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book was at first intended as the second edition of a 
small volume on Solution and Electrolysis published in 
the year 1895. It was, however, soon found necessary to 
rewrite such large portions of the text, and to incorporate so 
much fresh matter, that the result is in effect a new work. 

Our knowledge of the phenomena of solution is growing 
rapidly, and as yet there is considerable difficulty in producing 
a systematic treatise. Moreover, the use of modern thermo- 
dynamic methods, founded on the investigations of Willard 
Gibbs, is extending to almost all branches of the subject, while 
such methods are still unfamiliar to many students of physics 
and chemistry. An introductory chapter has therefore been 
prefixed, explaining the thermodynamic principles which are 
applied in the body of the work. 

Besides the papers of Willard Gibbs, the following books 
should be especially mentioned among those to which the 
writer is indebted: Buckingham's Theory of Thermodynamics, 
Roozeboom's Heterogenen Gleichgewicht, Bancroft's Phase Rule, 
Larmor's JEther and Matter, Duhem's Mecanique Chimique, 
Ostwald's Lehrbuch der Allgemeinen Chemie, Nernst's Theo- 
retische Chemie, the works on Physical Chemistry of van 't Hoff, 
Lehfeldt, and J. Walker, van 't Hoff's Chemical Dynamics, 
Le Blanc's Electrochemistry, and Das Leitvermogen der Elek- 
trolyte by Kohlrausch and Holborn. In these books the reader 
will find further details on particular points, while most of the 
new work on the subject can be studied, either in full or in 
abstract, in the pages of the Zeitschrift fur Physikalische 

1 30125! 


Chemie, the Journal of Physical Chemistry or the Zeitschrift 
fur Elektrochemie. 

Much that is of value in this book must be attributed to 
the kindly help of many friends. Notes and corrections of the 
older work referred to were sent by. Professors J. H. Poynting, 
L. R. Wilberforce, and W. McF. Orr, while the wise and 
suggestive criticism of the late Professor G. F. FitzGerald 
added another to the many kindnesses for which the writer 
will always gratefully cherish his memory. Mr F. H. Neville 
read the earlier part of the manuscript of the new book, and 
gave useful information and advice ; help on particular points 
was sought from the Earl of Berkeley, Professor J. J. Thomson, 
Dr J. N. Langley, Principal E. H. Griffiths, Mr S. Skinner and 
Mr G. F. C. Searle, while Professor Poynting read and criticized 
the chapters on osmotic pressure and allied phenomena. To 
all the writer offers his cordial thanks. Especially would he 
express his sincere gratitude and deep sense of obligation to 
Mr J. Larmor, whose kindness in reading some of the manuscript 
and all the proof sheets it is impossible adequately to acknow- 
ledge. Mr Larmor's wide knowledge and deep insight have 
enabled the writer to gain clear ideas on many points which 
before were doubtful, and the ungrudging way in which he has 
given time and trouble to this work has removed many 
blemishes which would otherwise have appeared therein. 
Finally the writer's thanks are due to his wife for preparing 
the index and for constant correction both of the manuscript 
and of the proof sheets which have developed into the following 

December, 1902. 



I. THERMODYNAMICS ... , . . 1 

Experimental basis of Thermodynamics. The first law. 
The second law. Work and energy. Complete cycles. 
Reversible processes. Reversible engines. The absolute 
scale of temperature. Generalized co-ordinates. Internal 
energy. Entropy. Thermodynarnic potential. Conditions 
of equilibrium. Application of thermodynamic potential. 
Free energy. Application of the free energy principle. 


Equilibrium. Equilibrium of phases. The phase rule. 
Non-variant systems. Monovariant systems. Divariant 
systems. Other systems. The principle of latent heat. 
Application of the phase rule. One component. Labile 
equilibrium. Allotropic solids. 


Compounds, mixtures and solutions. Anhydrous solutes. 
Hydrated solids. Concentration curves. Two liquid com- 
ponents. Alloys. Solid solutions. Two volatile components. 
Three components. The problems of solution. 


General problem of solubility. Supersaturation. Solu- 
bility of gases in solids. Solubility of gases in liquids. 
Henry's law. Solubility of gases in salt solutions. Solubility 
of liquids in liquids. Solubility of solids in liquids. Influence 
of pressure on the solubility of solids. Solubility of mixtures. 
Solubility in mixed liquids. Tables of solubility. 



V. OSMOTIC PRESSURE . . . ... .95 

Semi-permeable membranes. Osmotic pressure and 
vapour pressure. Perfect semi-permeable membranes. 
Theoretical laws of osmotic pressure. Osmotic pressure 
and heat of solution. Experimental measurements of 
osmotic pressure. 


Connection with osmotic pressure. The latent heat 
equation. The depression of the freezing point. Vapour 
pressures of concentrated solutions. Solubility of gases in 
liquids. Experimental measurements of vapour pressures. 
Boiling points. Determination of molecular weights. 
Freezing points. Osmotic pressure and freezing points of 
concentrated solutions. Experiments on the freezing points 
of solutions. Determination of molecular weights. Freezing 
points of alloys. Experiments on concentrated solutions. 


Thermodynamics as a basis for physical science. Appli- 
cation to the case of solution. Theory of direct molecular 
bombardment. Theory of chemical combination. Con- 


Introduction. Volta'spile. Early experiments. Faraday's 
work. Polarization. Faraday's laws. Electrochemical equi- 
valents. The electrolysis of gases. Nature of the ions. 


Ohm's law. Experimental methods. Experimental re- 
sults. Consequences of Ohm's law. Migration of the ions 
and transport numbers. Mobility of the ions. Experimental 
measurements of ionic velocity. Influence of concentration. 
Complex ions. Connexion between the mobility of an ion 
and its chemical constitution. Conductivity of liquid films. 


Introduction. Eeversible cells. Electromotive force. 
Effect of pressure. Concentration cells. Different con- 
centrations of the electrodes. Different concentrations of 
the solutions. Concentration double cells. Effect of low 
concentrations. Chemical cells. Oxidation and reduction 
cells. Transition cells. Irreversible cells. Secondary cells 
or accumulators. 




Volta's contact effect. Thermo-electricity. The theory 
of electrons. Single potential differences at the junctions 
of metals with electrolytes. Dropping electrodes. Electro- 
capillary phenomena. The theory of von Helmholtz. Electric 
endosmose. Single potential differences (continued). Electro- 
lytic solution-pressure. Electrochemical series. Polarization. 
Decomposition voltage. Polarization at each electrode. Evo- 
lution of gases. Electrolytic separations. 


Introduction. Osmotic pressure of electrolytes. Additive 
properties of electrolytic solutions. Dissociation and chemical 
activity. The mass law. Equilibrium between electrolytes. 
Thermal properties of electrolytes. Heat of ionization. 
Dissociation of water. The function of the solvent. 
Hydrolysis. Conclusion. 


Theory of diffusion. Experiments on diffusion. Diffusion 
and osmotic pressure. Diffusion of electrolytes. Potential 
differences between electrolytes. Liquid cells. Complete 
theory of ionic migration. Electrolytic solution pressure. 
Diffusion through membranes. 


The colloidal state. Process of gelation and structure of 
gels. Coagulative power of electrolytes. The nature of 
colloidal solutions. 

ADDITIONS .......... 403 



INDEX ... . 476 


Page 12, line 8, read: But consequences of the second law of thermodynamics 

which only hold for reversible systems have sometimes been applied, 


Page 26, footnote, read: Zeits. phys. Chem. xi. 289 (1893). 
Page 117, footnote 3, read: Zeits. phys. Chem. i. 481 (1887). 
Page 122, add to heading of chapter : Freezing points of alloys. Experiments 

on concentrated solutions. 

Page 162, the equation should be numbered (35). 
Page 193, line 19, read: 0'404xlO- n . 

Page 197, add to heading of chapter : Conductivity of liquid films. 
Page 206, the equation should be numbered (36). 
Page 231, add to footnote : Patterson, Phil. Mag. Dec. 1902. 
Page 241, line 18, read: to all kinds of reversible cells, etc. 
Page 248, line 18, in the denominator of the expression for E, 96440 should be 

9644, the ionic charge being here measured in c. G. s. units and not 

in coulombs. 
Page 272, line 15, read: Let us imagine a circuit composed of two wires of 

different metals surrounded by a dielectric, the two metallic junctions 

being maintained at different temperatures. In applying, etc. 
Page 328, the Figure should be numbered 64. 



Experimental basis of thermodynamics. The first law. The second law. 
Work and energy. Complete cycles. Reversible processes. Reversible 
engines. The absolute scale of temperature. Generalized co-ordinates. 
Internal energy. Entropy. Thermodynamic potential. Conditions of 
equilibrium. Application of thermodynamic potential. Free energy. 
Application of the free energy principle. 

THE subject of thermodynamics, which deals with the relations 
Experimental between heat and the other forms of energy 

dynan^cs^^hT possessed by material systems, rests ultimately, 
first law. like a ll physical sciences, on a basis of observa- 

tion and experiment. 

Careful measurements by Joule, Rowland, Griffiths and 
others have shown that, when various forms of mechanical and 
electrical energy are completely converted into heat' by friction 
and similar processes, the quantity of heat produced by a given 
amount of work is always the same in whatever form and by 
whatever means the work is applied. Heat is thus a form of 
energy, and one thermal unit must have its definite mechanical 
equivalent in other forms of energy. This experimental gene- 
ralization is one case of the principle of the conservation of 
energy, and constitutes the first law of thermodynamics. 

The best modern determinations show that the amount of 
heat necessary to raise one gram of water from 17 to 18 
centigrade, which is taken as the practical thermal unit and 
called the calorie, is always developed when 4'184 x 10 7 ergs 
of work are expended in heat. 

w. s. 1 


If we try to bring about the reverse change and convert 
heat energy into mechanical work, experience 

The second law. ' . . ... * 

shows that no heat engine will act if the whole 
of the available system is at a uniform temperature. Thus 
all steam engines have a boiler and a condenser, the atmo- 
sphere acting as condenser in the case of high-pressure engines. 
Oil engines work by the explosion of oil spray in the cylinder ; 
a high temperature is thus produced, the atmosphere again 
acting as condenser. It will always be found that there is 
some heat given up to the condenser besides that which is 
transformed into work, and, in all cases, the heat is absorbed 
by the engine from the hotter parts of the system. Such 
observed facts can be generalized in the statement that it is 
impossible by inanimate mechanical means to obtain a con- 
tinual supply of useful work by cooling a body below the 
temperature of the coldest of the surrounding bodies. It is 
possible to get a certain amount of work in this way, for in- 
stance, by allowing gas or vapour to expand, thus cooling 
itself and doing work ; but a continual supply cannot be so 
produced. When the gas or vapour is used in a heat engine, 
and put through complete cycles of changes, an external supply 
of heat at a high temperature must be constantly maintained, 
and the engine will continually give up some of this heat to 
the cooler parts of the system. 

It will now be seen that if a range of temperature is 
available, and a heat-engine be constructed to use it, the 
process will tend to diminish the difference of temperature 
between the parts of the system, heat passing from the hotter 
to the colder parts. Again, when heat flows by conduction, the 
transference always occurs in this same direction, and never in 
the reverse one. It is thus a general result of observation 
that heat cannot of itself, or by means of a self-acting 
mechanism, pass from a body of lower to a body of higher 

This statement will be found to be equivalent to that 
enunciated above in the form that a continual supply of 
mechanical work cannot be obtained from the heat energy of 
the coldest part of the system. Both statements embody the 


experimental generalization known as the second law of 
th erm ody namics. 

On the results_of _ experience, as formulated in these two 
laws, the whole subject of thermodynamics is founded. 

When a force X moves its point of application through a 
distance dx, the work done by the force is X.dx, 

Work and energy. 1 . p . 

and, if the force acts on a system, this amount of 
energy is added to the system. Take as an example the case of 
a quantity of fluid confined in the cylinder of an engine (Fig. 1). 
If the piston be forced inwards, work is done on the system, and 
if the piston be allowed to move outwards, work is done by the 
system on the environment. If the area of the 
piston be A, and the pressure be p, the force r^i-n 

is Ap, and if the piston rise through a small 
height dh, the work done by the contents of 
the cylinder is Ap . dh. But the change of 
volume of the working substance is A . dh, so 
that, representing the change of volume by dv, 
the work done is p . dv. In order to find the 
work done when a large volume change occurs, 
we must find the sum of all the separate values 
of p . dv. This process is termed integration, 

and is represented by the symbol fpdv. When Fig ^ 

certain simple relations hold between the pres- 
sures and the volumes, we can find the value of this integral. 
For instance, if p remains constant throughout the operation, 
the sum of all the p . dv's is the same as p multiplied by the 
sum of all the dvs, or p (v 2 v,) where ^ and v 2 represent 
the initial and final volumes. This constancy of pressure is 
practically attained when the working substance is a liquid 
in contact with its own saturated vapour: for instance, the 
water and steam in the boiler of an engine. If heat be 
supplied, evaporation goes on at constant temperature, and 
the pressure remains unchanged. Another important case 
arises when a gas, such as air, fills the cylinder. While the 
temperature keeps constant, the pressure is inversely pro- 
portional to the volume, and, as is well known, the whole 



pressure, volume and temperature relations have been found ex- 
perimentally to be represented very accurately by the equation 

pv = RT, 

where T is the temperature measured on the gas thermometer, 
of which the zero corresponds to 273 centigrade, and R is a 
constant, the numerical value of which can be calculated for any 
given mass of gas. If we take as our unit mass the number of 
grams equal to the chemical constant known as the molecular 
weight of the gas (a number which is commonly called the 
gram-molecule), the volumes of different gases under the same 
conditions of temperature and pressure will be equal, and the 
value of .R the same for all. The volume of the gram -molecule 
is found experimentally to be 22320 cubic centimetres when 
the pressure is that of the standard atmosphere (760 millimetres 
of mercury or T013 dynes per square centimetre) and the tem- 
perature is centigrade, or 273 on the absolute scale of the 
gas thermometer. These numbers give for the value of R 
corresponding to the chemical unit of mass or the gram- 
molecule, 8*284 x 10 7 ergs or T980 calories per degree centi- 
grade. In approximate calculations R may therefore be taken 
as 2 calories per degree for each gram-molecule of gas. 

As we have seen, the work done, while the volume changes 
from Vj to V 2) is 

fv 3 [ V *RT 

I p dv = I dv 

For isothermal changes, both R and T are constant, and can be 
put outside the sign of integration. Now 


and thus we know the value of the work done by the gas when 
its volume increases at constant temperature. Similarly, when 
the volume is diminished, the same integral gives the work 
done on the gas. 

These results can be well shown on a diagram (Fig. 2), in 
which the abscissae represent volumes and the ordinates pres- 
sures. On such a diagram the isothermal lines, defined by the 
relation that T and therefore pv is constant, will be rectangular 

CH. I] 


hyperbolas. Consider the work done while the gas passes from a 
state represented by the point A to a state represented by B. 

Fig. 2. 

For a small change in volume v to v a , the pressure can be taken 
as constant, and the work, which is pdv, is represented by the 
area of the narrow strip p a p a v a v^. The work from v a to v b is 
measured by the area of the corresponding strip pt>v a v b , and it is 
now obvious that the total work from v^ to v 2 is represented by 

the area of the figure ABv 2 v l under the curve AB, which there- 

rv z 
fore is equivalent to the value of the integral I pdv. Passing 

J Vi 

from A to B, the volume increases, and therefore work is done 
by the gas. If the process had been performed in the reverse 
order, from B to A, the work represented by the area would 

have been done on the gas, and the work done by the gas could 

be written I pdv. 

J V 2 

Now pv = constant ; 

hence differentiating, pdv + vdp = 0, 
or pdv vdp. 

Thus as regards the integrals 

fpdv = fvdp. 



[CH. I 

The latter integral is represented by the area ABp^ on the 
diagram, which is therefore equal to the area ABv^ as long as 
the curve is a rectangular hyperbola, that is, as long as Boyle's 
law holds good for the working substance. 

Now let us imagine that the working substance, whatever 
it may be, is carried round a complete cycle of 
complete cycles. c h a nges, so that in the end it is brought back 
to its original state, represented by A on the diagram. If the 
changes are not isothermal, the curve on the diagram can be 
made of any form we please. Thus on Fig. 3, let the cycle be 

Fig. 3. 

performed in the order ACBDA. The work done by the 
system along ACS is, as we have seen, measured by the area 
ACBHG, and that done on the system along EDA by the area 
BDAGH. The balance of work done by the system through- 
out the cycle is therefore the area ACBHG, less the area 
BDAGH, that is the area ACBDA enclosed by the curve 
representing the successive states of the substance as regards 
pressure and volume. This area is measured by the integral 
fpdv taken all round the cycle. The area ACBDA is also the 
difference between the areas ACBFE and BDAEF, it can 
therefore be also represented by fvdp, which thus measures 
the work done in a complete cycle of any kind ; while, when 
the cycle is not complete, this converse integral only measures 


the work in the case of the isothermal changes in an ideal gas, 
the properties of which are exactly described by Boyle's law, pv 
is constant. 

The relation 

pv = constant, 

as we have seen, describes the behaviour of an ideal gas under 
isothermal conditions. The corresponding 1 rela- 

Adiabatic rela- 
tions of an tion for adiabatic changes may be deduced by 

the help of the general equation which holds 
under all conditions for gaseous substances, 

pv = RT. 

We must also remember that experiment has proved that there 
is only a very small change in the internal energy of a gas 
when its volume is changed isothermally, or, in other words, 
that no appreciable work is absorbed or liberated in merely 
separating the parts of the gas from each other. This work 
becomes less as the gas approaches the ideal condition, and 
may, for an ideal gas, be considered to be zero. 

When a gas is heated at constant volume, no external work 
is done, and the whole of the heat energy is used in raising the 
temperature of the gas. On the other hand, if the pressure be 
constant, an amount of external work equal to pdv is done. 
Let us express everything in mechanical units, and denote the 
specific heats at constant pressure and constant volume by C p 
and C v respectively. Then, considering unit mass and unit rise 
of temperature, we have, since the internal work is negligible, 

C p C v =pdv. 

But the gas constant R is equal to pv/T, that is, to pdv/dT, or 
the pressure multiplied by the change in volume per degree. 

R = C P -G V . 

Now for the genera Incase, when a quantity of heat is allowed 
to enter an ideal gas, it is used in raising the temperature 
through a range which we will call dT, and in performing an 
amount of external work, which, for an infinitesimal change, 


may be denoted by pdv. Thus, for an adiabatic process, 
when there is no gain or loss of heat, 

C v dT + pdv = 0... (2). 

Again from the equation 

pv = RT, 

we have, by differentiating, 

pdv + vdp = RdT. 
Substituting for dT in (2), and replacing R by G p C v , we get 

C p pdv + C v vdp = 0. 
Then, denoting the ratio C P /C V by 7, 

y dV + ^ = 0. 

v p 

This ratio of the specific heats 7 is found by experiment to 
be nearly independent of the pressure, volume and temperature. 
The equation can therefore be integrated. Thus 

7 log v + log p = constant, 

or pvv = constant (3), 


the adiabatic relation required. 

Let us imagine a quantity of water in contact with its 
Reversible pro- vapour in an engine cylinder. It is known 
cesses - that, for a given temperature, there is one 

and only one pressure at which the system will be in equi- 
librium. If the pressure be slightly increased, vapour will 
condense till it again falls to its original value, or, if the excess 
of pressure be kept up, the whole of the vapour becomes liquid. 
Conversely, if the pressure be kept slightly below the equilibrium 
value, the whole of the liquid will evaporate. Either of these 
changes can be produced, theoretically at any rate, by a change of 
pressure infinitesimally small, if time enough is allowed. Similar 
changes can be produced if, instead of varying the pressure, the 
temperature be slightly altered, the least variation from the 
equilibrium value being enough to cause the system to move 
in one direction or the other. In such a case it is obvious that, 
when we have taken the system along a path AGB (Fig. 3) by 


such infinitesimal changes, we can by similar infinitesimal 
changes in the other direction of the external variables, pres- 
sure or temperature, cause the same path to be described in 
the reverse direction. Such processes are called reversible, 
and it is clear that in practice, though we can never use 
infinitely slow variations and thus get strictly reversible 
processes, we can make the processes which actually go on 
more or less nearly reversible by keeping the changes in the 
external variables more or less slow. 

It is evident that, to get reversible processes, we must 
keep the pressure and temperature indefinitely near their equi- 
librium values at all parts of the operations. If the temperature 
be kept constant, heat can be passed into the system while this 
condition is realized, provided that the temperature of the 
external source of heat, which must be higher than that of 
the working substance in order to produce a flow of heat at 
all, is made to differ from it by an amount infinitesimal only. 
The process is then very nearly reversible. Another case in 
which reversibility may be nearly attained arises when there is 
no passage of heat at all. The changes which then go on are 
called adiabatic. If the external pressure be kept indefinitely 
near that of the working substance throughout, such changes 
will be very nearly reversible. 

A good example of changes practically adiabatic is found 
in the alterations of pressure and volume which accompany 
the passage of a wave of sound through air, the vibrations 
being so quick that there is no time for heat to enter or 
leave the parts of the air affected. This case may also be 
used to illustrate what occurs when the changes are neither 
isothermal nor adiabatic. If, for instance, the air remained 
compressed long enough for a flow of heat to occur from parts 
of the air which have been heated by the compression to 
parts which have been cooled by expansion, the conduction of 
heat could not be reversed by an infinitesimal change of tem- 
perature, and the process becomes irreversible. 

It will be noticed that, for a process to be reversible in the 
thermodynamic sense of the word, it is not enough that it can 
be made to proceed in the reverse direction. It is also necessary 


that this change in the direction of the process should be effected 
by a change of an infinitesimal amount in the external conditions. 
No real process can be an exactly reversible one, though physical 
and chemical actions which are not accompanied by anything 
of the nature of friction, can be made almost reversible by 
keeping the conditions very near those of equilibrium, and the 
action consequently slow. Viscous forces, such as those which 
a liquid offers to the passage of a body through it, do not 
interfere with this result, for they may be made indefinitely 
small by reducing indefinitely the velocity of change. The 
existence of viscosity, then, does not prevent a system under- 
going reversible operations. Ordinary friction, on the other 
hand, such as that between solid surfaces, restrains a system 
from change till the driving forces reach a finite value, and 
entirely prevents even an approximation to a condition of re- 

Thus, although reversibility can never be attained in practice, 
systems can be divided into those which can be made very nearly 
reversible, and those which cannot. The directive forces of the 
former could be diminished without limit as the changes in 
them become indefinitely slow ; they are therefore called rever- 
sible systems. 

A similar distinction can be drawn between the equilibria 
of these two classes of systems. The weight of a body 
hanging by a spring is balanced by the force exerted by the 
spring, and the body will move in one direction or the other as 
the weight is increased or diminished by a very small amount. 
So a liquid in contact with its own vapour is in equilibrium 
when the rates of evaporation and condensation are equal. 
A solid is in equilibrium with its solution when the amount 
dissolved per second is equal to the amount precipitated. Such 
cases of true equilibrium are at once known by the fact that 
a small change in one of the external conditions, temperature 
or pressure, will at once cause a corresponding change in the 
factors of equilibrium ; more liquid will evaporate or condense, 
or more solid go into or out of solution. 

But equilibrium often exists which is not the effect of the 
balance of such oppositely directed active tendencies. A body 


can be kept on an inclined plane by the roughness of the 
surfaces in contact ; and so some physical and chemical trans- 
formations may possibly be prevented by forces analogous to 
friction. Such forces might be overcome by changing the condi- 
tions : for example, by heating some explosive substances which 
are unchangeable at ordinary temperatures ; but, as long as the 
frictional forces keep the system in equilibrium, it will not be 
disturbed by any small change in the external conditions. Thus 
it is thought a false equilibrium may be distinguished from a 
true one. On the other hand, viscous resistances, like those 
exerted on a moving body by fluids, delay but do not prevent 
motion, and will not affect the final conditions of true equi- 
librium. The equilibrium reached, then, will, if we wait long 
enough, be independent of all such viscous forces. 

Now the importance of this distinction between true and 
false equilibrium lies in the fact that, while the first law of 
thermodynamics, the principle of the conservation of energy, 
holds good for all processes whatever, the second law can 
only be applied to obtain quantitative results in a system 
which exists in true equilibrium. Such a system will respond 
to a slight change in external conditions. It is therefore 
strictly reversible and capable of being taken reversibly 
through a complete cycle of operations, and can finally be 
brought back to exactly that state from which it started, 
each part of the change being reversible. As we have seen, 
a very slow physical or chemical change is reversible when an 
indefinitely small alteration in one of the external conditions, 
such as temperature or pressure, is enough to reverse the 
direction in which the change proceeds. Thus the system 
must at each instant be indefinitely near its equilibrium 
condition. A good example of such an arrangement, described 
above, is seen when a liquid is in contact with its own saturated 
vapour at a given temperature. By bringing it into contact 
with a body at a temperature higher than its own by an infini- 
tesimal amount, heat slowly enters the system, liquid evaporates 
and external work can be done. On replacing the source of 
heat by a body the temperature of which is infmitesimally 
lower, the direction of flow of heat will change, and energy will 


be absorbed by the system, showing that the process is rever- 

Similar considerations apply to all processes where a true 
chemical or physical equilibrium exists. The systems are rever- 
sible, and can be carried through complete cycles of changes. 
Examples, such as the evolution of carbon dioxide from calcium 
carbonate or the solution of a solid in water, are numerous. 
But the second law of thermodynamics has sometimes been 
applied to cases of false equilibrium, equilibrium maintained 
by frictional forces, and to chemical actions, explosions and the 
like, which are not reversible, and cannot be carried through a 
cyclical process. Such applications are not legitimate, and the 
conclusions reached, though they may be correct expressions of 
tendencies, are not exact results. 

To study the laws which describe the transference of 
Reversible heat into work, it is best to examine the sim- 

engines. plest possible form of engine, consisting of a 

cylinder wherein is confined some substance, the volume of 
which depends on the temperature. The walls of the cylinder 
are perfect non-conductors of heat, and its bottom a perfect 
conductor. By putting the cylinder on a non-conducting 
stand, the contents are thermally isolated, and by transferring 
it to a conducting body of large size they are placed in 
isothermal conditions and an indefinite supply of heat can be 
admitted or abstracted. Two such bodies are needed, one at 
a high temperature, one at a low temperature. We now have 
an engine reduced to its simplest form. In order to draw 
valid conclusions about the heat absorbed and the mechan- 
ical energy developed, we must put the engine through a 
complete cycle, and bring the working substance back to its 
original state : its internal energy will then be the same as 
it was at first, and any work done must be due to the 
heat energy absorbed from the surroundings. This simplest 
theoretical form of engine was first described by Carnot, who 
revolutionized this branch of physics by calling attention to the 
importance of considering complete cycles of operations. 

We have already deduced the conditions of reversibility, 



CH. I] 

and have seen that the co-ordinates which determine the state 
of the substance must, at any instant, differ only infinitesimally 
from their equilibrium values. Now the simplest reversible 
cycle we can arrange consists of four processes, illustrated 
graphically by the diagram of Fig. 4. 

Fig. 4. 

(1) Allow the substance to expand isothermally in contact 
with the hot body from the state A to the state B. 

(2) Thermally isolate the substance, and continue the 
expansion adiabatically from B to G. 

(3) Transfer the cylinder to the cold enclosure, and com- 
press it isothermally from C to D. 

(4) Again isolate the cylinder, and compress it adiabatic- 
ally till the working substance again reaches the state denned 
by A. 

It will be observed that, since the temperature is, on the 
average, higher during the processes (1) and (2) than it is 
during (3) and (4), the pressure must be higher also, and 
therefore more work is done by the substance in expanding 
than is done on it while contracting. On the whole, then, 
a balance of useful work is performed by the engine, and this 
work has been obtained at the expense of some of the heat 
absorbed from the surroundings during process (1); for the 
quantity of heat given up to the environment during process (3) 


is less than that absorbed during (1) by an amount dynamically 
equivalent to the balance of work done. 

Now, in order that this cycle should be performed at all, the 
external conditions of temperature and pressure must differ 
appreciably from their equilibrium values, but to insure the 
reversibility of the cycle, they must only differ by infinitesimal 
amounts. Nevertheless, although the required conditions 
cannot be obtained, theoretically the cycle is a reversible 
one, and we can imagine each process performed in the reverse 
order, heat being taken in at the low temperature, a balance of 
work being done on the substance, and the thermal equivalent 
of this work added to the heat absorbed, and given out with 
it as a larger quantity of heat at the higher temperature. 
The whole cycle, and all the individual parts of it are theoreti- 
cally reversible. 

In no actual engine can a reversible cycle be obtained, and, if 
an indicator diagram, as it is called, be drawn to represent the 
relation at each instant between the pressure and the volume 
of the steam in the cylinder, and its form compared with the 
isothermal and adiabatic lines for saturated steam, it will be 
seen in what ways the engine fails. Since the working sub- 
stance must be colder than the source of heat and warmer 
than the condenser, and partly also in consequence of the 
unavoidable thermal losses which will occur, the top and the 
bottom of the indicator diagram will be nearer together than 
in the theoretical diagram of Fig. 4, the corners will be 
rounded off and the available area, that is the work done, 
will be less. 

In fact, it can be shown that a reversible engine is the 
theoretically perfect engine, and has the highest efficiency 
which an engine can possess: it will transform the greatest 
possible fraction of the heat absorbed into useful mechanical 
work. For, if possible, let an engine have a greater efficiency 
than a reversible engine, and let us use it in conjunction with a 
reversible engine in such a way that it works the reversible 
engine backwards over the cycle of Fig. 4, putting work into 
it, and forcing it to give up heat to the hot reservoir, which is 
common to the two engines. The more efficient engine is at 


the same time constantly taking a supply of heat from this 
reservoir, and, in virtue of its assumed efficiency, it can perform 
the work required, that is to keep the reversible engine in 
operation, by using a smaller quantity of heat than the rever- 
sible engine returns to the hot reservoir. This excess must be 
obtained from the cold reservoir, and therefore the combined 
machine enables heat to pass regularly and automatically from 
a cold to a hot body. Such a result is contrary to experience ; 
it proves that our hypothesis is false, and that no imaginable 
engine can possess a greater efficiency than a reversible engine. 
We have already seen that no actual engine can do the amount 
of work corresponding to a strictly reversible cycle. It therefore 
follows that no other engine can have as great an efficiency as a 
reversible engine. 

A reversible engine, then, has the maximum efficiency 
The absolute scale possible and we need not limit ourselves in 
of temperature. choosing the working substance. Any system, 
the volume of which depends on temperature, might be used. 
The efficiency of a reversible engine is thus independent of 
the nature of the working substance and of the kind of process 
employed. It depends only on the temperatures of the hot 
and cold bodies which are used as the source of heat and as 
the condenser of the engine. Now the efficiency of an engine, 
the fraction of the heat taken in which is transformed into 
work, can be expressed in terms of the heat changes only, 
for by the principle of the Conservation of Energy, if H^ is 
the quantity of heat absorbed from the hot reservoir, and H 2 
the quantity of heat given out to the cold reservoir, the work 
done must be equivalent to their difference, and the efficiency 
must be 

Thus HJH l} which is obtained by subtracting this expression 
from unity, must also depend only on the temperatures. But 
any property which depends only on the temperature can 
be used as a means of measuring temperature, just as the 
change in volume of mercury is used as a means of measuring 


temperature in the common mercury thermometer. We may 
thus agree to compare two temperatures by finding the ratio 
of the quantities of heat absorbed and ejected by a perfect 
reversible engine working between those temperatures. Then 
denoting by 0^ and # 2 the temperatures as thus defined, 

and this thermodynamic temperature scale, unlike those which 
depend on any one property of a particular substance, such as 
the volume relations of mercury or the like, is a true absolute 

Moreover this scale of temperature is a consistent one : for 
if a second reversible engine be coupled with the first, taking 
in as its supply of heat at # 2 the heat given out to its condenser 
by the first engine, the ratio of the heat taken in at 1 to that 
finally given out at S by the compound engine will be 

HI H% _ #1 #2 

H z H 3 2 3 ' 
giving the same formula as before, 

It remains to connect this absolute scale of temperature 
with some scale which can be practically constructed. Now, 
since all reversible engines have the same efficiency, to calculate 
the efficiency of any one such engine is to know that of all. 
It is easy to find the ratio of the heats taken in and given 
out by an engine using as its working substance a gas which 
is described by the laws of Boyle and Charles and suffers no 
changes of internal energy when its volume varies isothermally. 
Experiments can afterwards be made to determine how far any 
known gas departs from those ideal relations. We have seen 
that a simple reversible cycle may be performed by means of 
isothermal and adiabatic processes ; the experimental gaseous 
laws show that isothermal changes can be represented by the 

pv = constant, 


while we have already deduced the adiabatic relation, 
pv? = constant. 

Let us then take unit mass of an ideal gas through a simple 
cycle like that described above. As we see by equation (1) on 
p. 4, during an isothermal expansion at a temperature T l9 an 

amount of work is done by the gas equal to RT^ log . 

v i 

Similarly the work done on the gas during the isothermal 
contraction at T 2 is 

V 3 V 4 

If the gas absorbs no internal work, that is, if no energy 
is needed to separate or concentrate the molecules, these 
expressions for the external work done by the gas can also 
be taken as giving the heat absorbed and ejected during each 
process. Thus 

H t RT.logv./v, 

Now the change from v t to v 2 is isothermal and 

Similarly P^=p 3 v 3 . 

Dividing the first of these equations by the second 

Again the changes from v 2 to v s and from v 4 to Vi are 
adiabatic, and therefore 

Hence ^ = 1 = ^- y (6). 

Dividing (6) by (5) and clearing the indices 

- 1 = - 2 or - = 
We therefore have 

g 1 = jtr 1 io g yt> l = r lji __ (7)> 

w. s. 2 


and thus find that the thermodynamic temperatures are the 
same as those measured on an ideal gas thermometer. Ex- 
periments have shown that an air or hydrogen thermometer 
agrees very nearly indeed with the ideal gas thermometer. 

We may therefore take the air thermometer as giving a 
very near approximation to the absolute thermodynamic scale, 
and write indiscriminately lt 2 or T l} T 2 . 

The cycle of Fig. 4, consisting of two isothermal and two 
Complex adiabatic processes, is the simplest form of re- 

cycles, versible c^cle, but any curve on the pressure- 

volume diagram can represent a reversible cycle if the external 
conditions are kept throughout in- 
definitely near their equilibrium 
values. A closed curve, such as 
that in Fig. 5, can be described 
by taking the substance through 
small isothermal and adiabatic 
changes alternately, as indicated in 
the figure. If these changes are 
small enough, the lines representing 
them practically become the closed 

curve, and the cycle remains re- Fi 5 


We have hitherto expressed the work done on or by the 
Generalized system as the product of a force and a length 
co-ordmates. Qr o f a p ressure an( j a volume ; but the same 

energy dimensions can be obtained as the product of many 
other pairs of quantities, such as surface tension and area, 
electromotive force and quantity of electricity, etc. Each of 
these products consists of a coordinate defining some quantity 
in the system (volume, quantity of electricity, etc.) and a term 
often called an intensity factor (pressure, electromotive force, 
etc.), analogous to the force in the first case. 

Now, in the general case, the work done by a system may 
contain all such possible products, and its expression will then 
involve a series of terms 


The factors X l} X 2) etc. are the intensity factors or the 
"generalized forces," though, as we have seen, they are not 
all necessarily of the physical dimensions of real forces, and 
#!, # 2 > etc. may similarly be called the quantity factors or the 
" generalized coordinates." 

All the forms of energy thus considered are mutually 
convertible and, if perfect machines could be obtained, com- 
pletely convertible. Thus, the whole of a quantity of mechanical 
energy might, by the aid of a theoretically perfect dynamo, be 
transformed into electrical energy, while the electrical energy 
might drive a motor and be reconverted, theoretically without 
loss. All such forms of energy are therefore said to have the 
same value, and may be grouped in a single term, which may 
for convenience be written as 

2 (XSx). 

The fact that heat cannot in general be completely con- 
verted into other forms of energy, shows that it 

Internal energy. 

is not of the same value as they are, and should 
be represented by a separate term in the expression for the 
energy. The equation giving the increase in total energy e of a 
system which takes in a quantity of heat SZT, and also absorbs 
various kinds of external work, represented by 2,(X$x\ may 
therefore, in accordance with the first principle be written 

Now the internal energy of a body is, by the principle of 
conservation, the same when the body is in a given state, what- 
ever its previous history has been ; thus a change in energy 
can be expressed as the difference between the absolute values 
of the energy content of the system, and for finite changes we 
may write 

where the integral refers to any path of change between the 
states A and B. 



For a simple reversible cycle, between two 
temperatures 1 and 2 , we have seen that 

Treating heat taken into the system as positive and that given 
out as negative, this is equivalent to the statement that 

I +t- 

Similarly for any complex reversible cycle such as that illus- 
trated by Fig. 5, the same principle holds and 

If the operations are not reversible, the efficiency of the cycle 
must, as we have seen, be less, thus 

TT TT S\ S\ 

TT "^ 7) J 

tl\ "l 

or, subtracting unity from each side, we deduce 

and for a complex non-reversible cycle the corresponding 
relation is 


Now, if the system can pass from a state A to a state B in 
two different ways, each reversible, we can take the system 
from A to B along one of them and back from B to A along 
the other. Then for the complete cycle 


where the suffixes I and // refer to the two different paths. 
Thus f -f(f) =0, ' 

J A \ ^ 1 1 J A \ V Jll 

[ B /dH\ [ B fdfr 

the value of the integral being the same for all reversible paths. 

CH. I] 



This integral may therefore be taken as representing the 
difference in value of some definite quantity characteristic of 
the system in each of the states A and B, and we may write 


This function < is called the " entropy " of the system. 

For a small reversible change in any system, we have thus, 
as the expression of the second principle of thermodynamics, 

while, if the change is a non-reversible one, 

< in each case being a function of the constitution of the 
system in its given state and independent of its previous 

These relations are well illustrated by the diagrams already 
used. For an infinitesimal re- 
versible transfer of heat, the 
change from* the isentropic line 
along which the entropy is con- 
stant and represented by </> to 
that along which it is represented 
by (f) + (/> takes place by the iso- 
thermal along which the tempera- 
ture is 6 ; the heat absorbed, SH, 
being equal to the area 6B<f>. 

For any actual change, how- 
ever, the path will not be iso- 
thermal, and we get a non-reversible path, such as the dotted 
line of Fig. 6, in which case the area SH is less than OScf). 

For reversible changes, we see from the expression for the 
change in heat energy, 

Fig. 6. 

that the entropy c/> may be considered as the quantity factor in 
the heat energy of a system, just as x is the quantity factor in 


the generalized expression for the work XBx', it corresponds 
to the quantity of electricity, for example, in the expression for 
the electrical work ESq, or to the area in the expression for 
the surface energy S8A, where 8 is the surface tension. From 
this point of view, the temperature, 0, represents the intensity 
factor in the heat energy. 

If the system is isolated, SH is zero, and the condition 
of reversibility is 

while for non-reversible changes 

Thus the minimum possible value of OS^> is 0, while for all 
actual changes it has a greater value, and it follows that every 
possible change in the system is attended by an increase in the 
entropy. Therefore in an isolated system, stable equilibrium is 
attained when the entropy is at a maximum, for no further 
spontaneous change can occur. 

In order to obtain a clearer idea about the nature of entropy, 
we may write the equation 

in the form 

d(f> _ 1 dH 


which shows that the change of entropy per degree is equal to 
the specific heat of the substance under the given conditions, 
divided by the absolute temperature. In considering finite 
changes, it is necessary to notice that we no more want to 
know the absolute value of the entropy of a body than the 
absolute value of the energy. In each case it is with the 
changes in the value that we are alone concerned. The 
equation can be integrated in certain cases where the relations 
between the properties of the substance are simple, as, for 
instance, in an ideal gas. Here the internal work is zero, and 
any heat applied is used in raising the temperature and in 
doing external work. Now the specific heat, C vt of a gas 


at constant volume is the heat required to raise unit mass one 
degree without doing external work, thus 

__ V 
b< P-"-0-' -J- 

d9 dv 

= a 0- +jR T 

so that, integrating, 

&-<, = C, log f 

"l l 

From this equation it is easy to calculate the change in 
entropy corresponding to any given alteration in the state 
of the gas. 

When the system is not isolated, further considerations are 
Thermodynamic involved. The first law gives as the change 
potential. j n energy in a system 

which, by the second law, gives for a reversible transformation 

Subtract from each side 


8 (e - 6$) = - <#>S<9 + 2 (Z8a?). 
Again, taking the equation 



8 (e - 0$ - 2 (Za?)} = - 080 - 2 (a?8Z). 

If we write >/r for (e 6<j>} and f for {e - ^0 2 (Za?)} the two 
equations become 

a^~08#+2(Z&) ................... (8), 

8f=-080-2(8Z) .................... (9), 


these expressions again characterizing reversible changes. For 
non-reversible transformations the relation yields 

Let us apply these results to two special cases : 

(1) When the temperature and the generalized external 
coordinates x 1} # 2 ... are constant, and consequently SB and x 
vanish and 


that is, Sty must be negative, and a transformation of the 
system is only possible if it decreases ty. 

(2) When the temperature and the generalized external 
forces X ly X z ... are constant, so that S6 and SX vanish, we have 

and a change is only possible if it decreases f 

Thus in the first case, when 6 and x are constant, equilibrium 
is only possible when ty is a minimum ; in the second case, 
when 6 and X are constant, when f is a minimum. 

Now in dynamics a mechanical system is in equilibrium 
when its mechanical potential is a minimum ; thus ty and 
are functions analogous to potential in dynamics, and are hence 
known as thermodyiiamic potentials. 

When the generalized co-ordinates x 1} # 2 ... are constant, no 
external work is done, and the changes which occur involve 
internal variables alone ; for this reason ty is sometimes known 
as the internal thermodynamic potential. Constancy of the 
external generalized forces, however, does not prevent external 
work, which is equal to lEJT&c; thus f has been called the total 
thermodynamic potential. 

In many of the systems studied in thermodynamics, it is 
possible to exclude electrical and other similar actions. The 
only external co-ordinate is then the volume, and the only 
external generalized force which the system exerts is a uniform 
normal pressure p. The equations then simplify to 

S e -0<>-v = - 


In isothermal systems, equilibrium is reached at constant 
volume when ^r is a minimum, and at constant pressure when 
f is a minimum. On this account ^r has sometimes been 
called the thermodynamic potential at constant volume and ? 
the thermodynamic potential at constant pressure. 

We have thus deduced conditions of equilibrium in the 
Conditions of three cases which are of practical importance : 

equilibrium. ^ j n an j so i ate( j system, the entropy 

must be a maximum. 

(2) In an isothermal system, when the external coordinates 
are constant, ty must be a minimum. 

(3) In an isothermal system, when the external generalized 
forces are constant, f must be a minimum. 

The numerical values of the thermodynamic potentials can 
only be calculated for certain cases, but the 

Application of 

thermodynamic mere existence of such functions, as determining 
the conditions of equilibrium of isothermal 
systems, enables many useful deductions to be made. 

Let us, for example, consider the conditions of equilibrium 
between a solid and a solution of it in some liquid, the whole 
system being maintained at constant temperature and constant 
pressure. Since the function must be a minimum, the 
dissolution or precipitation of a small quantity of solid will not 
change its value. Now if the solid phase increase by a small 
mass Bm, and the liquid phase decrease by an equal amount, 
the rates of increase and decrease of the f functions for the 
two phases will be given by the partial differential coefficients 
di/3ra and d 2 /dm and the condition of equilibrium is that 

9?' = 9k. 
dm dm ' 

The common values of these differential coefficients give the 
amount of work necessary to introduce unit mass of each 
substance into any of the phases or states under the condi- 
tions of the system, and are termed by Gibbs 1 "the chemical 

1 Trans. Conn. Acad. vol. in. 1877, translated Thermodynamische Studien, 
Leipzig, 1892, and Equilibre des Systemes Chimiques, Paris, 1899. 



[CH. I 

potentials " or " the potentials " of the substances in the given 
phases or states. 

A graphical method, due to van Rijn van Alkemade 1 
enables us to treat the subject in a simple manner. 

Let the abscissae (Fig. 7) denote the number of gram- 
molecules of solvent in which one gram-molecule of the solid 
is dissolved, and the ordinates be proportional to the value of 
the f function for unit mass of the phase considered. 

Fig. 7. 

First consider the curve for the liquid phase, which gives 
the value of f for unit mass of the phase as the amount of 
solvent containing one gram-molecule of solute changes from 
zero to infinity. When m vanishes, f nas the value corre- 
sponding to the solute in its liquid form : a fused salt, for 
example. The direction of the curve is determined by the 
gradient d^/dm. When m is small, and the concentration of 
the water in the fused salt is therefore small, the amount of 
work required to introduce a further small quantity of water 
will be given by an expression analogous to that which holds 
for a gas, which is essentially only a dilute system (see p. 4). 
The value of d^/dm will therefore be of the form a + log bm 
where a and b are independent of m. When m is zero this 

1 Zeits. physikal. Chem. n. 289, 1893. 


expression becomes oo , and therefore the curve must at first 
touch the axis of f. As ra increases, it must leave the axis of 
f, and, when m is infinite, f reaches its value for pure water. 
The value of d/dm is then the work done in adding a gram- 
molecule of water to an infinitely dilute solution, and denotes 
the potential of pure water. For large values of m, then, d/dm 
must approach a constant value, and finally the curve must 
become a straight line. 

Let a curve which satisfies these conditions be drawn, 
and taken to represent the changes in f as the composition of 
the liquid phase changes. Such a curve is a(3CK. 

The composition of the solid phase does not vary, and is 
fixed by the condition that m is zero. 

At temperatures below the melting point of the solid, the 
fused substance will pass spontaneously into the solid form, and 
therefore the value of f for unit mass of the substance must 
be less in the solid than in the liquid form. Thus the point 
representing the value of f for the solid solute must lie below 
the point a, which represents the value for the fused solute. 
Let the value of f for the solid be represented by A. 

Now let us trace what happens when a dilute solution is 
isotherm ally evaporated at constant pressure. The potential 
of the liquid phase is represented for each concentration by the 
tangent to the curve. As long as these tangents cut the f axis 
below the point A, as in the case of the line PEG, the potential 
of the system as liquid is less than that of the other possible 
arrangement consisting of a certain amount of saturated solution 
and a certain amount of the solid, and no precipitation occurs. 
But when the tangent reaches A, as does the line DC A which 
touches the curve at the point C, the value of d/dm is the 
same for the liquid phase as for the mixture of saturated solution 
and solid, and equilibrium between solution and solid is therefore 
possible. Beyond the point C, the tangents cut the axis above A, 
and the potential of the liquid is greater than that of the system 
containing the solid. The solution is unstable, and the curve 
Cfta. represents supersaturated solutions, ending in the under- 
cooled fused solute, the value of f for which is represented by a. 
The values of the f function for unit mass of the system made 


up of the various mixtures of the solid with its saturated solu- 
tion are represented by the points on the straight line CA. At 
A there is no solvent, and thus no solution ; at C the amount 
of solvent is just enough to dissolve all the solid. 

We shall find later that a consideration of the possible 
forms of these f curves, throws much light on the special 
phenomena of the equilibrium of different phases. 

We have proved on p. 23 the relations 

Free energy. 

and S>/r< 

for reversible and non-reversible changes respectively. When 
the conditions are isothermal, (f>$0 vanishes and 

B^^^(XBx) ........................ (10). 

But ^(XBoc) denotes the external work taken in by, and there- 
fore 2 (X Bos) the external work which can be obtained from 
the system during an infinitesimal, isothermal variation. If we 
denote by W the work which the system can give in passing 
isothermal ly from a state A to a state B, 

thus W < ^r A -^ B , 

or, the work obtainable from the system during a finite 
isothermal change of state is equal to the decrease in its 
internal therm odynamic potential for a reversible process, and is 
less than that decrease for a non-reversible process. The decrease 
in the function ty therefore denotes the maximum amount of 
available energy which can be extracted as mechanical work 
from a system during isothermal processes, and -v/r has on this 
account been called by Helmholtz the free energy of the system. 
From the equation 

we have, when x, ... is constant, 


but \|r was defined by the relation 


thus, the equation of free energy may be put in the form 

*-+'& ........................ <"> 

expressing the relation between the free energy, the total 
energy and the temperature. 

If we know the expression for the work which the system can 

do in any case, the value of ^ can be determined. 

the free energy As an example, let us deduce the well-known 

latent heat equation. In the case of a change 

of volume, the external energy factors are pressure and volume, 

and, when the pressure is constant as in the isothermal evapora- 

tion of a liquid or fusion of a solid, the work done in that 

isothermal operation while the volume increases from v l to v. 2 is 

p ( V 2 ~~ v i) 8O that 

d\lr dp , . 


But the principle of the conservation of energy shows that the 
increase of internal energy of a system is equal to the difference 
between the heat absorbed and the work done by it, or 

e = H-W. 

For a reversible change W = -^ and we have H ty e. 
Substituting in the equation of free energy 

we find H = -e. 


The heat absorbed, H, may here be written as X, the latent 
heat of fusion or evaporation respectively of one gram-molecule 
of the substance ; thus 

Special problems of thermodynamics can often be treated 
by a direct application of the expression for the efficiency of a 


reversible cycle, which states that the balance of useful work 
obtained from the system when the temperature range be- 
tween its terminals is BO, is given by 

Now if the working substance is a liquid in contact with its 
own saturated vapour, the pressure during an isothermal change 
is constant. If the difference between the temperatures is 
infinitesimal, the indicator diagram becomes a narrow hori- 
zontal strip, the area of which is independent of the nature of 
its ends. The rate of change of the saturation pressure with 
temperature being dp/dd, the breadth of the strip is (dp/d&) &6 
and its area, measuring the effective work during a complete 
cycle is {(dp/dB) $6} (v 2 vj. The efficiency equation then 

We are therefore again led to the latent heat equation 

The process of evaporation involves, in general, an increase 
in volume, so that v 2 v l is a positive quantity. The sign of 
dp/d6, therefore, must be the same as that of X, the latent heat. 
Thus if, as is usual, heat must be supplied to evaporate a liquid, 
the sign of dp/dO is positive, and the vapour pressure increases 
with rise of temperature. 

When a solid is fused, however, the volume change is 
sometimes negative, that is to say, there is a contraction when 
the solid becomes a liquid, as in the case of water. A, being 
still positive, this makes dpfdO negative, and the pressure falls 
as the temperature rises. This pressure is, of course, the 
pressure under which ice is in equilibrium with water, and the 
negative sign of its differential coefficient shows that the 
freezing point of water is lowered by pressure. 

It is easy to calculate the numerical value of this lowering 
for the additional pressure of one atmosphere, that is, when 
&p is 760 x 13'6 x 981 or T014 x 10 6 dynes per square centi- 


metre. The freezing point of water is 273 on the absolute 
scale, X, the latent heat of fusion of one gram-molecule of 
ice, is 18 x 79'4 thermal units or 1431 x 4'184 x 10 7 ergs, and 
Vz Vi is 0'0908 x 18 cubic centimetres. Thus 

When the solid is denser than the liquid, as it is in most 
substances other than water, dp/dd is positive, and the freezing 
point is raised by pressure. 

Note. In deducing the latent heat equation by means of a reversible cycle, 
it is necessary to suppose that the range of temperature is infinitesimally small, 
for, if it be finite, it will not, in general, be possible to carry the sytem from 
a state of saturation at a temperature to a state of saturation at 0-30 by 
a wholly adiabatic process of cooling. Moreover, the quantity of work done 
during this operation will depend on the amount of the liquid left, which has 
also to be cooled by the expansion of the vapour. Thus, the system must 

(1) expand isothermally at till all the liquid is evaporated ; 

(2) expand adiabatically in dust-free space, so that there is no condensation, 
till the temperature falls to 6 - 86 (the pressure not being determined) ; 

(3) pass in one direction or the other along the lower isothermal till 
the saturation pressure p - (dplde) 50 is reached, and then condense to liquid 
along the same isothermal; 

(4) be compressed adiabatically as liquid till the temperature rises to 6 ; 
and, finally, as part of (1) pass along the isothermal till the original con- 
dition is once more attained. 

It is now obvious that only when the temperature range is infinitesimal can 
the work done in (2) and (4) be neglected compared with that done in (1) and 
(3), and the area of the indicator diagram be taken as independent of the shape 
of its ends. 



Equilibrium. Equilibrium of phases. The phase rule. Non-variant 
systems. Monovariant systems. Divariant systems. Other systems. 
The principle of latent heat. Application of the phase rule. One 
component. Labile equilibrium. Allotropic solids. 

IN studying the qualitative conditions of equilibrium for 
substances in contact with their saturated solu- 

Equilibrium. . . 

tions, the detailed examination ot a system 
consisting of a solid in contact with its own liquid and vapour 
is in the first place expedient. The results can then be 
extended to the more complex conditions introduced by the 
presence of a second substance. Both are cases of true 

Let us consider the single substance water. It can exist in 
three phases, the solid, the liquid and the vapour. A system 
containing two or more of these phases in equilibrium is 
still said to consist of a single independent component, 
namely, water, because the total mass of water being con- 
stant, the relative amounts of the phases are dependent on 
each other. For if, by changing the external conditions of 
temperature or pressure we increase or diminish the amount of 
liquid, we shall necessarily diminish or increase simultaneously 
the amount of one at least of the other phases. 

In a mixture of salt and water, the amounts of the salt 
and of the water are not mutually dependent decreasing the 
quantity of salt does not correspondingly increase the quantity 


of water. They are independent variables of the system, and 
each of them is a distinct component. Here again, we can 
have only one gaseous phase, and only one liquid phase ; for, 
when equilibrium is reached, the compositions of both vapour 
and liquid are homogeneous throughout. But we can have ice 
and crystals of salt, the latter in some cases being present in 
both hydrated and anhydrous forms. The number of solid 
phases is therefore only limited by the nature of the salt. 

Another case arises in the reaction between lime and carbon 
dioxide to form calcium carbonate, which is represented by 
the reversible chemical equation 

CaO + C0 2 7~~ CaO . CO 2 . 

The total quantity of CaO both free and combined does not 
depend on the quantity of C0 2 present ; they are independent 
variables, and therefore components of the system. But the 
amount of CaCO 3 does depend on the amount of lime and 
carbon dioxide, for if we decompose CaCO 3 more of these sub- 
stances is found. Thus CaCO 3 is not a component, but merely 
a solid phase in which the two components happen to exist in 
a definite proportion. A component in this sense need not be a 
chemical compound, though it usually is so, and, on the other 
hand, each compound is not necessarily a component. Again, the 
number of components in the same material system may change 
with the physical conditions, as in the case of the system 

2H 2 + O 2 =2H 2 O, 

at high and at low temperatures. 
We are thus led to define : 

(i) A phase as a mass chemically and physically homo- 
geneous, or, as a mass of uniform concentration. 

