 2 being still at . 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', v l and v% to v^ and v 2 ', and let - be the
t>
mean of the reciprocals of the temperatures of Vj', v 2 '. We have for the
bulb and manometer
FV,' F< FM PV, Pv,
!_ 4. . L j ij 1
e t e " e + e
and for the other system Y 2 and v 2 ,
FV PV PV.
e~ ' t
Now making V 2 = Vj, v 2 = v lt 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 ma ke 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 mence d 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 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 and 30, the mixture
is exceedingly near to 15 ; or the heat required to raise 1 gramme from
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 7oo 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 to 1. Another unit is the -j-J^ part of the heat raising 1 gramme
of water from 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= M X *
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 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, t v t 2 = 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 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. 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 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 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 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 - 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 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 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 t r 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 t 2 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 t 2 , \ 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 t v
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 t l 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 C 2 R, 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 C 2 IW 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
C 2 IU 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 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 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 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 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
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 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 to 20, and a steeper slope up after that than
the water sets. Thus, at 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 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 6 be the tempera-
ture at entrance, 6 l that at exit. Let s be the mean specific heat of the
liquid between and O v and J the mechanical equivalent of heat ; then
the work value of the heat gained by the water is JQs(O l - 6 ). 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!-0 ) = EC,
and measuring Q, p , 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 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 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 10 6 6
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 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 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
. -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 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 AB 2
C0 2 .
44
0-2169
9-5436
N 2 O ....
44
0-2262
9-9456
S0 2 . . . .
64
0-1554
9-9528
More easily condensable
H 2 . . .
18
0-4803
86454
H 2 S .
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 AB 2 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 was 0-2389, and he found no change between 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 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 + 0-340p 2 . 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.
CO 2 at 20 C., 0-20202; at 100 C., 0-22141.
Extrapolating, we obtain for air at 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, CaCO 3 , 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 ; 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, 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 d v d 2 from the
nearest faces differ from the temperature of the faces by 9 V # 2 , and let
the conductivity of water be /c 1 . We have
KA0
Q " d =: d l
If we observe the water temperatures at the distances d v d 2 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 d l + d 2 , but not sufficiently. Peclet aided
convection and virtually reduced d^ + d z 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 wa s 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 TT Vff 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 : k z
or, & 2 = k-^n 2 .
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 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 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. its value should be
6 '3 x 10 10 . 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- 10 Jfc/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 G l C 2 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 3 . 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
S 2
Copper
c,
",
Vj+Vz
e
v .
Substance X
Copper
c.
Uniform temperature enclosure
FIG. 70. Lees's Disc Experiments on Conduc-
tivity. CiC 2 C 3 , copper discs ; X, disc to be of the figure, and let their
tested; Viv 2 v 3 , 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 2 , we have
+ Jis 2 v 2
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 C 2 . But the heat flowing from C a
into X is equal to that emitted from X and C 2 , and that flowing from X
into 2 is equal to that emitted from 2 . 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 C a
Ebonite] Liquid X \Ebonite
Copper C 3
FlG. 71. Lees's Disc Experiments on Liquids. CiC 3 C 2 , 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 : v 2 v s 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 a g , that of its emitting surface s g ;
let its conductivity be K g , and let its thickness be g. We may take the
heat passing through its middle surface as
K a v i ~ V 2
9
The heat emitted by its surface is
and that emitted by the lower half is approximately
hs v i + v 2
M 9^
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 s c , 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
Q 2 = Q! - hs e v a
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(v 2 - 8 )
where A is some constant, and we may then put
fa V 3 )
If, then, we can determine A, we can find /c, since Q 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 K t = -00155(1 - 0048)
where, it must be remembered, the actual range of observation lies
between = 25 and tf = 45, Weber's results give K t = '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
mv 2 - mv' 2
- =mas
= force x distance travelled against it,
= work done against the force.
If v' = 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 energies 5
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 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 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
and 76 cm. is, according to Regnault, '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 to 1.
But it is only the excess of this over specific heat at constant volume
O r , which is the equivalent of the work done. Now C fl may be found
from Op from the relation (chap, xviii.)
G p Adiabatic Elasticity.
C v 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 O p -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 - 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 = C 2 R,
the product C 2 R is in mechanical units, if and R are in terms of the
above units.
But Joule showed that the heat developed in any resistance is
proportional to C 2 R, 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 C 2 R. 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 ;
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 10 7 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. absolute degrees. hydrogen scale.
5 4-212 xlO 7
6 4-209 4-203 x 10 7
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 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 10 7 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 10 7 .
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 E 2 /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 10 7 {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 10 7 on mercury thermometer of hard French glass.
4-1905 x 10 r on nitrogen scale.
4' 19 17 x 10 7 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 xlO 7
4*1895 xlO 7
4*180 xlO 7
4-199 xlO 7
4*194 xlO 7
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 10 7 ergs. Hence we
have the mechanical equivalent
4-1 9 x 10 7 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 10 7 . 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 V 3 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 V T in a cer-
tain direction against the face A of the vessel, and resolve this velocity
into three components u v v v w v perpendicular respectively to A, B, and 0.
Then V x 2 = 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 - u v or
there is a total change of momentum, 2mu r This, therefore, is the
momentum given to the wall by the impact.