(ii) The components of a phase or system, as the sub- 
stances contained in it which are of independently variable 

We saw in the introductory chapter that, to attain equi- 

Equiiibrium libriuiD in an isothermal system, it is necessary 

of phases. ^at e ^her the thermodynamic potential at 

constant volume ^ or the thermodynamic potential at constant 

pressure should be a minimum. These thermodynamic 

w. s. 3 


potentials will depend not only on 6 and p, or 6 and v respec- 
tively, but also on the composition of the phases. Let us 
imagine that we have a homogeneous phase containing n 
components, and that we increase one of these components 
by a small mass Sm. The consequent rate of change of the 
i|r function is given by the partial differential coefficient dty/dm } 
which measures the change in -^ per unit mass of the com- 
ponent added to the phase. This function, d-^r/dm or //,, is, as 
we have said, called by Gibbs the chemical potential or the 
potential of the component in the given phase. It is equal 
to the work required to increase by unity the amount of that 
component by means of a reversible process, under the con- 
ditions of constant temperature and volume. 

If we have several phases in contact with each other, any of 
the components may pass from one phase to the other. For 
instance, if solid CaO . C0 2 is in contact with gaseous CO 2 and 
solid CaO, some CO 2 may pass from the gaseous to the solid 
phase, and, at the same time, some CaO will pass from one solid 
phase to the other. 

In such a case the work done per unit change will be written 

8^ 2 8^ 

-^-^- or ^2-^. 
dm dm 

If this potential difference is negative, the component will 
pass from the phase 1 to the phase 2 ; and vice versa. If the 
potential difference is zero, there will be equilibrium. 

Thus the condition of general equilibrium is that the 
chemical potential of each component must have the same 
value in each phase in which the component is present. 

If the system is maintained at constant temperature and 
pressure, instead of at constant temperature and volume, the 
same reasoning can be applied to the f function, and a similar 
condition of equilibrium will hold with respect to the corre- 
sponding chemical potential. 

Let us now take the case of the equilibrium of r phases 
containing n components. In order to fix the 

The phase rule. 

composition of unit mass of each phase we must 
know the amounts of n 1 components which are contained 


sr v 


in it. The amount of the last component is then also known. 
In the compQsition of each phase there are n1 variables, 
and, since there are r phases, the number of variables, due to 
this cause, in the whole system is r (n 1). But, beside the 
compositions, the temperature and the pressure can also vary 
independently. Thus the total number of variables in the 
system is 

2 + r(w-l). 

In order to determine these variables, we must find 
equations between them. The potentials of the components 
are functions of the temperature, pressure and the composition 
of the phases, and will therefore give us the equations that we 
require. Now, as we have seen, the potential of each component 
in every phase must be equal and no other condition is necessary 
for equilibrium. Therefore, for each component, we shall have 
r 1 equations, or, for all the components, 

n (r 1) equations. 

The excess of the number of variables over the number of 
equations is thus 

2 + r (n - 1) - n (r - 1) = n + 2 - r, 

and this must therefore represent the number F of degrees 
of freedom of the system, or 


This equation is the expression of the Phase Rule, a genera- 
lization which we owe to Willard Gibbs 1 . 

If the number of phases in equilibrium with each other is 

Non variant ^ wo more ^ nan ^ ne number of components, or 

the number of degrees of freedom becomes zero, that is, the 
system is completely determined, and is therefore called a non- 
variant system. 

For example, if three phases of water can be obtained in 

1 Trans. Conn. Acad. vols. n. and in., 1875-8, translated Thermodynamische 
Studien, Leipzig, 1892, and de VEquilibre des Systemes Chimiques, Paris, 1899. 



equilibrium, the whole system is perfectly definite. Each 
phase must have a definite concentration, and the temperature 
and pressure each a definite value. That this is indeed the 
case, is a matter of experimental knowledge. Ice, water and 
steam can exist together in equilibrium at one temperature 
and pressure only. The usual freezing point of water is the 
temperature at which ice and water can permanently coexist 
under the normal atmospheric pressure. But when we have 
ice, water and steam isolated in a closed vessel, the pressure 
is that of the water vapour only, and in the neighbourhood of 
the freezing point this pressure is about 4*7 mm. of mercury. 
The freezing point of water is lowered by 0> 007 centigrade for 
each additional atmosphere of pressure, so that reducing the 
pressure from 760 mm. to 47 mm. will raise the freezing point 
by approximately that amount. We find that ice, water 
and steam can co-exist in permanent equilibrium only at a 
temperature of -f 0'007 and at a pressure of about 47 mm. of 
mercury. If either temperature or pressure be changed and 
kept at its new value, one or other of these three phases must 
eventually disappear. Thus, on raising the pressure, the vapour 
will all condense ; on lowering it, water and ice will evaporate 
and finally ice will be left in equilibrium with vapour at the 
slightly lower temperature corresponding to the new pressure. 

If the number of coexistent phases is n -f 1, i.e. one more 
Monovariant than the number of components, the system 

systems. ceases to be non-variant. To determine all 

the variables involved, it will be necessary to arbitrarily fix 
one of them. In our typical case of water, if we have only 
liquid and vapour in equilibrium with each other, an infinite 
series of temperatures and pressures are possible ; but, for 
each definite temperature, at one definite pressure, and at that 
pressure alone, is equilibrium attained. 

When the number of phases in equilibrium is n, so that it 

Divariant ^ equal to the number of components, we must 

systems. arbitrarily fix two variables before the system 

is determined. A divariant system with two degrees of freedom 


is then obtained. Our example in the case of water is now one 
phase alone, let us say the vapour. A gas or non-saturated 
vapour can possess any temperature at any pressure, and it is 
only when both these data are chosen that the other variable, 
the concentration, is determined. 

In the case of one component, there must at least be an 
equal number of phases, but with more than 

Other systems. . . 

one component it is possible to have cases in 
which a smaller number of phases, n 1, n 2, etc., is con- 
cerned. Such systems would be trivariant, tetravariant, etc., 
but they are too complicated to be of much present interest 
or importance. 

The foregoing paragraphs give a statement of Willard Gibbs' 
Phase Rule. It will be noticed that nothing is said about the 
possibility of the existence of non-variant systems with more 
than n 4- 2 phases in equilibrium. Since, however, the system 
must be definitely determined by n -f 2 phases, it is improbable 
that a greater number would coexist, and, as far as is known 
to the writer, none have been described, except those involving 
false equilibrium, maintained by passive resistance to change 
which is the analogue of frictional force. 

Again, the Phase Rule implies that the phases involved are 
homogeneous throughout. This excludes the consideration of 
disturbing factors such as gravity, which makes the lower layers 
of a gas or solution more concentrated than the upper ones, or 
surface tension, which differentiates the free surface layer from 
the bulk of a solid or liquid. Such effects are, however, usually 

Hitherto we have considered equilibrium only, and have not 

The latent taken into account the phenomena which accom- 

heat equation. p anv a cnan g e i n one o f the conditions. The 

relations governing such changes can be deduced from the 
latent heat equation 

dp X 

dO = eTv^v'Y 


obtained on p. 29, in which X and v refer to molecular quantities 
of the working substance. Let us take, as an example of its 
application, the equilibrium between a liquid and its vapour. 
If both X and v 2 v 1 are positive, a rise in the temperature 
will, as we have seen, cause a rise in the pressure also. This 
means that more liquid evaporates, and X being positive, this 
evaporation absorbs heat, and thus cools the system. 

On the other hand, a rise in the pressure produced by a 
compression of the vapour will obviously cause condensation ; 
X being positive, heat is evolved, and the temperature of the 
system rises till a new equilibrium is reached. 

Let us now take a case in which v 2 v l} and therefore dp/dO, 
is negative, for example : a mass of water and ice in equilibrium 
at the freezing point, under the pressure of the walls of an 
elastic vessel. The addition of heat causes ice to melt and the 
combined volume of the liquid and solid to contract. This 
causes the pressure to fall, arid since dp/dd is negative, 
the freezing point will rise and a new equilibrium be 

Thus in each case a change in the external conditions causes 
compensating changes within the system. 

This principle, originally due to James Thomson, was de- 
veloped in various directions about the year 1850 by William 
Thomson (Lord Kelvin), Rankine and Clausius. In recent 
times Le Chatelier has been prominent in illustrating its 
application to chemical systems, the reversible and therefore 
thermodynamic character of which has been clearly realized 
only since the work of Gibbs and Van't Hoff. ? ^ l 

These two principles, the Phase Rule and the theorem of 
latent heat, furnish a means of tracing all the qualitative 
phenomena of physical and chemical equilibrium. They involve 
no knowledge of the nature of matter or of the reactions which 
occur, and no distinction is drawn between physical and chemical 
changes. The Phase Rule is concerned merely with the relative 
number of components and coexistent phases, and the theorem 
of latent heat needs for its application only an experimental 
knowledge of changes of density and concentration, and the 
corresponding thermal phenomena. 


When systems of one component are studied in detail, it is 
convenient to represent the relations involved 
phase rule-one on a diagram. Thus Figs. 8 11 illustrate quali- 
tatively the phenomena which we have already 
briefly described as characteristic of water substance. The 
curves are diagrammatic, and are not drawn to scale. 



Fig. 8. 

The curve OA traces the connection between the temperature 
and the vapour pressure when a mass of water is in equilibrium 
with its own vapour. This curve divides the diagram into two 
areas. A mass of water substance the temperature and pres- 
sure of which are represented by any point in the space lying 
below OA must exist in the state of unsaturated vapour, and 
can none of it be liquid. Above OA the position of every 
point represents conditions of temperature and pressure which 
can exist only when all the substance is liquid. Thus the area 
below OA represents unsaturated vapour, and the area above 
OA, more or less compressed liquid, and along the line OA alone 
can there be liquid and vapour in equilibrium. 

If the temperature sinks below the freezing point and 
ice forms, the liquid will disappear and a curve OB, giving 
the vapour pressure of ice, can be traced a few degrees below 
the freezing point. This curve is not continuous with OA, for 
more heat is required to evaporate a gram of ice than a gram 



[CH. II 

of water, and our thermodynamic equation therefore shows 
that the rate of change of pressure with temperature, that is, 

Fig. 9. 

the slope of the curve, is greater for the solid than for the liquid. 
At the freezing point, however, the vapour pressures of solid 
and liquid have the same value, for otherwise transformation 
of one phase into the other would occur, and equilibrium be 

Like OA, the curve OB divides the diagram into areas 
representing two phases, in this case solid and vapour, which 
can be in equilibrium only along the dividing line. If the 

Solid si 


Fig. 10. 

system be compressed at the freezing point, the vapour dis- 
appears, and solid and liquid remain together in equilibrium. 


If the pressure be raised, as we have seen, the freezing point 
is lowered, and therefore the equilibrium curve is one which, 
springing from 0, runs upward with a slight inclination towards 
the axis of pressure. The latent heat equation shows that this 
is related to the fact that the volume becomes smaller on fusion. 
In substances, such as naphthalene, etc., where fusion causes 
expansion, the slope of the fusion curve is in the other direction. 

All these moriovariant curves start from the point 0. 
Their other limits have now to be considered. OA, the liquid- 
vapour curve, obviously ends at the critical point, where the 
distinction between liquid and vapour is lost. OB, the solid- 
vapour curve, has only been experimentally traced a few 
degrees below the freezing point of water, but there seems 
some ground for believing that the vapour pressure of the 
solid never quite vanishes, though it diminishes as the tem- 
perature sinks towards the absolute zero. The solid-liquid 
curve, OC, will be considered below ; the effect of high pressure 
is usually to make the properties of solids resemble those 
characteristic of liquids. This is indicated by experiments of 
Spring 1 , who found that metals subjected to a pressure of 
several thousand atmospheres assumed a structure similar to 
that possessed by them after fusion. 

The point 0, at which the three curves meet, represents 
a non-variant system. With only one component present it 
is always, as here, a triple point; but the number of curves 
which must meet in order to constitute a non- variant equili- 
brium will increase by one for each new component. Such 
points as are termed non-variant, transition or inversion 
points. The first name we have explained ; the second and 
third express the fact that it is at these points alone that 
unlimited conversion from one phase into another can occur 
without change of temperature. 

Each of the curves OA, OB, 00 represent equilibrium 
between two phases, i.e. for monovariant systems. Along them, 
if one variable, temperature or pressure, be chosen, the other 
is at once determined and can have only one value. They are 
the boundary curves for the areas AOO, COB and BOA. The 

1 Rapports Congres de Physique, Paris 1900, i. 402. 


curve OA is a liquid- vapour curve, OB is a solid- vapour curve, 
and OC a solid-liquid curve. Thus under the conditions of 
temperature and pressure represented by any point in the 
area AOC the substance will exist in the state of liquid, at 
points in COB in the state of solid, and in BOA in the state of 
vapour. In order to determine the position of a point in any 
one of these areas, two independent variables must be fixed ; 
the points in these areas represent divariant systems. 

Under normal conditions the three phases cannot exist 
Labile singly except under the temperatures and 

equilibrium. pressures represented by points inside their 

corresponding areas; but, as is well known, by preventing all 
disturbance a liquid can often, in the absence of its solid 
phase, be cooled considerably below its proper freezing point 
without solidification. Vapour, too, can, in the absence of 
liquid or dust nuclei, be cooled or compressed till it becomes 
supersaturated. Such cases are examples of substances in what 
is known as a labile or metastable state, and only occur in the 
absence of one of the phases whose equilibrium is represented 
by the boundary which has been passed. The curves of our 
diagram, for instance, illustrate the equilibrium between two 
phases. If one of these phases be not present, the condi- 
tions are entirely altered ; the initial formation of one of the 
phases is a different process, governed by different conditions, 
of which surface energy is one of the most important. By 
cooling water carefully, therefore, the vapour pressure curve can 
be traced below the freezing point, and experimentally proved 
to lie above the corresponding curve for ice. It is indicated by 
the dotted line OD in the diagram. 

If we begin with unsaturated vapour only, we have a 
divariant system indicated by some point on our diagram 
such as H. Cooling the vapour will cause a change of pressure 
depending on the nature of the substance. Let us take the 
system along some path represented by the arbitrary curve 
HKL. Eventually saturation is reached and the curve UK 
cuts the liquid-vapour curve at K. If no nuclei are present, 
however, cooling can take place without condensation and the 

CH. II] 



curve can be traced from K to L. The phenomena of super- 
saturation have been studied in the case of water vapour by 

Fig. 11. 

Aitken 1 and C. T. R. Wilson 2 . Aitken showed that air, care- 
fully freed from dust, by passage through cotton wool, could 
easily be cooled below its normal temperature of saturation 
without formation of moisture, and Wilson has found that the 
electric ions produced in air by the passage of Rontgen rays, 
by the incidence of ultra-violet light on a negatively electrified 
zinc plate, and by other methods, act as condensation nuclei 
like dust particles. The degree of supersaturation necessary 
for condensation is greater in the case of the ions than for 
dust particles, and greater for positive than for negative ions. 

These phenomena are closely connected with the surface 
tension between the liquid which would be formed and the 
substance surrounding it. The surface tension shows that a 
drop of liquid has, in addition to its mass energy, an amount 
of free superficial energy proportional to the area of contact 
between it and the air. Since the free energy tends to a 
minimum, a given volume of liquid will, other things being 
equal, assume the state which gives the minimum surface. The 
larger drops in a space saturated with water vapour will there- 
fore grow at the expense of the smaller ones, by means of 

1 Trans. E. S. Ed. xxx. 337, 1881 et seq. 

2 Phil. Trans. A. vol. cxcn. p. 403, 1899 and cxcm. p. 289, 1899. 


evaporation from the more convex surfaces where, as we shall 
see in a later chapter, the vapour pressure is higher, and con- 
densation on the less convex surfaces where it is lower. The 
precipitation of the excess of water from a supersaturated space, 
in the absence of dust or other nuclei, can only begin by the 
formation of very minute drops. The total area of these will 
be very large in proportion to their volume, and consequently 
the change might involve an increase in the total free energy 
of the system. When this is the case, spontaneous precipitation 
cannot occur. 

Similar cases of supersaturation are found in liquids, which 
can often be cooled far below the melting points of their solids, 
and still remain in the liquid state. If a crystal of the solid 
be then introduced, crystallization of the whole mass will 
usually at once occur. This subject has been extensively 
studied by Tammann 1 , who finds that the power of spon- 
taneous crystallization, as measured by the number of centres 
of crystallization started in unit time, increases to a maximum 
after the temperature sinks below the melting point, and then 
diminishes again rapidly as the surfused liquid is further cooled. 
The velocity with which the crystals grow when once in exis- 
tence has no relation to this power of starting the process, and 
is generally at a maximum at a very much higher temperature. 
When the surfused liquid is cooled much below the melting 
point, the power of spontaneous crystallization and the rate 
of growth both become so small that the liquid remains for an 
indefinite time in the surfused condition. Its viscosity also 
increases at an enormous rate, and finally the surfused system 
passes into the state in which it is called an amorphous solid. 
Thus, an amorphous solid, or a glass as it is often called, seems 
really to be an under-cooled liquid, and not a solid at all. 

Probably, in this case also, the surface tension between the 
crystals and the liquid surrounding them is one of the factors 
which cause supersaturation to occur. The case is not quite 
analogous to that considered above, in which drops of water 

1 Zeits. phys. Chem. xxi, 17, 1897. Wied. Ann. LXII, 28, 1897; LXVI, 473, 
1898 ; LXVIII, 553 and 629, 1899. Ann. der Physik, i, 275 ; n, 1, 1900. See 
also an abstract in the Rapports pres. au Congres de Phys., Paris, 1900, i. 449. 


are surrounded by their vapour, for the surfaces of crystals are 
plane and not curved, so that there is nothing of the nature of 
an increased pressure due to curvature inside them. Some 
suggestions on this subject have been made by Gibbs 1 , who 
points out that, on a dynamical view of the equilibrium between 
a crystal and the surrounding liquid, there need not necessarily 
be the same exact conditions of equilibrium between them as 
there are between a liquid and its saturated vapour. The 
equilibrium between a liquid and its vapour is explained as 
an equality between the number of particles evaporated and 
condensed in unit time, and, since the quantity of liquid can 
increase or diminish by infinitesimal amounts, the slightest 
change in the rates of evaporation or condensation, caused 
by the least variation in the temperature or pressure, is 
enough to alter the equilibrium. But in order that a crystal 
should increase in size, a whole new layer must be deposited on 
its faces, and therefore the thermodynamic potential of the 
liquid phase required for the growth of a crystal probably 
exceeds by a finite quantity that needed for its solution. At 
the edges of a crystal, however, the particles are probably less 
firmly held, and an exact balance of opposite processes may 
there occur. 

Phenomena analogous to those of the surfusion of pure 
liquids and of the growth of crystals in them will be found in 
solutions of one substance dissolved in another (two component 
systems), and a case similar to the condensation of vapour by 
gaseous ions will be found in the coagulation of colloidal 
solutions by the addition of electrolytes. 

When the solid can exist in more than one crystalline form 
we get more complicated phenomena, though 

Allotropic solids. & 

here also the non-variant systems are repre- 
sented by triple points. Thus Fig. 12 shows the equilibria in 
the case of sulphur, which, as is well known, is found both as 
monoclinic and rhombic crystals. The melting point of the 
monoclinic sulphur is 120 and its density 1*96, while the 
corresponding numbers for the rhombic variety are 114'5 and 
1 Trans. Connect. Acad. in, 494, 1874-1878. 



[CH. II 

2*05. Since the densities are different, it is easy to observe 
the temperature at which one modification passes into the 

Fig. 12. 

other by means of a dilatometer, and Reicher 1 has shown that 
the temperature of conversion is 95 0< 6. The change is a very 
slow one, hence the possibility of measuring the melting point 
of the rhombic crystals at 114'5, though they are of course in 
an unstable condition We thus have three points on our 
diagram; 1 (t=l2Q) at which liquid, vapour and monoclinic 
solid are in equilibrium, 2 (=114'5) the unstable meeting 
point of liquid, vapour and rhombic solid, and 3 ( = 95'4), 
where the two solids and the vapour coexist. Starting from 
O l is OT.A, the liquid- vapour curve, giving the relation between 
temperature and the vapour pressure of the liquid, 0-fl the 
curve showing the variation of the fusion point of monoclinic 
sulphur with the pressure, and Oi0 9 giving the equilibrium 
between monoclinic sulphur and its vapour. Below O s rhombic 
sulphur has the smaller vapour pressure, and is, therefore, the 
stable form. Thus starting from 3 besides 3 1 we have the 
stable curves S B } the vapour pressure curve of the rhombic 
solid, and S G showing the relation between the pressure and 
the transition point between the two forms of solid. The pitch 
of this curve has been determined by Reicher, and Roozeboom 

1 Van 't Hoff-Cohen, Studien zur Chemischen Dynamik, 1896, p. 185. 


has calculated the position of C, at which the two curves 3 C 
and 0-f! must meet if no other modification of sulphur exists, 
to be about 135 and 400 atmospheres. Here, the two solids 
would be in equilibrium with the liquid, and at higher pres- 
sures the less dense monocliriic sulphur would disappear and 
some curve such as CD would express the monovariant system 
rhombic solid and liquid. This curve would, if continued down- 
wards, be continuous with the unstable curve C0 2 . 

Thus neglecting labile equilibria, which are not completely 
described by the Phase Rule, we have three non-variant points 
Oi, 3 and C, at each of which three curves meet, each curve 
representing the series of states of a monovariant system. 
Moreover, as in the case of water, the areas of the diagram 
represent the states of divariant systems composed of one 
phase only. Thus the area below AO^B represents the 
vapour, the area to the right of AO^CD the liquid, the area 
to the left of B0 3 CD the rhombic solid, and the area enclosed 
by COfls the monoclinic solid. Outside its characteristic area 
each phase can exist only in a labile and unstable state. 

An interesting case of allotropy has recently been discovered 
by Tammann 1 , who finds that at low temperatures with very high 
pressures two new crystalline varieties of ice exist, both of 
which are denser than water. He has also traced parts of the 
curves giving the conditions of equilibrium between the three 
solid phases. 

Some solids are said to exist in allotropic forms one of 
which is amorphous. But, as pointed out by Lehmann in his 
Molekularphysik and confirmed by Tammann's work described 
above, amorphous bodies are probably surfused liquids. It 
seems likely (1) that a true solid is always crystalline, and 
(2) that the temperature and pressure necessary for its stability 
have definite limits on all sides. Thus the curve OC in Fig. 10, 
instead of ending at a critical point where solid and liquid 
become indistinguishable, bends to the left and eventually cuts 
the curve OB. It then incloses a definite area within which 
alone the conditions allow the permanent existence of the 
crystalline phase. 

1 Ann. der Phys. n. 1, 1900, also Paris Reports, 1900. 



Compounds, mixtures and solutions. Anhydrous solutes. Hydrated 
solids. Concentration curves. Two liquid components. Alloys. 
Solid solutions. Two volatile components. Three components. 
The problems of solution. 

A PHASE consisting of two or more components may be a 
mixture, a solution, or a compound. If it is 

Compounds, _ L 

mixtures and constituted according to the theorems of defin- 

solutions. 11-1 ... , 

ite and multiple proportions it is a compound ; 
if the relative quantities of the components can vary continu- 
ously between certain limits it is either a solution or a mixture. 
The distinction between the two latter is not sharp; though 
when the properties of the resultant are sensibly the sum of 
those of the components, as is nearly true for gases, it is usual 
to class it as a mixture. 

Solutions as thus defined need not necessarily be liquids. 
In so far as gases fail to conform to Dalton's law, mixtures of 
them must be treated as solutions, while instances of solid 
solutions are found in the alloys, the mixed alums, glasses, and 
perhaps in the substances formed when hydrogen and other 
gases are absorbed by metals. These cases will be considered 
more fully later. 

Two components which form a solution may be soluble in 
each other in all proportions, when they are called consolute, or 
only to a limited extent. In the latter case it is customary to 


distinguish between the medium or solvent, and the dissolved 
substance or solute. This phraseology, however, though con- 
venient, is of no import from the point of view of the 
Phase Rule, which draws no distinction in kind between the 
two components involved in an equilibrium. 

For the general case of the equilibrium of two components, 
when any disturbing effects of surface tension, gravity, electrifi- 
cation etc., can be neglected, the Phase Rule shows us that we 
must assemble four coexisting phases to get a non-variant, 
three for a mono variant and two for a divariant system. Thus 
in the case of sodium chloride and water, the system containing 
salt, ice, solution and vapour can exist at one temperature and 
pressure only at the freezing point ( 22 C.) of a saturated 
solution under the pressure of its own vapour. If heat be 
applied, the solids will eventually disappear, and conversely, if 
heat be taken away, the whole of the liquid will freeze; but 
the temperature, pressure and concentration will remain con- 
stant throughout. Such a constancy used to be considered as 
the characteristic of a pure chemical compound, and these 
mixtures of salt and ice with constant melting points were 
termed cryohydrates by Guthrie 1 . Considered in the light of 
the Phase Rule, however, it is clear from the constancy of the 
melting point that four phases must be present, i.e. that two 
different solids must separate, and therefore that the solid phase 
is not a chemical compound, but a conglomerate of two solids. 
It is obvious that the ^ryohydric point is the lowest tempera- 
ture which can be reached when the salt is used with ice as a 
freezing mixture, and thus, when a low temperature is required, 
a salt having a low cryohydric point must be used. Calcium 
chloride, for instance, gives a non- variant point at about 55, 
where the hydrate, ice, solution and vapour are in equilibrium. 

In passing from systems of one component to those of two, 
Graphic re- we introduce a new variable, namely, the com- 

EZS3S position of each phase. The obvious way to 
systems. represent this diagrammatically is to use a solid 

construction, the percentage composition of a variable phase 

1 Phil. Mag. (4) XLIX. (1875) ; (5) i. and n. (1876); vi. (1878). 
w. s. 4 


being measured along an axis at right angles to the plane of 
the paper, the pressure and temperature axes being used in 
that plane. The state of the system will now be represented 
by a point in space, fixed by three co-ordinates. The tri- 
variant systems are represented by volumes, the divariant by 
the surfaces separating those volumes, the monovariant by the 
lines in which the surfaces cut each other, and the non-variant 
by the points of intersection of the lines. 

For convenience, however, it is usual to represent these 
three dimensional diagrams by plane figures. In examining 
such figures it should always be borne in mind that they are 
only suggestions of solid diagrams, and that the lines seen in 
them do not in general really lie in one plane. 

The simplest cases of two component systems are furnished 
Anhydrous by solids which cannot crystallize in combina- 

soiutes. t | on vfifa the solvent, such as the anhydrous 

salts and water. Here we have a quadruple transition point, of 
definite pressure, temperature and concentration of solution, at 
which solid salt, ice, saturated solution and vapour are in 
equilibrium in any proportions. This point is shown by in 
Fig. 13 a diagrammatic sketch of the solid figure, in which 
the composition of the liquid phase is taken as the third 

Leaving the transition point by the addition of heat, we 
can, if salt be present in excess of ice, advance along the curve 
OA, which represents the monovariant system salt, solution 
and vapour. It resembles the corresponding curve of our 
former diagram, except that instead of showing the vapour 
pressure of a pure substance, it represents that of a solution, 
saturated with salt at each temperature. Since the presence of 
salt lowers the vapour pressure of a liquid, this curve lies below 
that for the pure solvent, unless the solute itself be volatile to 
an appreciable extent, a case which will be considered later. 
When the solubility increases with the temperature, the curve 
OA will also rise less steeply than that for the pure solvent, for 
the lowering of vapour pressure due to the solute will constantly 
increase. The curve will end at a critical point for the solution, 


at which the solid may melt and either give another quadruple 
point or mix perfectly with the solution. Another possibility 
is that by decreased solubility or increased vapour pressure the 
liquid and vapour may come to have the same composition. 

Salt Vapour 

Fig. 13. 

Again returning to the quadruple point on Fig. 13 let 
ice, instead of salt, be present in excess. The addition of heat 
causes ice to melt and thus solvent to be formed and salt dis- 
solved. This proceeds at a constant temperature till all the 
salt is dissolved. A further supply of heat melts more ice, and 
now the solvent formed from it dilutes the solution, which 
therefore ceases to be saturated. As long as ice remains, we 
have an equilibrium between it and an unsaturated solution, 
that is, we advance along a curve, giving the freezing points of 
solutions of constantly decreasing concentration, ending at B, 
the freezing point of the pure solvent, when the amount of ice 
which has been melted is enough to make the concentration of 




[CH. Ill 

the solution negligible. From the point B may be drawn the 
same liquid-vapour and liquid-solid curves which we drew for 
our one component system. 

The more important properties of the solubility curve OA, 
and the fusion curve OB, are those represented by their pro- 
jections on the temperature-concentration plane, the relations 
connecting pressure with temperature, best seen in our present 
diagrams, not being usually so prominent. 

Let us now again consider the non-variant system repre- 
sented by 0. Taking away heat, we shall freeze the whole of 
the liquid, and obtain the monovariant system salt, ice and 
vapour. For each temperature there is a definite vapour pres- 
sure, and we get a curve such as OC on the diagram. 

Once more starting from 0, we can by increase of pressure 
condense all the vapour and obtain the last possible mono- 
variant combination; that of salt, ice and saturated solution. 
The direction of this curve depends on the relative volumes 
of the salt and ice when solid and when forming a saturated 
solution, and on the signs of the heats of solution and fusion. 
Knowing these, the direction of the curve could be calculated 
from our equation ; but little has been done towards tracing it 


Fig. 14. 

As before, the areas of the figure represent divariant 
systems, although in this case each of these systems consists 
of two phases. A reference to Fig. 13 will show the pairs of 


phases which can exist in the different areas, and the meaning 
of the curves along which alone can three phases be in equi- 

If the solute is appreciably volatile as well as the solvent, 
we must add to our diagram CF, giving the vapour pressure of 
the pure solute, and CEB the vapour pressure of the pure ice, 
which will now not coincide with OC, the vapour pressure curve 
for the mixed solids. To complete the figure we may draw also 
EG the liquid-vapour curve, and EH. the solid-liquid curve for 
the pure solvent. If the solute were very volatile it would be 
possible for the curves OA and BG to cross each other, that is, 
for the vapour pressure of a saturated solution to be higher 
than that of the pure solvent. 

It is well known that many salts crystallize from solution in 
combination with one or more molecules of 

Hydrated solids. . 

water, forming solids which are called hydrates. 
It is possible also for solute and solvent to freeze out together 
in the form of a solid solution in which the proportions need 
not be related to those of the chemical equivalent weights. In 
either case our diagrams become more complicated, for each 
possible solid phase will have to be considered. 

Fig. 15 illustrates a case such as that of sodium sulphate, 
which usually crystallizes from water as Na 2 SO 4 . 10H 2 O. The 
crystals melt at 32'6 and pass into the anhydrous salt and 
water. Another hydrate, Na 2 SO 4 . 7 H 2 O can be obtained by 
adding alcohol to the solution in water, but it is unstable with 
respect to both the anhydrous salt and the other hydrate, and 
need not here be considered. 

The quadruple point represents a non-variant system in 
which ice, hydrate, saturated solution and vapour are in equi- 
librium. The monovariant curves springing from it are similar 
to those considered in the case of the anhydrous salt, and need 
not detain us. When, however, we heat the system composed 
of hydrate, solution and vapour, the anhydrous salt appears as 
a new solid at P, and another non-variant system is formed at 
a temperature of 32'6 and a pressure of 30'8 mm. of mercury 1 . 

1 Cohen, Zeits. f. physikal. Chemie, xiv. 90, 1894. 



[CH. Ill 

From this point P we can pass along PA, the monovariant 
curve representing salt, solution and vapour ; along PA repre- 
senting hydrate, salt and solution; along PO representing 



Fig. 15. 

hydrate, solution and vapour ; or along the new curve PC repre- 
senting the equilibrium between hydrate, salt and vapour. This 
latter can be realized experimentally by increasing the volume, or 
passing a current of air over the non-variant system, till all the 
solution has disappeared. The crystals of hydrate then " efflo- 
resce" as it is called, and give a definite vapour pressure for each 
temperature. In the case of sodium sulphate the vapour pres- 
sure of the anhydrous salt is inappreciable ; but, in the general 
case, the curve gives the sum of the vapour pressures of the 
constituents. The curve PC cannot lie above PO; for if the 
vapour pressure of the efflorescing hydrate were greater than 
that of the saturated solution at the same temperature, the 
hydrate would always be converted into solution, and this does 
not occur. For some systems the curves lie well apart, for others 
they seem nearly to coincide. Thus at 20 the vapour pressures 
of solution and hydrate are for calcium chloride 5'4 and 2'3, for 


sodium carbonate 16'0 and 101, for sodium sulphate 15'*7 and 
13'9, for cadmium bromide 10*0 and 9'0, and for barium iodide 
8-4 and 8'4. 

Hitherto, we have considered the qualitative phenomena 
concentration only, and no attempt has been made to draw 
the diagrams to scale. For the quantitative 
study of a monovariant system the general sketch of the solid 
model which has hitherto been used is conveniently replaced 
by a projection of the monovariant curve in question on one or 
other of the three planes, according to the pair of variables to 
be examined. Experiments on the relation between tempera- 
ture and solubility are illustrated by projecting the curve OA 
(Fig. 13) on the c-t plane. The pressure at each point should 
be that of the vapour, but, since the solubility of a solid does 
not change much with pressure, measurements under constant 
atmospheric pressure practically give the theoretical mono- 
variant curve. As before, the regions on each side of the 
curves represent the states of different systems; the curves 
themselves giving the conditions of equilibrium between them. 


100 99 98 97 96 95 94 H Z O 

Fig. 16. 

Thus Fig. 16 gives our quantitative knowledge of the 
equilibrium of sodium sulphate and water in this way, the 



[CH. Ill 

pressure throughout being taken as constant. The lettering is 
the same as that of the general diagram in Fig. 15. Thus 
is the cryohydric point, and OB the fusion curve, showing 
the gradual rise in the freezing point as the amount of salt 
diminishes. OP is the solubility curve of the hydrate 
NasSO 4 . 10 H 2 O. When the temperature reaches 32'6 and 
about 6 molecular per cents, of Na 2 SO 4 are dissolved in 
94 of water, the second non- variant point is reached, and 
the anhydrous salt appears. Beyond this temperature the 
measured solubility is that of the anhydrous salt, which, for 
some distance at all events, decreases with rising temperature, 
and has, therefore, a negative heat of precipitation. The 
solubility of the other hydrate Na 2 SO 4 . 7 H 2 is represented by 
the dotted curve RS ; solutions saturated with respect to this 

Fig. 17. 

hydrate are thus supersaturated with respect to the other, 
which will, therefore, crystallize out when a fragment of its 


solid is dropped in. Sodium sulphate has only one stable 
hydrate, but many salts can form several. Among these, 
calcium chloride and ferric chloride have been studied in detail 
by Roozeboom 1 . His results for ferric chloride are illustrated 
by Fig. 17. There are four hydrates, containing twelve, seven, 
five and four molecules of water, which with the anhydrous salt 
and ice make six possible solid phases. AB is the fusion curve 
giving the freezing points of solutions of concentrations varying 
from nothing at A to that of the cryohydrate of Fe 2 Cl 6 . 12 H 2 
at B. The cryohydric temperature is 55, and here we have 
a non- variant system, consisting of ice, the hydrate, saturated 
solution and vapour. From B onwards runs the solubility curve 
of this hydrate, and at C, where the curve shows a maximum, 
the solution has the same composition as the hydrate, which 
can, therefore, completely melt without change of temperature. 
Beyond C the liquid can take up more salt, and the solution 
contains more ferric chloride than the crystals. Thus C repre- 
sents the melting point of the hydrate, 37, the curve OB shows 
the lowering of melting point caused by the addition of water, 
and CD that due to the addition of ferric chloride. It is im- 
portant to notice this similar effect of changing the composition 
of a system in opposite directions on the two sides of a definite 
compound. At D, 27*4, a new hydrate, Fe 2 Cl 6 . 7 H 2 O, appears. 
This hydrate was actually discovered by Roozeboom from a 
study of the solubility curves. Its melting point, E, is 32*5. 

In a similar way the curve FGH, between 30 and 55, 
belongs to the hydrate Fe 2 Cl 6 . 5 H 2 O, and the curve HJK to 
the hydrate Fe 2 Cl 6 .4 H 2 O which melts at J, 73'5. At K, 66, 
begins the solubility curve of the anhydrous salt. 

In the light of this diagram let us trace the behaviour of 
a solution of ferric chloride which is evaporated to dryness 
at a constant temperature of 31. The phenomena will be 
represented by a horizontal straight line across the diagram. 
When the curve BC is reached, Fe 2 Cl 6 .12H 2 separates out, 
and the solution solidifies. Further removal of water will cause 
first liquefaction, and then re-solidification to Fe 2 Cl 6 . 7 H 2 when 
DE is cut. Again, the solid will liquefy, and again become solid 
1 Zeits. phys. Chem. iv. 31 (1889) ; x. 477 (1892). 



[CH. Ill 

as Fe 2 Cl 6 . 5 H 2 O. Further evaporation causes these crystals to 
effloresce and change to the anhydrous salt in the usual manner. 
As we have seen, the maxima of the various curve branches, 
at CEG and /, correspond to the melting points of the various 
hydrates at 37, 32*5, 56 and 73'5 respectively ; and at these 
points melting or solidification of the whole mass can occur at 
constant temperature. But we have before found this be- 
haviour to be characteristic of the transition points, which in 
this case are B, D, F, H y and K (- 55, 27'4, 30, 55 and 66). 
This clearly shows two ways in which a constant melting point 
can be accounted for. 

When each of the two components can exist as a liquid 
TWO liquid within the range considered, we shall have the 


phenomena on the left side of Fig. 16 repeated 
on the right, where the fusion curve of the second substance 
will appear. 

Thus in Fig. 18 the complete curve is given for mixtures of 
phenol and water. A is the melting point of ice, which is 




Fig. 18. 


gradually lowered to the cryohydric point, B, by the successive 
additions of phenol. BC is the solubility curve of solid phenol 


in water. At C a new liquid phase appears, consisting of a 
solution of water in liquid phenol, and at temperatures between 
C and D, we have two possible liquid phases : (1) phenol in 
water, the solubility of which is represented by CD, and 
(2) water in phenol, represented by DE. At D the composition 
of the two liquid phases becomes identical, and therefore, at 
temperatures above D, 68, the liquids are consolute, and only 
a single liquid phase can exist, G is the melting point of pure 
phenol, 40, and GE shows the lowering produced by the addition 
of water. At E we have a non-variant system, and here there 
is a constant melting point, phenol being the solvent. 

Thus phenol and water are two liquids which above a 
certain temperature are soluble in each other in all proportions, 
and below that temperature are not. The consolute tempe- 
rature varies greatly with different pairs of liquids ; with some 
it is too high to be experimentally reached, but since gases are 
always miscible with each other, liquids must all become con- 
solute at their critical points if not before. On the other hand 
the consolute point may be below the freezing point of any of 
the mixtures. We then have only one liquid phase, and the 
solubility curves on our diagram disappear, leaving two fusion 
curves only, which intersect each other at a point. This is 
illustrated by the case of phenanthrene and naphthalene, the 
fusion curves of which cut at 48, forming the non- variant 
system naphthalene, phenanthrene, solution and vapour. 

Precisely similar phenomena are shown by the mixtures of 
metals known as alloys, which have been ex- 
tensively studied by many observers including 
Guthrie, Le Chatelier, Roberts-Austen and Stansfield, Osmund, 
Stead, Charpy, Roozeboom, and by Heycock and Neville, who 
use the methods of platinum thermometry 1 . The simplest 
theoretical case of two consolute substances is realized by the 
behaviour of silver and copper, for which Heycock and Neville's 

1 Reports, with references, containing accounts of the work on this subject 
have been given by Roberts-Austen and Stansfield to the Congres International 
de Physique, Paris, 1900, i. p. 363, and by F. H. Neville to the British Asso- 
ciation, 1900, p. 131. 



[CH. Ill 

freezing point curves are shown by Fig. 19 1 . The melting 
point of silver is 960 and the addition of copper lowers it 

O 10 20 30 40 50 60 70 8O QO 100% 








Fig. 19. 

regularly and definitely in a monovariant manner till 40 atomic 
per cents, of copper are present. On the other hand pure 
copper melts at 1081*5 and the addition of silver lowers the 
freezing point till this curve cuts the other at a sharp angle 
at 777. Here therefore we have a non-variant system. The 
four phases which must there exist with a two component 
system are the two solid metals, the liquid consisting of the 
melted alloy, and the vapour of the mixed metals, the con- 
centration of the latter being very small. Thus the whole 
mass can fuse or solidify at 777 without variation of composi- 
tion and therefore without change of temperature when the 
atomic percentages are 40 copper and 60 silver. On account 
of its more uniform texture, as compared with that of other 
mixtures, this substance is called the Eutectic Alloy. 

When the composition of a mixture of two metals, A and B, 
is that of the eutectic, the two metals will crystallize simul- 
taneously but in separate crystals. Thus the solid eutectic 
alloy is a very minute conglomerate, while all other alloys 
contain large primary crystals of either A or B embedded in 
this conglomerate. This has been proved by the microscopic 

Phil. Trans. A, CLXXXIX. 25 (1897). 


work of Osmund 1 , Charpy 2 and Hey cock and Neville. A 
drawing illustrating it will be found in a future chapter 
(Fig. 43). 

A liquid whose composition is nearly that of the eutectic 
shows two changes in the rate of fall of temperature as it is 
allowed to cool. First a small quantity of one of the pure 
metals begins to crystallize out, and the rate of cooling is 
thereby diminished. This process continues till the composition 
of the liquid phase reaches that of the eutectic alloy, when the 
whole mass solidifies on further loss of heat without change of 
temperature, and gives a very definite freezing point. The 
process of cooling is thus represented by a path which runs 
vertically downwards till it cuts the freezing point curve, and 
then travels along it till the eutectic point is reached. In this 
way two temperatures are obtained, the higher giving a point 
on the equilibrium curve, the lower showing the eutectic. It 
will be noticed that this composition of the copper and silver 
eutectic corresponds to the formula Ag 3 Cu 2 . Nevertheless the 
nature of the process as shown by the curve proves nothing 
more than that a mixture of this composition is in chemical 
equilibrium with both pure metals at a certain tempera- 

The existence of definite compounds, however, is clearly 
brought out by the investigation of the fusion curve for gold- 
aluminium alloys, illustrated in Fig. 20 3 . It is quite analogous 
to the curve for ferric hydrate, and the maxima on the curve 
near D and at E and H, correspond to the definite compounds 
Au 5 Al 2 , Au 2 Al and AuAl 2 , the last named being the purple 
alloy discovered by Roberts-Austen. The breaks between the 
lines AB and BC at B, and between FG and GH at G, suggest 
two other possible compounds, perhaps Au 4 Al and AuAl. Thus 
the curve indicates that the following substances crystallize in 
succession from the melted alloy ; therefore, that it is these 
substances with which the liquid mixture is saturated in its 
successive states of equilibrium : 

1 Compt. rend, cxxiv. 1094 and 1234 (1897). 

2 Compt. rend, cxxiv. 957. 

* Phil. Trans. A, cxciv. 201 (1900). 



[CH. Ill 

along the branch AB gold pure at A, 

BG (?) Au 4 Al nearly pure at B, 

CD (?) Au 5 Al 2 or Au 8 Al 3 nearly pure at D, 

DEF Au 2 Al pure at E, 
FG (?) AuAl maximum not found, 

GHI AuAL 2 pure at H, 
IJ Aluminium pure at J. 

o 10 20 

40 5O 60 7O 80 OO IOO% 
- . - - 



7 oo l 

6oo c 




Fig. 20. 

Besides these pure compounds, we have systems of constant 
melting point at (7, F and /, which points correspond to eutectic 
alloys freezing at temperatures of 527, 569 and 647 respec- 

By the microscopic study of polished sections of these alloys, 
both by eye and by Rontgen ray photography, Heycock and 
Neville were able to identify the substances which crystallized 
out and thus confirm this explanation of the phenomena. 

In all cases hitherto considered, the addition of one com- 
ponent to a large excess of the other has 

Solid solutions. . n 

invariably, at first at any rate, produced a 
lowering of the melting point. This, as we shall see later, must 


always occur when it is the pure component that crystallizes 
out. When compounds are formed, the initial lowering will 
be followed by a change in the opposite direction if the 
compound which solidifies first is richer in the other compo- 
nent than is the liquid. The initial lowering may be confined 
to a very short length of curve, as in the gold-aluminium 
diagram in the neighbourhood of pure aluminium (Fig. 20); 
but when a definite compound is the cause of the main 
change, the initial lowering is always present. If, however, a 
solid solution of the two components is possible, that is, if the 
composition of the solid phase can vary continuously, the addi- 
tion of one component to the other may at once change the 
solid which first crystallizes from the solution, and may there- 
fore at once raise the melting point. This phenomenon has 
been observed in alloys for many pairs of metals. If the two 
components can mix with each other to form solid solutions of 
any composition, i.e. are consolute in the solid form as well as 
in the liquid, no non-variant system is possible, for we cannot 
assemble the necessary four phases. We shall expect to find 
for such cases all the types of curves that we shall consider 
later in detail as representing the boiling points of pairs of 
consolute liquids. If the composition of the solid solution which 
crystallizes out is the same as that of the liquid, the whole mass 
will fuse or solidify without change of temperature, in this respect 
simulating the behaviour of a definite compound, a cryohydrate, 
or a eutectic mixture. When the composition of the solid 
solution which freezes out is not the same as that of the liquid, 
more complicated cases arise. A general theory of solid solu- 
tions has, however, been recently developed by Roozeboom 1 
which seems likely to describe all such systems. 

In order to explain this theory, we must revert to the 
consideration of the thermodynamic potential at constant pres- 
sure by the graphic method of van Rijn van Alkemade, which 
we have described in the first chapter. The solid phase can 
now vary in composition as well as the liquid phase, and there 
will thus be two continuous curves running across the diagram. 
The abscissae in the figure represent the composition, the 
1 Zeit. phys. Chem. xxx. 385 (1899). 



[CH. Ill 

a b 


left-hand vertical corresponding to 100 per cent, of the com- 
ponent A and none of the component B, while the right-hand 
vertical gives 100 per cent, of B and 
none of A. The ordinates of the first 
four divisions of Fig. 21 represent the 
value of the f function for unit mass of 
the solid and liquid phases considered. 
In the first cases to be examined, the 
variation of f with the concentration is 
represented for each phase by a simple 
curve, like that of Fig. 7 on p. 26, with 
no changes in the direction of curvature. 

The first division of the figure repre- 
sents the two phases for a temperature 
above the melting points of both com- 
ponents. The curve for the liquid, which 
is then the stable phase, must lie below 
that for the solid. Each end of the curves 
must, like the left end of the curve in 
Fig. 7, touch the corresponding vertical 
axis. At the melting point of either 
component its solid and liquid will be in 
equilibrium, and the corresponding ends 
of the curves will coincide. At some 
temperature below the melting point of 
B, for certain compositions, the solid is 
the stable phase, and the curves will cut 
each other in the manner shown in the 
second division of the figure. The third 
division represents the state of affairs at 
a still lower temperature, while the fourth 
division gives the isothermals for a tem- 
perature below the melting point of A, 
when, for the kind of curves illustrated, 
the liquid phase is never stable, and therefore has everywhere 
a higher thermodynamic potential than the solid phase. Thus 
these four diagrams may be taken to be the successive sections 
of two solid figures, constructed after the fashion of Gibbs' 


Conc. n 
Fig. 21. 


thermodynamic surfaces, the three axes of which represent 
concentration, thermodynamic potential and temperature. Over 
each of these sections the temperature is uniform. 

It has already been pointed out (p. 25) that two phases, 
from one of which a given component can pass into the other, 
are in equilibrium when the potential of that component in 
each of them is the same, the potential being defined as the 
rate of change of the thermodynamic potential of the phase as 
the component enters it, that is as d%/dm, m being the per- 
centage amount of the component B. Now the value of this 
differential coefficient is given by the tangent to the curve at 
any point ; thus, the two phases will be in equilibrium when 
their compositions are such that their f-ra curves have a common 
tangent. In Division II. then, the phases will be in equilibrium 
when the liquid has the composition represented by a, and the 
solid the composition represented by b. At the lower tempera- 
ture of Division III. similar relations hold. 

From these curves the freezing point diagram can now be 
constructed by imagining a section cut at right angles to the 
others in such a plane that over it f is everywhere uniform. 
In Division V. of the figure, then, the abscissae, as before, 
denote the percentage composition of the phases, but the 
ordinates are now proportional to the temperatures. At the 
melting point of the pure component A, its solid and liquid are 
in equilibrium with each other ; thus at C, the ordinate on the 
diagram which corresponds to this temperature, the two phases 
are represented by a single point. The same thing holds good 
at D, the melting point of the pure component B. At the 
temperatures corresponding to Divisions II. and III. of the 
figure, however, the liquid of composition a is in equilibrium 
with the solid of composition 6, and we thus get two points on 
our new diagram in the same horizontal temperature line giving 
the compositions of the liquid and solid which are in equilibrium 
with each other. Putting in the corresponding points for 
intermediate temperatures, we finally obtain Division V., in 
which the upper curve refers to the liquid and the lower curve 
to the solid phases in equilibrium with each other at the 
temperatures denoted by the various horizontal straight lines, 
w. s. 5 


These curves are called by Roozeboom the liquidus and solidus, 
and by Neville the freezing point and melting point curves 

By the help of this freezing point diagram we can trace the 
changes which occur as a liquid mixture is cooled and solidified 
into either a mass of mixed crystals or a non-crystalline solid 
solution. As the temperature falls we pass along a vertical line 
in the diagram (say mnqz) corresponding to the composition of 
the given mixture. All points above n correspond to uniform 
liquid, and all below q to a uniform mass of mixed crystals. 
While solidification is in progress, the temperature falls from 
n to q, the solid forming alters in composition from o, in 
equilibrium with the liquid n, to q } and the liquid remaining 
at any instant unfrozen varies from the composition n t to the 
composition p which is in equilibrium with the solid q. Thus 
during solidification, the temperature of the whole and the 
composition of each phase vary continuously. 

Other possible forms of the f curves are shown in Figs. 22 
and 23. In Division I. of Fig. 22, the f curves for the solid 
and liquid are seen first coming into contact, and, in this case, 
touching each other at a point. The manner in which the 
curves intersect each other at different temperatures, and the 
method of deducing the freezing point diagram from them will 
be obvious from the figure. 

In Fig. 22, where the solid curve joins the liquid curve by 
touching it at a point, the composition of the two phases in 
equilibrium with each other is identical. This occurs at the 
highest temperature at which any equilibrium is possible, and 
the freezing point curve therefore reaches a maximum at this 
composition. In Fig. 23, where the curves touch each other 
at a point on separating, similar reasoning shows that the 
freezing point curve has a minimum at which the composition 
of the two phases is again the same. 

Now when the composition of the solid phase is the same 
as that of the liquid phase from which it separates and with 
which it is in equilibrium, the diagram also shows that there is 
no fall of temperature while solidification is going on nor any 
change in the composition of the two phases. The whole mass 


solidifies at a constant temperature, thus simulating the behaviour 
of either a pure substance, a compound, or a eutectic mixture. 




a' b b a 

a' b' 



Conc. n 

B A 

b' a' a b 


Conc. n 


a' I 1 

Fig. 22. 

Fig. 23. 


There is much danger of confusing these classes of bodies in 
examining the freezing point curves. A study of the composition 
of the solid and liquid phases in equilibrium with each other 
at different temperatures would serve to distinguish between 
them, and this method has been adopted by van Eyk 1 , Reinders 2 , 
and Hissink 3 , who have verified Roozeboom's theory for certain 
mixtures of two salts. In other cases, however, such as those 
in which the components are metals, there is great difficulty in 
separating the crystals from the liquid in which they form. 
The problem must then be attacked in other ways. 

If the rate of cooling of the liquid system be observed, the 
temperature range over which changes from liquid to solid 
occur will be evident by a decrease in that rate due to the 
latent heat evolved. Thus the temperatures at which solid 
begins to form and at which the mass completely solidifies can 
be determined for different compositions. Both curves of Rooze- 
boom's diagrams can thus be plotted, and the phenomena inter- 
preted. This method has been used by Roberts- Austen and 
Stansfield 4 and by Heycock and Neville 5 . If mixed crystals are 
not formed, a second halt in the rate of cooling will, if it exists, 
be found to coincide with the temperature of the eutectic point, 
and to appear at the same temperature as the composition is 
varied. This case has already been described on p. 61. 

Another mode of investigation depends on the microscopic 
study of polished surfaces of the solidified alloys on each side of 
the summit of the curve. Work on these lines has been done 
by Charpy 6 , Stead 7 and Heycock and Neville. 

Another interesting possibility in the form of the con- 
centration-f curves is illustrated by Fig. 24. Here the curve 
for the solid has changes of curvature. Between the points a 
and b the thermodynamic potential has higher values than it 
has beyond those points, and thus the compositions between a 

1 Zeits.phys. Chem. xxx. 430 (1899). 

2 Zeits. phys. Chem. xxxn. 494 (1900). 

3 Zeits. phys. Chem. xxxn. 537 (1900). 

4 Report to the Cong-res International de Physique, Paris, 1900. 

5 Neville, B. A. Reports, Bradford, 1900, and Glasgow, 1901. 

6 Comptes Rendus, cxxiv. p. 957 ; Soc. d 1 encouragement..., p. 384, 1897. 

7 Metallographist, n. p. 314. 


and b are unstable, the crystals passing spontaneously into 
mixtures of varying proportions of the solid solutions a and b. 
The freezing point curve which results will be apparent from 
the diagrams. At the temperature E, the liquid whose com- 
position is E is in equilibrium with both the solids F and G, 
and at this temperature a transition from one solid to the other 

Other examples of possible f curves, with their correspond- 
ing freezing point diagrams, will be found in Roozeboom's 
paper 1 . 

The importance of a knowledge of the properties of solid 
solutions in interpreting a complicated freezing point curve, 
such as that of the gold-aluminium alloys shown in Fig. 20, will 
now be apparent. Not only can mixed crystals of pure com- 
ponents exist, but, if compounds are formed, these compounds 
can form mixed crystals with each other, and the phenomena 
of solid solutions will also appear between the points on the 
freezing point curve corresponding to the compounds. 

As probable examples of alloys which form mixed crystals, 
Neville gives the following : copper-tin alloys ; alloys of lead- 
thallium, of bismuth-antimony and of gold-silver; alloys con- 
taining zinc or cadmium with either silver, copper or gold. 

The theoretical considerations are also well illustrated by 
some experiments on mixtures of mercuric iodide and silver 
iodide described by Roozeboom 2 . The freezing point diagram 
is represented in Fig. 25. Mercuric iodide melts at 257 and 
silver iodide at 526. The liquidus curve is the highest in the 
diagram, and consists of two branches meeting at 242 in a 
eutectic point. The solidus curves lie below it. While the 
composition of the system varies from pure HgI 2 to a mixture 
containing four molecular per cents, of Agl, a solid solution a, 
of varying composition, is formed, in which HgI 2 may be 
regarded as solvent. From 18 to 100 per cents, another series 
of solutions $, in which Agl is solvent, exist. Solid solutions 
of composition between 4 and 18 per cents, of Agl cannot be 
obtained, and in this part of the field we have a varying 

1 Zeits. phys. Chem. xxx. 385 (1899). 

2 Kon. Akad. Wetensch. Amsterdam, June 30, 1900. 



[CH. Ill 

conglomerate of the limiting examples of a. and fi. The upper 
boundary of these areas gives the solidus curve, which realizes 
the theoretical solidus of Fig. 24. 

7+D' D' + S 

do s'o 40 50 G'O 70 80 do Agl 

Fig. 25. 

If mixed crystals with 4 per cent, of Agl are cooled, they 
undergo a change near 127, owing to the transition of the 
HgI 2 from the rhombic to the tetragonal form. If, on the 
other hand, a mixture of the composition HgI 2 2AgI is cooled, 
at 157 the pink mixed crystals suddenly change into a red 
chemical compound having the same composition, represented 
by D on the diagram. This is exactly comparable with the 
solidification of a compound from a liquid solution, and, as in the 
compounds of ferric chloride and water of Fig. 17, or those of 
gold and aluminium of Fig. 20, the curve sinks on each side of 
this point, and forms eutectic mixtures of HgI 2 2AgI with HgI 2 
at 118 and with Agl at 135. Below these temperatures then 
all solid mixtures are transformed into conglomerates of double 
salt either with HgI 2 or with Agl. When these conglomerates 


are cooled to 45 the compound changes from red to yellow 
whether it is pure or mixed with one of the components. 

Similar phenomena have been traced in the case of alloys 
of copper and tin by Heycock and Neville 1 . Here also definite 
changes in crystalline structure take place on cooling at definite 
temperatures, long after solidification has occurred. The experi- 
ments consisted in measuring the freezing points of varying 
mixtures, and suddenly fixing the structure of the solid alloy 
at any temperature by plunging it into cold water. The surface 
was then polished, etched with acid, and examined under a 
microscope. The different crystalline species could thus be 
recognized by their characteristic forms. 

Since leaving the original qualitative diagrams we have 
TWO volatile studied in detail only the relations between 
components. concentration and melting point. Similar 
phenomena are found when boiling points are examined. 
Roozeboom's theory of solid solutions may be applied to con- 
solute liquids, the possible variations of f with concentration 
being plotted for the liquid and vapour phases. Boiling point 
curves will then be obtained similar to the freezing point curves 
of Figs. 21 to 24. Experimental results usually trace only the 
liquidus curve, analogous to the solidus curves given above ; few 
investigations of the composition of phases containing mixed 
vapours have been made. 

Another useful modification of method consists in measuring 
the vapour pressures of two components at varying concentra- 
tions. If the number of phases present is that giving a 
mono variant system, the changes might be represented by a 
projection of the mono variant p-t-c curve on the p-c plane. 
If the system is divariant, it is represented by a surface in the 
solid model, and we can draw a section of this surface in that 
same plane corresponding to any given constant temperature. 
While the projection curves serve to illustrate the varying 
vapour pressures of saturated solutions of solids at different 
temperatures, and therefore concentrations, the vapour pressures 
of mixtures of volatile liquids are often measured in divariant 
1 Proc. E. S. LXIX. 320 ; 1902. 



[CH. Ill 

systems, and are thus better set forth by section diagrams in 
the concentration-pressure plane. 

Pairs of liquids, as we have seen, must be divided into three 
classes (i) those which will not mix at all, (ii) those partially 
soluble in each other, (iii) those soluble in each other in all 
proportions. The laws of vapour pressure are different for 
each case. 

(i) With immiscible liquids the vapour pressure is equal 
to the sum of those of the constituents. This can be proved 
by passing the vapour of one boiling liquid into the other 
and examining the vapour which comes through ; in it the 
two substances will be present in the ratio of their own vapour 
pressures. The sum of the two pressures will, at the boiling 
point of the mixture, be equal to the atmospheric pressure, 
so the boiling point must be lower than that of either con- 
stituent. But this is usually masked ; for if one liquid forms 
a layer over the other, the mixture bumps violently if the 
more volatile liquid be below, while, if the positions are 
reversed, it is only the upper liquid which evaporates. 

(ii) The behaviour of partially miscible liquids has been 
studied by Konowaloff 1 , who found by experiment that the 
solution of a liquid A saturated with a liquid B exerts at a 


60 C 




Fig. 26. Percentage of alcohol. 

certain temperature the same vapour pressure as that which 
a solution of B saturated with A exerts at the same ternpera- 

1 Wied. Ann. xiv. 219 (1881) ; account with figures in Ostwald's Lehrbuch. 


ture. Konowaloff measured the vapour pressures of mixtures 
of two liquids of varying composition and at different tem- 
peratures. One result of his observations is shown by the 
form of the curve in Fig. 26, which gives the relation between 
percentage composition and vapour pressure of a mixture of 
water and isobutyl alcohol at 89 and at 60. While the 
percentage of alcohol is less than that required to saturate the 
water, the system is divariant and the vapour pressure in- 
creases with the percentage of alcohol. When the solution is 
saturated, the system becomes monovariant and the vapour 
pressure is independent of the excess of alcohol present. 
Such a mixture has then a constant boiling point, and the 
composition of the vapour is always the same. This constant 
vapour pressure is found to be smaller than the sum of 
those of the two constituents. When the percentage of 
alcohol is so large that all the water present can dissolve 
in it, the vapour pressure again alters with the composition 
of the solution, and finally sinks to its value for the pure 
alcohol. If a mixture represented by any point on either of 
the inclined portions of the curve be distilled, the composition 
of the vapour and the boiling point will gradually alter till the 
liquid present in large excess is finally left nearly pure. The 
curve of Fig. 26 is not directly derived from a complete p-t-c 
model, for it represents two variable liquid phases : the abscissae 
give the composition of the whole system, not of one phase only, 
(iii) Mixtures of liquids which are soluble in each other 
in all proportions give a single liquid phase. The vapour 
pressure curves can therefore be derived from the model 
by cutting sections of the corresponding divariant surfaces 
in the p-c plane. The following curves give KonowalofFs 
results. They at once show how the mixtures will behave on 
distillation. The tendency is (since there is no constancy in 
the composition of the vapour) for that particular mixture 
which has the greatest vapour pressure, and therefore the 
lowest boiling point, to come off first in greatest quantity, 
and in this way by repeatedly redistilling we at last get a 
distillate which has the composition corresponding to this 
lowest boiling point. Thus with water and propyl alcohol, 



[CH. Ill 

which mixture has a maximum vapour pressure when the 
percentage of alcohol is about 75, the final distillate obtained 
will have that composition. 



Fig. 27. Water and propyl alcohol. 

The curves for water with ethyl alcohol and with methyl 
alcohol show that in these cases no maxima are reached, so 
that by repeated distillation we get a nearly pure alcohol in 
the receiver, and pure water is left in the retort after the first 
boiliog. It is much easier to get water free from alcohol than 
alcohol free from water, because the influence of a little alcohol 

Fig. 28. Water and ethyl alcohol. 

Fig. 29. Water and methyl alcohol. 

on the boiling point of water is so much greater than that 
of a little water on the boiling point of alcohol. This case 
is of great importance in practice, for by such means mixed 
liquids of different boiling points are separated in the chemical 


laboratory by the process of " fractionation." We now see 
that this can only give perfect separation when the type of the 
vapour pressure curve is that shown in Figs. 28 and 29. 

A mixture of water and formic acid shows greater influence 
of the constituents on each other. The vapour pressure of 
the mixture is lower than that of either constituent, and 
reaches a minimum at a percentage of acid of about 73. 
All other proportions will therefore tend to distil over sooner 
than this, and finally we shall get a residue left in the retort 
containing 73 per cent, of acid. This will then distil over 

The last case really includes such liquids as aqueous solu- 
tions of nitric or hydrochloric acid, which were once thought 


Fig. 30. Water and formic acid. 

to show definite chemical combination in the proportions of 
the mixture which finally distilled over unchanged. Roscoe 1 
however proved that the composition of this distillate varied 
with change of pressure, and the facts are fully explained 
by the vapour pressure curves given above. 

Since the boiling point is higher as the vapour pressure is 
lower and vice versa, Figs. 26 to 30, if inverted, will quali- 
tatively represent the experimental relations between concen- 
tration and boiling point. Their connexion with the theoretical 
solidus curves of Figs. 21 to 24, as well as the analogy between 
the boiling points of the partially consolute liquids of Fig. 26 

1 Quart. Journ. Chem. XH. p. 128 (1859), or Treatise on Chemistry, vol. i. 
p. 138. 


and the melting points of the partially consolute solids of 
Fig. 25, will then be apparent. 

When three components are present the equilibria become 
Three com- much more complicated. The Phase Rule 

ponents. shows that for three components it is neces- 

sary to assemble five phases for a non- variant system, 
four for a monovariant and three for a di variant. Plane 
pressure-temperature diagrams can be constructed, the con- 
centration being assumed constant, but some modification is 
necessary when we require to plot concentration and tempe- 
rature or concentration and pressure. We can take a solid 
figure so as to get three axes, the temperature in one case 
or the pressure in the other being measured vertically, and 
then project this figure on to its base. The base may con- 
veniently be in the form of a triangle, the total mass of the 
components being kept constant. The corners of the triangle 
are taken to represent the three pure components and the com- 
position of any mixture in atomic percentages is represented 
by its position in the triangle, the position being the centre of 
mass of the three components placed at the corners. Thus 
Guthrie 1 has investigated mixtures of potassium, sodium and 
lead nitrates, and his results are shown in Fig. 31. At A we 

Fig. 31. 

have pure potassium nitrate melting at .340. Lead nitrate 

would exist pure at B, but since it decomposes before fusing, 

1 Phil. Mag. (5), xvn. 472 (1884). 


the temperature is unknown. At the other corner C we have 
sodium nitrate which melts at 305. The point D represents 
the composition of the eutectic mixture of the potassium and 
sodium salts ; its fusion point is 215. In like manner E (207) 
and F (268) correspond to the eutectics of the other pairs of 
substances. At (186) is found the non-variant system con- 
sisting of the three solid salts, the mixed liquid and the vapour. 
A fuller consideration of three component systems, together 
with many more examples of the phenomena of systems of one 
and two components will be found in Bancroft's book on " The 
Phase Rule 1 ." 

In this chapter we have studied the general phenomena 
The problems of f solutions, considered as mixtures of two 
solution. components, of which the equilibria are com- 

pletely determined qualitatively by thermodynamic principles 
and the phase rule. This treatment has already enabled us 
to survey our subject, and to examine its main outlines and 
its bearing on the general problem of the equilibrium of 
phases. The diffusion of dissolved substances and the elec- 
trical properties of solutions, since we are then studying 
phenomena which are not those of equilibrium, will involve 
other methods, but until we reach this part of our subject, we 
shall be chiefly concerned with the more thorough investigation 
of the relations we have already traced by aid of the phase 
rule. The detailed study of solubility will be but the quan- 
titative examination for certain cases of the solubility curves 
shown in the diagrams of this chapter ; while the experiments 
which have been made on the freezing points of solutions, 
dilute or concentrated, and on the vapour pressures, are merely 
more accurate delineations of their corresponding curves. 
Thus the phase rule is of importance, not only in placing 
the theory of equilibrium of phases on a sound basis, but also 
as furnishing a means of classification for the phenomena of 
solution, which enables us to arrange our material in logical 
and scientific sequence, and gives a standpoint whence we can 
trace the relations of the whole subject. 

1 Ithaca, New York, 1897. 



General problem of solubility. Supersaturation. Solubility of gases in 
solids. Solubility of gases in liquids. Henry's law. Solubility of 
gases in salt solutions. Solubility of liquids in liquids. Solubility of 
solids in liquids. Influence of pressure on the solubility of solids. 
Solubility of mixtures. Solubility in mixed liquids. Tables of 

THE solubility of a substance, which can be denned as the 
General problem quantity of solute required to saturate a given 
of solubility. mass of solvent, depends, as we have seen, on the 

change produced by the act of solution in the therm odynamic 
potential of the system, equilibrium being always reached 
when that potential is a minimum. Very little is yet known 
about the physical and chemical conditions which determine 
the solubility of a substance in a given solvent, in fact the 
essential nature of the process of solution must be regarded as 
at present uncertain. It has been noticed that solution is more 
likely to occur if the solute and solvent are somewhat alike 
chemically, but, even to this extent, no general rule can be 
framed. Still it is found that mineral salts and acids are in 
general readily dissolved by water, while benzene, for example, 
is a more likely solvent for organic substances. 

The phase rule and the principles of thermodynamics have 
enabled us to trace the course of the qualitative phenomena of 
the equilibrium of the phases of two components, including 
the problem of solubility as a special case. We have now to 
examine in greater detail the quantitative relations, and to 

CH. IV] 



give some further account of the experimental determinations 
which have been made on the subject. 

In a saturated solution there is equilibrium between the 
solid and the solution, and any structural change in either of the 
two will produce a change in the equilibrium. Thus, as we ex- 
plained on p. 53, the change in the nature of the solid crystals 
of hydrated sodium sulphate, N^SCX . 10 H 2 0, which are trans- 
formed into the anhydrous salt Na 2 S0 4 at a temperature of 32'6, 
causes a sudden change of direction in the solubility curve 
(Fig. 32). The transition point is the temperature at which 
the non- variant system salt-hydrate-solution-vapour can exist in 
equilibrium ; below this temperature the solubility of sodium 
sulphate increases with rising temperature, above it the solu- 
bility diminishes. This change was formerly explained by the 
supposition that below 32'6 hydrated salt is present in solution, 







Fig. 32. 

and above that temperature the liquid contains the anhydrous 
substance. Our present knowledge of the general problem of 
equilibrium shows at once that such a supposition is un- 
necessary ; moreover, there is evidence of a direct nature which 
proves it to be untenable. If this view were correct, we should 
expect the physical properties of the solution to differ from 
each other above and below the transition point, but in none of 
those properties has a sudden change been found 1 . 
1 Ostwald's Lehrbuch, or Solutions, p. 74. 


A similar continuity of properties holds good at the point 
of saturation. None of the curves indicating the 

Supersaturation. . _ 1-1 

variation with concentration of any physical pro- 
perty of the solution exhibit a sudden change of curvature as 
the saturation point is passed and a supersaturated solution 
obtained. This has been shown for the freezing points by 
Coppet 1 , for the electrical conductivities by Beetz 2 , Kohlrausch 3 
and Heim 4 , and for the densities, specific heats and heats of 
solution by Bindel 6 . 

It will thus be evident that there is nothing abnormal in 
the condition of a supersaturated solution; there is merely 
no solid present to induce equilibrium, and the case corre- 
sponds exactly to that of an under-cooled liquid. The facility 
with which a solid is spontaneously deposited by a liquid 
determines the conditions under which a supersaturated 
solution of the given substance can be produced. It is found 
that such solutions can with care be almost always obtained ; 
but it has been noticed that their production is usually easier 
in the case of salts which freely form large crystals, particu- 
larly when those crystals are hydrated. The phenomena of 
the supersaturation of air with water vapour, to which we have 
referred on p. 43, are closely analogous to those now under 
consideration. Just as a proper nucleus will at once induce 
precipitation of water from supersaturated air, so crystallization 
is started in a supersaturated solution by the presence of a 
fragment of a crystal, either of the substance with regard to 
which the solution is supersaturated or of any body isomorphous 
with it. 

In the case of supersaturated water vapour, we have seen 
that it is the existence of free energy due to surface tension, 
which is proportional to the area of contact between water and 
air, that makes it so difficult for very small drops of water to be 
formed spontaneously, and requires the presence of nuclei to 

1 Ann. Cliim. Phys. (4) xxm. 366 (1871). 

2 Pogg. Ann. cxvu. 1 (1862). 

3 Wied. Ann. vi. 28 (1879). 

4 Wied. Ann. xxvu. 643 (1886). 

5 Wied. Ann. XL. 370 (1890). 


induce precipitation. While, as seen on p. 45, the mechanism 
of the process is probably different, it is still likely that in the 
crystallization of solutions, the free energy of surface tension 
plays an important part, and it is probable that it is most 
difficult to start precipitation in those solutions where the 
energy of the surface between the liquid and the possible 
solid is greatest. 

The same free energy of surface tension causes the large 
drops of water in a collection of various sizes to grow at the 
expense of the smaller ones which have a larger area for a 
given volume, for the total free energy tends to a minimum ; 
and something of the same kind seems to occur in the case 
of solutions, the smaller crystals deposited from a saturated 
liquid having been seen to disappear while the larger ones 
have increased in size 1 . Gibbs has pointed out 2 that, in large 
crystals, the surface energy will not have the same value on the 
different faces, which will consequently be in equilibrium with 
solutions of different concentrations. Thus, in a saturated 
solution, certain faces will grow and certain other faces dissolve, 
and the crystal will eventually be bounded solely by the latter. 
In very minute crystals, the surface energy of a side will be 
affected by the other sides, and it seems likely that the form is 
then determined by the simple relation that the total surface 
energy must be a minimum for the volume of the crystal. In 
this case, the crystal will tend to possess a definite shape as 
well as definite angles. The small disturbing effect of gravity 
will make a crystal grow more readily in the upper or lower 
parts of the liquid, according as its growth causes expansion or 

Gases form dense films over the surfaces of glass vessels, and 
solubility of gases probably of all other solids. In certain cases, 
in solids. Slic j 1 as t h e ae tion of platinum and palladium 

on oxygen and hydrogen, the gas absorbed seems to penetrate 
the mass of the solid, though, if these solids be considered as 
porous, this again may be only a surface action. It is still 

1 For this interesting observation the author is indebted to Lord Berkeley. 

2 Trans. Connect. Acad. m. 494 (1878). 

W. S. 6 


uncertain whether the bodies thus formed are more of the 
nature of chemical compounds or of solid solutions. 

The film of air, etc., which covers the surfaces of glass vessels, 
is exceedingly difficult to remove. This is shown by the 
variations of pressure inside exhausted bulbs, particularly when 
electric discharges are passed. Even heating such a bulb to a 
high temperature seems to fail in completely removing the gas. 
The influence of the area and nature of the walls of the con- 
taining vessel, which is so marked in the reactions between 
gases 1 , may be largely or entirely due to the effect of the 
condensed film. For example, dry ammonia and hydrochloric 
acid gases will not combine, though, if moisture be present, a 
cloud of ammonium chloride is at once formed. A perfectly 
dry mixture of oxygen and hydrogen will neither explode when 
heated nor combine gently at temperatures of 400 or 500 
Centigrade as will the moist gas. There seems reason to believe 
that this action of moisture depends on the film of water or 
other concentrated substance formed on the surface of the 
vessel, or on the minute particles of water scattered throughout 
the volume. It is possible that chemical action can only occur 
between the gases when they are dissolved in this water, a 
possibility which might reduce all chemical actions to those 
taking place in solution 2 . 

The absorption of oxygen, hydrogen and other gases by 
such metals as platinum and palladium was discovered by 
Graham 3 , who gave the process its name of occlusion, and has 
since been studied by Deville and Troost 4 , Dewar 5 , Troost and 
Hautefeuille 6 , Berthelot 7 , Favre 8 , Roozeboom and Hoitsema 9 
and by Mond, Ramsay and Shields 10 . 

1 See, for instance, Van 't Hofif, Studies in Chemical Dynamics (1896), p. 43. 

2 See J. J. Thomson, Phil. Mag. xxxvi. 313 (1893), and C. T. E. Wilson, 
Phil. Trans. A. cxcn. 452 (1899). 

3 Proc. E. S. xv. 223 (1867), xvi. 422 (1868), xvn. 212 and 500 (1869). 

4 Comptes Eendus, LVII. 894. 

5 Phil. Mag. [4] XLVII. 324 and 342 (1874) ; Proc. Chem. Soc. CLXXXIII. 192 

6 Compt. Rend. LXXVIII. 686; Chem. Soc. Journ. xxvn. 660 (1874). 

7 Ann. de Chim. et Phys. [5] xxx. 519 (1883). 

Compt. Rend. LXXVII. 649, and LXXVIII. 1257 (1874). 

9 Zeits.phys. Chem. xvn. 1 (1895). 

10 Phil. Trans. A. CLXXXVI. 657, cxc. 129, cxci. 105 (1893-7 and 8). 


Both platinum and palladium can be prepared by chemical 
or electrolytic precipitation as very porous black deposits, and, 
in this state, they show the property of occlusion in the highest 
degree. A certain quantity of oxygen is occluded from the 
atmosphere, and any hydrogen admitted will first combine with 
this oxygen to form water. This fact was overlooked by the 
earlier observers of the absorption of hydrogen, and has vitiated 
their conclusions. Mond, Kamsay and Shields found that when 
platinum black is heated in an atmosphere of oxygen at 
ordinary pressure, absorption goes on till the temperature is 
raised to about 360, when the gas is expelled; the heat evolved 
by the occlusion being about 1100 calories, or 11 K., where K. 
denotes 100 calories, for each gram of oxygen absorbed, which 
gives 176 K. per atomic weight in grams. The same observers 
investigated the absorption of hydrogen by allowing it to enter 
the metal at the ordinary temperature and form water, which is 
pumped out at 184 together with the excess of gas, the latter 
being readmitted in an ice calorimeter. The heat of occlusion 
was found to be 68'8 K. per gram of hydrogen, and to be the same 
for the hydrogen which can be pumped out of the platinum at 
ordinary temperatures as for that which only comes off at 184. 
The heat of occlusion of oxygen is nearly the same as the heat 
of oxidation, thus suggesting that the process is a superficial 
oxidation. In the case of palladium, it is probable that the 
oxide PdO is formed ; on the other hand the only evidence for 
the existence of compounds with hydrogen is the approximate 
correspondence of the completely saturated palladium with the 
formula Pd 3 H 2 , while Hoitserna concludes, from a study of 
the vapour pressures of palladium containing hydrogen, that 
two immiscible solid solutions are probably formed. Palladium 
absorbs about 850 times its volume of hydrogen, the thermal 
evolution, which is about 46'4 K. per gram of hydrogen, remaining 
constant throughout the process. It is likely that the increase 
of absorbing power with a rise in temperature noticed in certain 
cases, is due to a corresponding increase in the rate of diffusion 
of the occluded gas from the saturated outside layers to the 
inner parts of the metal. The same difficulty of diffusion is 
probably the reason of the much less marked absorbing power 



of the metals in the form of dense foil, when the area exposed 
to the direct action of the gas is very much less than it is when 
they exist in the spongy black condition. 

There appear to be two different classes of solutions of 
solubility of gases gases in liquids. Sometimes, as in the case of 
in liquids. hydrochloric acid dissolved in water, the gas 

cannot be completely expelled either by lowering the pressure or 
increasing the temperature. On the other hand, air, oxygen, 
hydrogen and certain other gases can be thus removed, although 
the separation of the last traces of gas is a process of extreme 
difficulty. Nevertheless, in these cases, the solvent exerts a 
selective action, the gases differing from each other in solubility. 
It is found that, in general, the solution of a gas in water, 
even when the liquid is nearly saturated, is accompanied by an 
evolution of heat. From this it follows by the principles of 
thermodynamics that the solubility will decrease with rising 

In an experimental determination of solubility it is necessary 
to take precautions to ensure complete saturation, for the 
process of diffusion of matter from one portion of a liquid to 
another is very slow. Many forms of apparatus have been 
devised, the simplest being that used by Bunsen, who placed 
a measured volume of the gas in a graduated tube over mercury 
and added a certain volume of the liquid. The tube was then 
shaken in a water bath of constant temperature, the open end 
being screwed against an india-rubber plate. By repeatedly 
opening the end under mercury and then closing it again and 
shaking, saturation was obtained, the solubility being determined 
by measuring the volume of gas left over, the volume of the 
liquid, and the final pressure. 

Ostwald 1 has found an absorptiometer, constructed after 
a design due to Heidenhain and Meyer, to be both convenient 
and accurate. It is represented in Fig. 33. The vessel G is 
filled with the air-free liquid, and, running out a measured 
volume by raising the tube B, an equal volume of gas can be 

1 See Lehrbuch, or Solutions. 

CH. IV] 



introduced through a flexible lead tube from the graduated 
vessel A. The tubes A and C are immersed in a water bath, 
and G is constantly shaken to cause ab- 

The solubility of a gas has been 
denned by Ostwald to be the ratio of 
the volume of gas absorbed to the 
volume of the absorbing liquid, at any 
specified temperature and pressure, say 
S is equal to v/V. Bunsen used a more 
complicated constant, which he called 
the absorption coefficient. It is obtained 
from Ostwald's solubility by reducing 
the volume of gas absorbed to C. at 
the pressure of the experiment. In 

Fig. 33. 

certain cases we shall see that the volume of gas absorbed 
is independent of the pressure, so that if is Bunsen's absorp- 
tion coefficient, and a the coefficient of gaseous expansion, 

8 = 0(1 + at). 

Bunsen and others 1 have determined many absorption co- 
efficients for water and alcohol. The following are some of 
their results taken from Ostwald's Lehrbuch. 




In In 
Water Alcohol 

0-0215 0-0693 
0-0190 0-0673 


In In 
Water Alcohol 

0-0489 0-2337 
0-0342 0-2232 

Carbon Dioxide 

In In 
Water Alcohol 

1-797 4-330 
1-002 3-199 

Henry proved that the mass of a gas such as oxygen dis- 
solved in water is proportional to the pressure, 

Henry's Law. , r r . 

and established this law as an approximation 
by a series of experiments on five gases at pressures varying 
from one to three atmospheres 2 . Since the volume of a given 
mass of gas varies inversely as its pressure, it follows that the 

1 Bunsen, Pogg. Ann. xcm. p. 10, 1855; Winkler, Berichte, xxn. p. 1439, 
1899; Timofejeff, Zeitschr. f. physikal. Chem. vi. p. 141, 1890. 

2 Phil. Trans. 1803. 


volume which is dissolved, measured under the pressure it 
exerts above the liquid, is independent of that pressure. 

The reason of this result is suggested by the dynamical 
view of equilibrium, which imagines that saturation is reached 
when the quantity of gas going into solution per second is 
equal to the amount coming out. If the pressure is reduced, 
the number of gaseous molecules striking the liquid, and 
therefore the number per second retained by it, are reduced in 
the same proportion, while the rate at which they leave is at 
first unchanged. The concentration of the gas in solution is 
thus gradually lowered till equilibrium is again attained, and 
the concentration bears once more its old relation to the 
external pressure. At first sight it would appear that the 
solubility of a gas should be unaffected by an alteration of 
temperature, since the number of molecules impinging on the 
surface from within and without would vary in the same 
proportion. But here the influence of the solvent comes in, 
and the interaction between it and the gas is changed by 
increase of temperature so that the solubility becomes less. 

Bunsen confirmed Henry's law in a series of more accurate 
experiments, both by varying the pressure in his absorptiometer 
and by using a mixture of gases. If we have a volume of gas 
at atmospheric pressure, consisting of equal parts of two con- 
stituents, the total pressure is obviously due half to one and 
half to the other, so that, restricting our consideration to one 
gas, the pressure it exerts is half that of the atmosphere. In 
this way by using mixtures in which the proportion of one gas 
continually diminished, its pressure could be reduced from one 
atmosphere to zero, and it was found that the mass absorbed 
varied in the same proportion. 

In the case of many very soluble gases the phenomena are 
not so simple. For ammonia at 100 C. the law of Henry holds 
good 1 ; but if observations be made at lower temperatures, the 
mass of ammonia absorbed is not proportional to the pressure, 
and the curve drawn to show the variation of solubility with 
pressure when the temperature is kept at C. shows two 
changes of curvature. Sulphur dioxide behaves like ammonia, 
1 Sims, Annalen, cxvin. p. 345, 1861. 


the law only holding true above 40. Hydrogen chloride 
cannot be entirely removed from solution in water either by 
reducing the pressure or by boiling. If aqueous hydrochloric 
acid be distilled, its strength will either increase or diminish 
till a liquid of a certain composition remains, which distils over 
unchanged. This composition depends on the pressure at 
which the operation is carried on ; at normal atmospheric 
pressure the proportion of hydrogen chloride is 20*24 per cent., 
at 50 mm. of mercury pressure the proportion is 23'2 per cent., 
and at 1800 mm. it sinks to 18 per cent. As we have seen, the 
solution left is that mixture which possesses the highest boiling 
point under the condition of the experiment. 

The coefficient of absorption for a gas appears to be lowered 
Solubility of gases when a salt which does not act chemically on 
in salt solutions. ^ ^ ag ig p rev i ous iy dissolved in the water. 

In general, however, chemical action does occur, and the gas 
dissolved may be considered to consist of two parts one being 
held chemically by the salt nearly independently of the pressure, 
and the other varying with the pressure in accordance with 
Henry's law. Good examples of this are seen when carbon 
dioxide is dissolved in a solution of sodium carbonate or 
disodium phosphate. Solutions of similar salts of equivalent 
strength absorb nearly equal quantities of carbon dioxide e.g. 
the sulphates of zinc and magnesium 1 . 

The effect of mixing another liquid with the water is similar 
to that of dissolving a salt in it the absorption coefficient for 
a gas is reduced. This holds even with such substances as 
sulphuric acid and alcohol, which are themselves in the pure 
state as good as or better than pure water in absorbing power. 
Thus with sulphuric acid Setschenoff found for carbon dioxide 
a minimum absorption coefficient when the composition of the 
liquid was H 2 S0 4 . H 2 O. His results are as follows : 

H 2 S0 4 H 2 S0 4 + iH 2 H 2 S0 4 + H 2 H 2 S0 4 + 2H 2 H 2 SO 4 + 58H 2 O H 2 

923 719 -666 '705 '857 '932 

1 Setschenoff, Mems. de VAkad. Petersb, xxn. No. 6, 1875; Z. f. physikal. 
Chemie, iv. p. 117, 1889. 


These numbers show that a mixture of sulphuric acid and 
water absorbs less carbon dioxide than either liquid does when 
pure. Similar relations are found to hold good for other 
physical properties, such as the electrical resistance and the 

Solutions of liquids in liquids have been considered on pp. 58 
solubility of an( ^ ^1 under the head of two liquid components, 

liquids in liquids. frQm the point Q f yiew of the phase rule Ag 

we then saw, there are three classes into which pairs of liquids 
can be divided. Those in the first class are mutually soluble 
in all proportions ; thus mixtures of alcohol and water, or of 
water and sulphuric acid, can be prepared of any composition. 
Those in the second class are soluble in each other but not in 
all proportions ; thus water will dissolve about ten per cent, of 
ether, and ether about three per cent, of water, but if either 
substance be present in excess it separates out, forming a 
definite layer. The third class, which is small, consists of 
liquids which are insoluble in each other. The divisions 
between these classes are dependent on external conditions, 
for liquids which are only partially miscible at ordinary tem- 
peratures may mix in all proportions when heated, and it is 
probable that all liquids approach the condition of complete 
miscibility as they approach their critical points 1 . 

Measurements of the mutual solubility of liquids have been 
made by Alexejeff 2 , who placed weighed quantities in a sealed 
tube and noted the temperature at which the mixture became 

The form of the solubility curve for a pair of partially 
miscible liquids is shown in Fig. 34, in which the abscissae 
represent temperatures and the ordinates percentages of dis- 
solved substances in 100 parts of the solution. The curve a 
represents a solution of water and phenol ; the curve b water 
and aniline phenolate. At low temperatures there are two 

1 It is stated (Watts' Diet. Art. Solutions i.) that diethylamine and water, 
though miscible in all proportions at low temperatures, cease to be so when 

2 Wied. Ann. xxvin. p. 305, 1886 ; Chem. Centralblatt, pp. 328, 677, 763, 1882. 

CH. IV] 



definite states in which equilibrium is attained the lower 
branch of the curve representing a solution of phenol in water, 
the upper branch a solution of water in phenol. 






Fig. 34. 

It is fairly easy to make an approximate determination of 
Solubility of solids tne amount of a solid required to saturate a 
m liquids. given quantity of a liquid, but when accurate 

results are needed, the problem becomes one of extreme difficulty. 
There are two methods of procedure. The first is to keep the 
liquid in contact with an excess of the solid for many hours at 
as constant a temperature as possible, by immersing the vessel 
containing the mixture in a water bath with an automatic 
thermal regulator, while the apparatus is either constantly or 
intermittently shaken. The second method consists in heating 
the liquid with excess of solid to a temperature above that at 
which the determination is to be made, and then allowing it to 
slowly cool in contact with the solid till the temperature in 
question is reached. 

Whichever process is adopted, a quantity of the solution is 
then analysed either chemically or by evaporation, and the 
amount of solid in it determined. There is often a considerable 
difference in measurements made in these two ways, the second 
yielding higher results than the first. The attainment of the 
state of equilibrium seems to be a very slow process, even when 
the mixture is kept constantly stirred. 


The Earl of Berkeley, who has made many laborious 
experiments on this point, finds that the time needed before 
the saturation is approximately complete varies with the nature 
of the salt, the mass of solid present, and the temperature. 
Even with constant and rapid stirring, in certain cases as 
much as forty-eight hours may be required. 

The solubility is usually expressed as the parts by weight 
of the solid which dissolve in one hundred parts of the solvent 
to form a saturated solution. 

The solubility of solids, owing to the small changes in 
volume produced by solution, is only slightly 

Influence of pres- r * J E> J 

sure on the soiubi- affected by pressure, and accurate experimental 

lity of solids. , . . *L 

determinations are very difficult. The prin- 
ciples of thermodynamics indicate that the chief conditions 
determining the change of solubility with increasing pressure 
are the heat of solution of the salt in the nearly saturated 
solution, and the change in volume on solidification. The few 
experiments which have been made seem to confirm this con- 
clusion. Van 't Hoff states 1 that ammonium chloride, which 
expands on solution, loses solubility by 1 per cent, for 160 
atmospheres, while copper sulphate, which contracts, gains by 
3'2 per cent, for 60 atmospheres. 

The influence of temperature on the solubility of solids 
influence of tem- in liquids has been constantly studied from 
perature. fa Q ft me o f Q av Lussac to the present day. 

As a general rule solubility increases with temperature, 
though several exceptions to this rule are known, for example, 
calcium hydroxide, and sodium sulphate between the tempera- 
tures of 33 and 100. It is impossible, when studying the 
influence of temperature on solubility, to overlook the analogy 
between the solution of a solid in a liquid and the evaporation 
of a liquid into a closed vacuous space. Just as for every 
temperature there is a definite quantity of vapour present in 
the space when equilibrium is reached, so there is a definite 
quantity of solid dissolved. Increase of temperature causes in 

1 Lectures on Theoretical and Physical Chemistry, p. 34. 

CH. IV] 



the one case more liquid to evaporate, and in the other more 
solid to dissolve, till a new state of equilibrium is reached. 
We shall see hereafter that, while a liquid exerts a vapour 
pressure, a solid in solution exerts a pressure which can be 
recognized and measured by means of certain phenomena, to 
which the name of osmose has been given. The analogy 
between the two processes seems thus very close, and is borne 
out by the general similarity of the solubility curves (Fig. 35) 
to curves which show the variation of vapour pressure with 

As the vapour pressure curves end at the critical points of 
the liquids, so the solubility curves end at the melting points 
of the solids. The phenomena beyond depend on the miscibility 
or non-rniscibility of the liquid solvent and the fused solid, and 
are qualitatively described by the principles of the phase rule. 

70 -I 
5O - 

20 30 40 50 60 7O 

Fig. 35. 

80 90 100 TCente 

A long series of investigations on the influence of temperature 
on solubility has been made by Etard and Engel 1 , who examined 
various solutions at temperatures above 100, by heating them 
in sealed tubes. They find that solutions of many sulphates 

1 Comp. Rend. 1884-8, xcvm. pp. 993, 1276, 1432 ; crv. p. 1614 ; cvi. pp. 206, 


have solubility curves showing maxima at definite temperatures, 
while certain calcium salts have curves giving minimum values. 

If solubility be defined as the parts of salt in 100 parts of 
solution, instead of 100 parts of solvent, each part of the curve 
according to these observers generally comes out as a straight 
line. Thus the curve for copper sulphate consists of three 
straight lines which join each other consecutively at 55 and 105. 

As we have already remarked, the solubility increases or 
diminishes with rising temperature, the action depending on 
the absorption or evolution of heat when some of the solute 
dissolves in the nearly saturated solution, so that the thermal 
effect must change sign where maxima or minima occur in the 
solubility curve. 

When water is shaken with a mixture of two salts, the satu- 
soiubiiityof rated solution is usually found to contain less 

mixtures. Q f eac ] 1 substance than it would have done had 
the other been absent, though there are many exceptions to 
this generalization. 

In the case of salts which are not isomorphous and do not 
form double salts, the composition of the solution is independent 
of the proportion in which the solids are mixed, and of the 
method by which the solution is prepared. In the case of 
substances which form double salts, if we add excess of A to a 
saturated solution of B, the double salt separates out till 
a solution is formed which is saturated both as regards A and 
the double salt, and is not changed by a further addition of A. 
The third case, when the salts are isomorphous and can crystal- 
lize together in all proportions, gives saturated solutions of which 
the compositions vary continuously with the composition of 
the solid mixture. By adding successive quantities of A it is 
possible to completely displace the salt B from the solution. 
Much experimental work has been done in this subject by 
Rlidolf 1 , and Ostwald has pointed out the analogy between 
these phenomena and the vapour pressures of mixed liquids, 
the three cases given above corresponding to the cases (i) when 

1 Pogg. Ann. CXLVIII. pp. 456, 555, 1873; Wied. Ann. xxv. p. 626, 1885. 


the liquids do not mix, (ii) when they are partially miscible, 
(iii) when they are miscible in all proportions. 

Nernst 1 has remarked that the solubility of a slightly soluble 
salt like silver acetate must be greater in pure water than in 
a solution of any other electrolyte which contains either silver 
or the acetate group. A corresponding phenomenon is observed 
in the case of gases which, like the vapour NH 4 SH, decom- 
pose to a certain extent. The partial pressures of the products 
of decomposition are less in the presence of either ammonia 
or sulphuretted hydrogen. 

On adding a liquid to a solution with which it is miscible, 

solubility in the dissolved substance will be to some extent 

mixed liquids. precipitated if it is insoluble in the liquid 

added. Thus copper sulphate or sodium chloride can be 

precipitated from their aqueous solutions by the addition of a 

certain quantity of alcohol. No relation is yet known between 

the amount precipitated and the quantity of alcohol added. 

A dissolved body divides between two solvents in a constant 
ratio which is independent of the absolute concentration. This 
statement was verified for the solution of succinic acid in 
ether and water by Berthelot and Jungfleisch 2 . If the bodies 
have different molecular weights when dissolved in the two 
solvents, like benzoic acid in benzene and water, other laws 
hold good and have been investigated by Nernst 3 . 

Full tables of the solubilities of inorganic substances will 
Tables of be found in A Dictionary of Chemical Solubi- 


be found in Watts' Dictionary of Chemistry, Roscoe and 
Schorlemmer's Chemistry, and other similar works. The follow- 
ing selection of common substances has been made principally 
from the first-named book. The solubilities of solids are 
given in parts by weight of solute in 100 parts of water, but 
some of those of gases are expressed in terms of the volume of 
gas, reduced to 0, which dissolves in 100 volumes of water. 

1 Zeits. f. physikaL Chemie, iv. 372, 1889. 

2 Ann. de Chemie, xxvi. pp. 396, 408, 1872, [4]. 

3 Zeits. f. physikaL Chemie, vm. p. 110, 1891. 



[CH. IV 

3 02 02 02 

fcfcs 1*989 

Jt* CO 00 
I> 1>- 

S^ >o8 

00 (M ^ 

r-l 1> U? 

10 CO CO 



i>> Oi co 


C^ 9 


H -S>^$> ^J^O' 





Ammonium chloride 
Barium chloride 
Calcium chlorid 

1 1 -a 

! 1 Is 

S a ^ bo 


&p t>o 2 2 g> 

3 a 



5 8 



Semi-permeable membranes. Osmotic pressure and vapour pressure. 
Perfect semi-permeable membranes. Theoretical laws of osmotic 
pressure. Osmotic pressure and heat of solution. Experimental 
measurements of osmotic pressure. 

THE passage of liquids through animal and vegetable mem- 
Semi-permeabie branes has long been a subject of study owing 
membranes. to j ts bearing on physiological problems. If 

the mouth of a glass vessel filled with one liquid be closed 
by a membrane and immersed in some other liquid which 
passes more freely than the first liquid through the membrane, 
a pressure will be produced within the vessel until, by the slow 
processes of diffusion, the composition of the liquids inside and 
outside has become identical. If a membrane could be obtained 
which was quite impervious to one of the liquids, it is clear 
that this excess of pressure would be permanently kept up. 
The possibility of preparing such membranes was first suggested 
by Traube 1 , as the result of experiments on the methods of 
formation of organic cells. Solutions of certain substances, 
which precipitate each other, form insoluble pellicles when 
brought into contact ; these are at all events nearly im- 
permeable to the solutions forming the precipitate and to 
some other substances. 

Pfeffer 2 made a further study of these semi-permeable 
membranes, and measured the pressures obtained by their 

1 Archiv f. Anat. u. PhysioL p. 87, 1867. 

2 Osvwtische Untersuchungen, Leipzig, 1877. 


[CH. V 

means. The most usual method of preparing them is as follows. 
A porous pot of unglazed earthenware, six to eight centimetres 
high and two or three centimetres in diameter, has a glass tube, 
which just fits inside it, hermetically sealed to the top by 
means of sealing wax (Fig. 36). Having 
been thoroughly washed, it is filled with 
the solution of a salt such as potassium 
ferrocyanide, and the outside is then sur- 
rounded with the solution of another salt 
such as copper sulphate or ferric chloride, 
which gives an insoluble precipitate when 
in contact with the first salt. The two 
solutions gradually diffuse from opposite 
sides into the walls of the cell, and form 
an insoluble membrane over the surface 
on which they meet. The solutions are 
then washed away, and the wide glass 
tube is drawn out and sealed to a smaller 
one in the manner shown in the figure. 

Inside a cell thus prepared let us 
place the solution of some substance 
such as sugar in water, and surround the 
outside with a large volume of the pure 

solvent. Water will gradually enter the cell, and, by using 
the glass tube as a pressure gauge, it will be found that this 
influx will continue until a definite internal pressure is reached. 
This gives a measure of what is called the osmotic pressure of 
the solution as it finally exists in the cell. 

If the membrane has been well prepared, very little if any 
sugar will escape to the outside, and, although some simple 
salts and acids will often leak through to some extent, there 
seems little reason to doubt that membranes can be obtained 
which are practically impervious to solutions of complex 
chemical substances such as sugar. At first sight the obvious 
explanation appears to be that the membrane allows molecules 
of water to pass, but will not let through molecules of sugar. 
There is an experiment described by Pickering 1 which suggests 

1 Ber. Deut. Chem. Ges., xxiv. 3639 (1891). 

Fig. 36. 


another possible view. If a mixture of propyl alcohol and 
water be placed in a semi-permeable vessel and surrounded 
with water, it is found that water enters the cell, and no 
alcohol escapes. If, however, the same vessel with its same 
contents be placed in propyl alcohol, it is the alcohol which 
passes through and enters the vessel, while no water escapes. 
In this case it seems clear that the membrane is permeable 
to either water or propyl alcohol when pure, but is impervious 
to the combination of the two. Further experimental evidence 
is needed to decide whether or not similar phenomena occur 
with other solutions. The question is of great interest from 
its bearing on the problem of the fundamental nature of 

The mechanism of the passage of the liquid through the 
membrane is not fully understood. It may be a purely physical 
process dependent on the relative sizes of the molecules in the 
solution and the pores of the membrane, which in this case 
must be likened to an excessively fine sieve. On the other 
hand it is possible that loose chemical compounds are formed 
between the membrane and the solvent, and that these com- 
pounds, gradually saturating the membrane, again decompose 
on its other side where the concentration of the solvent 
is less. 

Another possible explanation depends on the difference in 
surface tension which may exist between the membrane and 
the solution or the solvent. In order to see how this may 
affect matters, we must remember that the existence of surface 
tension adds a term to the expression for the energy of any 
body proportional to the area of its surface. A body free from 
the action of other forces, such as a drop of rain or melted lead 
falling freely, will therefore assume the shape which makes this 
energy a minimum, that is, each drop becomes a sphere, the 
figure Jiaving the smallest area for a given volume. Similarly 
any change which causes a decrease in this portion of the 
energy of a system will, if it does not increase other parts of 
the energy, tend to occur. Now the surface tension of salt 
solutions is different from that of pure water. As usually 
measured, by capillary tubes, etc., it is the surface tension 
w. s. 7 


of the surface in contact with air, and, for solids which are 
wetted by the solution, the conditions of equilibrium at 
the line of contact of solid, liquid and air show at once that 
as this air tension increases, the surface tension between the 
liquid and the solid must decrease. It has been found that 
for most salt solutions, at any rate, the air tension is greater 
than for water ; wherefore the surface tension with which we 
are concerned must be less. The surface energy of a solid 
in contact with a solution is therefore less than with pure 
water, and thus the layer of liquid in contact with the solid 
will become richer in salt than the bulk of the solution 1 . As 
the solution flows through capillary tubes, the salt will collect 
along the walls, and the faster moving central regions will 
have their concentration diminished. The effect is, that the 
liquid which finally comes through is pure water. Thus 
J. J. Thomson and Monckman filtered potassium permanganate 
from its solution by passage through finely divided silica. The 
same thing can be seen by allowing dilute permanganate 
solution to run up into filter paper. The furthest regions 
reached by the liquid are colourless. 

Similar considerations may explain the behaviour of semi- 
permeable membranes. If they consist of tubes of molecular 
dimensions, the observed pressures could perhaps be obtained 
by differences in the surface tensions. At these dimensions, 
however, the usual principles of capillary action may cease to 
apply, the distinction between chemical and physical processes 
may break down, and all these suggested explanations may 
merely represent different aspects of the same thing. 

However this may be, the facts remain, and are of the 
greatest importance in the theory of our subject. 

Besides natural membranes and the artificial ones which we 
have described, there is a semi-permeable dia- 

Osmotic pressure 

and vapour phragm of the simplest and most perfect kind, 

consisting of the free surface of a volatile liquid 

in which some non-volatile substance is dissolved. The solvent 

in the form of vapour can pass in or out, but the solute is com- 

1 J. J. Thomson, Applications of Dynamics to Physics and Chemistry. 


pletely confined to the lower side of the surface. If the solvent 
were non- volatile and the solute a volatile gas, we should get the 
converse case, in which the solute can pass, but the solvent is 
confined to the lower side of the diaphragm. 

Let us imagine that a tall cylinder containing a solution is 
placed in an exhausted chamber which also contains a vessel 
filled with the pure solvent. The vapour pressure of the 
solvent being higher than that of the solution, liquid will 
evaporate from the solvent and condense on the solution, the 
level of which will therefore rise. The concentration of the 
solution tends to become uniform throughout by diffusion of 
the solute, and this process can be quickened and made 
complete at any instant by stirring the liquid. Distillation 
will then continue, and as the liquid condenses in the cylinder 
the height of its surface will rise. But the higher we go in 
the exhausted chamber, the less becomes the pressure of the 
vapour of the solvent which fills it, just as the pressure of the 
atmosphere grows less as we ascend. Thus, in the end, the 
vapour pressure at the surface of the solution becomes equal 
to that at the same level in the space outside it, and equilibrium 
results. The difference in pressure between the level of the 
solvent and that of the solution is equal to the weight of a 
column of vapour of unit cross section and height h, where h 
is the difference in level. If we assume the density, cr, of the 
vapour to be uniform, which involves the assumption that the 
solution is very dilute, and consequently that the height h is 
small, the difference in pressure can be written 

The lowest layer of solution is now under the pressure of 
the column of solution above it, so that, if the density of the 
solution is p, the hydrostatic pressure at the bottom of the 
column is 

Now imagine the bottom of the cylinder containing the 
solution to be put in connexion with the pure solvent through 
a semi-permeable membrane. There must still be equilibrium, 
for otherwise, liquid would enter or leave, the height of the 



column increase or diminish, and evaporation or condensation 
occur to compensate for this change. An automatic circulation 
which could yield an unlimited supply of work would thus be 
maintained in an enclosure which was originally at a uniform 
temperature throughout a process the possibility of which is 
contrary to experience as formulated in the second law of 
thermodynamics. It therefore follows that, in the apparatus 
described, no passage of liquid through the semi-permeable 
membrane can occur, and thus the hydrostatic pressure at the 
diaphragm must be equal to the osmotic pressure. The osmotic 
pressure of the dilute solution of a non-volatile substance is 
therefore connected with the lowering of vapour pressure in 
the simple manner expressed by the equation 

It seems that this result can be extended to the case of 
volatile solutes, for if we imagine the surface of the solvent to 
be covered with a membrane permeable to the vapour of the 
solvent but not to that of the solute, the pressure of the solvent 
vapour will be everywhere independent of the presence of the 
solute, and equilibrium will still exist as described. 

As FitzGerald indicated 1 , the relation between the vapour 
pressure of a solution and its surface tension is of importance 
in this connexion. It is well known that water will rise in a 
capillary tube which is wetted by the liquid, the upper surface 
being concave upwards, and the angle of contact with the tube 
zero. The weight of the column of liquid is hirr* (p a) g, 
where r is the radius of the tube, or the radius of curvature of 
the surface, p is the density of the liquid and & that of the 
atmosphere surrounding it, which we will suppose to be its own 
vapour 2 . This weight is supported by the upward pull of the 
surface tension $ between the liquid and the vapour, which 
acts across each unit of length of the circumference of the tube 

1 Helmholtz Lecture, Trans. Chem. Soc. Lon. Jan. 1896. 

2 Lord Kelvin, quoted in Maxwell's Heat, 5th ed. p. 290. See also J. J. 
Thomson, Applications of Dynamics to Physics and Chemistry, pp. 163, 164. 


where it is touched by the free surface of the liquid, and is 
therefore %7rrS. Thus 


or h = r . 

When there is equilibrium, it is evident from Fig. 37 that 
the vapour pressure from the flat surface of the bulk of the 
liquid A is greater than that from the concave surface in the 
capillary by the weight of the column of vapour hcrg. The 
decrease of vapour pressure due to the concavity is 2So-/r(p <r), 
or, since <r is small compared with p, SSa/rp. 

Now, by choosing the tube of the right diameter, we can 
obviously reduce the vapour pressure of a solvent in it till it 
sinks to the value of the vapour pressure of a given solution. 
The difference in vapour pressure of the solvent and solution 
is P(r/p, where, as we saw, P is equal to the osmotic 
pressure. Thus, 

rp p 

T> 28 
or > p= 7' 

an equation which gives the osmotic pressure of a solution 
having the same vapour pressure as the solvent has in the 
given capillary tube. 

If we have a tube which is not wetted by the liquid, 
the solution B will sink in it instead of rising and the free 
surface will be convex instead of concave. 

Outside a convex surface the vapour pressure is increased 
by an amount again given by the expression 

20- 8 
p or ' r' 

By properly choosing the diameter of the tube, remem- 
bering that the angle of contact now has a finite value, and 
placing the tube in the solution instead of the solvent, it is 
possible to obtain the reverse process, and increase the vapour 



[CH. V 

Fig. 37. 

pressure of the solution by capillary means till it is equal to 
that of the solvent. 

In Fig. 37, the conditions represented are those of 
equilibrium. The vapour pres- 
sure at the concave surface in 
the capillary rising from the 
solvent A is equal to that from 
the flat surface of the solution 
B at the same level, and the 
vapour pressure at the convex 
surface of the depressed solution 
B in the unwetted capillary at 
its side is equal to the vapour 
pressure of the flat surface of the 
solvent A at the same level. 

Thus vapour pressure, sur- 
face tension and osmotic pressure 
stand in an intimate connexion 
with each other, which can be traced by the principles of 

The phenomena of surface tension are of theoretical 
importance in yet another way. The artificial 

Perfect semi- -,.,. ^ 

permeable membranes prepared by precipitation often 

allow solutions of simple salts such as the 
chlorides of sodium or potassium to leak through them. 
Although they seem impervious to sugar and similar sub- 
stances, as far at least as experiment shows, this leakage 
indicates that they are not perfect. Now the chief importance 
of osmotic pressure from a theoretical point of view is the 
possibility it secures of the adaptation of thermodynamic 
reasoning to the case of solutions. But in doing so, we have 
to assume (1) that the diaphragms are completely impervious 
to the solute, (2) that the processes involved in using them are 
strictly reversible and that no irreversible heat effects occur 
in them when solvent is passed through. We know too little 
about the mechanism of the passage of liquids through semi- 
permeable membranes to be sure that it is not accompanied 


by some change in the structure of the walls which may 
eventually destroy them. Any such effect would render the 
transfer a non-reversible process and invalidate thermodynamical 
reasoning based on the second law except as giving an ideal 
limit which might never be practically realized. It becomes, 
therefore, of the utmost importance to imagine some diaphragm 
which is a perfect semi-permeable membrane and can be con- 
fidently used hypothetically for the purposes of thermodynamical 

Now, in the arrangement shown in Fig. 37, the vapour 
from the depressed surface of the solution B in the unwetted 
capillary is in equilibrium with the vapour of the flat surface 
of the solvent A at the same level. If the solvent be volatile 
and the solute perfectly non-volatile, solvent can freely pass as 
vapour from one vessel to the other, while the solute cannot 
pass at all. Another method of getting the same result is as 
follows. Let a plate of some substance which is not wetted by 
the solution be pierced by a number of capillary tubes, and 
placed on the surface of the solution. Pressure must be applied 
before the liquid will rise in the tubes, and, by making the 
tubes of the right diameter, we can arrange that the curvature 
of the surface of the solution in them is just enough to increase 
the vapour pressure to an equality with that of the flat surface 
of the pure solvent. There will then be equilibrium between 
the flat surface of the solvent and the solution held in the 
capillaries, and FitzGerald has pointed out that such an 
arrangement furnishes a perfect semi-permeable membrane for 
a non-volatile solute in a volatile solvent 1 . 

Van 't Hoff showed that the application of thermodynamics 
enables us to deduce from the observed exist- 

Theoretical Laws . i i ., i 

of osmotic pres- ence ot osmotic pressure its absolute value 
and the laws which describe its variation with 
volume and temperature 2 . We shall here treat this problem 
by an application of the principle of available energy. 

1 Helraholtz Lecture, Trans. Chem. Soc. Lon. Jan. 1896. 

2 Phil. Mag. vol. xxvi. p. 88, 1888. 



[CH. V 





Fig. 38. 

Let a volume of the solution of a volatile gas in a non- 
volatile solvent be confined between two semi- 
permeable pistons in an engine cylinder. Of 
these pistons, the upper BB' is permeable to 
the gas but not to the solvent, the lower CO' 
is permeable to the solvent but not to the 
dissolved gas. Two non-permeable pistons 
A A' and DD' are placed beyond the region 
occupied by the solution and confine a volume 
of gas above and a volume of pure solvent 

Let us at first confine ourselves to the 
consideration of solutions so dilute that 
the volume and thermal changes on further 
dilution are negligible. The piston BB', 
which defines the top of the solution, can then be fixed, as the 
total volume of the solution and solve tit together is constant. 

First, let the pistons A A', DD' also be fixed, so that the 
whole system is at constant volume, and let the moveable 
partition (7(7' have taken up such a position that there is 
equilibrium between the gas and solution, and between the 
solution and solvent. 

The hydrostatic pressure on the piston CC' must then be 
equal to the osmotic pressure in the solution (p. 100), while the 
gas is at the pressure determined by its solubility relations, 
which for many gases are those described by Henry's law (p. 85), 
equilibrium being reached when as many molecules enter the 
solution per second as leave it. 

Now when the outside pistons A A', DD' are fixed, the 
system is at constant volume. The condition of equilibrium 
then is that the internal thermodynamic potential, or the ty 
function, should be a minimum. 