Through the velocity M X the molecule would move to and fro w x 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 u v u 2 . . . u n and if the total momentum
imparted per second to A is equal to p,
Considering the face B we similarly obtain
p = m(v* + v 2 z +
and on
p = m(w 1 2 + w.?+
Adding these three together,
. . . . + m(u n * + w n 2
If V 2 is the mean of the squares of the velocities, then we may write
this in the form
3p = mnV 2
But mn is the total mass of gas in 1 cc., that is, is equal to the density p.
Hence 3 = />V 2 or V 2 = 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 V 2 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 p 1 and /a 2 , 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 m 2 that of a
molecule of the second gas, Maxwell's investigation shows that
or _
ra x ra 2
But if Nj N" 2 be the numbers of the two kinds of molecules in unit
volume before mixture,
whence Nj = N 2
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.
V 2 v
But ^s=- if p is constant.
2
Therefore, V, 2 = V 2 (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. 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 K e , 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 K p - K^ = op , the work done in expansion.
Putting =? = y , we have (y - 1 )K e = ap
&
TT increase in translational energy _ 3op _ 3(y - 1)K B 3/ _ , .
increase in total energy ~ 2 K ~ 2K B ~ 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 pV 2
momentum imparted to the face in one second is ~ x 2V = u_, Equating
b 3
this to the pressure, we get V 2 = .
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
P Y
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 12 P VS 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 p l V l : P 2 V 2 , and the volumes escaping in the ratio V l : V ;
or since V x 2 : V 2 2 = p 2 : p v when the pressures are the same, the volumes
are as
\/P 2 : \/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, 6 l 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 /3 2 V 2 respectively. If the pressures are equal to begin with,
and
whence
-VI
or if # 2 ># p p 1 V 1 >/) 2 V 2 , 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' p l ~V l = p 2 V 2
or putting V 2 =
P
until
142 HEAT.
But &
p 2 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
4s 2 . The effective area subtended at the centre of projection by the
molecule will be ?rs 2 + a fraction of 4s 2 , 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 Trs 2 , and let us represent the other molecules
at B, C, D, <fec., by points. Let Trs 2 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 ?rs 2 is Trs 2 Z. Let the number of molecules per c.c. be N.
Then if irsH is large enough the number of molecules in it will be Trs 2 ZN.
Equating this to n or Z/L, we have Trs 2 ZN = l/L, or L = l/irs 2 N = 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 Trs 2 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
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
P 3 ' 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 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 C. Air is regarded as a simple gas.
Gas.
Density at
and 760.
Coefficient of
viscosity ij.
M.F.P.
L=i7\/l
PP
Collision
frequency
V/L-P/*
Air .
129 -10 5
17 -10 5
82-=-10 7 cm.
59 x 10 8
H .
9-10 5
8-10 5
146-107 )}
125xl0 8
O .
143 -10 5
19 -10 5
87 -10 7
53 x 10 s
H 2 O
81 -10 5
9-10 5
55 -10 7
111 x 10 s
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 + 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 K V is the specific heat of constant volume, this will
dx
be equal to
pv de_ KvP vi. 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. = zK V) or putting TT~ =">}
the coefficient of viscosity, JL = ^K V) 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.S 3
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
9-10 5
7-=-10 2
130-10 6
146 10 7
11-4 10 8
3-4xl0 18
2-7-MO 23
O
143- 10 s
1-24
115- 10 5
87-10 7
6 -10 s
10-2 x 10 18
14 -^-lO 23
H 2
81 -10 5
1
81- 10 5
55- 10 7
2-7-10 8
8 x 10 18
10-hlO 23
Using the more correct formulae, Maxwell calculated that 1ST for gases
at and 760 should be 19 x 10 18 , say 2 x 10 19 (Papers, ii. p. 372),
whence m for hydrogen comes out 4'5 -4- 10 24 . 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 3'9xl0 19
(Thomson, Condndion of Electricity through Gases, p. 130). Maxwell
further obtained for s with hydrogen the value 5 -f 10 8 . 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 -=- 10 8 .
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-
s i na l 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 vrs 2
molecules coming to the hemisphere is
27TS 3
The average path added is, therefore,
27TS 3
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 **
N7TS 3
Here - = total volume of the molecular systems per c.c., and since this
is small we may put
V 2
where v = - is the volume of unit mass.
P
If we put
,N7TS 3 , , , x V 2
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 = =- = VNvrs 2 and Nwi = p where m = mass of a molecule.
L
therefore v = V/> .
m
THE KINETIC THEORY OF MATTER. 155
pV 2
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 V 3 7rs 2 T 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 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
other barometers, each with a small quantity of water at
the top. The height of the columns in B and 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.
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.
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
' 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 j s j us t 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
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 t z may be both turned off, and the air in G
be again expanded and cooled by turning on t 2 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
to 50 0. ; (&) below, and up to 0. ;
(c) from 50 upwards.
(a) From 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 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 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 differs from that
of water below 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 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 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 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., L = 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 L by boiling water at in a Bunsen calorimeter and measuring
the volume of ice formed by the subtraction of a given weight of
vapour. He obtained L = 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. to 100 0. This gives L 596'73,
closely agreeing with Dieterici's value, and L 100 = 536'6, closely agreeing
with Regnault's value L 100 = 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 + l
and t + # 2 . Then the heat given up in condensing is L + d^ and L + # 2 cr.
We may determine both of these quantities, and knowing 6 l 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 tu 