For isothermal changes (p. 23, equation 8), 

and when ^ is a minimum, Sty must vanish, so 


denotes the external work, which can be 


obtained from the system during an infinitesimal, isothermal, 
reversible change in the external co-ordinates. Thus under 
the condition of constant volume of the system the piston GO' 
will take up such a position that for a small displacement the 
work obtainable vanishes. 

Now if the piston A A' be allowed to rise reversibly, gas will 
come out of the solution and its osmotic pressure will therefore 
fall. The hydrostatic pressure below will therefore cause the 
piston CC' to rise. Besides the external work done or absorbed 
by the piston A A', external work can also be obtained from or 
given up to the piston CO' by connecting it with a rod running 
through the solvent and the wall DD'. 

If a volume Bv of gas be forced out of a volume SV of 
solution, the work done by the gaseous pressure on the piston 
A A' is p8v, and that done against the osmotic pressure by the 
piston CC' is PSF, so that 


By continuing the upward movement of the piston CC' 
under the constant pressure P, equal to the osmotic pressure, 
till the whole original volume V of the solution has disappeared, 
gas will be forced out of solution into the upper space, and the 
piston A A' will rise under a constant pressure p. This process 
can be continued till the whole of the dissolved gas is forced 
out of solution, and occupies a volume v. Both the pressures 
P and p keep constant throughout the whole operation. Thus 
the last equation becomes 

an equation which must hold good at any temperature. 

Thus the osmotic pressure of a volatile substance in dilute 
solution must obey the same volume and temperature relations 
as does gaseous pressure, and have the same absolute value as 
the pressure the same number of molecules would exert in an 
equal volume as a gas 1 . 

Another simple proof of this identity has been put in the 

1 For a discussion of the subject, see Larinor, Phil. Trans. A. cxc. 205, 1897. 


following form by Lord Rayleigh 1 . Let us suppose for simplicity 
that we have an involatile liquid solvent, that its volume is un- 
altered by dissolving in it a quantity of a certain gas and that 
the heat of dilution is negligible. The latter suppositions will 
again limit the strict accuracy of our result to the case of dilute 

We begin with a volume, v, of the gas under a pressure, p , 
and with a volume, F, of the liquid just enough to dissolve the 
gas under the same pressure. If we bring the gas at pressure 
p into contact with the liquid, we get an irreversible process of 
solution ; but by expanding the gas we can increase its rarity 
until no sensible dissipation of energy occurs when contact with 
the liquid is established. The gas is then gradually compressed 
and solution goes on under rising pressure until just as the gas 
disappears the pressure rises to p . By conducting the opera- 
tions at constant temperature, and so slowly that the conditions 
never deviate sensibly from those of equilibrium, the process 
can be made reversible. 

In order to calculate the amount of work in accordance with 
the laws of Boyle and Henry, we may conveniently imagine the 
liquid and gas to be confined under a piston in a cylinder of 
unit cross section. During the expansion, contact is prevented 
by a partition inserted at the surface of the liquid. If the 
distance of the piston from this surface be x, we have initially 
x v. At any stage of the expansion (x) the pressure, p, is 

IY\ rti 

given by p = *-^- , and the work gained during the expansion is 



x being large compared with v. The partition is then removed 
and during the condensation the pressure upon the piston in a 
given position x is less than before, for the gas is now partly in 
solution. If s denote the solubility, the available volume is 
practically increased in the ratio xix + sV, so that the pressure 
in the position x is now 

Nature, vol. LV. p. 253, 1897. 


and the work done during the compression is 
dx . x+sV 

On the whole the work lost during the two operations is 

f. x + sV , v } 
p Q vUog - + log-T^, 

( X 6 V ) 

and of this the first term can be neglected as x is indefinitely 
great. Since by supposition the quantity of liquid is such as 
to be just capable of dissolving the gas, sV = v and the second 
term also is equal to zero. The gas has therefore been dissolved 
reversibly without gain or loss of work. 

In order to complete the cycle, we must remove the gas 
from solution and bring it to its original state by an application 
of the osmotic process of Van 't Hoff. One semi-permeable 
membrane, permeable to the gas but not to the liquid 1 , is 
introduced just under the piston which rests at the surface of 
the liquid. A second, permeable to liquid but not to the gas, 
is substituted as a piston for the bottom of the cylinder and is 
in contact with pure solvent on its lower side. By suitable 
motions of the two pistons, the upper one being raised through 
the space v and the lower through the space F, the gas may be 
expelled at a constant pressure, p ', the solution which remains 
will keep a constant strength and therefore a constant osmotic 
pressure which we will call P. When the expulsion is com- 
plete the work done on the lower piston is PFand that done 
by the gas is p Q v. Thus the whole work done is PV p v, and 
this process as well as the first is reversible. 

Since the whole cycle has been conducted at constant 
temperature, it follows from the second law of thermodynamics 
that on the whole no work is gained or lost. Thus, 

PV-p v = or PV=p v. 

The osmotic pressure P is thus determined, and it is evident 
that its value is equal to that of the pressure which the gas, as 
a gas, would exert in a space F. 

1 The upper surface of an involatile liquid may itself be considered as such a 


This cycle could be performed at any temperature, and it 
must therefore follow that the temperature variation of the 
osmotic pressure of a dilute solution of constant concentration 
must be the same as that of a perfect gas. For dilute solutions, 
therefore, the osmotic pressure should be proportional to the 
absolute temperature. 

We have thus theoretically proved, without any assumption 
as to the real nature of a solution, that the laws of osmotic 
pressure are, for dilute solutions of volatile bodies, the same as 
those of gases. We may therefore collect our results in a form 
equivalent to the usual " gas equation " and write 


m denoting the number of gram-molecules considered, and the 
constant R having the same value as for gases. 

The extension of this theorem to the case of solutions of 
metallic salts and other substances not appreciably volatile 
involves a certain amount of assumption. It seems reasonable 
to suppose, however, that the distinction is one of degree rather 
than of kind, bodies of almost every grade of volatility being 
known. There is thus evidence to show that the results of the 
thermodynamic considerations we have given hold good for all 
solutions, and that the osmotic pressure of even non-volatile 
solutes has the same value as an equal number of molecules 
would exert could they be gasified and confined in the same 

Another point which must be noticed is the fact that the 
proofs assume that there is no change in the state of molecular 
aggregation as the solute is expelled from solution : that the 
same amount of substance yields the same number of integrant 
particles in each condition. Thus, if dissociation or association 
occurred either in the gaseous or the liquid state, the pressure 
exerted in that state would be changed to a corresponding 
amount. The importance of this remark in the case of electro- 
lytes will be evident later on, when we shall find that such 
bodies give abnormally high osmotic pressures if dissolved in 

It will be noticed that these deductions of the osmotic 


relations depend on the experimental result that the mass of 
gas in solution depends on its gaseous pressure. But the 
theory of osmotics can be placed on an abstract basis in- 
dependently of the law of the solubility of gases, which could 
then be deduced from it. The principles involved were laid 
down in a general manner by Gibbs as early as 1875 1 , and were 
also used in the equations given by von Helmholtz in 1883 2 , 
in a discussion of the energy relations of gases in connexion 
with the theory of galvanic polarization, which will be considered 
in a later chapter. The argument has been applied in a definite 
form to the problem in hand by Larmor, as quoted below 3 . 

Whatever the exact physical connexion between the solute 
and the solvent may be, " each molecule of the dissolved 
substance forms for itself a nidus in the solvent, that is, it 
sensibly influences the molecules around it up to a certain 
minute distance so as to form a loosely connected complex, in 
the sense not of chemical union but of physical influence. The 
laws of this mutual molecular influence are unknown, possibly 
unknowable ; but provided the solution is so dilute that each 
such complex is, for very much the greater part of the time, 
out of range of the influence of the other complexes, as for 
instance are the separate molecules of a free gas, then the 
principles of thermodynamics necessitate the osmotic laws. It 
does not matter whether the nucleus of the complex is a single 
molecule, or a group of molecules, or the entity that is called 
an ion : the pressure phenomena are determined merely by the 
number of complexes per unit volume. To determine the 
osmotic forces, we must know the change in available energy 
that is involved in dilution of the solution by further transpira- 
tion of the pure solvent into it. In finding that change, the 
laws of mutual action between molecules of the dissolved 
substance are not required : for there is actually no action 
between them, and as . soon as the solution becomes so con- 
centrated that such mutual action between the complexes 

1 Trans. Connect. Acad. m. 138 (1875) : (Equilibrium of Osmotic Forces). 

2 Abhandlungen, in. 101 (1883). 

3 Proc. Cambridge Philos. Soc. ix. 240 (1897) ; Phil. Trans. A. cxc. 205 


comes in, the theory is no longer exact. Nor are the laws of 
mutual action between the molecules of the dissolved substance 
and those of the solvent required, because the effect of trans- 
piration of more of the solvent into the solution is not in any 
way to alter the individual complexes. The change in available 
energy of the system, on dilution, thus solely arises from the 
expansion of the complexes into a larger volume ; and it can 
be traced into exact correlation with the change of available 
energy that occurs in the expansion of a gas. This argument 
meets the objection that a true theory should involve a know- 
ledge of the molecular actions between the various molecules. 
It would seem that with just the same cogency it might be 
argued that a real investigation of the connexion of the altera- 
tion of the freezing point of a liquid by pressure and its change 
of volume on freezing should involve a knowledge of the 
individual molecular actions in the liquid: and so it would, 
had we not the means of evading considerations of molecular 
constitution that is afforded by Lord Kelvin's great principle 
of dissipation, which is for this very reason at the basis of 
all physical theory. 

" There is however one point to be remembered, namely, that 
the theoretical osmotic pressure is a limiting value which may 
not be reached by an actual arrangement, unless we can be 
certain that it works reversibly and so without heating effects. 

"The remark has been made by Lord Kelvin, that the 
connexion between Henry's law and the osmotic law must 
break down when the solution of the gas is accompanied by 
change in its state of molecular aggregation. It is also probable 
from the fundamental ideas as to dissociation and aggregation, 
that such change would usually be partial, and not uniform 
over all the dissolved molecules ; so that it is not to be expected 
that Henry's law would in such circumstances hold good. The 
point in which the argument, as set forth in precise form by 
Lord Rayleigh, becomes then inapplicable, is that the gas 
expelled from solution by the osmotic process must be con- 
sidered as emerging in the actual state of aggregation differing 
from that of its free condition, and its return to the latter state 
involves further change of available energy." 


Thus the similarity between the laws describing the pres- 
sure of gases and those holding for the osmotic pressure of dilute 
solutions does not show that the real cause of these laws is the 
same in both cases. It simply depends on the fact that, in each, 
the molecules are so far apart that they are nearly always 
beyond each other's spheres of action. 

The foregoing investigations assume that both the heat of 
dilution and the change of volume on dilution of the solution 
considered are small. The results are therefore restricted to 
dilute solutions. In order to find a general expression giving 
the relation between the osmotic pressure of a solution and its 
concentration, let us take the free energy equation (p. 29), 

The free energy is the work obtainable by a reversible and 
isothermal process, and is thus equal to Pv, P being the 
osmotic pressure and v the increase in volume of the solution, 
when solvent is added isothermally and reversibly through a 
semi-permeable membrane. If the solvent had been added 
directly in a calorimeter, without a membrane, a certain 
amount of heat would have been evolved or absorbed in an 
irreversible manner. For small changes, this heat would be 
proportional to the increase in volume of the solution, and may 
thus be written as Iv, where I is the heat of dilution per unit 
change of volume. 

In this irreversible process, the internal energy of the 
system would decrease by an amount Iv, the external work 
being negligible. In the reversible osmotic process, the initial 
and final states of the system are the same as they are in the 
irreversible change, and thus the decrease in internal energy 
must be also the same, and e is Iv : we then get, 

.................... (13). 

'V, B 

Thus, in order to deduce the relation between osmotic pressure 
and concentration, we must experimentally determine the heat 
of dilution, and also know the rate of variation of the osmotic 
energy with the temperature. 


If we neglect the change of volume with temperature, 
equation (13) becomes 

o r> 

or P=l+6 ........................... (14), 

which gives a relation between the osmotic pressure and the 
heat of dilution of the solution. 

If the solution is so dilute that further addition of solvent 
produces no thermal effect, I is nothing, and the equation 


dP dd 
or, rearranging, -^ = -j . 

It can then be integrated and gives 

log P = log -f constant, 
or, P = 6 x constant, 

in accordance with our result on p. 108, which shows that, 
since the osmotic pressure of a very dilute solution always has 
the same absolute value as the gaseous pressure for the same 
concentration, it must be proportional to the thermodynamic 

Let us imagine a saturated solution, in contact with the 
solid crystals of its solute, to be confined in 

Osmotic pressure . i . 

and heat of a cylinder with the bottom formed of a semi- 

permeable membrane, and immersed in a large 
volume of the pure solvent. The piston of the cylinder is 
weighted till the pressure it exerts is equal to the osmotic 
pressure within the cylinder. The system is then in equili- 

By reducing the weight by an indefinitely small amount, 
the solvent can be made to enter from below and the piston 
to rise. The osmotic pressure is then performing work. As 
the solvent enters, more of the solid crystals will dissolve, 

CH. V] 



keeping up the saturation. This process will, in general, 

involve a (positive or negative) thermal change, and we must 

therefore let a (positive or nega- 

tive) quantity of heat enter to 

keep the system at a constant 


By increasing instead of dimi- 
nishing the weight, solvent can 
be forced out, and the whole 
process reversed. Thus, by keep- 
ing the pressure indefinitely near 
the osmotic pressure, and making 
the movements of the piston slow, 
the changes can be made rever- 
sible and isothermal. Fig. 39. 

In such a case the equation of free or available energy (p. 29) 

is applicable, just as it is to the case of the isothermal evapora- 
tion of a liquid or fusion of a solid. 

When the piston rises and a change in volume of the 
saturated solution from v l to v 2 occurs, the work done is 

P(,-,)- f, 

where P is the osmotic pressure. Thus the rate of increase of 
the available energy with the temperature at constant volume 
is given by 

d^r dP , . 

From the definition of ^r on page 23, 

^ = e - 0$, 
or for reversible changes, when H is equal to #</>, 

^ = 6-#, 

<, T/T, e and H denoting finite changes in the values of the 
usual functions. Thus, the free energy equation gives 

' .*- *='> 

w. s. 



If we suppose that the process of solution continues until 
one gram-molecule of the solid is dissolved, we may write X' for 
H, X' being the heat of solution of one gram-molecule of the 
solute when dissolved to form the volume, v, of saturated solu- 
tion in such a manner that the amount Pv of osmotic work 
is simultaneously performed. The result is the latent heat 

Instead of using the principle of available energy, this relation 
could have been deduced from the principle of entropy, or directly, 
by taking the system of cylinder, semi-permeable membrane and 
solvent through a complete reversible cycle, and writing down 
the expression for the efficiency of the process. The results 
expressed in equation (15) show that the rate of change with 
temperature of the osmotic pressure of a solution, kept 
constantly saturated, has the same sign as the heat of solution 
of the solid under the same conditions. 

The osmotic pressure of a solution depends on the concen- 
tration, and is approximately proportional to it. Thus the 
value of P in our equation is a function of the concentration of 
the saturated solutions, that is, of the solubility. 

The heat of solution in equation (15) is the heat which 
must be supplied to keep the temperature constant when one 
gram -molecular weight of solid is dissolved in a solvent to give 
a volume v of saturated solution, and a quantity of osmotic work 
Pv is done. The heat of formation as measured in a calorimeter, 
however, does not involve the performance of external work, 
and, since the internal energy of the solution must be the same 
by whichever process it is made, the difference between these 
two quantities of heat must be the thermal equivalent of the 
work done. Thus, in mechanical units, 

X' = X + Pv. 

The latent heat equation given above is general, but, if we 


limit the investigation to dilute solutions, we can write for one 
gram-molecular weight 

Pv = R6. 

The latent heat equation then becomes 

dPRO xfrdPI 

X= ^WP" := ^ -39?* 


~(lo ge P), 
that is, 

If we make the further assumption that the heat of solution is 
constant with changing temperature, a supposition which is not 
far from the truth in many cases for small ranges of tempera- 
ture, the equation can be integrated to give the result 

The calori metric heat of formation of the saturated solution 
X, is given by 

for dilute solutions ; the osmotic pressure is then 

where G is the concentration, that is the number of gram- 
molecular weights of solid dissolved in unit volume of the 
saturated solution. 

The latent heat equation gives, since R is constant, 


\ = R6 2 -j-g(\ogC) (17). 

Thus the calorimetric heat of formation of the saturated 
solution has the same sign as the temperature coefficient of the 



solubility, the heat of formation being reckoned positive when 
heat must be taken in by the system during the process of 
solution in order to keep the temperature constant. 

This heat will not, in general, be the same as the heat of 
solution in a large volume of the pure solvent, and very often 
has actually a different sign. Thus cupric chloride and also the 
hydrates of ferric chloride dissolve in much water with an 
evolution of heat, but when the solution is nearly saturated, it 
is cooled by taking up more of one of these solids. Heat must 
then enter the system to keep the temperature constant while 
solution is going on, and X is therefore positive. 

We can, from the equation, deduce X from the solubility 
curve, but the solubility cannot be deduced conversely from the 
heat of solution (though its temperature variation can), for if 
we integrate the equation we get 

log = J 

4- constant, 

and this constant, which determines the absolute value of the 
solubility, remains unknown. 
The equation 

\-JWj* (log OX 

however, like the other, can be integrated between limits on the 
assumption that the variation of X with temperature can be 
neglected, and then gives 

The osmotic pressure of an electrolyte, as we shall see later, 
is greater than the normal value, and must therefore be ex- 
pressed in the form 

where i denotes the ratio between the actual and the normal 
pressure. Assuming i to be constant, the equation then 


In order that the heat of solution and the ratio i should 
be sensibly constant, temperatures not too far apart must be 
chosen for experiment. Van 't Hoff, to whom these equations 
are due, gives a table of experimental verification from which 
we select the following examples 1 : 

ro X 00 calculated ^observed 

Boric acid 5*8 calories 5-6 calories 

Oxalic acid 8-2 8'5 

Potassium bichromate 17'3 17'0 

Amylic alcohol 31 2'8 

Phenol 1-2 2'1 

Alum 21-9 20-2 

Potassium chlorate 11 10 

Borax 27 '4 25-8 

For very slightly soluble salts such as barium sulphate, 
the saturated solutions are so dilute that the ionization may 
be taken as complete. The heat of solution is equal and of 
opposite sign to the heat of precipitation. This heat can be 
measured calorimetrically ; for example by treating a solution 
of barium chloride with one of sodium sulphate; the only 
change is the precipitation of barium sulphate, for the sodium 
and chlorine ions remain dissolved and unaffected 2 . 

l !/(?! 0. 2 1/C 2 X car 1 . X obs d . 

Barium sulphate 18 -4 50055 37'7 37282 5500 5583 
Silver chloride 13 -8 102710 26 -5 55120 15992 15850 

The importance of experiments on osmotic pressure was 
first pointed out by Van 't Hoff 3 , who called 

Experimental r . 

measurements of attention to the fact that Pfeffer's measure- 
osmotic pressure. . 

ments on cane sugar proved that the pressure 
varied as the concentration, i.e. that it was inversely pro- 
portional to the volume occupied by a given mass of sugar. 
This exactly corresponds to Boyle's law for gases. The following 
are some of Pfeffer's numbers as given by Van 't Hoff. 

1 Kongl. Svenska. Akad. Handl. xxi. p. 38, 1885. 

2 Lehfeldt's Physical Chemistry, London (1899), pp. 270, 271. 

3 loc. cit. ; Phil. Mag. xxvi. p. 81, 1888, or Zeits. f. physikal. Chemie, i. 
p. 481, 1897. 



[CH. V 

Percentage of sugar 
in solution 

Pressure in milli- 
metres of mercury 

Pressure calculated for 
one per cent, of sugar 






















The numbers in the last column are constant except for 
irregular experimental errors. 

In the case of gases, Boyle's law fails to represent the 
accurate relation between pressure and volume at high pres- 
sures, and it similarly fails for solutions when the concentration 
becomes considerable. We should expect the law of variation 
to be more complicated for solutions, since in addition to 
inter molecular forces similar to those brought into play in the 
case of gases, we shall here have forces between the dissolved 
molecules and the solvent. 

As we have seen, the theory shows that the pressure should 
increase as the temperature rises ; and that, for dilute solutions, 
the variation should follow the laws of gases and make the 
pressure proportional to the absolute temperature. This result 
has been examined experimentally by Pfeffer, who gives for 

cane sugar 

*! 14-15 P l 510 t 2 32 P 2 544 

15-5 520-5 36 567 

These numbers lead to a mean value for the coefficient of 
increase of pressure per degree of 1/234 of the pressure at 15. 

Again Bonders and Hamburger 1 found that the variation 
in pressure due to temperature was independent of the nature 
of the dissolved substance. This corresponds to the fact that 
the coefficient of increase of pressure is the same for all gases. 
The method used was a comparative one, and showed that 
solutions which were isotonic (i.e. gave equal osmotic pressures) 
at one temperature, 0, were also isotonic at another, 34. 

The protoplasmic contents of certain organic cells are 

1 Zeits. f. physikal. Chemie, vi. p. 319, 1890. 


surrounded by a membrane which seems to be very effective 
in only allowing pure water to pass. If such a cell be placed 
in a concentrated salt solution, the more dilute cell sap parts 
with water faster than the external liquid, the contents of the 
cell contract and shrink away from the cell walls. If on the 
other hand the cell be placed in water, liquid passes in, and 
the membrane becomes stretched. By staining the contents 
of the cell and having a graduated series of solutions of varying 
strength, it is easy to find, by observations with a microscope, 
what strength of solution gives equilibrium with the cell sap, 
and is therefore isotonic with it. Solutions of two different 
substances can thus be prepared so that both are isotonic with 
the contents of a given kind of cell, and, assuming that two 
solutions isotonic with a third are isotonic with each other, we 
can find the respective strengths of the two salt solutions 
which give equal osmotic pressures. De Vries 1 , who was the 
first to use this method, employed vegetable cells, and Bonders 
and Hamburger, in their investigation on the influence of 
temperature, used blood corpuscles. 

De Vries established the most important generalization, 
that solutions of different non-electrolytic substances containing 
the same number of gram-molecules in a given volume are 
isotonic. This is equivalent to saying that at equal pressures 
the solutions of all such substances contain, in a given volume, 
the same number of molecules, a statement which corresponds 
to Avogadro's law for gases. Tammann 2 confirmed this by 
allowing a drop of copper sulphate solution to fall into a 
solution of a ferrocyanide. A little membrane was at once 
formed round the drop, and the concentrations of the solutions 
were altered till, when this was done, no water entered or left 
the drop. Whether any such passage went on or not was deter- 
mined by noticing if there was any change in the index of 
refraction of the liquid just outside the little cell. 

It is important to observe that in the case of solutions 
which are electrolytes (that is to say, which have the power 
of conveying a current of electricity and of undergoing simulta- 

1 Pringsheim's Jahrbiicher, xiv. p. 427, 1884. 

2 Wied. Ann. xxxiv. p. 299, 1888. 


neous chemical decomposition), the osmotic pressure is greater 
than that given by the solution of a non-electrolyte containing 
the same number of gram-molecules in a given volume. Thus 
a table of the "isotonic coefficients" of some indifferent sub- 
stances given by De Vries is as follows, the isotonic coefficient 
being a number representing the osmotic pressure when that 
of an equimolecular solution of potassium nitrate is taken as 3 : 

Cane sugar 1*81 

Inverted sugar 1'88 
Glycerine 1'78 

while the coefficients of electrolytic solutions are greater: 

Potassium nitrate 3'0 

Sodium nitrate 3'0 

Potassium chloride 3'0 

Potassium sulphate 3*9 

Potassium tartrate 3'99 

Magnesium chloride 4'33 

Calcium chloride 4*33 

We shall examine this phenomenon in detail in a later 

When we pass on to the examination of the absolute value 
of the osmotic pressure, we find another striking relation to 
gaseous properties. We know that one gram of hydrogen or 
sixteen grams of oxygen, at normal atmospheric pressure and 
C., occupy a volume of about 11*16 litres. Therefore one 
molecular weight of a gas in grams (2 grams of hydrogen or 
32 grams of oxygen) occupies under these conditions a volume 
of 22*32 litres, or if compressed into one litre would, by Boyle's 
law, exert a pressure of 22'32 atmospheres. By Avogadro's law 
the same pressure would be exerted by any gas or vapour 
that was a considerable distance from its point of liquefaction. 
The absolute values of osmotic pressures have been found 
by Pfeffer, Adie 1 and Tammann. Pfeffer found that at 6'8 a 
one per cent, solution of sugar gave an osmotic pressure of 
505 mm. of mercury. The molecular weight of cane sugar 
is 342. Hence a one per cent, solution contains 

1 Chem. Soc. Jour. Proc. p. 344, 1891. 


^i_o_ of a gram-molecule in one litre. A volume of hydrogen 
or of any other gas, which contained % of a gram-molecule 
in one litre would at 6'8 exert a pressure of 

760 x -^ x 22-32 x ~ = 508 mm. of mercury. 

Thus we find that in dilute solutions of indifferent sub- 

(i) The osmotic pressure is proportional to the concentra- 
tion, that is, inversely proportional to the volume occupied by 
a given mass (Boyle's law). 

(ii) The coefficient of variation of pressure with tempera- 
ture is the same for all substances, and probably (though this 
is not fully established by experiment) the pressure is pro- 
portional to the absolute temperature (Gay Lussac's law). 

(iii) Solutions which exert the same pressures contain 
the same number of dissolved molecules in a given volume 
(Avogadro's law). 

(iv) The absolute value of the osmotic pressure of the 
solution of a non-electrolyte is the same as that of a gas or 
vapour containing the same number of molecules in a given 

Hence we find that the osmotic pressure of dilute solutions 
obeys all the gaseous laws, and has the same absolute value as 
it would have if the dissolved substance were transformed into 
a gas at the same temperature and confined in the same 
volume. Thus, the gaseous laws which we deduced theoreti- 
cally for dilute solutions of volatile substances are also 
established by direct experiment for non-volatile solutes. 

The direct determination of osmotic pressure is a very 
difficult process, but we shall show that there is a connection 
between this pressure and other properties of solutions 
their vapour pressures, and freezing points. This connection 
is independent of the particular view we take of the cause 
of osmotic pressure, and can be deduced simply from the 
principles of thermodynamics. For most purposes, therefore, 
it is better to make an experimental determination of the 
freezing point, and deduce the corresponding value of the 
osmotic pressure. 



Connection with osmotic pressure. The latent heat equation. The 
depression of the freezing point. Vapour pressures of concentrated 
solutions. Solubility of gases in liquids. Experimental measure- 
ments of vapour pressures. Boiling points. Determination of 
molecular weights. Freezing points. Osmotic pressure and freezing 
points of concentrated solutions. Experiments on the freezing points 
of solutions. Determinations of molecular weights. 

THE phase rule enables us to trace all the qualitative 

phenomena of the equilibrium of a solution 

ownotu^pressure. with ^ s possible gaseous and solid phases, 

but, in certain cases, a more detailed study 

of both the theory and the experimental investigation of the 

quantitative relations is possible. 

There is an intimate connection between the osmotic 
pressure of a solution, its freezing point and the saturation 
pressure of its vapour at a given temperature on which its 
boiling point obviously depends. 

Let us suppose that two vessels, one of which contains a 
solution of a non- volatile substance and the other pure water, 
are placed side by side under an exhausted bell-jar. Since the 
vapour pressure of the solution is less than that of the water, dis- 
tillation from one to the other will occur and the level of liquid 
in the vessel which contains the solution will rise. If the solu- 
tion be kept stirred, this process will go on until each liquid is 
in equilibrium with the vapour lying in contact with it. The 
difference in pressure of the vapour at these two levels will be 


equal to the weight of a column of the vapour equal in height 
to the difference between the heights of the two columns of 
liquid. Thus, if p be the vapour pressure of the solvent at the 
temperature of the experiment, and p r that of the solution, if <r 
be the density of the vapour, which we will at first assume 
uniform, and h be the difference in level of the two columns of 
liquid, which we must then take to be small, we get the 

The hydrostatic pressure in the solution at the level of the 
water is 

where p is the density of the solution. Thus 

P-P'^ ........ ............... (20). 

Here cr is the density of the vapour under its own pressure ; 
if <r be its density under the pressure of the standard atmo- 
sphere A, cr is o- p/A, and 

= = . ...(21). 

P A P 

The bottom of the solution vessel can now be replaced by a 
perfect semi-permeable membrane such as is described on 
p. 103, and put into connexion with the pure solvent ; equili- 
brium will still exist, for if not, water will enter or leave the 
solution, its level will change, the equilibrium with the vapour 
be upset and a constant circulation go on in an originally 
isothermal enclosure, a state of things which, contrary to ex- 
perience. would allow an unlimited supply of mechanical work 
to be obtained. Thus P can be taken as denoting the osmotic 
pressure of the solution, and we have a relation between it and 
the lowering of the vapour pressure. 

The freezing point of a solution is lower than that of the 
solvent if the solid which separates is the ice of the pure 
solvent, and it is easy to show that the vapour pressure of the 


solution at its freezing point is the same as that of the ice of 
the solvent at the same temperature. 

Let us imagine the system put through the following 
isothermal and reversible cycle, all the operations being per- 
formed at the freezing point of the solution. (1) Evaporate 
some solution, (2) compress or expand the vapour if necessary 
until it is in equilibrium with the solid of the pure solvent, 
(3) condense this vapour on to the solid, (4) allow the same 
mass of solid to melt in contact with the solution. This is 
a complete reversible isothermal cycle, and therefore no work is, 
on the whole, done. 

Let p' and p" be the vapour pressures of the solution and 
solid respectively, v the volume of vapour at pressure p', and v' 
its volume when the pressure is p' 1 ' . 

In process (1) an amount of work p'v is done by the vapour, 

in (2) the work (v v') is done on it, in (3) p"v' is done 

on it and in (4) no work is done. 
Equating the total work to zero 

p' v - P^ ( v - v ') - p " v ' = 0. 

The pressures involved are small, and during the second 
operation the vapour considered is not saturated and therefore 
the change may be taken as described by Boyle's law. 


p'v=p"v r . 

But - is a finite quantity, therefore 

v = v', 
and, by the Boyle's law relation, 


that is, the vapour pressure of the solution at its freezing point 
is the same as the vapour pressure of the solid of the pure 
solvent at the same temperature. 


We have seen that when a change of state is going on, the 
The latent heat relation of pressure to temperature at constant 
equation. volume can be described by the differential 


d6~~ d^-v^' 

If we confine ourselves at first to dilute solutions, the 
freezing points will only differ 
from that of the pure solvent by 
a very small amount. In Fig. 
40, then, which represents the 
equilibrium of the three phases 
of the pure solvent, solid, liquid 
and vapour, we need only consider 
the immediate neighbourhood of 
the triple point, T. The curve 
DTB denotes the vapour pressure 
of the liquid solvent, the part DT 
relating to the undercooled liquid, while the curve TO gives the 
vapour pressure of the solid. 

Along these two curves the equation may be written as 

H E 

Fig. 40. 

dp = X fa 
dd ~0v,- 


1 ' 

or, since we can neglect the small volumes of the liquid (I) or 
solid (s), as compared with the large volume of the vapour, 


Subtracting one of these equations from the other, and remem- 
bering that, at the triple point, the latent heat of the change 
from solid to vapour must be the sum of the latent heats from 
solid to liquid and from liquid to vapour, we have 

Sp 8 Spi = 


where a is the density of the vapour, and therefore equal to 
1/v, v being the volume of unit mass. 


Now, as we saw, the vapour pressure of a solution at its 

freezing point is equal to that of the solid phase 

Si h e e ffe e e P z?n S gpoint f of tne P ure solvent at the same temperature, and 

thus if we add to Fig. 40 the curve SF, giving 

the vapour pressure of the solution, the point F at which it cuts 

TC, the vapour pressure curve of the solid solvent, gives the 

freezing point of the solution. Now FG measures $p$8pi, 

and also denotes the lowering of vapour pressure of the solution 

as compared with that of the pure liquid solvent at the same 

temperature. It is thus equivalent to p p' in the equation 

, Pa- 
p-p=- . 

We may therefore write 

Xo- Per 

or S0 = ........................... (22), 


an equation which gives the connection between the osmotic 
pressure of a solution and the lowering of its freezing point in 
the limiting case where the dilution of the solution is pushed 
to the extreme. 

It is usual to obtain this equation by means of a thermo- 
dynamic cycle with a system composed of water separated from 
a solution by means of a semi-permeable membrane, but the 
above treatment of the problem, pointed out to the author by 
Professor Poynting, appears somewhat more direct, and has 
therefore been adopted. 

The vapour pressure equation 

Vapour pressures 

soi C ut 1 ions trated has been obtained by assuming that the density 
of the vapour in our exhausted bell-jar is every- 
where uniform. Such an assumption is only justified if the 
column of vapour of height h is short, that is, if the con- 
centration of the solution is exceedingly small. Where this 


is not the case, we must divide the height h of the vapour into 
a number of parts each equal to Sh and put 

Bp = gcrSh, 
or Bh . 

Let <r be the density of the vapour at the pressure, A, of 
the standard atmosphere, then 

T = r.J, 

and 8h = A . 

By integrating from to h we get 


p being the pressure at the level of the water, i.e. the vapour 
pressure of the pure solvent p, and p^ the pressure at the height 
h, i.e. the vapour pressure of the solution, p. 

Now h= , 


ff)\ P<r 
and therefore \og e ( -, } = - (23). 

This equation gives a necessary relation between the osmotic 
pressure and the lowering of the vapour pressure of any 
solution, and is quite independent of any assumption as to 
the physical nature of osmotic pressure. Be this what it 
may, we know that osmotic pressure exists, and it therefore 
follows that the vapour pressure must be lowered by the amount 
shown in our equation. The value of the osmotic pressure can 
thus be deduced from observations on the diminution of the 
vapour pressure, whatever be the concentration of the solution. 
It is easy to transform our equations into forms which 
give the concentration of the solution in terms of the ratio 
of the number of molecules of dissolved substance to the 
number of molecules of solvent, which simplifies the com- 
parison with experimental results. If one gram-molecule in 
a gas or solution fills a volume v at a pressure A, the osmotic 


pressure for a concentration of n gram-molecules in a volume 
v is 


Now the mass of the solvent is NM, where N is the number of 
gram-molecules and M its molecular weight, and the volume is 
the mass divided by the density, 


or v = - . 



The density <7 of the vapour under normal conditions of 
temperature and pressure is M'/v 0) where M' is the molecular 
weight of the solvent in the state of vapour. By substituting 
in the approximate equation (21) 

p-p' P<r* 

p Ap' 

p p' n M' 

We get r - - = irr- ^r 

p N M 

This equation shows that the relative lowering of vapour 
pressure depends on n, the number of molecules of the solute, 
but not on N, the number of those of the liquid solvent : if N 
be changed, M is changed in the inverse ratio, and the value of 
the expression is unaltered. If the molecular weight M' of the 
solvent as vapour is equal to M, the value assumed for its liquid 
in calculating the concentration of the solution in terms of the 
relative numbers of molecules, the equation becomes 

........................... <> 

If we treat equation (23), which gives the strict relation 
with the osmotic pressure, in the same way, assuming as before 
that P is proportional to the concentration, we get 

Both the ratios *- and , are independent of temperature, 
P P 


for heating the vapour in our bell-jar will change the pressures 
at the level and at the level h in the same proportion. The 
relative lowering of vapour pressure should thus be independent 
of the temperature, if no molecular change in the nature of the 
vapour takes place. 

These equations show that if solutions be prepared con- 
taining the same number of molecules of dissolved substance in 
the same number of molecules of solvent, the relative lowering 
of the vapour pressure will, on the assumptions specified, be equal 
in all cases. Thus if we have solutions in each of which there 
is one molecule dissolved in 100 molecules of solvent, the ap- 
proximate equation gives 

Raoult showed by experiment that if the same number of 
gram-molecules of various non-electrolytes were dissolved in 
water, or other solvent, the relative lowering of the vapour 
pressure was very nearly constant. He then took twelve 
solvents and, dissolving many bodies in each, proved that 
for a strength of solution of 1 molecule in 100 molecules the 
relative lowering of pressure was nearly constant, the mean 
value being about 0*0104. 

In 1890, however, he found 1 that, when acetic acid was used 
as a solvent, the number obtained was 0*0163. This seems to 
differ from the results of our equations, but it must be re- 
membered that in deducing them we assumed the molecular 
weight of the vapour to be the same as that which we took for 
the liquid. Now in reckoning the concentration of the solution 
the normal value of the molecular weight was taken for the 
liquid, and it is known that at moderate temperatures the 
vapour density of acetic acid is abnormal, showing that its 
molecular weight is also abnormal. At the boiling point, 
118 C., the ratio of the actual to the normal vapour density 
is 1*64, which makes the value of n/N 0'0164. We must 
always correct the theoretical number in this way by mul- 

] Eaoult and Recoura, Compt. Rend. ex. p. 402, 1890. 
W. S. 


tiplying it by the ratio of the actual to the normal vapour 

We have already seen (p. 84) that, with reference to their 

solubility in liquids, gases can be divided into 

S 1U i t n Hqu ids gaseS two classes : firstly, those which are removed by 

boiling the liquid or decreasing the pressure, 

and secondly those which cannot be so removed. 

In the first case, where the dissolved gas obeys Henry's 
law that the mass dissolved is proportional to the pressure, 
the laws of the vapour pressure are very simple. Let us 
consider a mass of air saturated with water vapour over a 
saturated solution of air in water. We know that if the 
external pressure be reduced, some air will at once come out 
of solution, while if the pressure be increased more goes in. If 
we have then some water with air dissolved in it over the 
mercury in a barometer tube, air will be expelled till that 
present in the barometric vacuum is in equilibrium with that 
dissolved, and whatever changes may occur in order that there 
may be equilibrium, the water must always keep saturated with 
air under the existing conditions of temperature and pressure. 
The pressure of aqueous vapour from the solution will obey the 
usual laws, and will therefore be less than that from pure water 
in accordance with our approximate equation 

or = . 

for the air in solution will exert osmotic pressure just like other 
substances. The total vapour pressure of the solution will be 
the sum of this and of the pressure due to the air, which, as we 
have seen, equals that in the vacuous space. This latter will 
depend on the relative volume of the solution and of the 
vacuous space, which takes air from the solution till there 
is equilibrium, so the measured vapour pressure would depend 
on the dimensions of the apparatus. We can however calculate 
the total vapour pressure in any given case if we know the final 


concentration of the solution. Thus if there are n gram-mole- 
cules of gas dissolved in N gram -molecules of solvent, the 
diminution of the pressure of aqueous vapour (due to osmotic 
pressure) is for dilute solutions 

If we know X , the solubility of the gas at the standard 
atmospheric pressure and C., we can find the vapour pressure 
of the dissolved gas, for 

\ ^ 

A*- y, 

where v is the volume of gas dissolved under normal conditions 
and V the volume of the solution. 

In a volume v c.c. there are v /22320 gram-molecules. Let 
us call this number n , then by Henry's law 

n _ p 

where p is the pressure of gas. 

. n A x 22320 n 
1 heretore p = A = . 

F, the volume of the solvent, contains -^ gram-molecules, 
where M = molecular weight and p the density of the solvent. 

v MN 
Hence V , 


A x 22320 on 
and = 

This gives the increase in the total vapour pressure due 
to the gaseous pressure, so the total increase in the vapour 
pressure is 

A x 22320 p 

In the second case of gases dissolved in liquids we have 
a solution like that of hydrochloric acid gas, which on distil- 
lation grows either richer or poorer in HC1 till a certain 
concentration is reached. The solution then distils over 



unchanged. This is exactly analogous to the solution of one 
volatile liquid in another and has already been considered 
qualitatively on p. 75. 

Determinations of the vapour pressures of solutions have 

been made by Faraday, Wullner, Tammann, 

measurements of Emden, Raoult, Walker, Beckmann, and others. 

vapour pressures. j^^j wag the firgt ^ examine solutions of 

organic substances, and to use solvents other than water. 
His method consisted in comparing the heights of three mer- 
curial barometric columns, the space over one being empty, 
and the others containing the vapours from the pure solvent 
and from the solution respectively. The depressions of these 
columns as compared with the first gave the vapour pressure of 
the solvent and of the solution. Raoult found that 

(i) The relative lowering of the vapour pressure 

is independent of temperature. 

(ii) For dilute solutions (p p')/p is proportional to the 
concentration n/N, but as the solutions get stronger it is more 
nearly represented by n/(N+ n), where n and N are the 
numbers of molecules of dissolved substance and of solvent 

(iii) The molecular lowering of vapour pressure (i.e. the 
lowering produced by 1 gram-molecule in 100 grams of solvent) 
is independent of the nature of the dissolved substance. Thus 
for ethereal solutions he found 

Molecular Molecular 

weight lowering 

Carbon hexachloride 237 '71 

Turpentine 136 -71 

Cyanic acid 43 -70 

Benzaldehyde 106 '72 

Aniline 43 -71 

Antimony chloride 228 -5 -67 

(iv) When the ratio of the number of molecules of the 
dissolved substance to the number of molecules of the solvent 

1 Compt. Rend. cm. p. 1125, 1886-7 ; civ. p. 1430. 

CH. Vl] 



is made the same, the relative lowering of vapour pressure is 
independent of the nature of the substance and of the solvent, 

and is measured by the value of -^ . 

N + n 

This is shown by the following experimental results : 


in degrees 



N + n 
(corrected for 
vapour density) 






Ethyl alcohol 






Carbon bisulphide 






Acetic acid 




We have already deduced all these results from the known 


values of the osmotic pressures, -~= being practically the 

same as n/N for dilute solutions. 

There are several objections to the barometric method. The 
quantity of vapour is so small that any impurity more volatile 
than the liquid would produce a large error, and since evapora- 
tion only occurs at the surface, the upper layers of the solution 
become stronger, and give too small a vapour pressure. Beck- 
mann 1 improved the method by allowing the solution to 
evaporate into a small flask. He then calculated the quantity 
of vapour produced from the decrease in weight of the solution, 
which was contained in a weighed bulb. 

A method applicable to low temperatures has been intro- 
duced by Ostwald and Walker 2 . A current of air is passed 
through two bulbs containing the solution, and is thus saturated 
with its vapour. It is then led through another bulb containing 
pure water. Since this gives a higher vapour pressure, the air 
takes up more water and again becomes saturated. Finally the 
whole of the aqueous vapour is extracted by passing the air 

1 Zeits. phys. Chem. iv. 532 (1889). 

2 Zeits. phys. Chem. n. 602 (1888). 



[CH. VI 

through pumice moistened with sulphuric acid. The gain in 
weight of the sulphuric acid gives the whole quantity of vapour 
evaporated, and the loss in weight of the water bulb gives the 
difference between the quantity furnished by it and that fur- 
nished by the solution. Thus the ratio (p p')/p is at once 

Tammann l has measured vapour pressures at 100 by 
noticing what decrease of external pressure was required to 
make the liquid boil at that temperature. He gives an immense 
number of figures showing the diminution of vapour pressure 
in millimetres of mercury, due to the solution of n gram- 
molecules in 1000 grams of water. We select a few of his 
results to which we shall have occasion to refer. 

n = 0-5 







Potassium chloride 
















Potash (KOH) 








Aluminium chloride 
















Succinic acid 








Citric ,, 















If we calculate the theoretical depression for a concentration 
of 0'5 gram-molecule in 1000 grams of water from equation 
(24) on p. 128 

P-P _ n 

P ~* T ' 
we get p p r 760 x y^ = 6'8 mm. of mercury. 

1 8 

Thus we see that bodies like lactic and succinic acids give 
a result w r hich agrees well with theory, while metallic salts are 
abnormal. Salts like potassium chloride, KC1, give numbers 
nearly double the figure deduced from theory, calcium and 
barium chlorides, CaCl 2 and BaCl 2 , produce nearly three times, 
and aluminium chloride, A1C1 3 , nearly four times the normal 

1 Mem. Acad. Petersb. xxxv. No. 9, 1887. Table in Ostwald's Lehrbuch. 


As we have seen in considering osmotic pressure, these ex- 
ceptions to the usual law all occur in the case of electrolytes. 
It is also important to note that KC1 contains two atoms, CaCl 2 
three atoms and A1C1 3 four atoms. The lowering of the vapour 
pressure by electrolytes seems then to be proportional to the 
number of atoms in the molecule. The discussion of these 
relations must be postponed for the present. 

Tammann's results show that in general the lowering of 
vapour pressure increases faster than the concentration for 
metallic salts, but appears to be nearly proportional to it for 
indifferent substances. The concentration of Tammann's 
solutions is expressed in terms of the number of gram- 
molecular weights of salt dissolved in 1000 grams of water. 
If we convert this into the number of gram-molecules in 
1000 grams of solution, the molecular lowering of vapour 
pressure will increase faster than Tammann's numbers do as 
the concentration gets greater. 

It is more convenient in some cases to measure the boiling 
point of a solution than its vapour pressure at 

Boiling points. o-i 

any other temperature. Since the effect of the 
dissolved substance is to reduce the vapour pressure at any 
given temperature, it must raise the boiling point, and the 
relation between the two is easily found. Let 1111 in Fig. 41 
be a portion of the vapour pressure curve of a solvent and II'II' 
a portion of that of a solution. If the solution is dilute, so 
that the change in the vapour pressure is small, we may 
consider the part of the curve for the pure solvent that we 
want to use to be a straight line. Any vertical line cutting 
IIII in A and II'IT in B will represent the change in vapour 
pressure at the corresponding temperature, and CB drawn 
horizontally from the point B to cut IIII in C, will represent 
the change in boiling point, BT. 

Now whatever be the direction and form of the solution 
curve II'IT, 

AB=CB tan ACS. 

Thus p-p' 

= ^. ........................ (26). 




If we observe 8T and know dpjdT for the pure solvent, we can 
at once calculate p p. The value of dp/dT can be experi- 
mentally determined by measuring the boiling point of the 
solvent first when the barometer is high and then when it is 
low, and dividing the difference in pressure by the difference in 


Fig. 41. 

Another method of getting dp/dT is to use the latent heat 


dp \ 

dT ss (v 9 -v 1 )T 9 

where X = latent heat, V 2 the volume of the saturated vapour 
and V-L the volume of the liquid. If we assume that the vapour 
obeys the gaseous law pv = RT, we get, since v l is small, 

dp _ \p 




Now we will assume that for 1 gram-molecule of the vapour 
the value of R is 1*980 calories : calling this 2, we can put 


CH. Vl] 



From this expression the relative lowering of vapour pressure 
can be calculated from observations on the rise of boiling point. 
In order to examine the validity of our theory, let us 
calculate 8T for a special case. Assuming that the law of pro- 
portionality holds at such concentrations, the equation on p. 129 
shows that, for a strength of solution of 1 molecule in 100 
molecules of solvent, (p p')\p is equal to '01 ; so for this 
concentration our equation gives 

09 T 2 
Sr= (29). 


The boiling point method was placed on a satisfactory 
footing by Beckmann 1 , whose apparatus in some form is now 
usually employed. It is necessary to measure the temperature 
of the solution, and not the temperature of its vapour which, 
although it comes off at the boiling point of the solution, soon 
cools to its saturation temperature ; condensation then begins 
and keeps the temperature of the vapour constant and equal to 
the condensing or boiling point of the pure solvent. To prevent 
"bumping" a piece of platinum wire is sealed through the 
bottom of the flask. Boiling then takes place exclusively 
from the end of this, and a constant and uniform stream of 
bubbles is given off. The following table gives the calculated 
values of '02 T 2 / A., and the mean results for the molecular rise 
of boiling point, deduced from observations on very dilute solu- 
tions in different solvents by means of Beckrnann's apparatus. 

02 T 2 


3T (observed) 

/ fk o 1 / i -i I f\ 4- g\ A \ 



4 to 5 



10 to 12 



17 to 18 



21 to 22 


Carbon bisulphide 

22 to 24 


Acetic acid 



Ethyl acetate 

25 to 26 



25 to 27 



35 to 36 


1 Zeits. phys. Chem. iv. 539 (1889). 


The problem of accurately determining the boiling point of 
a solution is different from that which arises when the boiling 
point of a pure liquid is required. Regnault's method of 
measuring the latter is described in most text-books of physics. 
It consisted in immersing as much of the stem of the thermo- 
meter as possible in a current of vapour from the boiling liquid, 
the vapour being contained in a copper cylinder the outside of 
which was also surrounded by a jacket of vapour. The thermo- 
meter passed through a cork and the top of the mercury column 
projected above in order that it might be visible. The un- 
certainty of temperature in this part of the stem causes an 
error both in graduating a thermometer and in determining 
boiling points. It has also been shown by J. Y. Buchanan 1 
that the jacket of vapour is unnecessary, the latent heat of the 
vapour keeping the inside of the vapour chamber, on which a 
film of liquid forms, accurately at the boiling point. It is 
therefore better to boil the liquid in a flask and to pass the 
vapour through a vertical glass cylinder about 4 centimetres 
in diameter which is narrowed below and fitted into the flask, 
the vapour escaping above directly into the air by a horizontal 
outlet. The thermometer can then be wholly immersed in the 
vapour, and read through the transparent walls of the cylinder. 
Even if involatile impurities are present in the liquid, the 
vapour which comes off is pure, and pure liquid will therefore 
condense on the bulb of the thermometer until its temperature 
is such that there is equilibrium between the film of pure 
liquid and its vapour. The thermometer must therefore show 
the boiling point of the pure liquid. 

Now let us pass to the consideration of the case of solutions. 
If a current of steam be passed into an aqueous solution of a 
salt, condensation occurs until the temperature rises beyond the 
temperature of the steam and reaches the boiling point of the 
solution. The explanation of this remarkable fact is seen if we 
remember that the boiling point of a solution is the temperature 
at which it is in equilibrium with the vapour arising from it. 
At lower temperatures, therefore, vapour will condense on the 

1 Trans. R. S. E. xxxix. 547 (1899) ; also Chemical and Physical Notes, 



surface of the solution, and its latent heat, which is of course 
really molecular energy transformed into heat, warms the liquid 
and vapour till the boiling point is reached. Thus a current of 
vapour passed through a solution must come out at the tem- 
perature of the boiling solution, and the same statement will 
hold good for the steam generated by the boiling of the solution 
itself: the steam from a boiling solution must come off at the 
boiling point of the solution. This, at first sight, appears 
contrary to the fact that a thermometer in the vapour registers 
the boiling point of the pure solvent, but, as soon as the vapour 
emerges from the solution, it is cooled by contact with the walls 
of the vessel, etc., and will therefore fall in temperature until 
the point of equilibrium with the pure solvent is reached The 
steam is then saturated, and the slightest further cooling will 
cause condensation on the walls or on an immersed thermo- 
meter. The latent heat set free by this condensation will warm 
the system and prevent it falling below the boiling point of the 
pure solvent. Thus the walls of the steam chamber and any 
thermometer hung in it will be constantly maintained at the 
temperature of the boiling point of the pure solvent, and, in 
order to determine the boiling point of a solution, the thermo- 
meter must be placed in the liquid itself. 

The passage of steam through a solution has been applied 
by Buchanan in a convenient method of 
measuring the boiling points of saturated 
solutions, or, when these temperatures are 
known, of checking the graduations of a 
thermometer at parts of its scale above the 
boiling point of water. A quantity of dry 
salt in small crystals is placed in the glass 
cylinder shown in Fig. 42, and a current of 
steam is then passed through, till in a few 
minutes, a mixture of boiling saturated brine 
and salt is produced, which remains at a 
constant temperature until nearly all the salt 
is dissolved. 

The boiling point of a solution, like that of 

Fig. 42. 

a pure liquid, depends on the barometric pressure to which it is 


subject. The change of boiling point for a given change of 
pressure is not the same for a solution as for its pure solvent, 
but the considerations detailed at the beginning of this 
chapter show that the ratio of the vapour pressures of 
solution and solvent, or the relative lowering of vapour 
pressure of a solvent produced by dissolving in it some 
solute, is constant at any temperature. 

A promising way of measuring the boiling points of non- 
saturated solutions has been described by E. B. H. Wade 1 . 
The essence of the method consists in determining the differ- 
ence in temperature between a solution and its solvent, boiling 
in similar vessels under the same atmospheric or artificial 
pressure. A difference in temperature can be easily and 
accurately measured by means of two platinum thermometers 
consisting of two equal coils of platinum wire wound on mica 
frames at the bottom of two long glass tubes. The difference 
between the electrical resistances of these coils gives at once 
the difference in temperature between them. Such duplicate 
measurements are free from errors due to changes in the pressure, 
which affect the results when the boiling points of solvent and 
solution are observed on different occasions. The liquids were 
heated by passing currents of steam through them, and at the 
instant when the measurement was made, solution was with- 
drawn for analysis. 

Like the depression of the freezing point, the lowering of 
vapour pressure has been used to determine 

Determination L x .... , . . _ 

of molecular the molecular weight of bodies in solution. It 

can be used for high temperatures, and for 
cases, such as for solutions in alcohol, when the freezing point 
method is not applicable. In this way Beckmann obtained 
the molecular weights of iodine, phosphorus and sulphur in 
solution. It was found that 1*065 grams of iodine, dissolved in 
30*14 grams of ether, raised the boiling point by 0*296. This 
concentration corresponds to (1*065 x 7400)/(30*14 x M) gram- 
molecules of iodine in 7400 grams (100 gram-molecules) of 
ether. Now it can be proved either by experimenting with 

1 Proc. E. S. LXI. 285, 1897 and LXII. 376, 1898. 


a body of known molecular weight, or by calculation from our 
formulae, that 1 gram-molecule of any non-electrolyte, dis- 
solved in 100 gram-molecules of ether, gives a change in the 
boiling point of 0*284. The above strength of solution must 
therefore be 296/284 gram- molecules. 

1-065 x 7400 296 
30-T4-- = 284' 

so that .=250-3. 

The atomic weight of iodine is 127, so that in ethereal solution 
the molecule consists of two atoms. 

In a similar manner it was shown that the molecule of 
phosphorus in carbon bisulphide contains 4 atoms, as it also 
does in the state of vapour, but that in the same solvent the 
molecule of sulphur consists of 8 atoms, whereas the vapour 
density gives a formula S 6 . 

The vapour pressures of amalgams have been examined by 
Ramsay 1 , who found that in nearly all cases the lowering of 
vapour pressure corresponded to that which would be produced 
by monatomic molecules. The value deduced for the molecular 
weight of potassium is however less than its atomic weight 
(29'6 instead of 39'1) and the numbers for calcium and barium 
(19'1 and 75*7) correspond to half their atomic weights. What 
this means it is as yet impossible to say. Aluminium and 
antimony tend to form more complex molecules. 

It is obvious from what has been said in Chapter II that the 
true freezing: point of a substance is the tem- 

Freezing points. , . , , i i T i , 

perature at which the three phases, solid, liquid, 
and vapour, are in equilibrium with each other, giving a non- 
variant system. In other words, it is the temperature at which 
solid and liquid are in equilibrium under the natural pressure of 
the vapour. As commonly understood, however, the pressure 
under which the equilibrium exists is not that of the vapour 
corresponding to the temperature of the freezing point, but 
that of the atmosphere. Thus, in the case of water, the 
saturation pressure of the vapour at the freezing point is about 

1 Chem. Soc. Journ. p. 521, 1889. 


4'7 mm. of mercury, and the true freezing point is, as we have 
seen, about 0'007 higher than the temperature of equilibrium 
under the atmospheric pressure of 760 rnm. 

Since the change in volume on fusion is usually small, the 
latent heat! equation shows that the change of freezing point 
with pressure is small also, and there is usually not much 
difference between the two freezing points. We shall, there- 
fore, for the sake of convenience, understand the freezing point 
to be determined under atmospheric pressure, remembering that, 
to deduce from it the true transition point of the substance, 
the pressure correction must be applied. 

The freezing point of a solution, as we have seen, is in 
general different from that of its pure solvent, the direction of 
the change depending on the nature of the solid which freezes 
out. When the solid separating is that of the pure solvent, 
the freezing point of the solution is always lower than that of 
the solvent, and for dilute solutions the depression is related 
to the osmotic pressure by equation (22) on p. 126 

If the solid is not the pure component, a study of the freezing 
point curve will generally, as we have seen in Chapter III, 
enable a knowledge of the nature of the solid to be deduced. 
In the case of salt solutions, the ice is usually that of pure 
water, and a proof of this fact has been given for solutions of 
sodium chloride by J. Y. Buchanan 1 . The ice as it freezes out 
always entangles some brine among its crystals, and this salt is 
so difficult to remove that it has often been thought to form an 
integral part of the solid phase. Buchanan showed, however, 
that the impure ice so obtained, when immersed in a salt 
solution, reduced it to exactly the same temperature as pure 
ice reduced a solution of the same final concentration, allowing, 
that is, for the salt added to the solution by the impure ice. 

The phenomena of the cryohydric point have already been 
considered in the light of the phase rule, which regards it 
as the non-variant point at which the system is completely 

1 Proc. R. S. E. xiv. 129 ; also Nature, xxxv. 516 and 608, xxxvi. 9, 1887. 

CH. Vl] 



determined, and a transition from liquid to solid or vice versa 
occurs without change of temperature. The subject must now 
be further considered, and some applications of the principles 
deduced made to the phenomena of freezing. Experimental 
determinations of cryohydric temperatures for many different 
salts have been made, among others by L. C. de Coppet 1 , from 
whose results we take the numbers which follow. 


temperature in 
Centigrade degrees 

Weight of 
anhydrous salt in 
100 parts of water 

Potassium chloride 




Sodium chloride 




Ammonium chloride 

NH 4 C1 



Strontium chloride 

SrCl 2 


Barium chloride 
Zinc sulphate 

BaCl 2 .2H 2 
ZnS0 4 .7H 2 

- 7-85 
- 6-55 


Copper sulphate 

CuS0 4 5H 2 

- 1-6 


Ammonium sulphate 

(NH 4 ) 2 S0 4 

- 19-05 


Potassium chrornate 

K 2 Cr0 4 



Sodium sulphate 

Na 2 S0 4 .10H 2 

- 1-2 


NaIS0 4 . 7H 2 O 

- 3-55 


Sodium carbonate 
Potassium nitrate 

Na.,COo. 10H 2 
KN0 3 

- 2-1 

- 2-85 



Sodium nitrate 

NaN0 3 - 



Ammonium nitrate 

NH 4 N0 3 

- 17-35 


Barium nitrate 

Ba(N0 3 ) 2 

- 0-7 


Strontium nitrate 

Sr(N0 3 ) 2 

- 5-75 


Lead nitrate 

Pb(N0 3 ) 2 

- 2-7 


If a solution be continually cooled ice will separate, and, 
since it is the ice of pure water, the residual solution becomes 
more and more concentrated, and its freezing point, therefore, 
lower and lower, till the cryohydric point is reached. Any 
further cooling will then deposit salt side by side with the ice, 
and the composition of the liquid phase remains constant as 
long as any liquid is left. At certain stages in the process, 
therefore, there will be crystals of ice separated by mother- 
liquid, the composition of which is that of the cryohydric 
mixture. The sizes of these crystals will depend on the nature 
of the freezing operation, its speed and the amount of stirring 
to which the liquid was subject during the process. Even the 
purest natural water contains a certain quantity of dissolved 
1 Zeits. phys. Chem. xxn. 239 (1897). 



[CH. VI 

matter, and thus natural ice has always a grained structure, the 
crystalline elements being separated by a film of liquid brine, 
the thickness of which depends on the amount of impurity 
originally present in the water. Only when the temperature is 
below the cryohydric point does the whole mass become solid, 
the cryohydrate acting as a kind of cement connecting the 
grains of pure ice. The dissolved air, which is always present 
to some extent in natural water, greatly increases the granular 
appearance of the structure. As the ice forms, this air is 
gradually expelled from solution; but the last traces of it, 
remaining in the films of liquid cryohydrate, are entangled as 
bubbles when the films solidify and thus further emphasize the 
lines of separation between the primary crystals. The grained 
structure is well seen if a block of natural ice, taken, say, from 
the inside of a glacier, is exposed to the sun's rays. The block 
is rapidly disintegrated, and is finally reduced to a heap of 
crystalline grains. 

The corresponding structure has been observed for mixtures 

Gold aluminium alloy. 
Magnification 450. 

Fig. 43. 

Magnification 100. 

Fig. 44. 

of metals. Microscopic photographs of alloys show clearly the 
crystals of the first solid formed (sometimes a pure component 
and sometimes a metallic compound) separated by channels of 
eutectic alloy which has solidified at a lower temperature than 
the crystals imbedded in it. Fig. 43 l is drawn from one of 

Phil. Trans. A. cxciv. 201, Plate 4 (1900). 


Heycock and Neville's photographs of an alloy of gold and 
aluminium containing 19'8 atomic per cents, of aluminium. 
Fig. 44, in like manner, shows a photograph by Ewing and 
Rosenhain 1 of a surface of cadmium cast on a glass plate. The 
thinness of the connecting lines here indicates that the metal is 
very nearly pure. The slight traces of other metals present, as 
well as any gases dissolved in the molten metal, are concen- 
trated in the channels left between the crystals as they slowly 

Returning to the case of water, we observe that, as Buchanan 
has pointed out, the power which ice in the form of a glacier 
possesses of flowing along a curved bed may be partly due to 
these channels of low-freezing liquid between the solid grains. 
The liquid films allow the ice to yield slightly under stress, and 
then the heat developed by the grinding of the crystals over 
each other raises the whole mass to the freezing point of pure 
water, and enables the regelation of ice (i.e. the melting of the 
solid under high pressure and its freezing again when that 
pressure is removed) to come into play and help the flow, in 
the manner suggested by the usual explanation of the plasticity 
of ice. The fracture of solid crystals and the sliding of the 
broken parts over each other under the action of a shearing 
stress has been described in the case of metals by Ewing and 

The phenomena of the freezing of sea water under the influ- 
ence of the intense cold of an arctic climate is an interesting 
example of the application of these principles. The process has 
been described by the explorer Weyprecht 2 , whose account is 
quoted by Buchanan. When a new surface of sea water is ex- 
posed to the cold air, in a short time the surface of the water 
begins to get thick, threads like a spider's web running out 
from the old ice. Brine is entangled in this structure, and its 
concentration continually gets greater as the quantity of ice 
increases. At this stage the ice is a pasty mass and follows 
every motion of the water on which it floats. With a tem- 

1 Phil. Trans. A. cxcm. 353, Plate 26 (1899). 

2 Die Metamorphosen des Polareises, Wien (1879). 

W. S. 10 


perature of 40 C. the new ice, even after twelve hours, is 
still so soft that, in spite of its thickness, a stick can easily be 
thrust through it. 

As soon as a layer of ice is formed over the surface, the 
cooling of the underlying water proceeds much more slowly, 
and less salt is entangled in the crystals. The lower layers of 
sea water ice therefore give, when melted, a much fresher water 
than can be obtained from the upper layers. Even when strong 
enough to walk on, the surface of new sea water ice frozen by 
air at 40 C. is still moist and soft, the residual liquid consist- 
ing of a concentrated solution of various salts, chiefly calcium 
chloride. The cryohydric point of calcium chloride, a very 
soluble substance, is very low, and that of a mixture of salts, 
unless they form mixed crystals, will be lower than that of 
either component, just as is the freezing point of a mixture 
of metals. This lowering of the cryohydric temperature was 
observed by Buchanan in experiments conducted in the 

It is obvious that the cryohydric mixture can be prepared 
by cooling a solution of any concentration and letting it freeze till 
the temperature becomes constant; the liquid remaining, if it 
be poured off, will freeze throughout at a constant temperature, 
and then give the cryohydrate. Another method of prepa- 
ration consists in mixing intimately snow or finely powdered 
ice with the powdered solid. Salt dissolves and melts some of 
the ice, the latent heat of which causes the temperature to fall 
until the cryohydric point is reached, when a saturated solution 
of the salt at its freezing point is in equilibrium with ice. It 
is evident, then, that ice and salt together form what is called 
a freezing mixture. In order to get the full cooling effect, the 
finely divided ice or snow should be dry, that is, well below the 
freezing point of water, otherwise the adherent moisture will 
soon be frozen when the mixture is made, and further contact 
between the parts prevented. The salt, too, should be cooled 
below before mixture. The temperature will sink more 
rapidly, also, if a salt be used which dissolves with an absorp- 
tion of heat, for then this cooling effect is added to the 


From equation (22) for indefinitely dilute solutions, deduced 
on p. 126, namely 


it is easy to calculate numerically the lowering of the freezing 
point of a solvent produced by dissolving in it a small quantity 
of some non-electrolyte. 

Let us take the case of a water solution of any body contain- 
ing one gram-molecule per litre. We have seen (pp. 104, 120) 
that the osmotic pressure is the same as the dissolved molecules 
would exert in the gaseous state. It is therefore 22'32 atmo- 
spheres, or 22-32 x 76 x 13'6 x 981 C.G.S. units. 

For water p = 1, T= 273 and X = 79'4 calories or 

79'4 x 4-184 x 10 7 ergs. 

If we calculate BT with these numbers we find that the freezing 
point of water should be lowered by one gram-molecule of 
dissolved substance per litre, by 

l-857 C. 

Raoult 1 made many experiments on this subject and his 
results give a mean value of 

l-85 C. 

for the same effect. Other observers have subsequently found 
results for cane-sugar differing considerably from this value; 
but just lately E. H. Griffiths, using the most exact methods of 
platinum thermometry, has found that for dilute solutions of 
cane-sugar in water, ranging in strength from 0'0005 normal 
to 0*02 normal, the molecular lowering of freezing point is 


It is easier to make a comparison with Raoult's results by 
changing the form of our equation, but the effects of dissolved 
bodies on any solvent can be calculated from (22) by using the 
values for T and X given on p. 149. This equation is the simplest 
expression of Van 't Hoff's theory, and the one which shows 
most clearly the connexion between the lowering of freezing 

1 Comp. Rend. xciv. p. 1517 (1882). 



point and the osmotic pressure ; another form however may be 

In our equation (22) let us put, since dilute solutions obey 

Boyle's law, 

Pv = RT, 

v being the volume of solution containing one gram-molecule of 
solute. The expression then becomes 


R is a constant of which the value for one gram- molecule of any 
gas or substance in dilute solution is, as we have shown on p. 4, 

R =77= 8*284 x 10 7 ergs per degree 
= T980 calories per degree, 

taking as the most probable value for the mechanical equivalent 
of heat 4-184 x 10 7 . 

XT 100 

Now v - , 


where n represents the number of gram-molecules per litre. 
We then get 

H)80 T 2 

" lOOOXp 

0-001980 T*n 

In the case of water this gives 8jP=l*86tt, and of course the 
value for other solvents can be deduced in a similar manner. 

Raoult expressed the concentrations of his solutions in terms 
of the number of gram- molecular weights of substance dissolved 
in 100 grams of the solvent. From observations on more dilute 
solutions, he calculated, on the assumption that the law of pro- 
portionality was still applicable, the depression of the freezing 
point which would be produced by one gram- molecule dissolved 
in 100 grams of solvent. 

We can at once throw our equation (22) into a form in 
which comparison with Raoult's results for different solvents is 

easy. The volume of 100 grams of solvent is - - . We have 

CH. Vl] 



seen that for dilute solutions, the osmotic pressure has the 
gaseous value 22*3 atmospheres per gram-molecule per litre. 
For such solutions, the density of the solution is the same as 
that of the solvent, and, if the law of proportionality still held 

good, when we dissolve one gram-molecule in c.c., we should 

get a pressure which is greater than that given by one gram- 
molecule per litre in the ratio of 

1000 : or lOp : 1. 

The value of n becomes lOp times greater than before and 
equation (31) assumes the form 

0-00198 r 1-98 T 2 2T* 

~V~ IOp= Wx = ToOA (32) ' 

The comparison between the values calculated from this 
equation by Van 't Hoff, and Raoult's observed numbers is 
given below. 










Acetic acid 





Formic acid 















The agreement between these results is sufficient to show 
that, at all events in dilute solutions, the theory of Van 't Hoff, 
according to which the osmotic pressure has the same absolute 
value as gaseous pressure, leads to results which agree with 
observation to a considerable degree of accuracy. 

Raoult stated that one molecule of a substance dissolved in 
100 molecules of solvent always gave a depression of the freezing 
point which was approximately equal to 0'63, and supported 
this generalization by experiments on solutions in formic acid, 
acetic acid and benzene. Theory gives no ground for such an 
assertion, but if we work out formula (32) for these particular 


cases, we shall find that, as a matter of fact, the numbers all 
happen to be nearly what Raoult gave. If the molecular weight 
of the solvent be M, the quantity represented by 100 gram- 
molecules is M times that represented by 100 grams, so that 
the solutions will be only l/M as strong as those we dealt with 
in the last table. The depression of the freezing point will 
therefore be not 2T*/WO\ but 2T 2 /IQQ\M. If we divide 
the figures given in the table by the molecular weights of 
the solvents we get for the depressions 

Formic acid = 0'62 

Acetic acid = 0*65 

Benzene = 0'68 

The approximate constancy of these numbers is however a 
pure accident, and does not hold for other bodies ; thus water 
gives 1'05. This point has been fully examined experimentally 
by Eykrnan 1 , who concludes, although his result for naphthy- 
lamine seems to be abnormal, that the evidence is conclusive 
in favour of Van 't Hoff. In Raoult's original generalization 
he was misled by a purely accidental agreement of numbers, 
and he has since accepted Eykrnan 's results and the accuracy of 
Van 't Hoff's formula. 

When the restriction limiting the treatment to the case of 
dilute solutions is removed, both the change of 
volume and the heat of dilution must be con- 
sidered. This has been done by T. Ewan 2 , 
from whose paper the following investigation 
has been adapted with simplifications. With the help of a 
semi-permeable membrane a reversible cycle can be performed. 
Beginning with a mass of solution m, in equilibrium at its 
absolute freezing point T 1} with an indefinitely small quantity of 
ice, carry the system through the following cycle : 

1. Allow dm gram of ice to freeze out, the heat which 
must be supplied to keep the temperature constant is 

LI dm + ^^ dm, 

1 Zeits. f. physikal. Chemie, m. 203, 1889. 

2 Zeits. /. phyaikal. Chemie, xxxi. 22, 1899. 



where L is the heat of fusion of one gram of ice and - J the 

heat of dilution, both at the temperature T lf The heat of 
dilution is measured at constant temperature and is therefore 
written as a partial differential. 

2. Heat the ice and solution separately to T the 
freezing point of the pure solvent on the absolute scale. The 
heat added is 

dm iGtdT + (m - dm) [*' (c t - ^ dm] dT, 
J T, J r, \ wn / 

Ci being the specific heat of ice, and GI that of the original 

3. Melt the ice at T . The heat added is L dm, where 
LQ is the heat of fusion of one gram of ice at T . 

4. Return the water to the solution through a semi- 
permeable membrane at a constant temperature T . The heat 
supplied to maintain this constancy of temperature is 

-^dm + Pvdm, 

where P is the osmotic pressure of the solution at T and v is 
the increase in volume which the solution undergoes per gram 
of water added to it at constant temperature and concentration, 
the volume of the original solution being large. 

5. Cool the solution thus formed to T^. The heat 
added is 

All these operations are reversible, and at the end the 
system is exactly in its original condition. We can therefore 
apply the second law of thermodynamics and write 



Collecting the terms and integrating we get (33) 
! Z, 13ft IdQ, T a dC, T a Pv 


In order to reduce this to a form which involves only terms 
which can be experimentally determined we must notice : 

(a) That a gram of ice at T^ can be changed into a gram 
of water at T by adding heat either 

L l + C w (T -T l ) or CiW-TJ + L,. 
Thus L, - Z = (C w - d) (T - T,) 

A A r /I I 

T 1 -T -^( l - 

(b) The difference in heat-contents of the solution 
between T and T is 

Therefore p - s = m f' (T a - 2\) 

dm cm dm 

TO dm T! dm dm \T Q TJ dm \ T l ) ' 

Substituting these values in equation (33) and expanding the 

Pv (I l\., r r} 

-nT = L (m - nT ) + W^ ~ *J 

T 1 IUr / y o- aft 
" 4 

Thu -/ 

- - 

(34) ' 

an equation which gives the relation between the osmotic 
pressure and the depression of freezing point, T () T 1 . 


It has long been known that the freezing point of a salt 
solution, such as sea water, is lower than that 

Experiments on . 

the freezing points of the water when pure, and in 1788 Blagden 1 
published some observations on the subject, 
which showed that the depression of the freezing point produced 
by dissolving a substance in water, was approximately pro- 
portional to the quantity of substance in solution, except when 
the concentration became considerable. 

More recent observations were made by Riidorff 2 and 
de Coppet 3 . -The latter noticed that if the lowering of the 
freezing point produced by chemically equivalent quantities of 
different salts was examined, it was found that the molecular 
lowering was nearly equal for salts of similar chemical con- 

The whole subject was first fully examined by Raoult 4 , who 
extended his observations to non-electrolytes, such as solutions 
in pure benzene, and solutions of organic compounds in water. 
He found that the depressions produced by equi-molecular 
quantities of different substances were nearly of the same value. 

Further measurements have been made by Arrhenius 5 , 
H. C. Jones 6 , Loomis 7 , Wildermann 8 , Archibald 9 , Barnes 10 , 
Pickering 11 , and many others. The theory of such determina- 
tions has been treated by Nernst and Abegg 12 , and since the 
publication of their results the precautions necessary to ascer- 
tain the true freezing point have been more fully understood. 

1 Phil. Trans., LXXVIII. p. 277. 

2 Pogg. Ann., 1861, 114 et seq. 

3 Ann. Chim. Phys., 1871, n. 23, 25, 26. 

4 Comp. Rend. (1882), xciv. p. 1517, xcv. pp. 188, 1030. Ann. Chim. Phys. 
(6), n. p. 66, (5), xxvm. p. 137, (6), iv. p. 401. Zeits. phys. Chem. xxvm. 617 
(1898) and Cryoscopie, Paris 1901. 

5 Zeits. phys. Chemie, n. 491 (1888). 

6 Zeits. phys. Chemie, xi. 110 and 529 (1893) ; xn. 623 (1893). 

7 Phys. Review, i. 199 and 274 (1893-4) ; m. 270 (1896) ; iv. 273 (1897) ; xn. 
220 (1901). 

8 Zeits. phys. Chemie, xix. 233 (1896). 

9 Trans. Nova Scotia Inst. Sci. x. 33 (1898). 

10 Trans. Nova Scotia Inst. Sci. x. 139 (1899) and Trans. R. S. Canada [ii.j 
vi. 37 (1900). 

11 Chem. Soc. Jour. 1893. 

12 Zeits. furphysikal. Chemie (1894), xv. 7, 681. 


The freezing point may, as we have said, be defined as the 
temperature at which an isolated mass of liquid can exist in 
permanent equilibrium with its own solid under the normal 
atmospheric pressure. It had been assumed that the stationary 
temperature assumed by a small quantity of a partly frozen 
liquid, contained in a vessel surrounded by a freezing mixture, 
gave at once the true freezing point, but Nerrist and Abegg 
pointed out that this limited volume of liquid, radiating to an 
outer enclosure, would, irrespective of freezing, tend to reach 
a convergence or equilibrium temperature, which depends on 
the amount of heat evolved by stirring and on the temperature 
of the environment ; and, unless this equilibrium temperature 
coincides with the freezing point, or unless the rate of approach 
to the freezing point is very great compared with the rate of 
approach to this temperature, the thermometer will not show 
the true freezing point. 

The corrections necessary on this account can be experi- 
mentally determined, and Nernst and Abegg obtained good 
agreement between the results of experiments performed under 
conditions so different, that the uncorrected numbers for the 
molecular depression of the freezing point of a one per cent, 
solution of sugar varied from T6 to 2']. Their mean corrected 
value is about 1'86 a number which agrees exactly with that 
calculated from the melting point and heat of fusion of ice 
(p. 147). 

In order to make the convergence temperature coincide 
with the freezing point, Ponsot 1 formed crystals of ice in his 
solution by surrounding it with a freezing mixture, and then 
removed the vessel containing the solution, placing it in an air 
jacket which was surrounded by a vessel filled with a mixture 
of ice and brine of such a concentration that its temperature 
was as nearly as possible that of the solution to be examined. 
The solution is then in an enclosure at the temperature 
of its own freezing point, and the only variation in the con- 
vergence point is due to the heat evolved by stirring. When 

1 Ann. de Chim. et de Phys., and Congelation des Solutions Aqueuses, Paris 

CH. Vl] 



the temperature becomes constant, therefore, it is very nearly 
indeed the true freezing point. 

The apparatus generally used for freezing point determi- 
nations when great accuracy is not 
required was introduced by Beck- 
maim, and is represented in Fig. 45. 

The solution to be examined is 
placed in a wide test-tube J., which 
is surrounded by a second larger 
tube B to serve as an air jacket. 
This is placed in a vessel (7, into 
which a freezing mixture can be 
introduced. There is one stirrer in 
(7, and another, made of a platinum 
wire, in A. A delicate thermo- 
meter graduated to hundredths of 
a degree, is also placed in A. It 
has a little reservoir at the top, into 
which some of the mercury can be 
driven, to make the instrument 
available for different solvents, which 
freeze at different temperatures. 

The method of using Beckmann's 
apparatus is as follows. A weighed 
quantity of the pure solvent is intro- 
duced into A, and its freezing point determined by placing in C 
some mixture whose temperature is just below the point to be 
reached. The tube A is then removed, and the solvent melted. 
A weighed quantity of the substance to be dissolved is intro- 
duced through the side tube D, and the tube replaced. It is 
better to cool it slightly below the temperature at which it will 
finally stand. This can be done if it be kept quite at rest. 
The undercooled liquid is then stirred by means of the platinum 
wire, when small crystals of ice form. The temperature rises to 
a certain point, and then keeps stationary, but will again begin 
to sink if we go on freezing the solution ; for as the solvent is 
frozen out, the remaining solution gets stronger, and so has 
a lower freezing point. The highest of these temperatures is 

Fig. 45. 


therefore the one giving the freezing point of the solution, the 
concentration being corrected for the volume of ice formed. 

An immense number of observations have been made with 
one of the many forms of this apparatus. Some of Raoult's 
results are given below. They represent what he calls the 
molecular depression, that is the lowering which would be pro- 
duced by one gram-molecule of the substance in 100 grams of 
the solvent. The numbers are calculated from observations on 
solutions of much less concentration than this, on the assump- 
tion that the law of proportionality is still applicable. 

Solutions in Acetic Acid. 
Van 't Hoff's formula gives 38-8. 

Methyl iodide 38-8 Butyric acid 37'3 

Chloroform 38'6 Benzoic 43-0 

Carbon disulphide 38-4 Water 33 '0 

Ethylene chloride 40 -0 Methyl alcohol 35-7 

Nitrobenzene 41-0 Ethyl 36 -4 

Ether 39-4 Arnyl 39-4 

Chloral 39-2 Glycerine 36-2 

Formic acid 36-5 Phenol 36 -2 

Sulphur dioxide 38*5 Stannic chloride 41-3 

Sulphuric acid 18*6 Magnesium acetate 18'2 

Hydrochloric acid 17 '2 

Solutions in Formic Acid. 
Van 't Hoff's formula gives 28-4. 

Chloroform 26-5 Potassium formate 28'9 

Benzene 29*4 Arsenious chloride 2 6 '6 

Ether 28-2 

Aldehyde 26*1 Magnesium formate 13*9 

Acetic acid 26*5 

Solutions in Benzene. 
Van 't Hoff's formula gives 53'0. 

Methyl iodide 50-4 

Chloroform 51 '1 Methyl alcohol 25'3 

Carbon disulphide 49-7 Ethyl 28-2 

Ethylene chloride 48*6 Amyl 397 

Nitrobenzene 48-0 Phenol 324 

Ether 49-7 Formic acid 2 3 "2 

Chloral 50-3 Acetic 25-3 

Nitroglycerine 49-9 Benzoic 25'4 

Aniline 46-3 


Solutions in Nitrobenzene. 
Van 't Hoff's formula gives 69-5. 

Chloroform 69 -9 Methyl alcohol 35-4 

Benzene 70'6 Ethyl 35-6 

Ether 67-4 Acetic acid 36-1 

Stannous chloride 71*4 Benzoic 37 '7 

Solutions in Water. 
Van 't Hoflf's formula gives 18 -9. 

Methyl alcohol 17 '3 Hydrochloric acid 39-1 

Ethyl 17-3 Nitric acid 35-8 

Glycerine 17'1 Sulphuric acid 38 ! 2 

Cane sugar 18 -5 Potash 35 -3 

Phenol 15-5 Soda 36'2 

Formic acid 19-3 Potassium chloride 33 '6 

Acetic 19-0 Sodium 35-1 

Butyric 18 -7 Calcium 49 -9 

Oxalic 22-9 Barium 48'6 

Ether 16 -6 Potassium nitrate 30-8 

Ammonia 19'9 Magnesium sulphate 19'2 

Aniline 15-3 Copper 18-0 

An examination of these tables at once shows that the 
molecular depressions produced by different substances in the 
same solvent are approximately constant. Leaving out of 
consideration, for the present, solutions in water, we find that in 
other solvents, besides a series of normal compounds, having 
molecular depressions which agree with the number deduced 
from Van 't Hoff's theory, there is in general a series of 
abnormal substances which give depressions of about half this 
value. Since on Van 't Hotf 's theory the effect is proportional 
to the number of dissolved molecules, and independent of their 
nature, it is at once suggested, that, in these cases, the number 
of molecules is halved owing to the formation of aggregates of 
two ordinary molecules, so that the molecular weight is doubled. 
There is further confirmation in that some of the compounds 
which show this effect (such for instance as the acids of the 
formic acid series, which give half values when dissolved 
in benzene or nitrobenzene) are known to form compound 
molecules in the gaseous state, and there is evidence from other 
sources (e.g. from the surface tensions) that these acids and also 
certain alcohols form polymeric molecules when liquid. 


The most accurate experiments yet attempted on the freez- 
ing points of very dilute solutions are those of E. H. Griffiths, 
who has adapted the most refined methods of platinum thermo- 
metry to this problem. The details of the apparatus have not 
yet been published, but its general features together with 
the few final results which have already been obtained were 
described to the British Association in 1901. In order to avoid 
any action of the solutions on glass, the vessels containing them 
are of platinum and the water used is finally distilled from a 
platinum still. The duplicate principle of compensation is 
adopted, simultaneous observations being made on water and a 
solution, contained in similar platinum vessels. These vessels 
are completely surrounded, except for tubes of entrance for the 
thermometers, etc., by air jackets ; the water apparatus is then 
immersed in a large bath of ice and water, and that holding the 
solution in a similar bath filled with ice and brine arranged to 
give, as nearly as possible, the anticipated temperature of the 
freezing point. The two sides are then frozen by evaporating 
ether in the air spaces; the local cooling produced by this 
operation soon disappears. Both the solution and the outer 
bath are kept constantly stirred by means of water motors, 
the heating effect due to the work thus done being the same 
on each side. Platinum thermometry is particularly sensitive 
when used in this differential manner, and about the hundred 
thousandth of a centigrade degree can be measured. A solution 
of cane sugar gave constant molecular depressions of the 
freezing point while the concentration was varied from 0005 
to 0'02 normal, the numerical value of the molecular de- 
pression being 1*858. A series of experiments on solutions 
of potassium chloride gave a limiting value of the molecular 
depression equal to 3*720, which, on the assumption that KC1 
produces twice the effect of a single molecule, gives for the 
characteristic number, 1'860, a result identical with that 
obtained for cane sugar within the limits of experimentaL error. 

It is evident then, that the determination of the freezing 
Determination point of a solution affords a means of controlling 
weight 601 the measurement of the molecular weight of 


the dissolved substance. If we do not know whether the 
molecular weight of a body is M or nM, we can see which 
of these values we must use in calculating the molecular 
depression in order to get a number nearly equal to the theo- 
retical value for the constant. It must be noticed that we 
only determine the molecular weight of a body in a certain 
solvent ; for the same substance may have different molecular 
weights in different solvents (as witness the alcohols in benzene 
and acetic acid) and of course these values may be all different 
from its molecular weight in the gaseous state, though in general 
this weight corresponds to one of the others. The nature of the 
solvent may affect the state of molecular aggregation, even as 
it is affected by conditions of temperature and pressure when 
the substance is a gas. The solvents of the benzene series 
appear to favour polymerisation, while formic acid and its 
analogues seem generally to produce simple molecules. 

In the case of aqueous solutions also we have two series, 
and, taken alone, we might be inclined to consider the higher 
numbers as normal, and to assign doubled molecular weights to 
those substances which give the lower values. But when we work 
out Van 't Hoff's formula for the case of water, it gives, as we 
have seen, a value 18'6 for the molecular depression. This at 
once shows that the lower numbers are the normal values, and 
that they can be explained on Van 't Hoff's theory. It is the 
higher series which requires some further explanation. Are 
we to suppose that, as in the case of certain gases at high 
temperatures, dissociation occurs, and increases the number of 
effective pressure-producing molecules, or are we to assume 
that some new cause is brought into operation ? In favour of 
the dissociation hypothesis it may be urged that the numbers 
for such salts as KC1, NaCl, etc., which can only be dissociated 
into two parts, never show values which are much greater than 
double the normal, while salts such as CaCl 2 , which can be split 
into three, sometimes give a molecular depression which is 
about three times the normal value. We must defer the fuller 
discussion of these phenomena till we are considering the 
electrical properties of solutions, but attention is here drawn 
to the important fact that all those substances which give 


abnormally great values for the molecular depression of the 
freezing point in aqueous solution, form, when dissolved in 
water, solutions which are electrolytes. Moreover their elec- 
trical conductivities bear at all events an approximate relation 
to the amount of dissociation which it is necessary to assume 
in order to account for the abnormal effect on the freezing 
point. Whatever is the cause of this abnormally great 
molecular depression, seems to be also the cause of electrolytic 

When metals are dissolved in mercury, they produce de- 
Freezing points pression of the point of solidification, just as 
of alloys. bodies dissolved in water produce depression of 

the freezing point. Tammann examined solutions of potassium, 
sodium, thallium and zinc, and found Raoult's laws approxi- 
mately true. These metals seem to form monatomic molecules. 

Hey cock and Neville 1 have used many metals as solvents, 
and found values for the atomic depressions of which we select 
the following : 

Solutions in Tin. Theoretical depression, 3'0. 

Silver 2 -93 Cadmium 243 

Gold 2-93 Mercury 2'39 

Copper 2-91 Calcium 2 '40 

Sodium 2-84 Indium 1-86 

Magnesium 2 - 76 Aluminium 1*25 

Lead 2-76 

Indium and Aluminium thus show a tendency to form more 
complex molecules when dissolved in tin. 

The importance of experimentally examining Van 't Hoif's 
theory has directed special attention to dilute 

Experiments on ; 'f , . 

concentrated solutions, but the effect of increasing concen- 

solutions. , . , , f , r 

tration on the freezing points or non-electro- 
lytes has been studied by many observers, among others by 

1 Chem. Soc. Journ. 1889, 1890. 


Beckmarm 1 , Eykman 2 , Raoult 3 and Ponsot 4 . They find that in 
almost all cases the curves drawn with the concentrations in a 
given mass of solvent as abscissae and the molecular depressions 
as ordinates are nearly straight lines, inclined at a small angle 
with the axis of the abscissae. In some few cases the molecular 
depression decreases fast as the concentration increases, and, at 
high concentrations, may even be reduced to half its former 
value. If we extend our method of calculating molecular 
weights to such solutions, the result indicates that the 
molecular weight has doubled at the high concentration, so 
that polymerisation must have occurred. These cases are 
few; they include such solutions as those of acetoxim and 
alcohol in benzene, and must be considered analogous to 
the polymerisation of gaseous nitrogen peroxide at moderate 

In general the change of molecular depression is far 
less than in these solutions, and is probably analogous to 
the variation from the usual laws shown by gases at high 
pressures, rather than to a case of gaseous polymerisation. 
The best value for the molecular weight at infinite dilution 
would obviously be obtained by producing the curve showing 
the depression of the freezing point till it cut the axis of 
no concentration. It is probable that the small deviations 
of Raoult's numbers for non-electrolytes from the calculated 
values would become still smaller if this correction for con- 
centration were applied to his observations. 

The variation from their ideal laws of gases at high 
pressures can be approximately expressed by Van der Waals' 

where the pressure p is changed by a term proportional to 
the molecular attraction (a) and inversely proportional to the 

1 Zeits.f. physikal. Chemie, n. p. 715 (1888). 

2 Zeits.f. physikal. Chemie, iv. p. 497 (1889). 

3 Comp. Rend., April and November, 1897 ; Cryoscopie, Paris, 1901. 

4 Ann. de Chem. et Phys. [7] x. 79 (1897). 

W. S. 11 



[CH. VI 

square of the volume, and the effective volume v is diminished 
by a constant b which, according to the theory, is equal to four 
times the actual volume occupied by the molecules themselves. 
An equation of the same nature has been developed by Ostwald, 
Bredig and Noyes, taking account of the molecular volumes of 
the solvent and of the substance dissolved, and of the inter- 
actions between them. In general these latter are very small, 
and the formula reduces to 


where the constant d expresses a correction for volume, which 
depends on the nature both of the solvent and of the substance 
in solution. On the assumption that the depression of the 
freezing point is proportional to the osmotic pressure, the 
results deduced from this equation give the linear relation for 
the freezing point curves found in the experiments described 

Experiments on the effect of concentration on the freezing 
points of electrolytes will be considered in a future chapter; 
but from the most recent results of Ponsot and Kaoult on the 
aqueous solutions of non-electrolytes the following examples 
may here be quoted. 

Cane Sugar. C 12 H 22 O U = 342-18. 
(Ponsot. Mean values.) 

Gram-equivalents of sugar 

Depression of 
freezing point 

8Tjn . 


per thousand grams 
of water (ri) 

per thousand grams 
of solution (n f ) 


























CH. Vl] 
























Alcohol. C,HO - 46-05. 






















The behaviour of very strong aqueous solutions has been 
examined by Pickering 1 who finds the following molecular 
depressions produced by n molecules dissolved in 100 mole- 
cules of solvent. 


n = l 








Solvent = Water 

Methyl alcohol 
Acetic acid 






Solvent = Benzene 

Methyl alcohol 



0-31 | 0-22 
0-33 j 0-22 







1 Chem. Soc. Journ. Trans. LXIH. p. 998 (1893). 



The difference between the results obtained by measuring 
the concentration by the number of gram-molecules of solute 
per 1000 grams / of solution and measuring it by the number 
of gram-molecules per 1000 grams of solvent, is well shown by 
the tables and diagrams given by Ponsot, who has determined 
the freezing points of many concentrated solutions. A higher 
value for the molecular depression is always obtained by using 
the former method, and as the concentration increases the 
difference becomes very great indeed. 




Thermodynamics as a basis for physical science. Application to the case 
of solution. Theory of direct molecular bombardment. Theory of 
chemical combination. Conclusion. 

THE results obtained in the last two chapters show that the 
osmotic phenomena can, by the aid of the prin- 
ciples of energetics, be deduced for volatile 

physical science. so i uteSj anc j hence extended to other cases. 

The investigation may start either from the experimental 
solubility law of gases, or from general molecular theory, 
which supposes the solute to exist as a number of discrete 
particles each immersed in and surrounded by the mass of 
the solvent. 

By the first of these methods it is possible to develop the 
theoretical relations of the subject without involving the 
molecular hypothesis. Such treatment, using as its sole 
principle of coordination the law of available energy, ulti- 
mately rests on the experimental impossibility of perpetual 

This way of treating physical science has recently been 
adopted by a certain number of chemists, as a means of 
presenting their subject without applying to it the language 
or conceptions of the atomic theory, in terms of which even 
its simplest experimental facts have come to be expressed. 
It may be granted that students have become too apt to 
ascribe purely hypothetical properties to atoms and molecules, 


and that it is often instructive to carry Dalton's atomic theory 
as far as possible merely as a principle of chemically equivalent 
weights. But a body of doctrine, based on the statical theory 
of energy alone, will be limited in its scope, and cases in which 
it ceases to be sufficient are soon reached. For instance, the 
phenomena of highly rarefied gases have only been successfully 
interpreted by the aid of strictly molecular conceptions. While 
the gases are dense enough to be treated as matter in bulk, 
their characteristic equations can be constructed from their be- 
haviour with regard to pressure, temperature, etc., and then 
their other relations can be deduced from the principles of 
thermodynamics. But this method offers no explanation of the 
identity of the physical constants for different gases, and also 
for substances in dilute solution. In such matters we are 
driven back to molecular theory, which offers an alternative 
method, equally definite, if in some ways more speculative, 
of correlating the phenomena. 

In considering the subject of osmotics, the same alternatives 
appear. The theory can be developed from ex- 

Application to rr . J 

the case of soiu- perimental facts by the principles of energetics 
alone, or it can be obtained by the application 
of the fundamental ideas of the molecular theory combined 
with the laws of energetics. Now, whichever method we adopt, 
the resulting relations do not depend in any way on the 
physical mode of action of the osmotic pressure ; conversely, 
therefore, the agreement of the results with observation throws 
no light on the physical cause of osmotic pressure or the 
fundamental nature of the state of a dissolved substance. It 
has often been supposed that the analogy between the laws 
of the gaseous and osmotic pressures in dilute systems, and 
still more the identity in the absolute values of those pressures, 
implies a corresponding identity in their physical nature. 
But it is now evident that no such conclusion can legitimately 
be drawn. Whatever the cause of the pressure or the nature 
of solution may be, they must, by the principles of thermo- 
dynamics, have the properties which have been theoretically 
deduced from known facts and experimentally confirmed. If 


we do not accept this result as a sufficient explanation, but 
wish to analyse the phenomena further, we must regard the 
exact physical method by which osmotic pressure is produced 
as still a subject of enquiry. 

Two possibilities have been suggested. First, that, like 
gaseous pressure, osmotic pressure is due to the impacts of the 
dissolved molecules on the walls of the membrane, which is 
impervious to them and permeable to the molecules of the 
solvent ; second, that the cause of the pressure is the force of 
chemical affinity between the solute and the solvent, which 
tends to make more solvent enter a solution. It may be, 
however, that these two views will shade into each other in 
course of development. 

On the theory of direct molecular bombardment, the 
phenomena of the osmotic cell are exactly 

Theory of J 

direct molecular analogous to those of the diffusion of gases. 

bombardment. - TT1 .. , , . 

When a mass or gas is placed in an empty 
vessel, it finally, if the small effects due to gravity are negligible, 
distributes itself equally throughout the volume. This result 
at once follows from the molecular theory, for the particles of 
which the gas is composed are imagined as always in rapid 
motion, though with very short free paths. If then we suppose 
that an imaginary partition is placed anywhere in the gas, the 
number of molecules crossing it in one second from left to right 
will be proportional to the number present in unit volume 
(i.e. the concentration) on the left-hand side, and the number 
crossing from right to left proportional to the number per unit 
volume on the right. If the concentration is greater on one 
side than the other, more molecules will leave that side per 
second than enter it, and thus the concentration will be reduced 
till it is equal on both sides. A similar process goes on in the 
case of a substance dissolved in a liquid : uniformity of distri- 
bution is finally reached, though here the difficulties put in the 
paths of the dissolved molecules by the presence of the denser 
solvent prevent their travelling fast, and make the process of 
diffusion very slow. 

In the case of mixed gases it is found that the final state 


of distribution of one gas is not affected by the presence of 
another. Thus the amount of aqueous vapour which diffuses 
from water into a vacuum, is sensibly the same as if the empty 
space previously contained air, though in this case the process 
of diffusion is slower. This too is obviously a necessary 
consequence of the molecular theory, for, provided the molecules 
are on the whole too far apart to exert mutual influence, the 
dynamical equilibrium of water and its vapour will not be 
affected by the presence of molecules of air. 

Encounters between the molecules of a gas are continually 
taking place, and the average energy of translation of each 
molecule becomes on the whole the same, though sometimes 
the molecule may be travelling faster and sometimes slower 
than the average. This can be proved to hold good even 
if the molecules are of different kinds, as in a mass of mixed 
gas the average energy of each is still the same; thus light 
molecules will travel faster than heavy ones and will therefore 
diffuse more quickly. This result can be illustrated by the 
familiar experiment of filling a closed porous pot with air and 
surrounding it by an atmosphere of hydrogen or coal gas. The 
molecules of hydrogen enter more rapidly than the heavier ones 
of air go out, and a pressure gauge will show that the pressure 
inside the pot becomes greater than outside. 

If we could in any way entirely prevent the air from 
ultimately becoming equalized inside and out, we could get a 
permanent increase of pressure, for the hydrogen would enter 
till its concentration was the same within as without. The 
corresponding phenomenon actually occurs in the case of 
liquids and is shown by osmotic pressure, which can, as we 
have described, be demonstrated by the use of membranes 
which are practically semi-permeable in the manner required. 

Let us place a solution of some substance, cane sugar for 
example, inside a semi-permeable cell, and immerse it in pure 
water. The molecules of liquid will strike the walls of the 
membrane on both sides, but since there are both sugar and 
water molecules inside, fewer water molecules will, in a given 
time, hit the wall inside than outside. More water molecules 
pass in therefore than go out, and since none of the sugar can 


escape, an internal pressure is produced which can be measured 
by any convenient gauge. The process will go on until the 
pressure due to the water is the same on both sides : the 
excess of pressure may then be regarded as due to the sugar 
alone. Sugar is here chosen because little or no contraction in 
volume occurs when it is dissolved, or when the solution is 
diluted, which makes the theory of the subject much less 
complicated than in other cases. The simple physical explana- 
tion of colliding molecules gives, at any rate, some idea of a 
possible mode of action of the phenomena. 

In most cases, even on this theory, the osmotic pressure, as 
experimentally measured, must involve other properties which 
cause a diminution in the available energy of the system on 
dilution. There may be, for example, a change of volume, or 
a certain amount of chemical action between the solvent and 
the dissolved substance, as well as the pressure due to the 
bombardment of the molecules in solution. When equilibrium 
is attained, the available energy of the whole system must 
have reached a minimum value. 

For very dilute solutions, however, the cause of osmotic 
pressure is, on this hypothesis, referred simply to bombardment ; 
and Boltzmann, on special assumptions required for the extension 
to liquids of the methods of the kinetic theory of gases, has 
offered a demonstration of the law of osmotic pressure on the 
basis that the mean energy of translation of a molecule shall 
be the same in the liquid as in the gaseous state at the same 
temperature 1 . Such an extension of the bombardment theory 
to liquids seems however vague and speculative, and, as has been 
often pointed out by Lord Kelvin and others, the similarity in 
the mathematical laws of gases and dilute solutions does not 
necessarily connote identity of physical nature. 

The alternative theory of the nature of solution already 
Theor of mentioned, refers osmotic pressure to something 

chemical com- resembling chemical affinity, which tends to 

bination. J 

make solvent enter the osmotic cell and combine 
with the solution. There are two varieties of this theory to be 

1 Zeit. phys. Chem. vi. 478 (1890). 


considered. There is what is often called the hydrate theory ; 
and there is the view that each particle of solute unites with or 
influences in some way a large and uncertain number of solvent 
molecules, thus forming a mobile and somewhat loosely con- 
structed molecular complex, which constantly interchanges its 
parts with those of other similar complexes. 

The hydrate theory imagines that definite hydrates exist in 
solution, the hydrates being chemical compounds of the solute 
with water, which, like other chemical compounds, agree with 
the laws of definite and multiple proportions. As more solvent 
is added, new compounds containing a larger number of water 
molecules are formed, and the mixture of these different hydrates 
allows the continuous variation of composition which is found 
in solutions. 

Theories based on these ideas have been recently framed 
by H. E. Armstrong 1 , S. U. Pickering 2 and others. Pickering 
supposes that, when solvent is frozen out, some of the existing 
hydrate is decomposed, and the next lower one formed. From 
the heats of dilution of solutions of sulphuric acid of different 
strengths, he calculates the work required to do this, and, 
adding it to that required to compress the molecules dissolved, 
deduces the lowering of freezing point 3 . The agreement of his 
numbers with observation shows that the excess of freezing 
point depression can be calculated from the heat of dilution, 
but does not decide whether that heat of dilution is due to the 
combination with additional molecules of water or (partly at 
any rate) to the resolution of some sulphuric acid molecules 
into their ions. 

Pickering's main argument for the existence of hydrates in 
solution is however based on the sudden changes in curvature, 
first noticed by Mendeleeff, which appear in the lines drawn to 
represent the variation of some physical property with the 
concentration. He has made, for instance, a long and careful 
determination of the densities of sulphuric acid solutions of 
different strengths, and drawn a curve to show his results. 

1 Proc. E. S. No. 243 (1886). 

2 For general account see Watts' Diet., Art. Solutions, n. 

3 B. A. Report, 1890, p. 320. 


Changes of curvature appear at points corresponding to defi- 
nite molecular proportions (e.g. H 2 S0 4 . H 2 and H 2 SO 4 . 4H 2 O). 
These changes can be more readily seen if a new curve is drawn 
connecting the concentration with the rate of change of density 
with concentration (i.e. with the slope at different points of the 
first curve). By this process of " differentiation " a series of 
straight lines is obtained with breaks at the positions where, 
in the first curve, changes of curvature appeared. Similar 
figures were drawn for electric conductivity, expansion by heat, 
contraction on formation, heat of dissolution, heat capacity, 
refractive index, magnetic rotation, and freezing point, and 
changes of curvature were found at the same points for all. 
Ostwald however says 1 that the position of the breaks alters 
with change of temperature. With weak solutions it is impos- 
sible to say whether such points correspond to definite molecular 
proportions, owing to the smallness of the change in percentage 
composition which would be caused by the addition of another 
water molecule to H 2 SO 4 ; but the breaks are found of precisely 
the same character as in the case of stronger solutions, and are, 
apparently, due to the same cause. The thermal change, result- 
ing from dilution of a strong solution, is usually of the same sign 
as that obtained by dissolving the solid in the first instance, and 
this also indicates that, if hydrates are present in concentrated, 
they are also present in dilute, solutions. If we allow this, it 
follows that one acid molecule is able to combine with, or at all 
events to influence in some way, an enormous number of water 

Several hydrates, before unknown, were indicated by the 
presence of these breaks, and subsequently obtained in the 
solid form. Thus Pickering isolated H 2 SO 4 . 4H 2 O, HBr . 3H 2 O, 
HBr.4H 2 0, HC1.3H 2 0, HN0 3 .H 2 O and HNO 3 .3H 2 O. He 
considers that the crystallization of a definite hydrate is 
strong evidence that it exists in solution, for bodies suddenly 
formed at the instant of precipitation come down as amorphous 
substances a common observation in the processes of chemical 
analysis. Dilute sulphuric acid, dissolved in acetic acid, pro- 
duces a smaller depression of the freezing point than the sum 
1 Watts' Diet., Art. Solutions, i. 


of those due to the acid and water separately, hence Pickering 
argues that no dissociation, but rather chemical union, result- 
ing in a reduction in the number of molecules, has occurred. 

The combination of large numbers of solvent molecules 
with one molecule of a body in solution may produce forces 
equal in all directions and thus secure the mobility of 
the dissolved molecules. Certain definite numbers of solvent 
molecules will be capable of more symmetrical arrangement 
than others, and will form hydrates, but their parts are 
freely interchangeable with each other. A dissolved molecule 
will be able to pass through a crevasse only when the number 
of solvent molecules requisite to keep it in solution can pass 
simultaneously, and this may explain the action of semi-per- 
meable membranes. Pickering, as described on p. 97, found 
that, when a mixture of propyl alcohol and water was placed 
in a porous pot, and the whole immersed either in pure water 
or pure alcohol, the volume of liquid inside the porous pot 
increased, showing that the phenomenon is due, not to the 
impermeability of the pot to either constituent alone, but to 
its impermeability to the solution as a whole. 

On the other hand, it may be argued that the evidence in 
favour of the existence of definite hydrates in the liquid phase 
is inconclusive, for the study of saturated solutions as special 
cases of systems in equilibrium, which has been made in the 
early chapters of this book, shows that it does not follow 
because a definite solid crystallizes from a solution, that it 
must necessarily exist in the same state of molecular aggre- 
gation in the liquid phase. 

The general analogy between the process of solution and 
cases of definite chemical action is, nevertheless, very close ; and 
it was accepted as a real identity 1 till the development of 
osmotic theory by Van't HofT showed the similarity between 
solutions and gases, and thus caused more stress to be laid on 
that aspect of the subject. There is evidence to show that 
chemical action does not always result in the formation of 
compounds in which the usual valencies of the elements 
present are exactly satisfied. The fact that salts often combine 
1 Tilden, B. A. Report, 1886, p. 444. 


with one or more molecules of water to form definite crystalline 
hydrates is an instance of this property, and the phenomena 
have been extensively studied by chemists, sometimes under 
the name of residual affinity, the resultant substances being 
usually known as molecular compounds. From such bodies as 
these to the mobile aggregations required by the molecular 
complex view of solution is no impossible step. It is easy to 
imagine a loose kind of chemical union in which the continuously 
variable compositions and the general mobility characteristic of 
solutions might be realized, but the chief difficulty in the way 
of such a chemical theory has been its inability to suggest a 
probable mechanism by which the equality in absolute values 
of the osmotic and gaseous pressures would necessarily follow. 

The same difficulty has confronted the theory of definite 
chemical compounds, but in the year 1896 Poynting showed 
that, if certain assumptions were made, the observed result would 
follow 1 . Let us consider the effect of combination on the vapour 
pressure. " If the molecules of salt were simply mixed with 
those of the solvent, or if they combined to form stable non- 
evaporating compounds with the solvent, which compounds 
were simply mixed, then the mixture should have the same 
vapour pressure as the pure solvent. For we might regard the 
salt or compound molecules at the surface as equally reducing 
the effective evaporating and the effective condensing area, 
somewhat as a perforated plate or gauze laid on the surface 
would do. But the salt probably combines with the solvent to 
form unstable molecules which continually interchange consti- 
tuents, so that when near the surface they may serve equally 
with those of the pure solvent to entangle the molecules of 
vapour coming downwards, these descending vapour molecules 
taking the place of molecules attached to the salt. Probably, 
however, they are less energetic than the pure solvent molecules 
and do not contribute so much to evaporation. We shall make 
the supposition that they do not contribute at all. " It may be 
observed that the same result will be reached if each salt 
molecule diminishes the facility for evaporation of x solvent 
molecules by the l/#th part. 

1 Phil. Mag. XLII. 298 (1896). 


If N is the number of gram molecules of solvent per litre 
and n the number of gram-molecules of solute, the number of 
solvent molecules left unaffected is N - n. There are then N 
molecules active for condensation, and only Nn active for 
evaporation. Hence the vapour pressure is reduced in the ratio 
(N-n)/N. Thus 

p p n 
and - = -TT, 

p N 

which is the relation already deduced on p. 130, from the 
known value of the osmotic pressure. Conversely, this last 
result yields the true osmotic law by an inversion of the process 
there used. 

Reasons have already been stated for believing that the 
osmotic pressure is proportional to the number of spheres of 
influence of solute particles immersed in the solvent, and there- 
fore that, in solutions of electrolytes, which have abnormally 
great osmotic pressures, partial dissociation must occur, resulting 
in an increase in the number of such effective particles. Later, 
we shall find that a similar dissociation is indicated by the 
facts of electrolysis, which lead to the conclusion that some of 
the molecules of salts, etc. are resolved into two or more parts 
by the act of solution, and that these parts, or ions as they are 
called, travel through the liquid in opposite directions under 
the action of an electromotive force and are therefore charged 
electrically. The two independent lines of enquiry thus lead to 
the same hypothesis of electrolytic dissociation, and the evidence 
for and against this theory will have to be fully considered in 
future chapters. 

On Poynting's view of osmotic pressure then, as the writer 
has previously indicated 1 , the supposition of combination be- 
tween the solute and the solvent has to be extended to include 
the case where the solute is resolved into its ions. We must 
imagine that each ion itself destroys the facility for evaporation 
of one solvent molecule, or diminishes that facility in like pro- 
portion in a group of solvent molecules, just as each molecule 
1 Nature, LIV. 571 ; LV. 33 (1896). 


of a non- electrolytic solute does 1 . With this extension, the 
theory of chemical combination seems to agree with the facts. 

There are thus different views as to the nature of solution, 
each offering a reasonable explanation of the 

Conclusion. . _ . . . . . _ 

phenomena. At first sight, the idea ot mo- 
lecular bombardment on the walls of the membrane by solute 
particles which are dynamically independent of the solvent 
molecules seems diametrically opposed to the hypothesis of 
chemical combination between them; but we know too little 
about the nature of chemical affinity to be quite sure that it is 
not due to some relation in the dynamical properties of the 
reagents, and the two views of solution may after all be 
different statements of the same truth. 

However this may be, the two theories at present stand 
opposed, and each seems capable of explaining the ordinary 
facts of osmotic pressure. These phenomena, therefore, are 
unable to provide a crucial experiment to decide between the 
hypotheses. It will, however, be noticed that Pickering's ex- 
periment, in which either propyl alcohol or water enters as 
solvent an osmotic cell containing a mixture of these two 
liquids, seems to show that it is to a combination that the 
membrane is impervious, and is thus in favour of the view that 
solution is due to something analogous to chemical action. 

It must be clearly understood that an enquiry about the 
nature of solution and the physical mode of action of osmotic 
pressure is a problem entirely distinct from that of the essential 
difference between an electrolyte and a non-electrolyte. The 
hypothesis of ionic dissociation is quite independent of the 
direct bombardment theory of osmotic pressure, with which it 
has often been confused, and is perfectly consistent with the 
view that solution is a process of the same ultimate nature as 
ordinary chemical action. Stress is laid on this point, because 
criticisms of the direct bombardment theory of osmotic pressure 
have sometimes been adduced as reasons for refusing to accept 
the idea of the ionic dissociation of electrolytes 1 . 

1 For a controversial discussion of these questions see Nature, LIV., LV., 
indexed under "Osmotic Pressure," "Ions, theory of," etc. 



Introduction. Volta's pile. Early experiments. Faraday's work. 
Polarization. Faraday's laws. Electrochemical equivalents. The 
electrolysis of gases. Nature of the ions. 

THE origin of the study of electrolysis is to be found in the 
work of Gal van i at Bologna. About the year 


1786 he noticed that the leg of a frog con- 
tracted under the influence of a discharge from an electric 
machine. Following up this discovery, he observed the same 
contraction when a nerve and a muscle were connected with 
two dissimilar metals, placed in contact with each other. 
Galvani attributed these effects to a so-called animal electricity, 
and it was left for another Italian, Volta of Pavia, to show that 
the essential phenomena did not depend on the presence of an 
animal substance. In 1800 Volta invented the pile still known 
by his name, which, by reason of the greater intensity of its 
action, provided a means of investigation that was at once put 
into use by himself and his contemporary workers in other 

Volta's pile consisted of a series of little discs of zinc, copper 
and blottingf-paper moistened with water or 

Volta's pile. 

brine, placed one on top of the other in the 
order zinc, copper, paper, zinc, etc., finishing with copper 1 . 

1 Volta thought that the origin of the effects was at the junction of the two 
metals, hence the order of discs in the pile, and the terminal metal plates in air 
in the crown of cups. These plates are now known to be useless. 


Such an arrangement is really a primitive primary battery, 
each little pair of discs separated by moistened paper acting as 
a cell, and giving a certain difference of .electric potential, the 
differences due to each little cell being added together and 
producing a considerable difference of potential or electro- 
motive force between the zinc and copper terminals of the pile. 
Another arrangement was the crown of cups, consisting of a 
series of vessels filled with brine or dilute acid, each of which 
contained a plate of zinc and a plate of copper. The zinc of 
one cell was 'fastened to the copper of the next and so on, the 
isolated copper and zinc plates 1 in the first and last cups forming 
the terminals of the battery. 

Volta arranged the metals in an electromotive series so 
that, when placed in a solution, a metal is always positive to 
any of those below it in the series and negative to those 
above it. J. W. Ritter pointed out that the order of this list 
is also the order in which the metals precipitate each other 
from solution, an important connexion between electrical and 
chemical phenomena only appreciated long afterwards. 

Volta also discovered that the same difference of potential 
is given by two metals, whether they are directly connected, 
or joined by means of a third metal. Thus, in any complete 
circuit made up of a number of different metals, the total 
electromotive force mu^t- vanish. 

Using a copy of Volta's original pile, Nicholson and Carlisle 2 

Early Experi- found that when two brass wires leading from 

ments. its terminals were immersed near each other in 

water, there was an evolution of hydrogen gas from one, while 

the other became oxidised. If platinum or gold wires were 

used, no oxidation occurred, but oxygen was evolved as gas. 

They noticed that the volume of hydrogen was about double 

that of oxygen, and, since this is the proportion in which 

these elements are contained in water, they explained the 

phenomenon as a decomposition of water. They also noticed 

1 See note, p. 176. 

2 Nicholson's Journal, iv. p. 179 (1800). 

w. s. 12 


that a similar kind of chemical action went on in the pile 
itself, or in the cups when that arrangement was used. 
Cruickshank 1 soon afterwards decomposed the chlorides of 
magnesia, soda and ammonia, and precipitated silver and 
copper from their solutions an observation which afterwards 
led to the process of electroplating. He also found that the 
liquid round the pole connected with the positive terminal 
of the pile became alkaline and the liquid round the other 
pole acid. In 1806 Sir Humphry Davy 2 proved that the 
formation of the acid and alkali was due to impurities in 
the water. He had previously shown that decomposition 
of water could be effected although the two poles were 
placed in separate vessels connected together by vegetable 
or animal substances, and established an intimate connexion 
between the galvanic effects and the chemical changes going on 
in the pile. The identity of " galvanism " and electricity, which 
had been maintained by Volta, and had formed the subject 
of many investigations, was finally established in 1801 by 
Wollaston, who showed that the same effects were produced 
by both, while in 1802 Erman measured with an electroscope 
the potential differences furnished by a voltaic pile. 

In 1804 Hisinger and Berzelius 3 stated that neutral salt 
solutions could be decomposed by electricity, the acid appearing 
at one pole and the metal at the other, and drew the con- 
clusion that nascent hydrogen was not, as had been supposed, 
the cause of the separation of metals from their solutions. 
Many of the metals then known were thus prepared, and in 
1807 Davy decomposed potash and soda, which had previously 
been considered to be elements, by passing the current from 
a powerful battery through them when in a moistened con- 
dition, and so isolated the metals potassium and sodium. 

The remarkable fact that the products of decomposition 
appear only at the poles was perceived by the early experi- 
menters on the subject, who suggested various explanations. 

1 Nicholson's Journal, iv. p. 187. 

2 Bakerian Lecture for 1806, Phil. Trans. 

3 Ann. de Chimie, LI. p. 167 (1804). 

CH. VIIl] 



Grotthus 1 in 1806 supposed that it was due to successive 
decompositions and recombinations in the substance of the 
liquid. Thus if we have a compound AB in solution, the 
molecule next the positive pole is decomposed, the B atom 
being set free. The A atom attacks the next molecule, seizing 






Fig. 46. 

the B atom and separating it from its partner which attacks 
the next molecule and so on. The last molecule in the chain 
gives up its B atom to the A atom separated from the last 
molecule but one, arid liberates its A atom at the negative 

Grotthus, and other pioneers in the subject, thought that 
Faraday's the decomposition was due to a direct attrac- 
work ' tion exerted by the poles on the opposite 

constituents of the decomposing compound, which varied as 
the square or some other power of the distance. This explana- 
tion of electrolytic action, as framed by the early experimenters, 
was finally disproved by Faraday 2 , who placed two platinum 
strips, kept at a constant difference apart and connected through 
a galvanometer, at different positions in a trough of dilute acid 
through which a current was flowing. The deflection of the 
galvanometer was the same for all positions of the strips, thus 
showing that the electric forces were the same everywhere 
between the poles. He also showed that chemical decom- 
position could be produced without the presence of any metallic 
pole. An electric discharge from a sharp point connected with 
a frictional machine, was directed on to a strip of turmeric 

1 Ann. de Chimie, LVIII, p. 54 (1806). 

2 Experimental Researches, 1833. 

12 2 


paper moistened with sulphate of soda solution, the other end 
of the paper being joined to the other terminal of the machine. 
Alkali appeared on the paper opposite to the discharging point. 
Another experiment showed that insoluble hydrate of magnesia 
was produced at the junction between a strong solution of 
sulphate of magnesia and pure water when a current was 
passed across it. Faraday accepted the idea of Grotthus' chain, 
but held that there were chemical forces between atoms of 
opposite kinds in neighbouring molecules as well as in the 
same molecule, and that when the electric force was added to 
these they became strong enough to overcome the attractions 
between the atoms in the same molecule, so that a transfer of 
partners occurred. We shall see later that transfers of part- 
ners are probably always going on in solutions, whether a 
current is passing or not, and that the function of the electric 
forces is merely directive, but Faraday's account of the conse- 
quences of this interchange still holds good. He pointed out 
how it explained all the facts, including the passage of acids 
through alkalies under the influence of the current, a pheno- 
menon which had created great surprise when discovered by 
Davy. Faraday remarked that the presence of the alkali not only 
facilitated the passage of the acid, but was even necessary, for, 
without something with which to combine on its way, the acid 
would be unable to travel. Thus Faraday's view amounts to 
supposing a constant stream of acid in one direction and of 
alkali in the other. 

Faraday introduced a new terminology which is still used. 
Instead of the word pole which implied the old idea of 
attraction and repulsion, he used the word electrode, and called 
the plate of higher electric potential, by which the current is 
usually said to enter the liquid, the anode, and that by which it 
leaves the liquid, the cathode. The parts of the compound 
which travel in opposite directions through the solution he 
called ions cations if they went towards the cathode and 
anions if they went towards the anode. He also introduced 
the words electrolyte, electrolyse, etc., which we have already 

Faraday pointed out that the difference between the effects 


of a frictional electric machine and of a voltaic battery 
lay in the fact that the machine produced a very great 
difference of potential, but could only supply a small quantity 
of electricity, while the battery gave a constant supply, much 
larger in quantity, but only produced a very small difference 
of potential. 

The diminution with time of the intensity of the voltaic 
pile or cell was noticed by the early observers, 

Polarization. . . J J 

and was investigated by Davy and Faraday. 
The researches of the latter physicist brought out its con- 
nexion with the accumulation at the electrodes of the products 
of the decomposition of an electrolyte through which a 
current is passed. Faraday showed that a definite minimum 
" intensity," depending on the nature of the electrolyte, was 
necessary for the ions or their products to be liberated at 
the electrodes. He arranged certain substances in the order 
of what we should now term their decomposition voltages, and 
pointed out the relation between this order and that in which 
the same bodies could be placed with reference to the intensity 
of secondary current they would furnish when disconnected 
from the primary battery and then joined with a galvanometer. 
This phenomenon, originally observed by Ritter, lies at the base 
of the action of the accumulator. 

When the intensity of the primary current is not enough 
to visibly decompose an electrolyte, Faraday showed that a 
small current still passed. Whether this leakage current really 
flows without chemical separation at the electrodes, or is kept 
up by the removal of the products of the action as fast as they 
are formed, is a question to be considered later. 

When the nature of the electromotive force of a battery 
was more generally understood, it was evident that Faraday's 
work showed that the reverse electromotive force of " polariza- 
tion," as the phenomenon under consideration was named, must 
be subtracted from the primary electromotive force of the 
battery, before the effective electromotive force of the system 
could be calculated. 


The injurious effects of polarization in primary batteries 
led to many attempts to overcome it. The methods in use 
in the common form of cell are well known. They can be 
classed in three groups, according as their action is : 

(1) mechanical, as in Smee's cell, where the silver plate 
is covered with crystals of platinum, the sharp edges of which 
aid the escape of the hydrogen evolved ; 

(2) chemical, as in the bichromate cell, where the hydrogen 
is converted into water by an oxidising agent ; or 

(3) electrochemical, as in Daniell's cell, where the hydrogen 
ions enter a solution of copper sulphate, and are therefore 
replaced on the electrode by copper, which has a lower de- 
composition voltage. 

Davy had previously shown that there was no accumulation 
of electricity in any part of a voltaic circuit, 

Faraday's Laws. . 

and that a uniform now or current existed 
throughout. Faraday set himself to examine the relation 
between the strength of this current and the amount of 
chemical decomposition. He first proved by observations on 
the decomposition of acidulated water, that the amount of 
chemical action in each of several cells was the same when 
the cells were joined together and a current passed through 
them all in series, even if the sizes of the platinum plates were 
different in each. The volume of hydrogen was unchanged 
even if electrodes of different materials such as zinc or 
copper were used. He then divided the current after it 
had passed through one cell into two parts, each of which 
passed through another cell before being reunited. The 
sum of the volumes of the gases evolved in these two cells 
was equal to the volume evolved in the first cell. The 
strength of the acid solution was then varied, so that it 
was different in the different cells in one series, but the 
chemical action still remained the same in all. Thus the 
induction known as Faraday's first law was made : 

The amount of decomposition is proportional to the 
quantity of electricity which passes. 


An apparatus for the decomposition of water can therefore 
be used to measure the total quantity of electricity which 
has passed round a circuit. Such instruments are termed 

The same law was then shown to be true for solutions 
of various metallic salts, and also for salts in a state of 
fusion the weight of metal deposited being always the same 
for the same quantity of electricity. A second law also was 
formulated : 

The mass of an ion liberated by a definite quantity of 
electricity is proportional to its chemical equivalent weight. 

In the case of elementary ions this equivalent weight is 
the atomic weight divided by the valency, and in the case 
of compound ions it is the molecular weight divided by the 

It was then proved that the amount of zinc consumed in 
each cell of the battery was identical with that deposited by 
the same current in an electrolytic cell placed in the external 

Faraday's work laid the foundations of the modern quan- 
titative science of electrolysis. His results can be gathered 
into one statement, as follows: 

The quantity of a substance which separates at an elec- 
trode is proportional to the whole amount of electricity 
which passes and to the chemical equivalent weight of the 

This statement implies that no current flows without a 
corresponding amount of chemical separation at the electrodes. 
Faraday himself thought that, in certain cases, a small current 
could leak through electrolytes without causing separation, 
a point which cannot yet be regarded as settled. In the case 
of the electrolysis of solutions in water of metallic salts, such 
as those of silver, copper, etc., experiments seem to show that 
there is no leakage current, and that the deposition of metal 
or the evolution of gas is strictly proportional to the electric 
transfer as long as the electromotive force is high enough to 
overcome the reverse force of polarization, which is generally 
present in cases of electrolysis. When a smaller electromotive 


force than this is applied, the current flows at first, but its 
strength gradually diminishes, until finally it almost vanishes. 
The cause of the slight leakage current that then remains will 
be considered later. 

In connexion with this, it is interesting to note that 
Nernst has recently investigated mixtures of the oxides of 
certain metals, such as magnesium, zirconium, etc., which 
conduct well when hot, and give very little chemical decom- 
position. These substances however show signs of polarization , 
and are also transparent to light, a property considered in- 
compatible with true metallic conductivity 1 . Again, metallic 
sodium dissolves in liquid ammonia, giving a conducting solution 
which shows no polarization and seems to undergo no chemical 
changes 2 . It is as yet uncertain whether metallic and electro- 
lytic conductivity are ever associated in the same substance, and 
further experiments are necessary to decide the point. 

The confirmation of Faraday's law for solutions of silver 
Electrochemical sa ^ s nas been incidentally effected in the 
equivalents, course of many experimental determinations 
of the electrochemical equivalent of silver. If the value 
obtained for the silver deposited by unit quantity of electricity 
is the same when the strength of current and the other 
conditions of the experiment are varied, the quantity of elec- 
tricity and the mass of silver deposited must be proportional 
to each other. An exact knowledge of the electrochemical 
equivalent of silver is of great importance, since, given this 
constant, a silver voltameter can be used as a means of 
measuring accurately the total quantity of electricity, or the 
average current, which has passed through a circuit. This 
method has been adopted in the determination of the electro- 
motive force of the standard Clark cell, and in several 
measurements that have been made by electrical means of 
the thermal equivalent of the unit of mechanical energy. 
In order to determine the electrochemical equivalent, a 

1 Zeits. Elektrochem. vi. 41 (1899). 

2 Cady, Jour. Phys. Chem. i. 710 (1897). 


constant current of known strength is passed for a measured 
time through a solution of some silver salt. The most constant 
results are obtained when a neutral solution of the nitrate is 
used containing about fifteen parts of salt to one hundred 
of water, and the current has an intensity of about one 
hundredth of an ampere to the square centimetre. The silver 
may be deposited on a platinum bowl used as cathode, the 
anode being a silver plate wrapped in filter paper to catch 
any particles disintegrated. The electrochemical equivalent 
is expressed as the number of grams of silver deposited by a 
current of one ampere in one second. The following are 
perhaps the best determinations of this constant: 

Lord Rayleigh and Mrs Sidgwick 1 ... 0-00111795. 

F. and W. Kohlrausch 2 0'0011183. 

Pellat and Potier 3 0'0011192. 

Patterson and Guthe 4 0'0011192. 

Richards, Collins and Heimrod 5 0'0011172. 

Thus the mean result is about O'OOlllS or 0'001119 grams 
per ampere-second. The electrical measurements of the thermal 
equivalent agree better with the mechanical ones if the higher 
value is taken, and we shall therefore consider that the most 
probable value in the present state of our knowledge is 
0'001119 grams of silver per ampere-second. The correspond- 
ing constant for other elements or compounds can be calculated 
from this number by dividing it by the chemical equivalent of 
silver, viz. 107 '9, and multiplying by the chemical equivalent 
of the substance required. The value for hydrogen thus comes 
out 1'045 x 10~ 5 , its atomic weight being taken as 1*008, when 
oxygen is 16. 

It will be noticed that the chemical constant involved is 
the equivalent, and not the atomic weight. Therefore, in the 

1 Phil. Trans. CLXXV. 411 (1884). 
, 2 Wied. Ann. xxvu. 1 (1886). 

3 Journal de Physique [2] ix. 381 (1890). 

4 Phys. Rev. vn. 257 (1898). 

5 Zeit. Phys. Chem. xxxn. 301 (1900). 


case of substances like iron, which form two series of salts, the 
amounts deposited will be different when solutions of the 
different salts are used. The two amounts will be in the 
proportion of the two chemical equivalents ; if a current is 
sent through solutions of a ferric and a ferrous salt in series, 
the resultant weights will be as 56/3 : 56/2. 

With no substance other than silver have such accurate 
experimental results been obtained, though many observations 
have been made on copper and other metals in aqueous 
solutions of their salts. In all cases, Faraday's law has been 
found to be true within the limits of experimental error, the 
apparent variations which sometimes appear, especially with 
copper, having been traced to known causes 1 , such as the 
solubility of the metal in the solution. 

The experimental errors are much greater when gases are 
evolved, as in the electrolysis of acidulated water. The gases 
are to some extent soluble in the liquid, and may be absorbed 
in the substance of the electrodes ; oxygen is often liberated 
partly in the condition of ozone, while gases like chlorine 
attack the liquid or the electrodes, forming chemical com- 
pounds with them. Although several direct measurements of 
the electrochemical equivalent of water have been made, on 
account of these sources of uncertainty, none of them can be 
considered as very accurate. Since the general evidence for 
Faraday's law is very strong, it is better to calculate electro- 
chemical equivalents from the measured value for silver and 
the known chemical equivalents of the different ions. Kohl- 
rausch and Holborn 2 give a list of equivalent and electrochemical 
equivalent weights, the experimental value for silver being 
taken as 1'118 mg./amp.-sec. It is now probable that this 
number should be raised by one part in a thousand, and the 
electrochemical equivalents in the following table have all been 
increased in the same proportion. 

1 W. N. Shaw, B. A. Report, 1886, p. 411. 

2 Leitvermogen der Elektrolyte, Leipzig, 1898. 




Equivalent weights A (JO = 8*00), and electrochemical equivalents 
E in mg. I (amp. -sec.) of mono- and di-valent ions. 

Cations Anions 









































NH 4 



N0 3 






C10 3 






Br0 3 







CH0 2 






C2H 3 2 
























) 4 











) 4 







> 3 



Solvents other than water, for example acetone and pyri- 
dine, have often been used. Faraday's laws also hold good 
in such cases, and the electrochemical equivalents seem to 
be identical with those obtained when the solvent is water 1 . 

Faraday's laws have also been demonstrated for fused salts, 
many of which are good electrolytes, with conductivities of the 
same order as those of aqueous solutions 2 . 

Again, in recent years it has been shown that the discharge 
of electricity through gases is an electrolytic process accom- 
panied by chemical decomposition. Here also the same laws 
describe the phenomena, the electric charge associated with a 
gaseous ion being the same in amount as the charge on an ion 
in solution. J. J. Thomson 3 has found that the sign of the 

1 Kahlenberg, Journ. Phys. Chem. iv. 349 (1900) ; and Skinner, B. A. Report, 

2 Faraday, Experimental Researches, and Helfenstein, Zeits. Anorg. Chem. 
xxm. 255 (1900). 

3 Proc. R. S. LIII. 90 (1893) ; LVIII. 244 (1895). 


charge depends on the nature of any other ion present. More- 
over, it may even be changed by varying the conditions of the 
experiment : the electrodes at which hydrogen and oxygen are 
liberated in the electrolysis of steam are reversed when an arc 
instead of a spark discharge is used. 

In every case of electrolysis Faraday's laws seem to apply, 
and the amount of a given substance liberated by a given 
transfer of electricity appears to be the same under all 
conditions. This result leads to an exact view as to the 
nature of the process. Since the amount of substance de- 
posited is proportional to the quantity of electricity which 
passes, it follows that a definite charge of electricity is 
associated with a definite mass of the substance. We are 
thus led to look on electrolysis as a kind of convection, each 
ion carrying with it a fixed charge of electricity, positive or 
negative, which is given up to the electrode under the influence 
of an electromotive force above a certain limit. It is clear that, 
on this convective view of electrolysis, the conductivity of a 
solution must be proportional to the charge on each ion, to 
the number of ions, and to the velocity with which they move 
through the solution. 

Whenever one gram-atom or gram-molecule of any mono- 
valent ion is separated at an electrode, the same quantity of 
electricity passes round the circuit ; if the ion is divalent, the 
quantity is twice as great, and so on. All monovalent ions 
must therefore be associated with the same charge, all divalent 
ions with twice that charge, etc. 

The quantity of electricity involved is easily calculated by 
considering an example. If a current of one ampere flows for 
one second, experiment shows that 0*001119 grams of silver are 
liberated from the solution of one of its salts. Thus, when 
the equivalent weight in grams is deposited, the quantity of 
electricity passing is 107-92/0-001119 or 96440 ampere-seconds 
or coulombs. The same result is of course true for the gram- 
equivalent of any other substance, the gram-equivalent being 
the gram-molecule or gram-atom divided by the valency. 
Whenever a gram- equivalent of a substance is decomposed t 
therefore, 96440 coulombs of electricity pass round the circuit, 


and, as we shall prove later, this is the amount of charge 
actually transported through the electrolyte by one gram- 
equivalent of any ion. 

It is possible to calculate approximately the absolute electric 
charge carried by a single monovalent ion, since the number of 
molecules in a given volume of gas can be estimated by the 
kinetic theory from the viscosity and diffusion constant. At 
C. and normal atmospheric pressure, there exist about 
2 '5 x 10 19 molecules in one cubic centimetre of any gas. 

As we have seen, one electromagnetic unit of electricity 
evolves 1*045 x 10~ 4 gram of hydrogen, which at normal tem- 
perature and pressure fills a volume of 116 c.c., and therefore 
contains about 3 x 10 19 molecules or 6 x 10 19 atoms, and yields 
the latter number of ions when dissolved as a hydrogen salt. 
Each ion is then associated with 1'7 x 10~ 20 electromagnetic 
units. The ratio between the units of electric quantity being 
3 x 10 10 , the ionic charge is about 5 x 10~ 10 electrostatic units. 

Another value for the number of molecules of gas can be 
deduced from the measured variations of the gases from Boyle's 
law. The kinetic theory here leads to the result 1 '2 x 10 19 , and 
the corresponding ionic charge is about 10~ 9 electrostatic units. 

The charge on the ions produced by the passage of Rontgen 
The electrolysis ra y s through gases has been investigated by 
of gases. j j Thomson 1 . If N is the number of ions 

in unit volume of the gas, e the charge on each of them, and 
v the mean velocity of the positive and negative ions under a 
given electromotive force, the product Nev can be determined 
by exposing a gas to the action of Rontgen rays and measuring 
the current produced through it by a known electromotive 
force. Rutherford 2 has determined v for a considerable number 
of gases, and the number N can be estimated by a method due 
to C. T. R. Wilson, who, as described on p. 43, has shown that 
gaseous ions act as condensation nuclei in air saturated with 
aqueous vapour. From the velocity of fall of the cloud so 
formed, it is possible to calculate the approximate size of the 

1 Phil. Mag. XLVI. 528 (1898). 

2 Phil. Mag. XLIV. 422 (1897). 


resulting drops of water, and from the weight of the drops 
precipitated their number is known. Assuming that each ion 
acts as a centre of condensation, this result gives the number 
of ions present. The measure of the current through the 
ionized gas then furnishes a value for e the ionic charge equal 
to about 6'5 x 10~ 10 electrostatic units. This result, it will be 
seen, lies between those reached by the aid of the kinetic 
theory of gases, and indicates that the ionic charge is the 
same in this case as it is in liquid electrolytes. 

This identity was established in another way by J. S. 
Townsend 1 . At 15 and normal pressure one electromagnetic 
unit evolves 1'23 c.c. of hydrogen : if the number of molecules 
in one cubic centimetre is N, the number of atoms or ions 
associated with the unit of electricity is 2-46^, so that, if E is 
the charge on an ion in the liquid electrolyte, 

2'46 NE= 1 electromagnetic unit 

= 3 x 10 10 electrostatic units. 

NE= 1-22 x 10 10 electrostatic units. 

Now, by investigating the rates of diffusion of the gaseous 
ions produced by the action of Rontgen rays, and using 
Rutherford's values for the corresponding ionic velocities, 
Townsend deduced the following results for the product NE 
in gases: 

Air 1-35 x 10 10 , 

Oxygen T25 x 10 10 , 

Carbonic acid 1'30 x 10 10 , 

Hydrogen I'OO x 10 10 , 

numbers agreeing with that calculated for liquid electrolytes. 
Since N is a constant for all gases, it follows that the charges 
on the ions in these gases are all the same, and equal to the 
charge on a monovalent ion in a liquid electrolyte. 

In all cases in which gaseous ions are produced by the 
action of Rontgen rays and similar agencies, their charges seem 
to consist of the same quantity of electricity as that associated 
with a monovalent ion in a liquid electrolyte. When steam is 

1 Phil. Trans, cxcm. A, 129 (1899). 


electrolyzed by an electric spark, the ions of divalent substances 
like oxygen possess a double charge as they do in liquids, but 
these larger charges are always simple multiples of the mono- 
valent charge. The quantity of electricity on a monovalent ion 
seems to be a natural unit, and the results summarized above 
lead to an atomic theory of electricity. This natural unit of 
electricity is called an electron. The effect of magnetic and 
electrostatic forces on their paths gives evidence to show that 
the negative ions produced by Rontgen rays, etc., are of much 
smaller mass than hydrogen atoms, while the positive ions are 
comparable in mass to ordinary molecules. Thomson 1 holds 
that these negative ions or corpuscles constitute the funda- 
mental basis of all chemical atoms, and are likewise identical 
with electrons or free charges of negative electricity, a positive 
ion being produced when one of these corpuscles is removed 
from any neutral atom or molecule. According to this view, 
the ions in liquid electrolytes, or in gases through which an 
electric spark or discharge passes, consist of separated parts of 
molecules possessing an excess or defect of one electron, and in 
this way being negatively or positively charged. On the other 
hand, the ionizing action of Rontgen rays, etc., causes one 
particle to be detached from the un dissociated molecule of the 
gas, leaving that molecule positively electrified and furnishing 
a free corpuscle, which constitutes an isolated negative charge 
or electron. 

Speculation as to the nature of the ions began at an early 
The nature of period in the history of electrolysis. In the 
the ions. early years of the nineteenth century, Berzelius, 

from a prolonged study of the electrolytic decomposition of 
neutral salts, enunciated a theory that all chemical action was 
the result of electric forces between oppositely charged atoms. 
When two atoms united, he supposed that the charges were 
not exactly neutralised, and the group of atoms was left with 
a balance of positive or negative electricity, and so could still 
combine with other atoms or groups of atoms. He regarded 

1 Phil. Mag. XLIV. 293 (1897) ; XLVIII. 547 (1899). 


each chemical compound as formed by the union of an electro- 
positive group with an electro-negative group, and held that 
the action of the electric current in producing acid round the 
anode, and alkali round the cathode, of a neutral salt solution, 
was to be explained simply as a direct separation of the salt 
into acid and base. When the attention of experimenters 
Avas directed to organic chemistry, the dualistic conception of 
Berzelius was perforce abandoned, and even from the physical 
side his theories were soon found to need alteration. Thus 
Daniell showed that in the electrolysis of a solution of sodium 
sulphate an equivalent of hydrogen was produced as well as an 
equivalent of acid and base. This is at once reconciled with 
Faraday's law if we suppose that the parts of the salt, from an 
electrolytic point of view, are Na and S0 4 , and that the 
hydrogen results from a secondary action of the sodium on the 
water of the solution. 

With simple salts, acids and alkalies, there is seldom any 
doubt about the character of the ions ; the cation is the metal 
or hydrogen, the anion is the halogen (chlorine, bromine or 
iodine), a compound acid group (such as SO 4 ), or hydroxyl HO 
when the electrolyte is an alkali. 

A study of the products of decomposition does not neces- 
sarily lead directly to a knowledge of the ions actually involved 
in the passage of the current through the electrolyte. Since 
the electric force is active throughout the whole solution, all the 
ions must come under its influence and therefore move, but 
their separation from the electrodes is determined by the electro- 
motive force needed to liberate them. Therefore as long as 
every ion of the solution is present in the layer of liquid next 
the electrode, the one which responds to the least electro- 
motive force will alone be set free until the amount of this ion 
becomes too small to carry all the current across the junction 
layer, when the other ions begin to appear. In aqueous solu- 
tions, a few hydrogen and hydroxyl ions derived from the water 
are always present, as we shall see later, and will be liberated 
if the other ions require a higher electromotive force and the 
current be kept small. 

The issue is also obscured in another way. When the ions 


are set free at the electrodes, they may unite with the sub- 
stance of the electrode or some constituent of the solution 
and form secondary products. The h} 7 droxyl mentioned above 
decomposes into water and oxygen, and the chlorine produced 
by the electrolysis of a chloride may attack the metal of the 

This leads us to examine more closely the part played by 
water in electrolysis. It was at first thought to be the only 
active body, and to be necessary in every case of electrolytic 
decomposition. The dilute acid or alkali which was always 
added when water was to be decomposed, was supposed merely 
to allow the passage of the current by reason of its conductivity, 
and it was imagined that the current then directly decomposed 
the water. Now pure water is known to be a very bad 
conductor, though when great care is taken to remove all dis- 
solved bodies, there is evidence to show that some part of the 
small trace of conductivity remaining is really due to the water 
itself. Thus F. Kohlrausch 1 has prepared water the conduc- 
tivity of which as compared with that of mercury was 1*8 x 10~ u 
at 18 C. Even here some little impurity was present, and 
Kohlrausch estimates that the conductivity of chemically pure 
water would be 0'36 x 1Q- 11 at 18 C. As we shall see later, 
the conductivity of very dilute salt solutions is proportional to 
the concentration, so that it is probable that in most cases 
practically all the current is carried by the salt. At the elec- 
trodes, however, the small quantity of hydrogen and hydroxyl 
ions from the water are first liberated in cases where the ions 
of the salt have a higher deposition voltage. The water being 
present in excess, the hydrogen and hydroxyl are at once 
re-formed, and therefore constantly liberated. If the current 
is so strong that new hydrogen and hydroxyl ions cannot be 
formed in time, other substances are liberated; in a solution 
of sulphuric acid, a strong current will evolve sulphur dioxide, 
the more readily as the concentration of the solution is in- 
creased. Similar phenomena are seen in the case of a solution 
of hydrochloric acid in water. When the solution is weak, 
hydrogen and oxygen are evolved ; but, as the concentration is 

1 Wied. Ann. LIII. 209 (1894). 

w. s. 13 


increased, and the current raised, more and more chlorine is 
liberated. We shall return to this point in a later chapter in 
connexion with the study of decomposition voltages. 

An interesting example of secondary action is furnished by 
the common technical process of electroplating with silver from 
a bath of potassium silver cyanide. The operation has been 
studied by Hittorf l among others, who holds that the cation is 
potassium, and the anion the group AgCy 2 . Each K ion, as it 
reaches the cathode, precipitates silver by reacting with the 
solution in accordance with the equation 

K + KAgCy 2 = 2KCy + Ag, 

while the anion AgCy 2 dissolves an atom of silver from the 
anode, and re-forms the complex cyanide KAgCy 2 by combining 
with the 2KCy produced in the reaction described by the above 
equation. If the anode consists of platinum, cyanogen gas 
is evolved thereat from the anion AgCy 2 , and the platinum 
becomes covered with the insoluble silver cyanide AgCy, which 
soon stops the current. The coating of silver obtained by the 
process described above is coherent and homogeneous, while 
that deposited from a solution of silver nitrate, as the result 
of the primary action of the current, is crystalline and easily 

The corresponding cyanide process in the case of gold is now 
extensively used for the extraction of gold from its ores. The 
rock, containing small quantities of gold in a state of very fine 
division, is treated with potassium cyanide, and the solution 
of the double cyanide obtained in this way is electrolysed 
between steel anodes and lead cathodes. Prussian blue, which 
is again worked up into potassium cyanide, is formed on the 
anodes, and the gold is removed from the lead cathodes by 

In the electrolysis of a concentrated solution of sodium 
acetate, hydrogen is evolved at the cathode and a mixture of 
ethane and carbon dioxide at the anode. According to Jahn 2 , 

1 Pogg. Ann. cvi. 517 (1859). 

2 Grundriss der Elektrochemie, p. 292 (1895). 


the processes at the anode can be represented by the equa- 

2CH 3 . COO + H 2 = 2CH 3 . COOH + 0, 

2CH 3 . COOH + = C 2 H 6 + 2C0 2 + H 2 0. 

The hydrogen at the cathode is developed by the secondary 

2Na + 2H 2 = 2NaOH + H 2 . 

Many organic compounds can be synthetically prepared 
by taking advantage of secondary actions at the electrodes, 
such as reduction by the cathodic hydrogen, or oxidation at 
the anode 1 . 

Our knowledge of the nature of the ions has been profitably 
extended by another method. The changes in the concen- 
tration of a solution which occur near the electrodes are in 
some cases very marked, and it seems necessary to assume that 
unaltered salt is attached to one of the ions, forming a complex 
ion. In alcoholic and concentrated aqueous solutions of cad- 
mium iodide, some of the anions appear to be CdI 4 or I 2 (CdI 2 ), 
and are perhaps derived from double or Cd 2 I 4 molecules 2 . It 
has even been suggested that molecules of solvent may be 
attached to ions and be carried along with them under the 
influence of the electric forces 3 . 

It is sometimes possible to study the question by examining 
the conductivity of a solution and its variation with the 
concentration. The rate of variation with concentration of the 
equivalent conductivity of an electrolyte (that is, the conduc- 
tivity divided by the concentration) is much less for salts of 
monovalent acids than when the valency of the acid is higher. 
The conductivity curve of potassium permanganate, for example 4 , 
indicates that the acid is monovalent, and the formula of the 
salt consequently KMn0 4 . 

Again, it is possible to distinguish between double salts and 
salts of compound acids. Thus Hittorf showed that when a 
current was passed through a solution of sodium platinichloride, 

1 Liipke's Elektrochemie, Eng. Trans, p. 29. 

2 Hittorf, see below. 

3 W. N. Shaw, B. A. Report, 1890, 201; see below, Electric Endosmose. 

4 W. C. D. Whetham, Phil Trans. A, cxciv. 321 (1900). 



the platinum appeared at the anode. The salt must therefore 
be derived from a compound acid, and have the formula 
Na 2 PtCl 6 , the ions being sodium and PtCl 6 , for if it were a 
double salt it would decompose as a mixture of sodium chloride 
and platinum chloride, and both metals would go to the 

Kohlrausch 1 has found that, in the electrolysis of solutions 
of the salt PtCl 4 . 5H 2 0, the weight of the cathode remains 
unaltered for small current densities ; he therefore concludes 
that no platinum is deposited primarily. With greater current 
densities a grey deposit is obtained, which loses weight on 
heating and probably contains hydrogen. At the anode oxygen 
is first evolved, but, as time goes on, it is replaced by chlorine, 
the solution becoming darker in colour and acquiring a higher 
conductivity, showing the formation of the acid H 2 PtCl 6 . 
Kohlrausch explains the facts by assuming the existence of 
compound ions of the formula PtCl 4 O. 

Osmond and Houllevigne have studied the electrolysis of 
solutions of salts of iron, using iron containing carbon for the 
electrodes. The latter observer has found that, while carbon 
dissolved in steel is not carried with the current but remains as 
a muddy deposit at the surface of the anode, combined carbon 
forms with the iron a complex ion, and is carried with it in the 
direction of the current 2 . 

1 Wied. Ann. LXIII. 423 (1897). 

2 Journ. de Physique, 3rd series, vn. 708 (1898). 



Ohm's law. Experimental methods. Experimental results. Consequences 
of Ohm's law. Migration of the ions and transport numbers. Mobility 
of the ions. Experimental measurements of ionic velocity. Influence 
of concentration. Complex ions. Connexion between the mobility of 
an ion and its chemical constitution. 

THE current through a metallic conductor is, to a very great 
degree of accuracy, proportional to the electro- 
motive force applied. This relation, known as 
Ohm's law, may be expressed in the form that C = EjR, where 
R is a constant for any given conductor under fixed conditions, 
and is called its resistance. The law is verified if the re- 
sistance is shown to be independent of the current passing 
through it. The early experimenters, in the course of their 
investigations, made efforts to discover whether electrolytes 
also conformed to Ohm's law. It was known that, owing to 
the reverse force of polarization, no permanent current of 
moderate intensity could be maintained through an electrolyte 
unless the electromotive force exceeded a certain limit ; but 
polarization occurs, primarily at any rate, at the electrodes, and 
it remained to see, when all reverse forces were eliminated, 
whether the flow of the current in the body of the liquid was in 
accordance with the law. Eventually F. Kohlrausch, in experi- 
ments to be described below, clearly proved that solutions have 
a real resistance, which remains constant when measured with 
various currents and by different methods. 


The current in a circuit containing an electrolytic cell can 
therefore be calculated by Ohm's law if from the total electro- 
motive force of the circuit be subtracted the reverse electromotive 
force due to the polarization of the electrodes and to any changes 
produced by the current in the nature and concentration of 
different parts of the solution. 

Many attempts were made to measure the resistances of 
Experimental electrolytes before a satisfactory method was 

methods. discovered. Horsford 1 passed a current between 

two electrodes in a rectangular trough, then moved them nearer 
together, and determined the resistance of a wire which, when 
interposed in the circuit, reduced the current to its former 
value. Assuming that the polarization is equal in the two 
cases (which, owing to migration, is difficult to insure) the 
resistance of the wire is the same as that of a column of 
solution equal in length to the difference of the distances 
between the electrodes in the two positions. The method 
was improved by Wiedemann, who used as electrodes plates 
of the metal present in solution, and thus reduced polarization. 

Beetz 2 used an ordinary Wheatstone bridge arrangement, 
getting rid of nearly all polarization by making his electrodes 
of amalgamated zinc placed in a neutral solution of zinc 

Since the electromotive force between any two points of a 
given circuit is proportional to the resistance between them, 
the resistance of two parts of a circuit can be compared by 
comparing the electromotive forces between their ends. In 
this way Bouty 3 examined many solutions. He placed them 
in inverted U tubes and passed a current through two of them 
in series. Tapping electrodes were constructed by putting zinc 
rods in zinc sulphate solution, with thin siphon tubes, filled 
with the same solution, to make contact where required. The 
electromotive forces between the ends of the two tubes were 
thus compared. The only polarization is at the surfaces of 
contact of the different solutions. 

1 Fogg. Ann. LXX. p. 238 (1847), 2 Pogg. Ann. cxvu. p. 1 (1862). 

3 Ann. de Chemie et de Physique, 1884, in. 


Another way of eliminating the effects of polarization and 
migration has been used by Stroud and Henderson 1 . Two of 
the arms of a Wheatstone's bridge are composed of narrow 
tubes filled with the solution, the tubes being of equal diameter 
but of different length. The other two arms are equal coils, 
and metallic resistance is added to the shorter tube till the 
bridge is balanced. Equal currents then flow through the 
two tubes ; the effects of polarization and migration are the 
same in each ; and the resistance added to the shorter tube 
must be equal to the resistance of a column of liquid the length 
of which is the difference in the lengths of the two tubes. 

At present the resistance of electrolytes is most frequently 
determined by means of alternating currents. This method 
was first successfully adapted to the purpose by Kohlrausch 2 , 
who employed the alternating currents from a small induction 
coil, and used a telephone as indicator. The electromotive 
force of polarization in the electrolytic cell is thus rapidly 
reversed, and never reaches its full magnitude. But, unless 
proper precautions are taken, a small amount of chemical 
decomposition can produce so much effect that, even with 
alternating currents, the polarization is appreciable, and the 
resistance as measured is found to depend on the rate of 
alternation. The products of the decomposition of -fa milligram 
of water on two platinum plates, each having an area of one 
square metre, will give an electromotive force of about one volt. 
The electromotive force of polarization is proportional to 
the surface density of the deposit ; its effect can therefore be 
diminished by increasing the area of the electrodes, a condition 
obtained by coating them with platinum black. This is done 
as follows. A current from two accumulators or two or three 
Daniell cells is passed backwards and forwards between the 
electrodes through a solution of platinum chloride, which is 
now usually prepared by dissolving 1 part of platinum chloride 
(i.e. H 2 PtCl 6 ) and O'OOS part of lead acetate in 30 parts of water. 

1 Phil. Mag. [5] XLIII. 19 (1897) ; Proc. Phys. Soc. Lond. xv. 13 (1897). 

2 Pogg. Ann. cxxxvm. CLIII. (1869 1874) ; Wied. Ann. vi. LXIV. (1877 
1898); also Kohlrausch und Holborn, Leitvermogen der Elektrolyte, Leipzig, 



[CH. IX 

The strength of the current is adjusted to give a moderate 
evolution of gas. The platinized plates obtained by this method 
have very large effective surfaces, and are quite satisfactory for 
the examination of strong solutions. They have the power, how- 
ever, of absorbing a certain amount of salt from the solutions 
and of giving some of it up again when water or a more dilute 
solution is placed in the cell. The investigation of very dilute 
solutions, hereby made difficult, has been successfully carried 
out by first platinizing the electrodes and then heating them 
to redness. This process gives a gray surface which has enough 
area to prevent polarization from interfering with the results, 
while it does not absorb any appreciable quantity of salt 1 . 
Various causes of disturbance must be taken into account or 
eliminated by adjustment of the arrangements ; both the self- 
induction of the circuit 2 and its electrostatic capacity 3 may 
become appreciable. 

The most usual arrangement of apparatus is shown diagram- 
matically in Fig. 47. The metre bridge is adjusted till no 

Fig. 47. 

sound is heard in the telephone, when the well-known relation 
between the resistances of the four arms of the bridge holds 

1 Whetham, Phil Trans. A, cxciv. 329 (1900). 

2 Encycl. Brit., Art. Electricity, or B.A. Report, 1886, 384. 

3 Chaperon, Cornpt. Rend. cvm. 799 (1889), and Kohlrausch, Zeits. phys. 
Chem. xv. 126 (1894). 

CH. IX] 



The telephone is not a very pleasant instrument to use in 
this way, and a modification of the method, used by MacGregor 1 , 
Fitzpatrick 2 and the present writer 3 , is more rapid and also 
more accurate. The current from one or more dry cells is led to 
an ebonite drum, turned by a hand- wheel and cord, on which 
are fixed brass strips with wire brushes touching them in such 
a manner that the current is reversed several times in each 
revolution. The wires from the drum are connected with an 
ordinary resistance box in the same way as the battery wires of 
the usual Wheatstone's bridge. A moving coil galvanometer 
is used as indicator, and on the other end of the drum there is 
another set of strips, arranged to periodically reverse the con- 
nexions of the galvanometer, so that any residual current which 
flows through it is direct and not alternating. These strips are 
rather narrower than the first set, and thus the galvanometer 
circuit is made just after the battery circuit is made and broken 
just before the battery circuit is broken. The high moment of 
inertia of the galvanometer coil makes its period of swing very 
slow compared with the period of alternation of the current, 
and therefore the slight residual effects of polarization and other 
periodic disturbances are prevented from sensibly affecting the 
galvanometer. When the measured resistance is not altered 
by increasing the speed of the commutator, or changing the 
ratio of the arms of the bridge, the disturbing effects may be 
considered to be eliminated. 

Fig. 48. 

Fig. 49. 

1 Trans. Roy. Soc. Canada, 1882, 21. 

2 B.A. Report, 1886, p. 328. 

3 Phil. Trans. A, cxciv. 330 (1900). 



[CH. IX 

The form of vessel chosen to contain the electrolyte depends 
on the order of resistance to be measured. For dilute solutions 
the shapes of figures 48 and 49 will be found convenient, while for 
more concentrated solutions, those indicated in figures 50 and 51 
are suitable. 

Fig. 50. 

Fig. 51. 

The absolute resistances of certain solutions have been de- 
termined by Kohlrausch by comparison with mercury, and by 
using one of these solutions in any cell, the constant of that cell 
can be found once for all. From the observed resistance of any 
given solution in the cell, the resistance of a centimetre cube, 
or the specific resistance, can then be calculated. The reciprocal 
of this, or the conductivity, is a more generally useful constant ; 
it is conveniently expressed in terms of a unit equal to the 
reciprocal of an ohm. This unit is sometimes written as a 
" mho," a name it is not intended to use in this book. 

As the temperature coefficient of conductivity is large, 
usually about two per cent, per degree, it is necessary to place 
the resistance cell in a paraffin or water bath, and observe its 
temperature with some accuracy. 

Kohlrausch expresses his results in terms of equivalent 
Experimental conductivity, that is the conductivity k divided 
results. k v the number of gram-equivalents of electro- 

lyte per litre n. He finds that, as the concentration of solutions 
of monovalent salts, such as potassium chloride, sodium nitrate, 
etc., diminishes, the value of k/n approaches a limit, and, if the 
dilution is carried far enough, becomes constant, that is to say, 
at great dilution the conductivity is proportional to the concen- 
tration. In establishing this result, Kohlrausch used very pure 

CH. IX] 



water prepared by careful distillation. He observed that the 
resistance of the water continually increased as the process 
of purification proceeded. The conductivity of the water, and 
of the slight impurities which must always remain, was 
subtracted from that of the solution, and the result, divided by 
n, gave the equivalent conductivity of the substance dissolved. 
This method of calculation appears justifiable, for, as long as 
conductivity is proportional to concentration, it is evident that 
each part of the dissolved matter produces its own independent 
effect, so that the total conductivity is the sum of those of the 
parts, and when this relation ceases to hold, the conductivity of 
the solution has, in general, become so great that the part due 
to the solvent is negligible. 

The general result of these experiments can be graphically 
represented by plotting k/n as ordinates, and n* as abscissae ; n* 
is a number proportional to the reciprocal of the average dis- 
tance between the molecules, to which it seems likely that the 
equivalent conductivity will be closely related. The general 
forms of the curves for the neutral salt of a monovalent acid 
and for a caustic alkali or monovalent acid (like HC1) are shown 
in Fig. 52. The curve for the neutral salt comes to a limiting 
value, while that for the acid or alkali attains a maximum at 
a certain very small concentration, and falls again when the 


dilution is pushed to extreme limits. This fall has usually been 
considered to be due to chemical action between the acid and 


the residual impurities in the water, which, at such great 
dilution, are present in quantities quite comparable with the 
amount of acid. The phenomena however seem too regular to 
be due to the action of such impurities, for the fall begins at 
about the same dilution whatever the amount of impurity 
present. An explanation is suggested if we consider that the 
cases in which the fall occurs are those in which one of the ions 
(H or OH) of the solute is present in the solvent water 1 . 
Whatever be the cause of the phenomena we must take the 
maximum value of the equivalent conductivity to be the limit 
in the case of acids though it is possible that this method may 
give too low a result. 

It will be seen from the tables in the appendix that the 
values of the equivalent conductivities of all neutral salts 
are, at great dilution, of the same order of magnitude, while 
those of acids at the maximum are about three times as 

Passing to salts of divalent acids and other more complicated 
electrolytes, Kohlrausch found it impossible to reach such 
definite limiting values for the equivalent conductivity as 
were given by monovalent salts. Moreover, the influence of 
increasing concentration was more marked, the curves sloping 
at much larger angles. These changes in the phenomena were 
still greater when, as in copper sulpTiate, both metal and acid 
were divalent, and greatest of all in such substances as ammonia 
and acetic acid, which have very small conductivities when 
dissolved in water. We shall presently return to this subject. 

One of the most important results of Kohlrausch's work 
consisted in the proof that the resistance of a given electrolyte 
had a definite value, which was independent of the particular 
method used to determine it. This amounts to a demonstration 
of Ohm's law within the limits of the conditions of the experi- 
ments. A more direct proof of the law for strong currents was 
given by FitzGerald and Trouton 2 , who showed that the measured 
resistance was independent of the strength of the current. 

1 Whetham, Phil. Trans. A, cxciv. 353 (1900). 

2 B. A. Report, 1886, p. 312. 


The conformity of electrolytes with Ohm's law is most 
consequences of instructive. Since any electromotive force, 
ohm's law. however small, is able to produce a corre- 
sponding current, there can be no appreciable reverse electro- 
motive forces in the interior of an electrolyte, and no 
measurable amount of chemical work can be there done by the 
current. It follows either that the function of the current is 
merely directive, controlling the direction of the motions of the 
ions which it already finds in a state of mobility, or else that 
the work done in splitting up one molecule is exactly equal to 
that given back in the formation of the next. 

The first of these hypotheses was advanced by Clausius 1 to 
explain electrolysis, and, as it is the one generally adopted, we 
will examine the evidence for it in some detail. If two solutions 
containing the salts AB and CD are mixed, double decomposi- 
tion is found to occur AD and CB being formed, till a certain 
part of the first pair of substances has been transformed into 
an equivalent amount of the second pair. The proportions 
between the four salts AB, CD, AD and CB, which finally 
exist in solution, are the same under similar conditions of 
temperature and pressure whether we begin with AB and CD 
or with AD and CB. The phenomena were found by Guldberg 
and Waage to be fully represented by a theory which supposed 
that both the change from AB and CD into AD and CB, and 
the reverse change from AD and CB to AB and CD were always 
going on, and that the quantities transformed per second were 
proportional to the product of the active masses of the original 
substances and to a coefficient k, depending on the temperature 
and pressure, which expresses the rate at which the action 
proceeds when the active masses of the reagents are each unity, 
and measures the affinity producing the reaction. If the active 
masses of AB, CD, AD, CB are p, q, p', q' respectively, and k 
and k' the two coefficients of affinity, we get for the rate of 
transformation of AB and CD into AD and CB 

and for the velocity of the reverse change 

1 Pogg. Ann. ci. p. 338 (1857). 


When there is equilibrium, these two rates of transformation 
must be equal and opposite, and we get 


This equation can, as we shall see later, be obtained for dilute 
solutions by the principles of thermodynamics, and its results 
have been experimentally confirmed for many cases. It may, 
however, be explained as above, by a kinetic theory of the phe- 
nomena, and this view of double decomposition is universally 
admitted to be a true one. But in order that this process of 
chemical change in opposite directions should continually go on, 
it is necessary to have perfect freedom of interchange between 
the parts of the molecules, and to imagine that separations and 
reunions are perpetually occurring among them. This hypo- 
thesis was first advanced from the chemical side by Williamson 1 
in order to explain the process of etherification. 

A study of chemical changes shows that it is always the 
electrolytic ions of a salt that are concerned in the reactions. 
The tests for a salt, potassium nitrate for example, are the 
tests not for KNO 3 , but for its ions K and NO 3 , and in cases 
of double decomposition, it is always these ions that are 
exchanged. If an element is present .in a compound other- 
wise than as an ion, it is not interchangeable, and cannot be 
recognized by the usual tests. Thus neither the chlorates, 
which contain the ion C1O 3 , nor monochloracetic acid, show 
the reactions of chlorine ; and the sulphates do not answer 
to the tests which indicate the presence of sulphur as 

It seems certain, then, that the parts of the molecules in 
solution are continually interchanging, that the electrolytic 
ions are also the parts which enter into chemical combinations, 
and that the effect of a current is merely so to control the 
direction of these decompositions and recompositions, that, on 
the whole, a stream of positively electrified ions travels in one 
direction, and a stream of negatively electrified ions in the 
other. As far as we have gone, there is no evidence to show 
that the ions remain dissociated for any appreciable time, and 

1 Chem. Soc. Journal, iv. 110 (1852). 


the reasoning given above merely proves that there is freedom 
of interchange. This freedom may only exist in the case of 
those molecules which, according to the kinetic theory, at any 
instant happen to be moving with a velocity so much greater 
than the average, that, on colliding with another molecule, they 
produce sufficient impact to cause dissociation, and make rear- 
rangement possible. So much seems to follow from the truth 
expressed in Ohm's law and the phenomena of chemical action. 
There is, however, further evidence, which we shall discuss later, 
that the ions remain dissociated, or at all events keep a certain 
amount of freedom, throughout a considerable fractional part of 
their existence. 

Kohlrausch's work on solutions of simple salts of mono- 
valent acids also drew attention to the 

The migration of . . . 

the ions and trans^ additive nature of their conductivity. The 

port numbers. , j ,. ., , , 

equivalent conductivity in such cases can be 
represented as the sum of two independent quantities, one de- 
pending solely on the anion, and the other on the cation. To 
examine the meaning of this result, we must remember that, as 
we saw in the last chapter, the experimental relations sum- 
marized in Faraday's laws indicate that electrolysis is to be 
considered as a process resembling convection, a constant 
stream of cations moving with the current, and a stream of 
anions in the opposite direction. The quantity of electricity 
thus conveyed will be proportional to the number of carriers 
and to the speed with which they travel. 

If we pass a current between copper plates through a 
solution of copper sulphate, the colour of the liquid in the 
neighbourhood of the anode becomes deeper, and in the neigh- 
bourhood of the cathode lighter in shade. This is well seen 
if the electrodes are arranged horizontally with the anode 
underneath. When the electrodes are of copper, the quantity 
of metal in solution remains constant, since it is dissolved from 
the anode as fast as it is deposited at the cathode, but if 
we use platinum electrodes, the amount in solution becomes 
continually less, since more salt is taken from the neighbour- 
hood of the cathode than from the anode, and the colour of 


the solution, therefore, becomes pale more rapidly near the 
cathode than near the anode. 

This subject was first systematically investigated by Hittorf l , 
who examined many solutions in a manner which enabled the 
liquid round the two electrodes to be separately analysed after 
the passage of the current. 

Two explanations of these changes in concentration seem 
possible. It may be that the ions are really complex, unaltered 
salt being attached to the anion or solvent to the cation, so 

that some of the anions have the composition Cu(CuS0 4 ) 

or some of the cations the composition SO 4 (H 2 O) ; in this 
way salt would be drawn to the anode or solvent to the 
cathode. It may be that the velocities of the ions are different, 
the anion, in the case of copper sulphate, travelling faster than 
the cation. It is possible that, in many cases, both these 
effects occur; and indeed, as we shall see later, the evidence 
indicates that such is the case. In developing the hypothesis 
of different ionic velocities it is certain that if the opposite 
ions move with equal velocities, the result of the passage of 
the current will be that, while the composition of the middle 
portion of the solution remains unaltered, the products of 
the decomposition, which appear at the electrodes, are taken 
in equal proportions from the solution surrounding the anode, 
and from that round the cathode. If, however, one of the ions 
travels faster than the other, it will get away from the portion 
of the solution whence it comes more quickly than the other 
ion enters. When the electrodes are of non-dissolvable material, 
therefore, the concentration of the liquid in this region will fall 
faster than in that round the other electrode. 

Let us assume that the cation drifts to the right with a 
velocity u, and the anion to the left with a velocity v. The 
velocity of the cation can be resolved into (u + v) and J (u v), 
and the velocity of the anion into ^(v + u) and \ (v u). On 
pairing these components, we have a drift of the two ions right 
and left, each with a speed \ (u + v\ involving no accumulation 

1 Pogg. Ann., LXXXIX. 177, xcvm. 1, cm. 1, cvi. 337, 513 (1853-9). 

CH. IX] 



at the electrodes, and a uniform flow of the electrolyte itself 
without separation with a speed ^(u v) to the right 1 . 

Thus at the cathode there is a gain of electrolyte equal to 
(u v), and a loss, due to electrolytic separation, of (u 4- v) : 
a total loss of v. At the anode there is a loss of \ (u v) and 
a loss of (u + v), a total loss of u. The initial losses of 
electrolyte at the two electrodes, then, before diffusion sensibly 
affects the result, are in the same ratio as the velocities of the 
ions travelling away from them. 

The process can be clearly illustrated by a method due to 


























Fig. 53. 

Hittorf. In Fig. 53 the black dots represent the one ion, and 
the white circles the other. If the black ions move to the 
left twice as fast as the white ions move to the right, the black 
ions will move over two of our spaces while the white ones 
move over one. Two of these steps are represented in the 
diagram. At the end of the process it will be found that 
six molecules have been decomposed, six black ions being 
liberated at the left and six white ions at the right. Looking 
at the combined molecules, however, we see that while five 
remain on the left side of the middle line, only three are still 
present on the right. The left-hand side, towards which the 
faster ions moved, has lost two combined molecules, while 
the right-hand side, towards which the slower ions travelled, 
has lost four just twice as many. Thus we see that the ratio 
of the masses of salt lost by the two sides is the same as the 
ratio of the velocities of the ions leaving them. Therefore, on 
the assumption that no complex ions are present, by analysing 

Larmor, Aether and Matter, Cambridge, 1900, p. 290. 

w. s. 




[CH. IX 

the contents of a solution after a current has passed, we can 
calculate the ratio of the velocities of its two ions. A long 
series of measurements of this kind has been made by Hittorf 1 , 
Kuschel 2 , Lenz 3 ,Loeb and Nernst 4 , Bein 5 ,Hopfgartner 6 ,Kiimmel 7 , 
Kistiakowsky 8 and others, who used various forms of apparatus 
arranged so as to enable the anode and cathode solutions to be 
separately examined after the passage of the current. One such 

Fig. 54. 

apparatus used by Bein is shown in Fig. 54. Hittorf called the 
phenomenon the "migration of the ions," and expressed his 
results in terms of a transport number, or migration constant, 
which gives the amount of salt taken from the neighbourhood of 
one electrode as a fraction of the whole amount that disappears. 
If there are no complex ions, it also expresses the ratio of the 

1 loc. cit. p. 208. 

* Wied. Ann. xm. 289 (1881). 

3 Mem. Petersb. Acad. ix. 30 (1882). 

4 Zeits. f. physikal. Chemie, n. p. 948 (1883). 

5 Wied. Ann. XLVI. 29 (1892) and Zeits. phys. Chem. xxvu. 1 (1898). 

6 Zeits. phys. Chem. xxv. 115 (1898). 

7 Wied. Ann. LXIV. 665 (1898). 

8 Zeits. phys. Chem. vi. 97 (1890). 


velocity of one ion to the sum of the opposite ionic velocities. 
Many results on the subject were collected by T. C. Fitzpatrick 
in his tables of " The Electro-Chemical Properties of Aqueous 
Solutions," published in the British, Association Report for 
1893, and reprinted by permission in the appendix to this 
book. A more recent list appeared in Kohlrausch and 
Holborn's book Das Leitvermogen der Elektrolyte 1 , from which 
is taken the table on the next page. In it results in italics 
are considered by Kohlrausch to have been obtained under 
uncertain conditions. The numbers represent the migra- 
tion constants for the anions. Thus CuS0 4 '632 means that 
the amount of salt taken from the cathode vessel is to the 
whole amount decomposed as *632 : 1, and is therefore to 
the amount taken from the anode vessel as '632 : '368. The 
concentration n gives the number of gram equivalents of salt 
per litre of solution. 

The transport numbers for cadmium iodide, which, for 
solutions of more than half normal concentration, are greater 
than unity, show that the cathode vessel loses more salt than 
the whole solution does. It follows that some unaltered salt 
must travel through the solution towards the anode, and this 
result at once led to the conception of complex ions of the type 
I (CdI 2 ). The changes with concentration in the transport 
numbers of many other substances, such as calcium chloride 
and copper sulphate, seem too great to be explained by a 
different rate of variation for the two ions of the quasi-f fictional 
resistance which the solution offers to their passage, and 
suggest that complex ions may exist in many solutions. Other 
evidence in favour of this supposition will be given later. 

Bein has shown that, if a membrane be used to separate the 
anode and cathode solutions, a considerable effect, varying with 
the nature of the membrane, is produced on the transport 

A further step was taken in the year 1879 by Kohlrausch 2 , 

Mobility of wno showed that a knowledge of the conduc- 

the ions. tivity of a solution enabled the sum of the 

opposite ionic velocities to be calculated. We have seen that 

1 Leipzig, 1898. 

2 Wied. Ann. vol. vi. p. 160 (1879). 




[CH. IX 

I I I I I I s I I 

1 1 111 

i i 



t- t CO >n Co 



CO C- . , O . O ^ C- 

CO rH O rH ^ CO 

ITS O$ 1C O5 CO <>* i-H 

r-t ^* Tt< C- O C~ CO ?3 

10 c> < <M 


00 rH CO 
05 O* CD t- 
t>i 00 00 rH 

C<lrHO5rHC<lCOf<t) CO<3<J^C 
yJt^TfCplpCp^W t-QjDODrH 


*O CO ^3 

f5 ^0 <> p ep <p CO ^ 

il i 


1 ? 1 ll 1 1 1 



'P'P 1 ?>> 1 1 1 1 

O O 


we can represent the facts by considering the process of elec- 
trolysis to be a kind of convection, the ions moving through the 
solution and carrying their charges with them. Each mono- 
valent ion may be supposed to carry a certain definite charge, 
which we can take to be the ultimate indivisible unit of elec- 
tricity ; each divalent ion carries twice that amount, and 
so on. 

Let us consider, as an example, the case of an aqueous 
solution of potassium chloride of which the concentration is 
m gram-equivalents per cubic centimetre. There will then be 
m gram-equivalents of potassium ions and the same number 
of chlorine ions in this volume. Let us suppose that on each 
gram-equivalent of potassium there reside 4- q units of elec- 
tricity, and on each gram-equivalent of chlorine ions q units. 
If u denote the average velocity of the potassium ions, the 
positive charge carried per second across unit area normal to 
the flow is mqu. Similarly, if v be the average velocity of 
the chlorine ions, the negative charge carried in the opposite 
direction is mqv. But positive electricity moving in one direc- 
tion is equivalent to negative electricity moving in the other, 
so that the total current, C, is mq (u + v). 

Now let us consider the amounts of potassium and chlorine 
liberated at the electrodes by this current. At the cathode, if 
the chlorine ions were at rest, the excess of potassium ions 
would be the number arriving in one second, viz. mu. But, 
since the chlorine ions move also, a further separation occurs, 
and mv potassium ions are left without partners. The total 
number of gram-equivalents liberated is therefore m(u + v). 
Now, by Faraday's law, the liberation of one gram-equivalent 
of any ion involves the passage of a definite quantity Q of 
electricity round the circuit. Thus, in one second, the total 
quantity passing, that is the current, is mQ (u + v). On com- 
paring this result with the first expression for the same current, 
it follows that the charge, q, on one gram-equivalent of either 
ion is equal to the quantity of electricity passing round the 
circuit when the gram-equivalent is liberated. 

We know that Ohm's law holds good for electrolytes, so 
that the current C is also given by kdP/dx, where k denotes 


the conductivity of the solution, and dP/dx the potential 
gradient, i.e. the fall in potential per unit length along the 
lines of current flow. 

Thus mq (u + v) = k -=- - ; 

k dP 
or u + v = --- .-=- ..................... (37), 

mq dx 

an equation in which everything may be expressed in centi- 
metre-gram-second units. By measuring \jk in ohms (an ohm 
being 10 9 c.G.s. units), q in coulombs (10" 1 ), and writing n for 
the number of gram-equivalents of solute per litre instead of 
per cubic centimetre, we get 

nq x 

Now q is 96440 coulombs (p. 188), so that for a potential 
gradient of one volt per centimetre (10 8 C.G.S. units), we have 

Mj + t^ 1-037 x 10-" x - ............... (38), 


which gives the relative velocity (or the sum of the opposite 
velocities) of the two ions in centimetres per second under unit 
potential gradient. These numbers, u^ and v l , measure what we 
may call the mobilities of the two ions. 

Since the transport numbers give us the ratio of the ionic 
velocities if no complex ions are present, we can deduce the 
absolute values of u^ and v x from this theory. Thus, for in- 
stance, the conductivity of a solution of potassium chloride 
containing one-tenth of a gram-equivalent per litre is 0'01119 
reciprocal ohms at 18 C. Therefore 

Ul + v l = 1-037 x 10~ 2 x 0-1119 
= 0'001165 cm. per sec. 

Hittorfs experiments show us that the ratio of the velocity 
of the anion to that of the cation in this solution is "51 : '49. 
The absolute velocity of the chlorine ion under unit potential 
gradient is therefore 0*000595 cm. per sec., and that of the 
potassium ion 0*000570 cm. per sec. Similar calculations can 
be made for solutions of other concentrations. The following 
table gives the ionic mobilities of three chlorides of- alkali 



metals as multiples of 10~ 6 cm. per sec., per volt per cm. at 
18 C. 






M i 

v i 

u^ + v l 


v i 

u 1 + v 1 

M i 

v i 






































































































These numbers clearly show the increase in ionic mobility 
as the dilution gets greater. Moreover, if we compare the 
values for the chlorine ion obtained from observations on these 
three different salts, we see that, as the solutions get very 
weak, the mobility of the chlorine ion becomes the same in 
all of them. Similar phenomena appear in other cases of 
simple monovalent salts ; and, in general, we may say that, 
at great dilution, the velocity of an ion in the solution of 
such a salt is independent of the nature of the other ion 
present. From this result we may deduce the existence of 
specific ionic mobilities, the values of which are given in the 
following table for different monovalent ions in centimetres per 
second per volt per centimetre. 


67xlO~ 6 


70xlO~ 6 









NH 4 






C 2 H 3 2 




C 3 H 6 2 


Having once obtained these numbers, we can calculate the 
equivalent conductivity of the dilute solution of any salt con- 
taining .the ions referred to, arid the comparison of such values 


with observation furnished the first confirmation of Kohlrausch's 
theory. Some exceptions, however, are known. Thus, acetic 
acid and ammonia give solutions of much lower conductivity 
than is indicated by the sum of the specific mobilities of their 
ions as determined from other compounds. 

Oliver Lodge was the first to directly measure the velocity 
of transport of an ion 1 . In a horizontal glass 

Experimental . 7 

measurements of tube connecting two vessels filled with dilute 
sulphuric acid, he placed a solution of sodium 
chloride in solid agar-agar jelly. This solid solution was made 
alkaline with a trace of caustic soda to bring out the red colour 
of a little phenol-phthalein added as indicator. A current was 
then passed from one vessel to the other along the tube. The 
hydrogen ions from the anode vessel of acid were thus carried 
along the tube, and decolorized the phenol-phthalein as they 
travelled. By this method the velocity of the hydrogen ion 
through a jelly solution under a known potential gradient 
could be observed. The results of three experiments gave 
0-0029, 0*0026, and 0'0024 cm. per sec. as the velocity of the 
hydrogen ion for a potential gradient of one volt per centimetre. 
Kohlrausch's number is 0'0032 for the dilution corresponding to 
maximum conductivity. Lodge does not mention the concen- 
tration of his solution, but it was probably large enough to 
appreciably reduce the velocity. Experiments in which the 
motion of other ions was traced by the formation of precipitates, 
gave results differing considerably from the theoretical numbers, 
probably owing to the indeterminate values of the potential 

When the current density at the cathode in a solution of 
copper sulphate exceeds a certain limit, copper is deposited 
as a brown or black hydride. C. L. Weber 2 attributed this 
to the inability of the copper ions to migrate fast enough 
to keep up the supply for carrying the current, part of which 
will consequently be conveyed by sulphuric acid formed by the 
action of S0 4 ions on the water. By measuring the limiting 

1 British Association Report, 1886, p. 389. 

2 Zeits. phys. Chem. iv. 182 (1889). 

CH. IX] 



current density and the conductivity of the solution, he esti- 
mated the speed of the copper ions when they could travel 
just fast enough to carry all the current, and hence he deduced 
their specific velocity. Similar methods were used for solutions 
of cadmium sulphate and zinc nitrate. The copper sulphate 
measurements were repeated with an improved apparatus by 
Sheldon and Downing 1 . This method does not appear to be a 
very good one, for the dilution of the liquid round the cathode 
makes it impossible to accurately determine the conductivity 
of the solution concerned. This source of error will make the 
deduced velocities too great. 

The velocities of a few other ions have been directly deter- 
mined in another way by the present writer 2 . 
Two solutions, having one ion in common, of 
equivalent concentrations, different densities, 
different colours, and nearly equal specific re- 
sistances, were placed one over the other in a 
vertical glass tube. In one case, for example, 
decinormal solutions of potassium carbonate and 
potassium bichromate were used. The colour 
of the latter is due to the presence of the bi- 
chromate group, Cr 2 O 7 . When a current was 
passed across the junction, the anions C0 3 and 
Cr 2 O 7 travelled in the direction opposite to that 
of the current, and their velocity could be de- 
termined by measuring the rate at which the 
colour boundary moved. Similar experiments 
were made with alcoholic solutions of cobalt 
salts, in which the mobility of the ions was found to be much 
less than in water. The behaviour of agar jelly was then 
investigated, and the mobility of an ion was shown to be very 
little less in a solid jelly than in an ordinary liquid solution. 
The velocities could therefore be measured by tracing the 
change in colour of an indicator or the formation of a pre- 
cipitate. Thus decinormal jelly solutions of barium chloride 

1 Physical Review^ i. 51 (1893). 

2 Phil. Tram. A, CLXXXIV. 337 (1893) ; Phil Mag. Oct. 1894 ; Phil. Trans. 
A, CLXXXVI. 507 (1895). 

Fig. 55. 



[CH. IX 

and sodium chloride, the latter containing a trace of sodium 
sulphate, were placed in contact. Under the influence of an 
electromotive force, the barium ions moved up the tube, and 
their presence was shown by the trace of insoluble barium 
sulphate formed. By keeping the conductivities of the two 
solutions nearly the same, discontinuity of potential gradient 
was avoided, and the gradient could then be calculated from 
the area of cross section of the tube, the conductivity of the 
solution, and the strength of the current as measured in a 

In dilute aqueous solutions of simple salts, the direction of 
motion observed at the junctions was always normal ; but 
as the concentration was increased, in some cases, such as 
that of alcoholic cobalt solutions, more than one boundary line 
appeared, and the direction of some of these lines was occasion- 
ally even reversed. In order to explain these results it seems 
necessary to assume the existence of complex ions, unaltered 
salt being attached to one or other of the simple ions. 

The following table shows the velocities of the ions which 
have been experimentally determined by the methods of Lodge 
and Whetham. A comparison is given with their values as 
calculated, for the same concentration, on Kohlrausch's theory. 

Specific ioni 

s velocity in 



per second 

tion of solu- 

Name of Ion 

tion in gram- 


Calculated from 

per litre 




Hydrogen in chlorides 



., in acetates ... 




Copper (in chlorides) ... 



Barium ... 












Sulphate group (S0 4 ) ... 
Bichromate group (Cr 2 7 ) 




Cobalt (in alcoholic CoCL) 

,;;( Co(tfo 3 ) 2 ) 



Chlorine (in alcoholic CoCl 2 ) 



Nitrate group (NO 3 ) (in alcoholic 

Co(N0 3 ) 2 ) 




NOTE. The migration data for solutions of copper chloride are not 
known. The specific ionic velocity of copper at infinite dilution (when it 
would be independent of the nature of the combination) is given by 
Kohlrausch as 0-00031, but in a solution of the strength used it would be 
considerably less. The sum of the ionic velocities of cobalt chloride in 
alcohol, as calculated from the conductivity, is O'OOOOGO cm. per sec., and 
that of cobalt nitrate 0*000079. These numbers are to be compared with 
the sum of the observed velocities given in the table namely, 0-000048 
and 0-000079 respectively. 

These experiments, it will be noticed, depend on the pheno- 
mena which occur at the junction of two solutions when a 
current is passed across it. It was observed by Gore 1 that in 
such a case the surface of contact sometimes remained clear, 
giving a sharp boundary, and sometimes became blurred and 
indistinct. Similar results were obtained in the experiments 
under consideration, and shown by the writer to depend on 
the relative conductivities of the two solutions. The electro- 
motive force between two points of a circuit is proportional 
to the resistance, as Ohm's law indicates, and the potential 
gradients in the two solutions are proportional to their specific 
resistances. Since one ion, let us say the anion, is the same in 
each solution, a solution of high resistance means one in which 
the cation has a low velocity, and a solution of low resistance 
contains a fast moving cation. Now, if the current pass from 
the liquid of high to that of low resistance, a cation which 
chances to get in front of the boundary will find itself in a 
region of lower potential gradient, and will, therefore, drop 
back again into line, and if one of the faster ions find itself 
behind the boundary, it will have entered a region of higher 
potential gradient and will be once more pushed forwards. The 
boundary therefore keeps sharp and distinct while moving with 
the current. On the other hand, if the current flow from the 
low resisting to the high resisting liquid, a straggling slow ion 
will drop behind into a region of smaller potential gradient, 
and be still further retarded, while a wandering fast ion will 
enter a region where the higher electric forces will still further 
hasten it. The boundary will therefore become blurred and 
indistinct. Thus the condition necessary for the existence and 
1 Proc. R. 8. 1880 and 1881. 


permanence of a sharp boundary is, that a specifically slower 
ion must follow a specifically faster ion. The general theory 
of such boundaries has been considered by Kohlrausch 1 and 
H. Weber 2 . 

Orme Masson has applied these results to obtain a more 
accurate method of experimentally determining ionic velocities 3 . 
From what has been said, it follows that a current passing from 
a solution of high to a solution of low resistance, adjusts the 
potential gradients so that the actual velocity of the specifically 
slow ion in the region of high potential gradient is equal to 
that of the fast ion in the region of low potential gradient. 
Masson placed a jelly solution of a colourless salt, potassium 
chloride for instance, in the central region of a horizontal glass 
tube, the ends of which were filled with jelly solutions of salts, 
one with a coloured anion and one with a coloured cation, these 
ions being specifically slower than the ions of the potassium 
chloride which they respectively adjoined. The solutions used 
for this purpose were potassium chromate with a yellow anion, 
and copper sulphate with a blue cation. The chromate ion and 
the copper are slower than chlorine and potassium respectively, 
and thus, if a current be passed from the copper end through 
the chloride to the chromate, at each end a specifically slower 
follows a faster ion, and the condition of stability of the 
boundary is fulfilled. The potential gradient is the same 
throughout the chloride solution, and can be calculated from 
the conductivity and the current strength, and therefore the 
speed of the colour boundaries at each end gives the velocity of 
potassium and chlorine iinder the same potential gradient. By 
measuring the relative velocity of these two margins, therefore, 
the ratio between the velocities of potassium and chlorine can 
be determined, and compared with Hittorf's migration constant. 
Other salts were examined in the same way, and the relative 
mobilities of different ions, thus measured, were found to agree 
well with Kohlrausch's values. 

1 Wied. Ann. LXII. 209 (1897). 

2 Sitz. Akad. Wiss. Berlin, 936 (1897). 

3 Phil. Trans. A, cxcu. 331 (1899). 

CH. IX] 



The following table gives the mobilities of the ions, rela- 
tively to the value for potassium, which is put equal to 100, as 
determined by Masson, Kohlrausch's theoretical values for one- 
tenth normal solutions being appended for comparison. 




n = -5 

n = l 

n = 2 

n = -5 

n = l 

n = 2 

n = 'l 








' Na 















NH 4 












|S0 4 




B. D. Steele 1 has extended Masson's method by the dis- 
covery that, under certain conditions of concentration and 
potential gradient, the boundary between two colourless solu- 
tions, owing to the difference in refractive index, is clearly 
visible. He has also freed the method from the disturbing 
influence of jellies by placing the solution to be examined in 

Fig. 56. 
1 Phil Trans. A, cxcvm. 105 (1902). 



[CH. IX 

the limbs of the glass apparatus of figure 56, and confining it 
between two partitions of jelly, containing the indicator solu- 
tions, aqueous solutions of which are also poured into the tubes 
above the jelly walls and contain the electrodes. When the 
current flows, the indicator ions leave the jellies, and enter the 
liquid columns, after which their velocities cannot be influenced 
by the presence of the jelly. If the indicator solutions have 
densities greater than that of the other, the rubber stoppers 
closing the bottom of the apparatus are removed, and the tubes 
shown at the sides are inserted. The indicator ions can thus 
be made to enter the solution from below. 

Steele's results for the migration constants agree well with 
the best of those obtained by the method of Hittorf, and 
generally with those obtained by the method of Masson. From 
appreciable differences in certain cases it is, however, concluded 
that the jelly of Masson's experiments affects the two ions 

The following selection from Steele's results may be given : 



Migration constant 



Hittorf, etc. 


























BaCl 2 










MgS0 4 


















Steele has also calculated from his results the absolute ionic 
mobilities of some ions and compared the numbers with those 

CH. IX] 



of Kohlrausch, obtaining in most cases a satisfactory agreement. 
As examples : 



u i 

v i 

















BaCl 2 






MgS0 4 







The agreement with theory of all experimental measure- 
influence of ments of the ionic mobilities of simple mono- 
Concentration. va ] ent sa i tg) as ma( }e by different observers, 
is a striking confirmation of the truth of the fundamental ideas 
which underlie Kohlrausch 's treatment of the subject. As the 
concentration of solutions of these salts increases, both the 
theoretical and the experimental mobilities are seen to dimi- 
nish, and still to show a satisfactory agreement. Whatever 
the cause of the decrease of equivalent conductivity with 
increasing concentration may be, Kohlrausch's theory still gives 
the true value of the actual velocities with which the ions on 
the average move through the liquid under the conditions of 
the experiment, though these velocities are less than those 
acquired by the action of the same electric forces in dilute 

If we still wish to express the results in terms of the 
specific ionic mobilities, that is, in terms of the ionic velocities 
(u x and v x ) at infinite dilution under unit potential gradient, 
we must, for these more concentrated solutions, introduce 
a factor a measuring the ratio of the actual to the limiting 


values of the sura of the ionic mobilities. Then, from equation 
38, page 214, we have 

fa. + * )= 1-037x10-*- 

or - = 96'44 a (u^ + v* ). 

The coefficient a is thus given by the ratio between the 
actual value of the equivalent conductivity of the solution and 
its value at infinite dilution, and can readily be determined 

Now there seem to be two causes which could reduce the 
velocities of the ions. If we look on the passage of the ions 
through the solution as analogous to the motion of bodies 
through a viscous medium, we see that the frictional forces 
will increase with the velocity till they become equal and 
opposite to the driving forces producing the motion. The 
ions will then travel with constant velocity, arid the resistance 
for such minute bodies being relatively enormous, this limiting 
velocity will be readied practically instantaneously. An in- 
crease in this viscosity, or a decrease in what may be called the 
ionic fluidity, would therefore diminish the velocity of the ions, 
and consequently the conductivity of the solution. Chiefly to 
this cause is to be assigned the variation of ionic velocity, and 
therefore of conductivity, with temperature. Heating a solu- 
tion seems to increase the ionic fluidity to about the same 
extent as it diminishes the ordinary or molar viscosity. Never- 
theless Arrhenius has shown that there is no sudden change in 


the conductivity of a jelly solution at the moment when, by 
cooling or by the addition of more gelatine, the jelly "sets." 1 
While this result certainly proves that no exact connexion 
exists between the ionic fluidity and the molar viscosity, it 
does not imply that the ionic fluidity is not affected by the 
addition of more of the electrolyte, which might affect the 
molecular condition of the system. This leads to the consider- 
ation of another method in which the ionic velocities might be 
reduced. In developing the theory, the assumption is made that 
all the substance dissolved is actively concerned in conveying 
1 B. A. Report, 1886, p. 344. 


the current, though it is possible that such is not always the 
case. It may be that, under certain conditions of temperature 
and concentration, a certain fraction of the solute is in a state 
of inactivity, which must mean that its ions do not drift in 
opposite directions under the influence of electromotive forces. 
If, for the present, we exclude the consideration of complex 
ions, these inactive molecules will be unaffected by the electric 
forces, and will have no drift in either direction. Now, what- 
ever be the cause of the activity or non-activity of the solute, 
it is certain that the equilibrium between active and inactive 
molecules must be a mobile equilibrium, molecules continually 
passing from one state to the other 1 . Each ion will sometimes 
be active and sometimes be inactive ; while active it will move 
and while inactive be stationary, and the net result will be that 
its effective velocity will be reduced in the ratio of the active 
time to the whole time. 

Thus the velocities of simple ions may be reduced by an in- 
crease in frictional resistance, by a diminution in the fraction of 
the dissolved substance which is, at any moment, active, or by 
a combination of both these causes. In dilute solutions, the re- 
sistance offered by the liquid to the passage of the ions through 
it is probably sensibly the same as in pure water ; but when the 
proportion of non-ionized molecules becomes considerable, we 
cannot assume that this is the case. If, however, no complex 
ions are present and the solution is dilute enough for the friction 
to be taken as constant, the coefficient a can be given a very 
simple physical meaning. The fraction which expresses the 
ratio of the actual to the limiting velocity of the ions must 
then also express the fraction of the dissolved substance which 
is, at any moment, electrolytically effective, and consequently 
the fraction of its time during which, on the average, any ion 
remains active. This fractional number may be called the 
coefficient of ionization. 

Thus, although we can, if we like, always put Kohlrausch's 

theory in the form shown in our last equation, the constant 

a will only have a definite physical meaning when no complex 

ions are present, and the solution is so dilute that the ionic 

1 Whethain, Phil. Mag., July, 1891; Phil. Trans., A, CLXXXIV. 340 (1893). 

W. s. 15 


viscosity keeps constant. This caution is necessary, for it seems 
to be often assumed that a, as deduced from the ratio of the 
actual to the limiting equivalent conductivity, always ex- 
presses the ionization of the solution, whatever its concentration 
may be, although for fairly strong solutions no convincing 
evidence has been adduced in favour of the assumption made. 
On the other hand, equation (38) given on p. 214, 

MJ + Vl = 1*037 x 10-* x - 

in which u t and v l denote the actual mobilities of the ions 
under the conditions of the experiment, probably holds good 
whatever be the concentration of the solution, and gives the 
simplest and most certain form of Kohlrausch's theory. 

Hittorf himself recognized that the migration constant of 
cadmium iodide requires the supposition of 

Complex ions. . L L 

complex ions, some unaltered salt migrating in 
company with the iodine, as a complex anion. There is 
considerable evidence besides that already described that 
similar ions exist in many other solutions in water and other 
solvents 1 . This evidence may be summarized as follows : 

(1) In the case of simple salts such as potassium chloride, 
Hittorf 's transport number is independent of the concentration, 
but this is not so for more complicated salts such as barium 
chloride or magnesium sulphate. The change is so great that 
it is not easy to explain it by a difference in the variation of 
the mobility of the two ions with concentration. 

(2) While it is possible to assign a definite specific mobility 
to the ions of potassium chloride and similar salts, the velocities 
of the ions of more complex salts depend on the nature of the 
other ion present, until the dilution becomes almost infinite. 

(3) In direct measurements of ionic mobilities by the 
method of moving boundaries, the results agree better with 
theory for the simple salts, and when the solutions of the more 
complex salts are of considerable concentration, the phenomena 
at the boundaries become very complicated. 

1 Whetham, Phil. Tram. A, CLXXXIV. 358 (1893) ; Steele, Phil. Trans. A, 
cxcvni. 133 (1902) ; Schlundt, Jour. Phys. Chem. vi. 159 (1902). 


(4) As we shall see later, the ionization as calculated from 
the electrical conductivity agrees better with that deduced from 
the freezing point in the case of simple salts than for more 
complicated ones. 

All these relations are easily explained by the supposition 
that, as the concentration increases, many solutions, especially 
of such salts as magnesium sulphate, contain complex ions, 
formed by the union of some unaltered salt molecules with the 
anion or cation. 

Such molecules of salt will be dragged forwards with the 
ions and may increase the effective resistance to their motion, 
thus reducing the velocity below the value given by the 
fraction indicated above, which expresses the ratio of the active 
to the total solute at any moment. The ratio of the actual to 
the limiting velocity then ceases to be equal to the ratio of the 
average active time to the whole time for each ion. The 
equilibrium will still be dynamical, however, and these attached 
molecules must in turn become inactive stationary molecules 
and active molecules, the parts of which are moving ions. The 
life of an ion can then be divided into four parts, (1) the time 
during which it is active as a simple ion, and therefore moving 
with nearly the velocity it would have in pure water, (2) the 
time when it is part of an inactive molecule at rest, (3) the 
time it has an inactive molecule attached to it, and is therefore 
moving with a velocity smaller than that referred to above, and 
(4) the time during which it forms part of an inactive molecule 
dragged along by an active ion, when it moves with the same 
diminished velocity but is ineffective as far as carrying current 
is concerned. 

The effective or resultant velocity of an ion is found by 
dividing the average distance it travels during the periods (1) 
and (3) by the whole time considered, for during the periods 
(2) and (4) it does nothing towards carrying the current. The 
effective velocities, as thus calculated, will be correctly deter- 
mined by Kohlrausch's equation 

7, fjp 

^ + 0! = 0-01037 x- x^-. 
n dx 

But when we wish from this result to calculate the 



individual values of u and v lt we must use the migration 
constant for the given electrolyte, which has been determined 
by the method of Hittorf. Now the theory of Hittorf 's method 
(page 208) assumes that the difference produced in the con- 
centrations of the liquids round the two electrodes is, in 
general, entirely due to a difference in the velocities of the two 
ions; though, as we stated, Hittorf recognized the action of 
complex ions in exceptional cases. But the differences in 
concentration might always be explained by the supposition 
that inactive solute or solvent molecules were attached to one 
or other of the ions. If this were the case, the division of the 
value of M!+ V! in the ratio of Hittorf 's number would lead to 
an erroneous result for the individual ionic velocities. The 
calculated velocities would then differ from those experimen- 
tally determined by a greater and greater extent, as such 
complex ions became more numerous owing to an increase in 
concentration or to other causes. We may thus explain the 
fact that the experimental results agree less nearly with the 
calculated ionic velocities in solutions such as those of 
magnesium sulphate than in solutions of potassium chloride 
and similar salts. 

It will now be evident that, if complex ions are present, the 
mobility of an ion calculated from observations on solutions of 
different salts containing it will not be constant, since different 
numbers of complex ions may exist in the different solutions. 
Moreover, in the solutions of any one substance, the number 
of complexes depends on the concentration, as the change in 
the transport number indicates, and therefore the mobility 
at infinite dilution cannot be calculated unless the transport 
number has been determined for a solution dilute enough to 
secure the absence of complex ions. Experiments on transport 
numbers have not usually been made in very dilute solutions, 
and consequently the values for the mobility of such an ion as 
barium, found by experiments on different solutions of two or 
more of its salts, do not in general agree with each other. 
Steele points out, in this connexion, that recent transport 
experiments by Noyes on solutions of barium chloride and 
nitrate at a concentration of 0*02 normal give the same 
mobility to the barium ion in the two solutions. At greater 


concentrations, the relative amount of salt taken from the 
neighbourhood of the cathode (p. 212) is increased. This result 
might be explained by the assumption that some double 
molecules of composition 2BaCl 2 exist, which yield the ions 

Ba, Cl and (BaCl 2 ) Cl. The effects of complex ions will again 
be considered in Chapter XII. 

We may conclude, from the experimental confirmation 
described above, that the velocity of an ion of a 

Connexion be- ^ 

tween the mobility simple salt, as calculated by Kohlrausch s theory 

of an ion and its , ... ,, , , . 

chemical constitu- from the conductivity, really does represent the 
actual speed with which, on the average, the 
ion makes its way through the solution. We may therefore 
apply the theory with confidence to cases in which the experi- 
mental confirmation would be difficult or impossible. 

If we know the specific velocity of any one ion, we can, 
from the conductivity of very dilute solutions, at once deduce 
the velocity of any other ion with which it may be combined, 
without having to determine the migration constant of the 
compound, a matter often involving considerable trouble. 
Thus, taking the specific ionic mobility of hydrogen as 0*0032 cm. 
per sec. per volt per cm., we can, by determining the conduc- 
tivity of dilute solutions of any acid, at once find the specific 
velocity of the acid radicle involved. Or, again, since we know 
the specific velocity of the silver ion, we can find the velocities 
of a series of acid radicles at great dilution by measuring the 
conductivity of their silver salts. 

By these methods Ostwald, Bredig, and other observers have 
found the specific velocities of many ions both of inorganic and 
organic compounds, and examined the relation between consti- 
tution and ionic mobility. A full account of such data has been 
given by Bredig 1 . The velocities given by him are relative 
numbers calculated from the conductivities measured in terms 
of mercury units, and so must be multiplied by 110 x 10~ 7 if 
they are wanted in centimetres per second per volt per centi- 

The mobility of elementary ions is found to be a periodic 

1 Zeits. phys. Chem. xra. 191 (1894). 


function of the atomic weight, similar elements lying on similar 
portions of the wavy curve. The curve much resembles that 
giving the relation between atomic weight and viscosity in 
solution. For compound ions the mobility is largely an additive 
property ; to a continuous additive change in the composition of 
the ion corresponds a continuous but decreasing change in the 
mobility. Ostwald's results for the formic acid series give 

Diff. for CH 2 

Formic acid HCO' 2 51 -2} _i2-9 

Acetic H 3 C 2 2 38'3i _ 4 - 

Propionic H 6 C 8 0' 2 34-3J " 3>5 

Butyric H 7 C 4 2 30-8 

Valeric H 9 C 5 0' 2 28-8 

Caprionic H U C 6 0' 2 27-41 

Bredig finds similar relations for every such series of 
compounds which he examined. Isomeric ions of analogous 
constitution have equal mobilities. A retarding effect is, in 
general, produced by the replacement of H by Cl, Br, I, CH 3 , 
NH 2 or N0 2 : of any element by an analogous one of higher 
atomic weight (except and S) ; of NH 3 by H 2 ; of (CN) 6 by 
(C 2 4 ) 3 ; by the change of amines into acids ; of sulphonic acids 
into carboxylic acids ; acids into cyanamides, dicarboxylic into 
monocarboxylic acids ; and by monamines into diamines. The 
additive effect is, however, largely influenced by constitution. 
Thus in metamerides the mobility increases with the symmetry 
of the ion, especially as the number of C N unions gets greater. 

Reinold and Riicker have investigated the electrical re- 
Conductivity of sistance of thin soap films 1 . The thickness was 
liquid films. measured by optical means, depending on the 

interference of two parts of a beam of light. One part of the 
light passed through a tube across which several films were 
stretched, and the consequent optical retardation was deter- 
mined. On the assumption that the refractive index of a film 
is the same as that of the liquid in bulk, an assumption for 
which reasons are given, these measurements enable the ag- 
gregate thickness of the films to be estimated. It was found 
that when the films became too thin to reflect light and there- 
fore, like the central spot of a system of Newton's rings, looked 
1 Phil. Trans. A. CLXXXIV. 505 (1893). 


black by reflected light, no further reduction in thickness could 
be obtained, and the thickness remained constant for any given 
liquid. If some salt was added to the liquid, the thickness 
decreased ; thus the following table shows the thickness in 
micro-millimetres (metres x 10~ 10 ) of films of 1 part of hard 
soap in 40 parts of water with varying amounts of potassium 

Optical method. 

Percentage of KN0 3 3 1 0-50 

Thickness in ^i 12'4 13'5 14'5 22'1 

If the conductivity of the film is the same as that of the 
liquid in bulk, the electrical resistance of a film should give 
values for the thickness which agree with these numbers. It 
was found that, as long as the amount of salt present was 
greater than about 2 or 3 per cent., the results of the two 
methods agreed, but that, if the amount was less, the electrical 
method gave a result greater than that obtained optically. 

Electrical method. 

Percentage of KN0 3 32 10-50 

Thickness in /z/i 10'6 127 24'4 26'5 148 

These results indicate that the conductivity of a thin film is 
much greater than that of the liquid in bulk when the concen- 
tration of the dissolved electrolyte is very small, but that the 
conductivities become identical as the concentration increases. 

The phenomenon cannot be explained by supposing that 
the effect of the surface energy is to increase the ionization, 
because it is in dilute solutions, where the ionization is already 
nearly complete, that the difference is most. marked. Unless 
the presence of the soap has a disturbing influence, it seems 
that the ionic friction must be less, and the ionic mobilities 
greater, in the film than in the bulk of the liquid. It is worthy 
of note that there is evidence to show that the conductivity of 
a thin metallic film is less than that of the metal in bulk. On 
the electron theory this is explained by the interference with 
the motions of the corpuscles which results when the thickness 
of the conductor becomes comparable with the mean free path 1 . 
1 Longden, Amer. Journ. Sci. ix. 407 (1900) ; Phys. Rev. July & Aug. (1900). 



Introduction. Reversible cells. Electromotive force. Effect of pressure. 
Concentration cells. Different concentrations of the electrodes. 
Different concentrations of the solutions. Concentration double 
cells. Effect of low concentrations. Chemical cells. Oxidation and 
reduction cells. Transition cells. Irreversible cells. Secondary cells 
or accumulators. 

SINCE the invention of Volta's pile in the year 1800 many 
forms of battery have been introduced. An 

Introduction. < 

account of those now in use, and of the pur- 
poses to which each is specially adapted, may be found in 
any book on practical electricity. We shall here confine our- 
selves to the theory of the production of the electric current to 
be obtained from such cells. 

When two metallic conductors are placed in an electrolyte, 
a current will flow through a wire connecting them provided 
that a difference of any kind exists between the two conduc- 
tors in the nature either of the metals or of the portions of 
the electrolyte which surround them. A current can be 
obtained by the combination of two metals in the same elec- 
trolyte, of two metals in different electrolytes, of the same 
metal in different electrolytes 1 , or of the same metal in solutions 
of the same electrolyte at different concentrations. 

1 An effective difference in the electrolytes can be secured by dissolving 
either different substances in the same solvent, or the same substance in 
different solvents. 


In order that the current should be maintained, and the 
electromotive force of the cell remain constant during action, 
it is necessary to insure that the changes in the cell, chemical 
or other, which produce the current, should neither destroy the 
difference between the electrodes, nor coat either electrode with 
a non-conducting layer through which the current cannot pass. 
As an example of a successful cell of fairly constant electro- 
motive force we may take that of Daniell, which consists of the 
electrical arrangement 

zinc / zinc sulphate solution / copper sulphate solution / copper, 

the two solutions being usually separated by a pot of porous 
earthenware. When the zinc and the copper plates are con- 
nected through a wire, a current flows, the conventionally 
positive electricity passing from copper to zinc in the wire and 
from zinc to copper through the cell. Zinc dissolves, and zinc 
replaces an equivalent amount of copper in solution, copper 
being simultaneously deposited on the copper electrode. The 
internal rearrangements which accompany the production of a 
current do not cause any change in the original nature of the 
electrodes, and, as long as a moderate current flows, the only 
variation in the cell is the appearance of zinc sulphate on the 
copper side of the porous wall. As long as the supply of copper 
sulphate, is maintained, copper, being more easily separated 
from its solution than zinc, is alone deposited at the cathode, 
and the cell remains constant. On the other hand, if no current 
be allowed to flow, slow processes of diffusion, unchecked by 
migration in the opposite direction, will cause copper to appear 
in the anode vessel, and finally to be deposited on the zinc. 
Little local galvanic cells are thus formed on the surface of the 
zinc, which then dissolves even though the circuit of the main 
cell is not completed. Till this deposition occurs, the cell can 
be left on open circuit without waste, and no zinc will dissolve 
if it is chemically pure. If however commercial zinc, which 
contains iron, be used, local action is again set up. This 
action can be prevented by amalgamating the zinc ; probably 
because that process produces a uniform surface, iron being 
insoluble in mercury. 


The conditions necessary for the continuous production of a 
current are well illustrated in an experiment described by 
Ostwald 1 . Plates of platinum and amalgamated zinc are sepa- 
rated by a porous pot, and are each surrounded by some of the 
same solution of a neutral salt of a metal more oxidizable than 
zinc, such as potassium sulphate. When the plates are con- 
nected together by a wire, no permanent current flows and no 
appreciable quantity of zinc is dissolved, for any current must 
primarily liberate potassium at the platinum, the potassium 
secondarily decomposing water. This primary process would 
absorb more energy than is supplied by the solution of the zinc. 
If sulphuric acid be added to the vessel containing the zinc, 
these conditions are unaltered, and still no zinc is dissolved. 
On the other hand, if a few drops of acid be added to the 
vessel in which is the platinum plate, bubbles of hydrogen at 
once appear, a continuous current flows, and zinc is simul- 
taneously dissolved.' This experiment illustrates two conditions 
necessary for the production of a current. In order that posi- 
tively electrified ions may enter a solution, an equivalent amount 
of other positive ions must be removed or negative ions be 
added; and, 'for the process to occur spontaneously, the possible 
actions at the two electrodes must involve a decrease in the 
total available energy of the system. 

Considered thermodynamically, galvanic cells must be 
divided into reversible and non-reversible sys- 

Reversible cells. . 

terns. If the slow processes of diffusion be 
ignored, the Daniell cell already described may be taken as 
a type of a reversible cell. Let an electromotive force exactly 
equal to that of the cell be applied to it in the reverse direction. 
When the applied force is diminished by an infinitesimal 
amount, the cell produces a current in the usual direction, and 
the ordinary chemical changes occur. If the external electro- 
motive force exceeds that of the cell by ever so little, a current 
flows in the opposite direction, and all the former chemical 
changes are reversed, copper dissolving from the copper plate, 

1 Phil. Mag. [5] xxxn. 145 (1891). 


while zinc is deposited on the zinc plate. The cell together 
with this balancing electromotive force is thus a reversible 
system in true equilibrium, and the thermodynamical reasoning 
applied to such systems in the first chapter can be used to 
examine its properties. 

Another reversible cell of similar type is the arrangement 

zinc / zinc sulphate / zinc sulphate with mercurous sulphate / 


due to Latimer Clark. It is used as a standard of electro- 
motive force, giving T434 volts at 15 C. The very slightly 
soluble mercurous sulphate acts as depolarizer, depositing 
mercury on the cathode, when the cell works in its natural 
direction. Here also a reversal of the current reverses all the 
internal changes of the cell. 

Cells from which gas is lost into the atmosphere, such as 
Volta's original zinc /dilute acid /copper couple, and others in 
which irreversible processes of reduction occur, such as the 
Grove arrangement, zinc / dilute suphuric acid / nitric acid / 
platinum, form essentially irreversible systems. Moreover, it 
does not follow that, because an accumulator can be used to 
give a current in the reverse direction to the charging current, 
it is in the thermodynamic sense a reversible cell. This is only 
the case when an electromotive force greater by an indefinitely 
small amount than the secondary electromotive force of the* cell 
will reverse the current through it and the chemical actions in 
it also. For this to be possible, it is necessary that the whole 
of the energy of the charging current should be put into 
available energy of chemical separation, which can all be 
regained when the cell is discharged. 

Let us now apply the thermodynamic relations, which we 

have established in Chapter I., to investigate 

Electromotive the e i ectromot i v e force of reversible cells. The 

solution of this problem was given in different 

ways by Willard Gibbs and von Helmholtz. For us the 

simplest method will be to use the available energy equation 

which was obtained on p. 29. 


Let E denote the electromotive force of the cell at a 
temperature 6, and let a quantity q of electricity pass reversibly 
through the cell in the natural direction. The external work 
done is then equal to Eq, which therefore represents the decrease 
in the available energy of the system. Thus the equation (11) 
of available energy 


The decrease e of the internal energy of the cell will be the 
same if the final state of the system is reached in any other 
way, as for instance by direct chemical action, the energy 
equivalent of which can be found by measuring the heat evolved 
by the reactions. Let V be the heat of reaction per gram- 
equivalent corresponding to the chemical changes which occur, 
and let q denote the number of electrical units simultaneously 
passed through the cell ; then we get 



Writing X for we have 

as the expression for the electromotive force of the cell, 
where X denotes the calorimetric heat of reaction which would 
correspond to unit electric transfer. 

The same equation can of course be obtained in other ways, 
as for instance by putting the cell through a complete ideal 
reversible cycle of changes in the manner of Carnot's engine, 
the external work here being done by the energy of the current. 

It will be observed that if the temperature coefficient 
dEjdO is zero, the equation shows that the electromotive force 


is equal to the heat of reaction. The earliest formulation of 
the subject, due to Lord Kelvin, assumed that this relation 
was true in all cases; as, calculated in this way, the electro- 
motive force of the Daniell cell, which has a very small tempera- 
ture coefficient, agreed with observation. The heat of reaction 
when one electrochemical equivalent of zinc replaces copper 
in sulphate solution, which is the effective process of the cell, 
is 2*592 calories. Multiplying by the mechanical equivalent 
of the calorie, 4*18 x 10 r , we have 1*09 x 10 8 electromagnetic 
units, or 1*09 volts, a number agreeing with that observed. 

In cases in which the temperature coefficient is appreciable, 
the exact expression must be used. It has been experimentally 
confirmed by Czapski 1 and Gockel 2 , and quantitatively by 
Jahn 3 ; it has been verified for the Grove gas cell by Smale 4 , 
and for cells in which fused salts instead of solutions are used 
as electrolytes by L. Poincare 5 , J. Brown 6 , and Buscemi 7 . It 
is clear, since the electrical energy is not equal to the heat of 
reaction in the equation, that there must be a reversible 
evolution or absorption of heat energy in the cell per unit 
electric transfer equal to the thermal equivalent of the expres- 

3 TT1 

sion 0-yTf. This reversible heat is to be distinguished from 

the irreversible heat produced in a cell by the passage of a 
current through it against the resistance. The latter depends 
on the square of the current, and can therefore be reduced to 
any extent, as compared with the reversible heat, by lowering 
the strength of the current. Jahn compared the reversible heat 
thus calculated from the electromotive force and its tempe- 
rature coefficient, with that found by means of experiments with 
an ice calorimeter. 

1 Wied. Ann. xxi. 209 (1884). 2 Wied. Ann. xxiv. 618 (1885). 

3 Wied. Ann. xxvin. 21 (1886). 4 Zeits. phys. Chem. xiv. 577 (1894). 

5 Paris Eeparts, n. 411; Compt. rend. ex. 339 (1890) ; Ann. Ghim. et Phys. 
[6] xxi. 344 (1890). 

6 Proc. R. S. LII. 75 (1892). 7 Att. Accad. in Catania, xn. (1900). 


The following table gives some of his results: 



at 0. 


energy in 

Heat of 

Reversible heat effect 



Cu/CuS0 4 . 100H 2 O/ ) 
ZnS0 4 . 100H 2 0/Zn ] 





- 428 

- 416 

Ag/AgCl/ ) 
ZnCl 2 .100H 2 0/Zn J 





+ 5148 


Ag/AgN0 3 .100H 2 0/ ) 
Pb(N0 3 ) 2 .100H 2 0/PM 




+ 7890 

+ 7950 

Ag/AgN0 3 .100H 2 0/ ) 
Cu(N0 3 ) 2 .100H 2 0/Cui 




+ 8920 

+ 8920 

Certain mercury cells gave results not so concordant with 
theory, but this want of agreement was afterwards shown by 
Nernst to be due to an erroneous value for the heats of 
formation of mercury compounds. 

Attempts have been made by Jahn 1 and Gill 2 to localize 
this reversible heat by measuring the Peltier effect at the 
junctions in the cell. They find that the usual thermo-electric 
equation, which we shall consider in the next chapter, giving 
the sum of the Peltier effects 

holds good within the limits of experimental error. 
The relation 

X E=\ + 

then becomes 

so that 


The relation thus verified has been applied by Jahn to the 
determination of heats of formation 3 . 

1 Wied. Ann. xxxiv. 755 (1888) and L. 189 (1893). 

2 Wied. Ann. XL. 115 (1890). 

3 Wied. Ann. xxxvn. 408 (1889). 


Whenever the action of a cell causes change of volume, the 
electromotive force must depend on the external 
pressure 1 . In cells where metals only are de- 
posited or dissolved, the changes iu volume are 
small; but when gases are evolved or condensed at either 
electrode, a considerable amount of external work is done. In 
treating this problem from the point of view of thermodynamics, 
we naturally employ the thermodynamic potential at constant 
pressure instead of that at constant volume (pp. 23 and 24). 

The two thermodynamic principles give, as we have seen, 
the relation 

since, in this case, the external work comprises a term pBv as 
well as the electrical term ESq. Subtracting S (0<f> +pv) from 
each side, 

S(e-0(f> -pv) = EBq - vSp - 
or, writing f for e 6$ pv, 

The right-hand side is a perfect differential, and we may write 

3? F 3? at . 

*- E - a--> w = -*< 

hence we have relations such as 

(d_E\ d_ /ar\ _a/af\ _(to\ 

\dp) q dp\dqJ dq\dp) \dq) p ' 

which prove that the rate of increase of the electromotive 
force with the pressure is equal to the decrease in volume at 
constant pressure per unit quantity of electricity passing, when 
the temperature in each case remains constant. Faraday's law 
shows that the volume change is proportional to the quantity of 
electricity, so that if Vi and v 2 be the initial and final volumes, 

and we get 

1 See Duhem, Le potentiel Thermodynamique, p. 117 ; and Love, Report of 
the Australasian Association, Sydney, 1898, p. 84. 


For solids and liquids Vj and v 2 are sensibly independent of 
the pressure, and we get by integration for the change of 
electromotive force with change of pressure 


In the case of gases, if Boyle's law be assumed, we can 
again readily integrate the equation. Let us as usual denote 
by H the gas constant for one gram-molecule, so that the value 
of R is the same for all gases. Let each molecule of gas dissolve 
as n ions ; let the valency of each ion be y, and let q be the 
amount of electric charge on one gram-atom of a monovalent 
ion. The electric transfer required to liberate one gram-molecule 
of the gas is then qny. We thus obtain 

P^ = 

J Pl p 

Pi qny qny J Pl p qny "# 

The decomposition of water with platinized electrodes is a 
reversible process, so that this equation also determines the 
effect of pressure on the decomposition point of water. 

These two relations (42) and (43) have been experimentally 
confirmed by Gilbault 1 throughout a range of pressure 
extending from 1 to 500 atmospheres. The effect for a 
Daniell cell is about the hundredth part of that for a gas 

Many of the results here deduced thermodynamically can 
be obtained in other ways. Thus J. J. Thomson 2 has found the 
effect of pressure on the electromotive force of gas cells by an 
application of the Lagrangian function in a strictly dynamical 
way, and, by making a probable assumption, has also obtained 
Helmholtz's equation in a similar manner. Again, as we shall 
see later, Nernst and Planck have developed a theory of galvanic 
cells from a knowledge of the velocities of the ions and the 
osmotic pressures. 

1 Ann. Fac. des Sci. de Toulouse, v. A.S. 1891. 

2 Applications of Dynamics to Physics and Chemistry, pp. 86, 98. 


As stated above, an electromotive force is produced whenever 

there is a difference of any kind at two electrodes 

Conc cen t s ration immersed in electrolytes. In ordinary cells the 

difference is secured by using two dissimilar 

metals, but an electromotive force also exists if two plates of 

the same metal are placed in solutions of different substances, 

or of the same substance at different concentrations. Another 

method is to use in the same solution electrodes of different 

concentration. Such electrodes can be constructed by taking 

hydrogen in contact with platinized platinum, and making the 

pressure different at the two ends. 

In all such cells the electrical energy is not obtained from 
chemical changes, but from the energy of expansion of substances 
from greater to smaller concentrations. For the cases in which 
very dilute substances, gaseous or dissolved, alone are used, the 
gaseous laws are obeyed, and there is consequently no heat of 
dilution. Thus in Helmholtz's general equation, which is appli- 
cable to all kinds of cell, namely 

X vanishes, and we get 

E _ e dE dE_d0, 

so that integrating, 

log E = log 6 + constant, 

E=C0 (44). 

The electromotive force is therefore proportional to the absolute 
temperature. This relation, it will be noticed, depends on the 
absence of chemical action or heat of dilution, and is only 
true, even for concentration cells, when the substances are 
so dilute that no sensible heat is evolved on further dilution. 
Concentration cells, in which it holds, are really heat engines, 
and work by using the heat energy of their surroundings. 
These remarks apply to all concentration cells for which the 
gaseous laws hold, whether the difference in concentrations is 
in the electrodes or in the solutions. 

w. s. 16 


The nature and theory of concentration cells were first fully 
discussed by von Helmholtz by an application of the principles 
of thermodynamics and a knowledge of the phenomena of 
vapour pressure 1 , without any special electrolytic hypotheses, 
and the general accuracy of his theory was confirmed by the 
experiments of Moser 2 . 

Let us consider the example mentioned, hydrogen electrodes 
at different pressures. If these electrodes are 

Different concen- . j i , P -j 11 T 

trations of the immersed in a solution of acid or alkali, a 
current will flow, gas dissolving at the electrode 
of high pressure and appearing at that of low pressure. Now a 
thermodynamic cycle can be performed at constant temperature 
by allowing such a current to flow reversibly against a balancing 
electromotive force, taking out the gas evolved, slowly com- 
pressing it, and then passing it into the other electrode vessel 
till everything is in its initial condition. 

The work gained from the gas during its escape at constant 
pressure from the first electrode vessel is p^. In compressing 

it from p l to p 2 the work gained is I pdv, as was shown on 


p. 3. Finally in passing it into the second electrode vessel 
the gas does work p 2 v 2 . The total work may be written 

\pv] -Tpdv. 

L J2 J Pi 


so that 

I vdp \ pv\ - Ipdv. 

Thus the work done during the process under consideration is 

always measured by I vdp] and only if Boyle's law holds, so 


that pv vanishes, can it also be expressed as I pdv. 

L J2 J Pi 

1 Wied. Ann. in. 201 (1878); Ges. Abhand. i. 840, n. 979 ; Sitzungsber. Berl. 
Akad. Juli 1882. 

2 Wied. Ann. HI. 201 (1878). 


Then, as before, let each molecule of the gas dissolve as n 
ions, the average valency of which is y\ q being the electric 
charge on one gram-atom of a monovalent ion, the electric 
transfer required to liberate one gram-molecule of the gas is 
qny. In the complete cycle of the concentration cell, both the 
electrical process and the reverse operations can be performed 
isothermally, so that the balance of work gained must be zero, 
and we may write 

Eqny + I vdp = 0. 

J Pl 

This result is general ; but if the gas obeys Boyle's and Charles' 

laws we may put 

and obtain 

= _ , ............ 

qny Pl p qny p, 
an equation which shows that the electromotive force of the 
cell described is proportional to the logarithm of the ratio of 
the pressures at the two electrodes. It seems that no quanti- 
tative experiments have yet been made on such cells, and this 
relation therefore remains without practical confirmation. The 
method of deducing it, however, will serve later on to elucidate 
other more complicated cases. 

A cell similar in theory, in which the hydrogen is 
replaced by mercury, has been experimentally realized. The 
electrodes consist of a long and a short column of mercury, each 
separated from the solution of a mercurous salt by parchment 
paper, which is impervious to the mercury in bulk but 
apparently allows it to pass in the form of ions. Mercury 
dissolves from the column of high pressure, and is precipitated 
beneath the column of low pressure, a corresponding electric 
current passing through the cell. The process can be me- 
chanically reversed by raising the required quantity of mercury 
through a height h equal to the difference in level of the two 
columns, and the electrical work gained is equal to the work so 
expended. Thus, if A is the atomic weight in grams, 

Eqny = Agh, 
n being in this case equal to unity. 




[CH. X 

Des Coudres 1 arranged such a cell, and obtained the 
following results: 

Pressure in 

E (calculated) 

E (observed) 


7-2 x lO- 6 volts 

7-4xlO- 6 volts 


9'3 ,, 





Another method of varying the concentration of the elec- 
trodes is to use amalgams of a metal, of different proportions. 
Here again, the passage of material is from a concentrated to a 
dilute condition; and, if we suppose that metals dissolved in 
each other exert osmotic pressure like that of ordinary solutions 
(a hypothesis which is supported by the experiments of Ramsay 
on the vapour pressures of amalgams and those of Heycock and 
Neville on the fusion points of various alloys), we can calculate 
the osmotic work needed to undo the changes produced by the 
current in exactly the same way in which we calculated the 
mechanical work in former cases. Assuming that the osmotic 
pressure is proportional to the concentration c, we get 

7"? 1 o 

E = - - lo e - 

qny * e Cl 

The electromotive forces of such cells have been determined 
by G. Meyer 2 , who finds a good agreement with theory for amal- 
gams of zinc, cadmium and copper. Thus for zinc amalgam in 
zinc sulphate solution : 

Temp. Cent. 


c i 

E (observed) 

E (calculated) 







0-0419 volts 

0-0416 volts 

1 Wied. Ann. XLVI. 292 (1892). 

2 Zeits. phys. Chem. vii. 447 (1891). 


In calculating these numbers, the value for R was taken 
corresponding to one gram-molecule. Now for metals dissolved 
in mercury, the vapour pressures show that their molecules 
consist of one atom each, and therefore the gram-molecule of 
zinc was taken as Zn or 63*5 grams. The concordance with the 
observed values therefore confirms the monatomic nature of the 
molecule of a metal dissolved in mercury. 

In both kinds of cell, it is seen from the equation that the 
electromotive force is independent of the nature of the electro- 
lyte. Again, the equation shows that the electromotive force 
should be proportional to the absolute temperature, and this 
result also is confirmed by the experiments. The conditions 
necessary to secure this result have been already considered on 
p. 241. 

Of more practical importance is the case of a concentration 
cell when two plates of the same metal are 

Different con- . . 

centrations of immersed in solutions of the same salt at diffe- 

the solutions. A . m , ., , . 

rent concentrations. Take for example the cell 
silver / dilute silver nitrate / concentrated silver nitrate / silver. 

Here metal dissolves in the more dilute solution and is 
deposited from the more concentrated solution. When one 
electrochemical unit of electricity passes, one gram-equivalent 
of silver dissolves at the anode and an equal quantity is 
deposited at the cathode. In this manner the anode vessel 
must gain one gram-equivalent of salt and the cathode vessel 
lose the same amount. Now consider the motion of the ions 
through the solution. The current, which is exclusively carried 
by silver ions at the electrodes, is shared between silver ions 
and NO 3 ions in the body of the liquid. If the ionic velocities 
were the same, therefore, half a gram-equivalent of each would 
pass across the surface of contact of the solutions. In the 
general case, when the transport ratio of the anion is r, and 
that of the cation 1 r, the anode vessel will, on the whole, 
gain 1 (1 r) or r gram-equivalents of silver and therefore of 
salt, while the cathode vessel must lose an equal amount, the 
difference between this case and that considered on p. 208 
consisting in the fact that we now have a dissolvable anode. 


In order to return these r equivalents of salt from the 
dilute to the concentrated solution in a reversible manner, 
osmotic operations can be performed analogous to those re- 
quired to effect a similar change in the hydrogen electrodes 
described on p. 242. Let us place the more dilute solution, 
which has received additional salt by reason of the electric 
transfer, in an osmotic cylinder, of which the piston is im- 
pervious to the salt in question, and is backed by a large 
volume of the pure solvent. Let the pressure on the piston 
be that of equilibrium. Allow this pressure to fall by an 
infinitesimal amount, so that solvent enters the solution till 
the concentration is again exactly as it was before the electric 
transfer. The change in concentration is very small if a large 
volume of solution is present, so that the process practically 
occurs at constant pressure and the work gained is P^, where 
Vj_ denotes the change in volume, and P x the constant osmotic 
pressure. Now separate bodily from the solution that volume 
of it which contains the amount of salt transferred by the 
current, and reversibly compress this quantity in an osmotic 
cylinder till its osmotic pressure rises to P 2 , that of the more 
concentrated solution of the cell. The work done by the 

osmotic forces is / Pdv. Finally place this liquid in contact 

J Pi 
with the stronger cell solution, connect it through a semi-per- 

meable wall with the reservoir of the pure solvent, and squeeze 
out solvent till the solution regains its initial volume by the 
expenditure of work equal to P 2 v. 2 . The thermodynamic cycle 
is then complete. 

Both the electrical and the osmotic processes of this cycle 
can be made reversible and isothermal; then the balance of 
external work must vanish. Denoting the electromotive force 
by E, and considering the electric transfer q, we may write 

- ( 


which gives, as on p. 242, 

Eq=- l"vdP. 


Now, as before, let q be taken to represent the electric 
transfer needed to liberate one gram-atom of two opposite 
monovalent ions at the electrodes, and therefore to decompose 
one gram-molecule of a monovalent salt. If the salt does not 
yield two opposite monovalent ions, let y be the total valency 
of the anions or of the cations obtained from one molecule ; for 
instance, y will be 2, whether the cations be two monovalent 

ions such as the two H ions of a molecule of sulphuric acid, or 

one divalent ion such as the Cu of copper sulphate. The total 

electric transfer corresponding to the decomposition of one 
gram-molecule of salt and the liberation of one gram-atom of 
each of the ions is then qy, and we have 

- vdP (46). 

It has been shown above that while one gram-atom of an 
ion is liberated at the electrode, the transfer of salt from the 
concentrated to the dilute solution is r, where r is the transport 
ratio for the anion. Again, as explained on p. 159, the osmotic 
pressures of electrolytes are abnormally high, so that, when the 
solutions are dilute, the usual gaseous equation gives 

Pv = riET 

for the amount of salt under consideration, where i is van 't 
Hoff's osmotic factor. Substituting for v in equation (46) we 

* (47). 

0y J PI p 

In general, the factor i is a complicated function of the 
concentration and therefore of P, so that this integral cannot 
be directly calculated. A similar expression has been con- 
sidered in detail by Lehfeldt 1 , and made the basis of a method 
of determining the osmotic pressures of concentrated solutions. 
If the two concentrations are not very different from each 

1 Phil. Mag. [6] i. 377 (1901). 


other, and the solutions moderately dilute, in certain cases no 
serious error will be involved in the assumption that ri is 
constant. The last equation then becomes by integration 

Again, for these dilute solutions, the osmotic pressures are 
proportional to the concentrations c 2 and c 1} and we get 

r, riRT, c a //irkX 

E = -- -log e - .................. (49). 

qy 5e c x 

Finally for very dilute solutions, i, the ratio of the actual 
to the non-electrolytic value of the osmotic pressure, becomes 
equal to n the total number of ions given by one molecule of 
salt. We thus reach the result 

^ rnRT , c 2 


which is strictly applicable to very dilute solutions only. 

This expression can be calculated numerically. For deci- 
and centi-normal solutions of silver nitrate the transport 
number r is the same, and has the value 0'52S (p. 212). In 
a cell containing these liquids, 

0-528 x2x 8-28 x 10 7 x 291 

E = -96440-xT- x 2-303 xlo glo 10 

= 0-060 x!0 8 C.G.s. units 
= 0-060 volts. 

Nernst measured the electromotive force of this cell experi- 
mentally and found the value O'Ooo volt 1 . Considering the 
restrictions made in developing the equation, this number is 
in remarkable agreement with the theoretical result. 

It will be noticed that the electromotive force of the 
concentration cells just described, of which the arrangement 

silver / dilute silver nitrate / concentrated silver nitrate / silver 
is an example, depends on the migration ratio for the anion. 

1 Zeits. phys. Chem. vn. 477 (1891). 


A second type of cell can be constructed, the formula for which 
involves the migration number for the cation. In the system 

silver / silver chloride / concentrated potassium chloride / 

dilute potassium chloride / silver chloride / silver, 

the silver chloride is very insoluble, so that the mass of it 
dissolved, which alone is electrolytically active, is constant, and 
the two silver junctions are kept always in the same condition. 
When an electrochemical unit of electricity passes through 
the cell in the direction from left to right as above, a gram- 
equivalent of silver dissolves at the first electrode. This metal 
displaces some of the silver in the chloride, and the silver so 
liberated forms fresh silver chloride with an equivalent of the 
chlorine ions of the potassium chloride. A gram-equivalent of 
this salt is thus removed from the more concentrated solution. 
At the other end of the chain, silver is deposited from the 
silver chloride, and a gram-equivalent of potassium chloride 
must therefore appear in the more dilute solution. But mean- 
while chlorine ions have been migrating against the current 
from the dilute to the concentrated solution; and if r is, as 
before, the migration ratio for the anion, this process involves a 
loss of r gram-equivalents of salt at the cathode (see p. 245). 
The dilute solution, therefore, on the whole, only gains 1 r 
gram-equivalent, and the concentrated solution must lose an 
equal amount. Now 1 r is the migration ratio of the cation. 
It will now be evident that, when we imagine the cycle of 
operations completed by the osmotic process described on 
p. 246, we shall arrive at the result 

On the approximate assumptions there made we shall get 
or, for very dilute solutions, 

.? (53), 

\LV c i 

with the same notation previously used. 



[CH. X 

From an equation equivalent to (52) the following table was 
constructed by Nernst 1 , giving a comparison between the 
observed and calculated values of the electromotive force of 
concentration cells. c : and c 2 denote the concentrations of the 
two solutions in gram -equivalents per litre. 



C 2 

E in volts 

E in volts 































NH 4 C1 










Na0 2 C 2 H 3 





NH 4 OH 







double cells. 

Some of these results have been recalculated by Lodge, 
with later values for the migration numbers 2 . In some cases 
the agreement is improved, in others it is made worse. The 
general result of the comparison is unaltered. 

In the cells hitherto described, the process is complicated 
by the effects of migration, but these effects 
can be eliminated in a manner due to von 
Helmholtz. If a calomel cell, 

zinc / dilute zinc chloride / mercurous chloride / mercury, 

be coupled in the reverse direction with a similar cell in 
which the zinc chloride is concentrated, the arrangement is 
equivalent to the chain 

Zn / dilute ZnCl 2 / HgCl / Hg / HgCl / concentrated ZnCl 2 / Zn. 
In this double cell there is no migration from one solution of 
zinc chloride to the other, but a diminution of the amount of 

1 Zeits. phys. Gliem. iv. 128 (1889). 

2 Lond. Phys. Soc. Proc. xvn. 369 (1900) ; Phil. Mag. [5] XLIX. 351 and 454 


mercurous chloride in the first cell, and an increase of it in the 
second. Mercurous chloride is very insoluble, and hence its 
active or dissolved mass remains constant, and the mercury 
surfaces in the two cells keep always in the same state. The 
double cell is therefore equivalent to a simple concentration cell 

Zn / dilute ZnCl 2 / concentrated ZnCl 2 / Zn, 

in which the effects of migration are eliminated. Von Helmholtz 
originally solved the problem of the concentration cell indepen- 
dently of ionization hypotheses by imagining the thermodynamic 
cycle to be completed by evaporation from the one solution and 
condensation on to the other. 

In this way thermodynamic data only are needed, but it is 
now simpler to treat the subject by an application of the 
principles of osmotic pressure and electric ionization as above. 
An investigation similar to that already used holds good, but, 
in this case, when one gram-atom of zinc dissolves, one gram- 
molecule of salt is formed in the dilute solution and decomposed 
in the more concentrated solution, and this result is not com- 
plicated by migration. The transfer of salt, corresponding to 
unit electrochemical transfer, is therefore unity instead of r, 
and equation (52) becomes 

iRT P 

j i t J t Z i J. *> 

^= logep; 

If we use very dilute solutions of an electrolyte yielding 
n ions, the electromotive force of the concentration cell is 

_ n x 8-28 x 10 7 x 291 c, 

E = 96440x^x10* * 2-303 xlo glo - 

= -x 0-0575 xlog 10 - (55). 

y c i 

In the double calomel cell described above, the number of ions 
is three and the valency of the zinc is two, so that when the 
ratio of concentrations is 10 the electromotive force is 0'0863 volt. 
For very dilute solutions of any salt giving two monovalent 
ions, whatever the absolute concentrations, if the ratio of the 
concentrations is 10, the electromotive force is theoretically 
0-115 volt. 



[CH. X 

When the concentrations though still small are too great 
for the ionization to be taken as complete, an approximate 
result may be obtained from equation (54), 
iRT, P 2 

*= logep. 

when the actual values of the osmotic pressures P x and P 2 and 
of van 't Hoff's osmotic factor i are known. 

Experimental investigations on these double cells have been 
made by Goodwin 1 ; the following are examples of his results. 

Zinc chloride / calomel and zinc chloride / silver chloride 

cells at 25. 

Concentration of 
the ZnCl 2 solutions 
in fractions of normal 

Observed E.M.F. 
of calomel cells 
in volts 

Observed E.M.F. 
of silver chloride 
cells in volts 

Calculated E.M.F. 
in volts 





















Zinc sulphate / lead sulphate cells. 

Concentration of 
the ZnS0 4 solutions 
in fractions of normal 

Observed E.M.F. 
in volts 

Calculated E.M.F. 
in volts 









By the use of the concentration double cells described in 
this section, the effects of migration are eliminated. Another 
class of concentration cells, invented by Nernst, eliminates all 
effects except those of migration, and thus enables measurements 
to be made of the potential difference which exists at the junction 
of two solutions, differing either in the nature or the concen- 
tration of their contents. These cells will be considered in a 
future chapter under the head of the diffusion of electrolytes. 
1 Zeits. phys. Chem. xm. 577 (1894). 


The logarithmic formulae for all these concentration cells 
indicate that theoretically their electromotive 
concentrations force can be increased to any extent by di- 
minishing without limit the concentration of the 
more dilute solution ; logP 2 /Pi then becomes very great. This 
condition can to some extent be realized in a manner that throws 
light on the general theory of the subject. 

Let us consider the arrangement 

Ag / AgCl with normal KC1 / KNO 3 / deci-normal AgN0 3 / Ag. 
The concentration of silver chloride is very small in saturated 
aqueous solution ; from the electric conductivity it has been 
estimated as 0'0000117 normal. It is still further reduced by 
the presence of the large excess of chlorine ions of the potas- 
sium chloride. According to principles to be explained in a 
later chapter, the product of the concentrations of the ions 
divided by the concentration of the non-ionized molecules should 
be a constant at each temperature, so that the lowering of 
solubility produced by a solution of potassium chloride of given 
strength can be calculated. The normal solution used in the 
example has a coefficient of ionization 0'756 ; and so the final 
concentration of the silver ions, in presence of deci-normal 
potassium chloride, which determines the amount of silver 
chloride dissolved, is 0'0000117 2 /0'0756 normal. Putting in 
this value, allowing for the ionization 0'82 of the silver nitrate 
solution, and assuming that the presence of the potassium 
chloride does not affect the osmotic work done by the cell, the 
electromotive force is calculated as 0'52 volt. This number was 
experimentally confirmed by Ostwald 1 who also examined other 
cells with similar electrodes giving high electromotive forces. Thus 


/ silver chloride in 
Deci-normal silver nitrate / . ,, ., 
/ potassium chloride 




)3 33 33 

/ ammonia 


/silver bromide in 


33 33 33 

/ potassium bromide 




33 33 33 

/ sodium thiosulphate 



/silver iodide in 


33 33 33 

/ potassium iodide 




33 33 35 

/ potassium cyanide 




53 33 35 

/ sodium sulphide 



1 Lehrbuch, n. 882. 


The effective concentration of the silver can also be reduced 
by adding some substance which, by combining with the silver 
ions, removes them as such from solution. This is shown by the 
high electromotive forces of the cells Nos. 2, 4, and 6 in the 
above list. 

Other metals have been used as electrodes by Zengelis 1 , who 
showed that, in many cases, cells with electrodes of copper, 
lead, nickel, or cobalt, possessed electromotive forces which 
were greater the more the concentration of the ions round 
one electrode was depressed by the addition of a salt. 

Hittorf 2 has even shown that the effect of a cyanide round 
a copper electrode is so great that copper becomes an anode with 
regard to zinc. Thus the cell 

Cu / KCN / K 2 S0 4 / ZnS0 4 / Zn 

furnishes a current which carries copper into solution and 
deposits zinc. In a similar way, silver could be made 
to act as anode in presence of cadmium. 

If we know the concentration of the ions round one 
electrode, it is possible to calculate it round the other from 
observations on the electromotive force, and this has been 
done by Behrend 3 . 

The success of the theory of such cells as we are now 
considering confirms the natural hypothesis made in the in- 
vestigation, namely, that the osmotic pressure to be used in 
calculating the osmotic work is simply that of the migrating 
substance, one of the ions of which is the metal of the electrode. 
In the cell containing silver chloride in potassium chloride, for 
example, the osmotic pressure which appears in the logarithmic 
formula is that due to the silver chloride alone, not the total 
osmotic pressure of the solution round the electrode due to 
potassium chloride as well. Moreover, if, when the silver ions 
are dissolved, nearly all of them are at once converted into 
compound ions, such as the KAgCy 2 ions of potassium silver 
cyanide, the effective concentration and the effective osmotic 

1 Zeits. phys. Chem. xn. 298 (1893). 

2 Zeits. phys. Chem. x. 592 (1892) ; see also next chapter. 

3 Zeits. phys. Chem. xi. 466 (1893). 


pressure are those due to the slight trace of Ag ions left 1 , and 
not the values due to the total concentration of the silver in 
the solution, whether present as simple or compound ions. It 
seems that in deducing the formulae by the processes described 
on pp. 246, 249, we should imagine the osmotic work done 
against a piston permeable to everything except the actual 
salt, one of the ions of which is the dissolving electrode. This 
is probably legitimate, for although such a semi-permeable 
membrane cannot in every case be practially constructed, its 
existence would violate no known natural principle, and the 
thermodynamic reasoning based on its imaginary use would 
therefore still be valid. 

The ideas used in developing the theory of concentration cells 
have been applied to the usual type of galvanic 

Chemical cells. . J f . . . 

cell by Nernst and others, though in this case 
the basis of the investigation is more speculative. When a 
metal is placed in contact with the solution of one of its salts, 
and a current is passed across the junction and metal dissolved, 
changes occur in the chemical, osmotic, and electrical energies 
of the system. As the osmotic pressure of the solution rises, 
the tendency of the metal to dissolve as electrolytic ions 
becomes less, and it is suggested that eventually at a certain 
pressure no further tendency to dissolve would exist. Above 
this pressure the metal would tend to come out of solution and 
be deposited. This critical pressure bears no relation to the 
limit set to the osmotic pressure of a solution by reason of the 
finite solubility of the salt. With some metals it may be much 
too high to be ever reached, with others it may be too low. If 
the concentration of the solution giving the critical pressure 

1 According to Morgan (Zeits. phys. Chem. xvn. 513 (1895)), potassium 
argento- cyanide undergoes ionization in three steps. The first, KAgCy 2 = 

K + AgCy 2 , is nearly complete. A small number of the complex ions AgCy 2 
give AgCy + Cy, while to an almost infinitesimal extent occurs the third process 

AgCy = Ag + Cy. The mass of silver in the ionic state in a litre of a one- 
twentieth normal solution of potassium silver cyanide is estimated as four 
millionths of a milligram, whereas in a solution of silver nitrate of equivalent 
concentration, the quantity is 10 times as great. 


could be obtained, so that there would be no tendency for ions 
of the metal to enter or leave the liquid, it is fair to conclude 
that the metal and solution would be electrically neutral to 
each other, and that no difference of potential would exist 
between them. This critical osmotic pressure has been called 
the electrolytic solution pressure of the metal in the given 
solution. Nernst identifies it with the osmotic pressure of the 
ions of the metal in the substance of the metal itself. Such an 
idea is perhaps suggested by the osmotic pressure of certain 
metals when dissolved in mercury to form amalgams ; the use 
of these amalgams to give electrodes of different concentrations 
has already been described. On this view the osmotic work 
done in transferring a gram-molecule of metal from the elec- 
trode to the solution may be calculated in the same way as on 

p. 246, where we calculated it when salt passed from a dilute 

r JP 
to a concentrated solution. It will have the value I vdP, 

J p 
where P is the osmotic pressure of the ions in the solution, 

and P m the electrolytic solution pressure of the metal. If E l is 
the potential difference at the surface, the electrical work is 
E^qy, where q is the electric transfer corresponding to the 
solution of a gram-equivalent, and y the valency of the ions. 
Thus as before, 

It is usual to go further, and make the assumption that 
both in the solution and in the metal the osmotic pressure 
obeys the gaseous laws. If this be done, we get equations (47) to 
(50) p. 247, in order of increasing inaccuracy, r being now unity. 

So far we have been considering the solution of metal at 
the anode. In a Daniell cell, which we may take as example, 
there are three junctions to be considered, two metal-liquid, 
and one liquid-liquid. The effect on the electromotive force 
of the surface of contact of the two solutions will be considered 
in the chapter on the diffusion of electrolytes; it is very 
small compared with the potential differences at the surfaces 
of the two metals, and may here be neglected. On the assump- 
tion explained above, we may apply the logarithmic formulae to 


the two metal-liquid junctions and express the electromotive 
force of the cell with the usual notation as 


If we write the expression for the potential difference at one 
of these junctions in the form 


we see that - log P m , which includes the so-called electro- 

lytic solution pressure, is a mere constant for the metal at the 
given temperature. Writing this as M^ we eliminate some of 
the assumptions of the preceding investigation, and apply the 
gaseous laws to the solutions only. The expression for the 
electromotive force of a Daniell cell then becomes 

E = M, - M, - (log P, - log P 2 ) 


-jr,-jr.-??k* ..................... (56). 

qy -LI 

This equation may be derived directly from the principles of 
energetics by observing that the electric work of the cell must 
be equivalent to the algebraic sum of the following terms : 

(1) the work done in dissolving an equivalent of zinc from the 
electrode, its ions being produced at a standard pressure P ; 

(2) the osmotic work I vdP required to expand or compress 

' PO 

the zinc ions so obtained ; (3) the corresponding reversed work 

vdP required to reduce the copper ions to the standard 

pressure ; (4) the work of depositing the copper on the cathode. 

The equation shows that the electromotive force of a 
Daniell cell can be raised by diminishing the concentration of 
the zinc sulphate, or by increasing that of the copper sulphate. 
Since the third term in equation (56) is small compared 
with M l and M 2 , this effect is slight. 

We shall return to the consideration of the electrolytic 
solution pressure of the metals in the next chapter, under the 
head of single potential differences. 

w. s. 17 


In the chemical processes of oxidation and reduction, there 
Oxidation and occur changes in the valency of the ions, in- 
reduction cells. dicating changes in their electric charges. The 
energy of these processes can be made to supply an electric 
current. For example, two platinized platinum plates may be 
placed, one in a solution of stannous chloride, and the other 
in a solution of ferric chloride 1 . If the two be metallically 
connected, a current passes within the cell from the tin solution 
to that of the iron, stannic and ferrous chlorides being formed. 
The divalent stannous ion, taking up a third positive electric 
unit from the anode, becomes a trivalent stannic ion, while 
the equivalent amount of positive electricity is removed at the 
cathode by the conversion of the trivalent ferric into the di- 
valent ferrous ion. 

The gas cells with hydrogen and oxygen, or hydrogen 
and chlorine, as electrodes, may be classified in this group, the 
hydrogen being "oxidized" by its conversion into positively 
electrified hydrogen ions; in fact it is possible to regard all 
chemical cells from this point of view. 

Assuming that the cell may be treated simply as a reversible 
heat engine, Gibbs has deduced another expres- 

Transition cells. . , , 

sion for the electromotive force 2 . Let 1 be the 
transition point, the temperature at which the chemical action 
which gives rise to the current would go on reversibly in either 
direction, and let X be the heat of reaction per electrochemical 
unit of mass if carried on outside the cell. Let 6 be the tem- 
perature of the cell. Now X heat-units at 6^ are equivalent to 

s\ s\ _ n 

X -r units of heat at 9, together with X ^ units of external 

U l Ui 

work, as is indicated by the formula for the efficiency of a 
reversible engine. Thus for each unit of electricity which 

passes, a reversible cell being of maximum efficiency yields 
/t _ a f\ 

X -^ units of electrical work, and X -^ units of reversible 

heat. Looked at in this way, the reversible evolution of heat 
is seen to be of the essence of the problem. 

1 See Le Blanc's Electrochemistry, Eng. Trans, p. 235. 

2 B. A. Report (1886), 388. 


Now, as we know, the available electrical work, when one 
unit of electricity passes, measures the electromotive force, so 


This result of Gibbs' is of great interest, for if two of the 
quantities X, lt and 6 be known, the third can be calculated. 
Cohen 1 has verified the equation experimentally, and used it as 
a means of determining transition temperatures, obtaining values 
which agree well with those found in other ways or by direct 

Differentiating the equation with respect to 0, we get, since 

#! is constant, 

dE _ _\ 

~dB~ ft' 
eliminating ft by means of the equation (57) this gives 

dE _E -\ 
dd~ 6 


Thus we recover von Helmholtz's equation. 

Returning to equation (57), we see that at the transition 
point, where becomes ft, the electromotive force vanishes. 
On this fact depends one of the methods used by Cohen 2 for 
determining transition points. Let us take, as an example, 
the case of the two hydrates of zinc sulphate, ZnS0 4 .7H 2 O 
and ZnSO 4 . 6H 2 0. Two vessels are filled with powdered zinc 
sulphate moistened with water, and connected by means of 
a tube filled with cotton-wool saturated with a solution of zinc 
sulphate. The vessels contain electrodes of zinc and are finally 
sealed up. The contents of one of them are now converted 
into ZnSO 4 . 6H 2 O by heating it for an hour to a temperature 
higher than the transition point. The whole cell is then placed 
in a thermostat, and connected in series with a galvanometer, 

1 Zeits. phys. Ghem. xiv. 53 and 535 (1894). 

2 Zeits. phys. Ghem. xiv. 53 and 544 (1894), or see Van 't Hoff, Studies in 
Chemical Dynamics, Eng. Trans, p. 193. 



which is deflected, since the saturated solutions, in contact with 
different solids, are of different concentrations. The tempera- 
ture is lowered and then allowed to rise slowly. The deflection 
falls, and finally is reversed, the exact point at which it vanishes 
being the transition temperature from the higher to the lower 
hydrate. When the temperature is maintained above the 
transition point for some time, the meta-stable form of salt 
passes completely into the stable form, the solutions become 
identical and the electromotive force gradually sinks to zero. 

When a salt, for instance sodium sulphate, the metal of which 
cannot be used as an electrode, is to be examined, an electrode 
such as mercury in mercurous sulphate, which is unpolarizable 
with respect to the anion, is used. 

Another method of determining transition points electri- 
cally by the use of a concentration cell is of special value 
when the meta-stable form of a substance is difficult to keep 
for any length of time. The electromotive force of a con- 
centration cell depends, as we have seen, on the difference 
in concentration of the two solutions. Thus, if one solution 
be kept at a constant strength, and the other be kept 
saturated with salt, as the temperature slowly rises, any change 
in the solubility is shown by a corresponding change in the 
electromotive force. Now, as we saw on p. 40, the tem- 
perature-solubility curves for the two phases of a component 
cut each other at an angle at the transition point ; so, although 
the solubility itself suffers no sudden change there, its tem- 
perature coefficient does. The temperature coefficient of the 
electromotive force, therefore, will also show a sudden change 
at the transition point, and the temperature-electromotive 
force diagram will consist of two branches, cutting each other 
at a sharp angle at that temperature. The cell is made up in 
open vessels, the solutions being kept stirred in order to insure 
the constant saturation of the one that is in contact with the 
solid. The diagram in Fig. 57 shows the electromotive forces of 
saturated sodium sulphate combined in a concentration cell 
with normal, half normal and quarter normal solutions, the 
weaker solutions giving the higher electromotive forces. The 
transition points estimated from these three cells are 33 '8, 

CH. X] 



33'0 and 32'9, the value found by other methods averaging 
about 33. The theory of this second form of transition cell 






Fig. 57. 

40 6 

50 C 

has been considered by Van 't Hoff, Cohen and Bredig 1 , who 
show that the electromotive force can be calculated from the 
equation by using the known value of the heat of inversion. 

This change in the direction of the solubility curve at the 
transition point, it will now be clear, affects the temperature 
coefficient of standard cells like that of Latimer Clark, which 
contain a saturated solution of zinc sulphate 2 . The transition 
point from ZnSO 4 .7H 2 to ZnS0 4 . 6H 2 is 39, and at this 
temperature there is a sudden change in the temperature co- 
efficient of the electromotive force. When using the cell as a 
standard, a knowledge of the temperature coefficient is needed, 
and the cell would be unsatisfactory above 39. The Weston 
cell, another standard, of the form 

cadmium / saturated cadmium sulphate 

/ mercurous sulphate / mercury, 

has been recommended as having a much lower temperature 
coefficient than the Clark, and an electromotive force of ap- 
proximately one volt (1-019). It has been shown, however, by 

1 Zeits.phys. Chem. xvi. 453 (1895). 

2 Barnes, Jour. phys. Chem. iv. 1 (1900). 


solubility measurements 1 , and by using the Weston as an in- 
version cell 2 , that cadmium sulphate has an inversion point at 
15, which seriously interferes with its trustworthiness as a 
standard of electromotive force. 

Many of the cells in common use are essentially irreversible, 
and it is necessary to enquire how far the prin- 

Irreversible cells. . , , IP- , , ,1,1 

ciples we have used ior investigating the theory 
of reversible cells may be extended to others. 

It has sometimes been held that irreversible cells have no 
definite electromotive force 3 , the measured value depending 
among other things on the number of the ions of the metals 
used as electrodes which happen to be present in the liquids. 
On the other hand it may be argued that single-liquid 
polarizable cells of the type 

zinc / potassium sulphate / copper, 

are limiting cases of Daniell cells 4 . In accordance with the 
expression for the electromotive force of Daniell cells given on 
p. 257, namely 

the electromotive force is independent of the absolute osmotic 
pressures of the two solutions ; it will therefore be unchanged 
if the solutions be equally diluted. Now the remarks on 
p. 255 make it probable that the electromotive force is un- 
affected by the presence of a salt not containing the electrode 
metal, and if so, dilution with potassium sulphate solution will 
be equivalent to dilution with water. On this view, the initial 
electromotive force of the potassium sulphate cell, before 
polarization sensibly intervenes, should be equal to that of 
a Daniell, and a similar result should hold for other such cells. 
The errors of experiment on polarizable cells are considerable, 
but, as an ideal limit, the general results seem to be consistent 
with the required equality. 

1 Kohnstamm and Cohen, Wied. Ann. LXV. 344 (1898). 

2 Barnes, Jour. phys. Chem. iv. 339 (1900). 

3 Ostwald, Lehrbuch, n. 815. 

4 Bancroft, Zeits. phys. Chem. xii. 289 (1893) ; Phys. Rev. m. 250 (1896) ; 
Taylor, Jour, phys.- Chem. 1. 1 (1896). 


Any reversible cell can theoretically be employed as an 
accumulator, though in practice, conditions of 
general convenience are more sought after than 
strict thermodynamic reversibility. 

The accumulator commonly used can be made by placing 
two lead plates in dilute sulphuric acid and passing a current 
between them. Hydrogen is evolved at the cathode, while the 
anode becomes covered with a layer of insoluble lead peroxide. 
As long as the metallic lead of the anode is in contact with the 
solution, it has been shown by C. J. Reed 1 that hydrogen is 
evolved at the cathode under a total electromotive force of about 
0'5 volt, and a considerably, volume of the gas can be collected 
if the area of the anode is large. Eventually the voltage 
necessary for the generation of hydrogen rises to about 2*3. 
This suggests that the first action at the anode is the 
formation of a coating of insoluble lead sulphate, which be- 
comes the effective electrode and yields' sulphuric acid and lead 
peroxide on further action of the current. The cell is now in 
a condition to give a current in the reverse direction, during 
which process lead sulphate is formed at both electrodes until 
these become identical in constitution. In a second charging, 
the lead sulphate at the cathode is reduced to spongy lead, 
while at the anode it again gives peroxide as before. Not even 
at the beginning of the second charging is the anode a lead 
electrode, and there is no action until the voltage reaches about 
2'3. The mass of spongy material at the electrodes is increased 
by continual charging and discharging, which adds to the 
effective capacity of the cell ; and the whole preliminary process 
of forming the cell can be greatly hastened if the plates receive 
in the first place a coating of red lead, Pb 2 3 . 

The main chemical action of a fully formed accumulator 
seems to be in accordance with the equation 

Pb0 2 + Pb + 2H 2 SO 4 Z 2PbS0 4 + 2H 2 O, 

which read from left to right describes the discharge, and 
from right to left the charging of the cell. Although ozone, 
hydrogen peroxide, persulphuric acid, and traces of lead per- 
sulphate have been detected, it seems likely that the above 

1 Jour. phys. Chem. v. 1 (1901). 


equation represents the chief part of the changes. The concen- 
tration of the acid solution is an important factor in determining 
the electromotive force, which increases with increasing con- 
centration, since part of the available energy of the reaction 
is due to the dilution of the residual acid by the water formed. 

It is found in practice that the effective electromotive force 
of a secondary battery is less than that required to charge it ; 
the energy efficiency of a lead accumulator is from 75 to 85 
per cent., although from 94 to 97 per cent, of the current used 
in charging it can be regained. This drop in the electromotive 
force has led to the belief that thermodynamically the cell 
is only partly reversible. Dolezalek 1 however has attributed 
the discrepancy to mechanical hindrances, which prevent the 
equalization of acid concentration in the neighbourhood of the 
electrodes, rather than to any essentially irreversible chemical 

On the provisional hypothesis that the system may be 
treated as reversible, the Gibbs-Helmholtz equation 

has been applied. The discharge reaction for dilute solutions 
gives a calorimetric heat-evolution of 87,000 calories, which, 
on the assumption that the energy is all available, is equi- 
valent to an electromotive force of about 1*88 volts. This 
number agrees with that observed for cells filled with weak acid, 
and indicates that the temperature coefficient is very small, a 
result which has beeu experimentally confirmed by Streintz 52 . 
The quantitative measurements, by the same observer, of dE/dB 
for different concentrations of the acid, can be used to calculate 
the increase of electromotive force for a given change in the 
concentration. Dolezalek calculates the same increase from the 
vapour pressure method applied by von Helmholtz to concen- 
tration cells. To accomplish this we must imagine two lead 
accumulators, one cell A containing more concentrated acid 
than the other cell B. Let them be arranged to work in 
opposite directions. Since the electromotive force of A is 
greater than that of B, the combination acts as a double cell 
and will produce a current in the direction natural to A. The 
1 Zeits. Elektrochem. iv. 349 (1898). 2 Wied. Ann. XLVI. 454 (1892). 


chemical changes of the lead and its compounds are equal and 
opposite in the two cells, and the effective reaction consists in 
the transfer of two molecules of sulphuric acid from A to B 
while two molecules of water pass from B to A. The acid con- 
tents of the two cells thus tend to equality, and the double 
arrangement may be looked on as a concentration cell. The 
available energy is the difference between the work of mixture 
of pure sulphuric acid with water in the proportions of A and 
of B. This work can in either case be calculated if we imagine 
water distilled from the cell to the acid till the resulting 
liquid has the same composition as that in the cell, when it 
can be added to the cell without change of energy. Let p l and 
p 2 denote the constant vapour pressures of water, from the 
liquids in A and B respectively, and p the variable pressure 

over the isolated acid. The work of distillation from the 


cell A to the acid is I vdp (p. 242), or, if we assume that the 
J p 

vapour conforms to the gaseous laws, RTlog per gram- 

C n\ in 

molecule, and for n 2 gram-molecules of water RT log dn. 

.' o p 
Thus the work of transferring one gram-molecule of H 2 S0 4 from 

A to B is 

TF 2 - W, = RT p log ^ dn - RT [*' log ^ dn 

(rn z \ 

w 2 log^ 2 - H! logp, - logpdn . 
J m / 

The actual changes in the double cell also involve the transfer 
of one gram-molecule of water from B to A, a process which by 

distillation would involve the work RTlog . Finally we have 
for the electromotive force of the double cell 

&E = RT (n z \ogp. 2 - n log p l + log^ 2 - J logpdn\ . 

The vapour pressures of sulphuric acid of various concentrations 
have recently been measured accurately by Dieterici 1 , and from 
his results the above equation can be solved numerically. 
Dolezalek has measured the electromotive force E of lead cells 

in ice, as follows : 

1 Wied. Ann. L. 61 (1893). 



[CH. X 



of acid 

/ H 2 S0 4 

Grams of water 
to 1 grm.-mol. 
H 2 S0 4 

pressure in 
mm. Hg 































The differences between these observed values of the electro- 
motive force were compared with the results of theory in two 
ways : by the vapour pressure equation deduced above, and by 
the use of von Helmholtz's equation combined with a knowledge 
of the heat of dilution of sulphuric acid. As measured by 
Thomsen, this heat of dilution may be expressed as 



17860 calories 


when a gram-molecules of sulphuric acid are mixed with b 
gram- molecules of water. The results of the comparison are 
given in the following table. 


Double cell 



From H 









I- IV 




































The agreement of these numbers not only confirms the 
theory given, but also indicates the general conformity of the 
lead accumulator to the thermodynamic properties of reversible 



Volta's contact effect. Thermo-electricity. The theory of electrons. 
Single potential differences at the junctions of metals with electro- 
lytes. Dropping electrodes. Electrocapillary phenomena. The 
theory of von Helmholtz. Electric endosmose. Single potential 
differences (continued). Electrolytic solution-pressure. Electro- 
chemical series. Polarization. Decomposition voltage. Polarization 
at each electrode. Evolution of gases. Electrolytic separations. 

THE source of the energy of a galvanic cell is certainly the 
voita's chemical action, a correction being applied for 

contact effect. anv rev ersible heat which the cell absorbs 
from or gives up to its surroundings. The exact seat of the 
difference of potential, however, has remained undetermined 
for a century, and proved a fruitful subject of discussion. Volta 
located it at the junction of the unlike metals; while Faraday's 
work, which showed the regular and fundamental part played 
by the chemical processes, seemed to indicate the surfaces at 
which the metals were in contact with the liquids. These two 
views of the nature of the phenomena have continued till the 
present 1 , though it seems, from the evidence described in the 
last chapter and for other reasons that will be given, that a 
considerable difference of potential probably exists at the 
surface of separation between metals and electrolytes or 

1 For a description of the phenomena of the contact effect, and an account 
of the Volta theory, see Lord Kelvin, Phil. Mag. July, 1898. For the other 
point of view, see Sir Oliver Lodge, Proc. Phys. Soc. Lond. xvn. 369 (1900); 
and Phil. Mag. XLIX. 351 and 454 (1900). 


dielectrics such as air. The facts to be explained, besides 
those of the galvanic cell, are as follows. 

Dry zinc and copper brought into contact with each other 
in dry air become oppositely charged, and, if their surfaces are 
arranged parallel and very close to each other, so as to form a 
condenser of large capacity, these charges may be consider- 
able. They can be exhibited by separating the plates; the 
capacity is then diminished and the difference of potential is 
thereby increased, so that it can be indicated by an electro- 
scope or measured by an electrometer. By making the con- 
nexion between the metals through part of a potentiometer, a 
difference of potential in the direction opposite to that natural 
to the junction can be applied. Adjusting the potentiometer 
till, on separating the plates, the electrometer is not deflected, 
the natural potential difference can be determined. Another 
method consists in making the actual quadrants of an electro- 
meter of the two metals to be examined. On connecting 
them through a wire a deflection is observed which can be 
destroyed by applying an external electromotive force in the 
opposite direction. An electromotive force of about three 
quarters of a volt neutralizes the natural potential difference 
produced by the contact of zinc and copper. 

Many experiments have been made on this subject ; to 
some of them we shall refer below. Ayrton and Perry have 
examined many metals, obtaining among others the following 
potential differences in volts 1 . 

Zinc j o-210 
Lead { 0-069 

f n f 0-313 

Iron J 

Copper } 
Carbon } 0-113 

By the summation of potential differences, a principle ex- 
perimentally established by Volta, we can find the contact 
effect between any two metals in this list by adding together 
the values for all the pairs of intervening metals. 
1 Phil. Trans. CLXXI. 15 (1880). 


It will be noticed that in all the phenomena described it is 
the difference of potential in the air surrounding the two 
metals which is experimentally observed. Nevertheless, many 
physicists, following Volta, have held that this potential differ- 
ence in the air is due to, and measures, a natural potential 
difference between the metals themselves. When the metals 
are surrounded, except at the area of contact, with the non- 
conductor air, this potential difference is maintained, and can 
be demonstrated by means of an electrometer. In a galvanic cell 
it is supposed that the metallic contact between the electrodes 
constantly keeps up a potential difference, which is constantly 
tending to sink to zero by the action of the electrolytic liquid. 

The theory of the cell given in the last chapter suggests 
that the chief potential difference is to be sought at the liquid- 
metal surface ; but it is clear that, before any such interpretation 
can be accepted, it must be reconciled with the phenomena of 
contact electricity just described. On the analogy of the cell, 
the most natural explanation is that the potential difference is 
due to the action of the oxygen of the air ; and this hypothesis 
receives support from the possibility of approximately calcu- 
lating the observed Volta force as the electrical equivalent of 
the difference between the heats of oxidation of zinc and copper. 
It is perhaps not necessary to imagine actual oxidation; a 
sufficient cause might possibly be found in some slight modifi- 
cation of the film of condensed gas, which, as we have seen, 
seems to exist on all solid surfaces, and to be so difficult to 

The chemical affinity of the oxygen for the zinc can be 
represented by supposing the film of gas to be electrically 
polarized, perhaps by the similar orientation of the electrically 
bipolar oxygen molecules. Such polarization would produce a 
layer of oxygen atoms straining to attack the zinc but prevented 
from reaching it by want of a way of escape for the correspond- 
ing negative charge from the metal, or of a means of approach 
for an equivalent positive charge. Another metal such as copper 
will have a different affinity for oxygen, and thus the electrical 
potential difference between it and the surrounding air will be 
different from that shown by zinc. If contact be made between 


them, the potentials of the metals are equalized, or at any rate 
reduced to the small difference of true metallic contact, by a 
flow of electricity, which, looked at in another way, may be 
referred to the greater force of attraction for oxygen shown 
by the zinc than the copper. Modifying double electric layers 
are thus produced at each interface, analogous to those caused 
by electrolytic polarization. This process may perhaps be ac- 
companied by incipient chemical combination between positively 
electrified zinc atoms at the surface of the metal and negatively 
electrified oxygen atoms in the film of air. The corresponding 
positive atoms of oxygen would then no longer be neutralized, 
and would give the film of air and its neighbourhood a positive 
potential. The zinc and copper themselves are at the same 
potential, but since the outside of the condensed film on the 
zinc is more intensely positive than that on the copper, there is 
a potential gradient through the air between them. It is this 
difference of potential that is observed in experiments on contact 

A modification of the above hypothesis has been suggested 
by Lodge, who imagines that, when contact is made between 
the metals, the negative atoms of the oxygen film facing 
the zinc move nearer to the metal, while the film outside the 
copper recedes further from it. A change is thus produced in 
the thickness of the two condensers formed by the zinc-oxygen 
and the copper-oxygen layers. Their capacities are altered in 
opposite senses, and an electric transfer must take place from 
one to the other. Now the capacities must be large, since the 
separating space is of molecular dimensions, and Lodge has 
shown that a change of about the hundred thousandth of the 
original thickness will produce enough electric transfer to give 
the observed charges to the parallel metal plates, which form 
a condenser of relatively enormous thickness, and hence of very 
small capacity. 

In passing from either metal to the surrounding air, there is 
-a sudden rise of potential, but this rise is greater for the zinc 
than for the copper. We can calculate the magnitude of each 
step of potential on the assumption that all the heat of oxidation 
passes into the energy of electrical contact, and that the method 


of calculating the electromotive force of a cell, given on pp. 235, 
237, is applicable to each electrode considered separately. If 
oxygen were removed by the metal from the film of air, its 
place would be supplied from the free atmosphere, so that the 
effective process which is possible is the oxidation of zinc by 
gaseous oxygen ; it is therefore the ordinary heat of oxidation 
that is involved. The value of this heat for a gram-molecule of 
zinc is about 85,800 calories, aod for copper about 37,200 calories. 
For the electrochemical equivalents, the mechanical values 
correspond to 1*85 and 0'80 volt. We can experimentally 
determine only the difference of these effects. Observation 
indicates about 0'7 or 0*8 volt, which is appreciably less than 
the calculated result. 

When, instead of the insulator air, the plates are surrounded 
by an electrolytic conductor, the slope of potential is accom- 
panied by a current through the solution. At the contacts of 
the liquid with the metals, the natural potential difference is 
constantly tending to be again set up by the chemical affinity ; 
thus a constant current is maintained and zinc is actually 

The probability that the contact effect depends on the 
chemical action or affinity of the surrounding medium will 
be much increased if it can be shown that the magnitude 
of the effect depends on the nature of the medium. Many 
experiments, such as those of Bottomley, indicate that no 
change is produced by working in vessels at high exhaustion, 
or by placing the metals in an atmosphere of hydrogen, though 
a reversal of the sign of the potential difference was obtained 
by J. Brown by replacing the air by hydrogen sulphide or 
ammonia 1 . The film of gas which clings to a solid is ex- 
ceedingly difficult to remove, and it now seems likely that its 
persistence explains the negative results so often found. From 
a recent research by Spiers 2 , in which extraordinary precau- 
tions were taken to remove the film of gas, it is clear that the 
difficulties of getting rid of it have been greatly undervalued, 
and that, when it is really disturbed, large changes in the 

1 Proc. E. S. XLI. 294 (1887). 

2 Phil. Mag. XLIX. 70 (1900). 


magnitude of the Volta force are produced. More work on this 
point is very desirable, but in such a case, a few experiments 
that yield a positive result, and indicate a probable reason for 
the negative results of others, seem to carry great weight. The 
effect of small changes in the nature and condition of the 
surfaces has been recently studied by Erskine-Murray 1 , who 
showed that the potential was increased by polishing and 
burnishing, and diminished by a film of oxide. There would 
certainly be less affinity between gas and a partially oxidized 
metal than between gas and a clean metal, and we should 
naturally expect the potential difference to be reduced by 

The phenomena of thermo-electricity have an intimate 
connexion with the subject now under conside- 
Sectricity. ration 2 . Let us imagine a condenser composed 
of two plates of different metals, separated by 
a layer of dielectric and connected by means of a wire of a 
third metal. In applying the principles of thermodynamics 
to this system, there are two irreversible processes to be 
considered ; the conduction of heat along the metals, and the 
frictional generation of heat by the flow of the current. The 
latter effect, being proportional to the square of the current, 
will be negligible when the current is very small, and can 
therefore be considered to be eliminated under ideal conditions. 
The conduction of heat may be neglected if it proceeds inde- 
pendently of the current, except for the reversible Thomson 
effect considered below ; this independence it is necessary to 

In order to explain the phenomena of the reversal of 
thermo-electric currents when one of the junctions of certain 
metallic circuits continually rises in temperature, Lord Kelvin 
has imagined a convection of heat by the passage of the current. 
Thus the heat absorption per unit quantity of electricity may 
be written as a&T for a current passing up the temperature- 
gradient, where cr may be called the specific heat of electricity, 

1 Phil. Mag. XLV. 398 (1898). 

2 See for instance, Larmor, Aether and Matter, p. 306. 


and denotes possibly a differential effect, depending on an 
inequality in the properties of streams of oppositely moving 
ions. The corresponding heat absorption for a second metal 
may be expressed as <r'ST. 

Now let us consider a complete circuit of these two metals, 
Tj and T 2 being the temperatures of the junctions. Let ITj 
and TT 2 be the heat evolved by unit electric transfer at the 
junctions of the temperatures 7\ and T 2 respectively ; IIj and 
n 2 are called the Peltier effects. Then, considering unit electric 
transfer round the circuit, the energy and entropy principles 
lead to the results 

\(T-<r')dT ............. (58), 


and _ + .ir.o ............ (59) 

By differentiation, equation (59) gives 

1/m <r-<r' 
dT\T)* ~T~ 

whence we obtain 


Thus, while the Peltier effects depend on the temperature 
coefficient of the total electromotive force of the circuit, in 
equation (58) for E y the Peltier effect appears as a local 
electromotive force at the junction. Each electron as it passes 
across, introduces an energy effect qll which involves a re- 
versible evolution or absorption of heat. The Peltier effects 
have been experimentally determined, and their electrical 
equivalents, which measure the contact potential differences at 
the metallic junctions, are calculated as a few millivolts only, 
values much too small to explain by themselves the observed 
Volta effects, without taking account of similar effects at the 
surfaces of the surrounding dielectric. 

w. s. 18 


According to the corpuscular theory of electric conduction, 
the passage of a current through a metal 
is accompanied by the transfer of electrons 
in its line of flow. In each metal, the cor- 
puscles or electrons are present in a certain concentration 
on which depends the conductivity of the material, and may 
perhaps produce something of the nature of osmotic pressure. 
On the analogy of Nernst's conception of electrolytic solution 
pressure at the junctions of metals and electrolytes, a similar 
characteristic pressure has been imagined at the surface of one 
metal in contact with another. The better conducting metal 
in which the corpuscles are more concentrated, will send 
electrons into the other metal till the equilibrium of the 
various tendencies prevents further transfer. The electrostatic 
effect thus produced is the explanation on this view of the 
potential difference of contact. 

We may thus imagine two metals in contact to be in a 
certain sense a concentration cell, the difference of concentration 
being that of the electric corpuscles in the two materials. On 
this view, the contact potential difference is analogous to the 
electromotive force of the cells previously described. There 
is however a vital difference between the two cases. A con- 
centration cell is a system in an unstable state ; if a current 
passes through it, the difference of concentration is changed, 
and the electromotive force altered. Two metals in contact, on 
the other hand, possess a constant difference of corpuscular 
concentration, which must re-establish itself if disturbed. In 
this case, then, there is no source of energy available for the 
production of a current ; and, consistently with this result, the 
total electromotive force of a circuit of several metals at the 
same temperature is found to vanish. Nevertheless, if we 
accept the idea of corpuscular osmotic pressure, the transfer of 
corpuscles from one side to the other of a metallic interface 
will involve the loss or gain of osmotic work. The energy 
change thus produced may be but another aspect of the contact 
difference of electrical potential ; it may be a complicating 
effect, which will alter the relation of that potential difference 
to the Peltier effect of reversible heat production. 

CH. Xl] 



Many attempts have been made to determine experimentally 
single potential tne sm ^ e potential differences at the individual 
differences at the junctions in a circuit containing electrolytes as 

junctions of . J 

metais with well as metals, in a galvanic cell, for example, 

there must be at least two such junctions, and 
the problem is to separate their effects and measure the step of 
potential at each. The measured electromotive force gives the 
sum of all the single potential differences, but the impossibility 
of directly connecting the electrolyte with an electrometer 
without introducing another metallic junction throws difficulties 
in the way of observing them individually. If a method could 
be devised capable of application to one such junction, the 
combination of that junction with any other would enable the 
value for the other to be calculated from the total electromotive 
force as observed in the usual manner. 

Two possible diagrams of the distribution of potential in a 
simple galvanic circuit are exhibited in Fig. 58. The diagrams 
are supposed to be drawn completely round the surface of a 

Zn Cw 



Zn CM 




In C 






Fig. 58. 

Fig. 59. 

cylinder and then unrolled, so that the two vertical lines marked 
Zn denote the same zinc plate, considered to be at the zero 
of potential. Beginning at the left end of the figure, there 
is a sudden rise of potential at the surface of the zinc, the 



potential difference of the electric double layer being denoted 
by the vertical rise of the dotted line. Passing through the 
cell there is a downward potential gradient due to its resistance 
to the current. At the surface of the copper plate, there is another 
discontinuity of potential, upwards or downwards according as 
the natural potential difference at the copper-electrolyte surface 
has a sign opposite to or the same as the zinc-electrolyte 
junction. These two possibilities are indicated in the two 
diagrams of the figure ; the external electromotive force of the 
cell, the same in each case, being represented by the vertical 
height EE'. Since both metals are oxidizable, though not with 
the same readiness, we should expect the potential difference with 
the liquid to have the same sign for copper as for zinc, though 
generally received experiments described below indicate opposite 
signs. Leaving the copper plate, there is another potential 
gradient conforming to Ohm's law in the external circuit, till, 
at the outside surface of the zinc, the potential again falls to 
zero. If the external circuit be broken, the natural potential 
differences at the surfaces of contact remain unaltered, but the 
Ohmic potential gradients are destroyed. The resulting dia- 
grams are shown in Fig. 59, in which EE' now denotes the 
total electromotive force of the cell, as measured on open 
circuit. Thus it will be seen that a determination of the 
electromotive force of the cell only tells the value of EE' , 
the algebraic sum of two effects. The absolute values of these 
two effects remain undetermined ; it is even uncertain whether 
they have the same or opposite signs. Since the current passes 
from metal to electrolyte at the surface of the zinc, and from 
electrolyte to metal at the surface of the copper, both junctions 
help to drive the current if the signs of the two effects are 
different, while, if the signs are the same, the copper plate 
reduces the natural electromotive force of the zinc electrode. 

For circuits containing metals and dielectrics only, Volta's 
law of the summation of potential differences is a necessary 
consequence of the principle of the impossibility of perpetual 
motion, but if electrolytes connect the metals, the possible 
chemical action gives a source of energy, and Volta's law cannot 
be assumed to hold good without experimental demonstration. 


Again, the potential difference between copper and zinc in air 
is probably due to the more or less complete transverse orienta- 
tion of bipolar molecules at the metal / air interfaces, but for a 
metal / electrolyte junction, besides the corresponding molecular 
orientation, due to the essential difference in the nature of the 
two materials, there is a superposed potential difference due 
to the presence of a double sheet of electrolytic ions in the 
neighbourhood of the surface. If a small external electromotive 
force be applied across the junction in the direction opposite to 
that of the natural potential difference, the number of ions in 
the double layer is altered, and, by proper adjustments, the ionic 
effect might be destroyed. The natural potential difference, 
due to the molecular orientation, however, will probably not 
sensibly be affected. On the other hand, when a current flows, 
chemical changes occur, and the molecular layer may in time 
be disturbed. The electromotive force of a galvanic cell may 
possibly involve the potential differences due to the molecular 
arrangement at the various junctions, as well as those due to 
the distribution of ions. It seems probable, however, that 
the methods commonly used for determining single potential 
differences at the junctions of metals and electrolytes give the 
differences due to the ionic layers only. If so, before the 
experimental methods can be considered to solve the problem 
of the location of the effective potential differences in a galvanic 
cell, it is necessary to prove that Volta's summation law holds 
with regard to the molecular potential differences, so that its 
total effect in the circuit vanishes. 

We must now pass to the consideration of the attempts that 
have been made to determine these single potential differences. 
The results are still doubtful and unsatisfactory, but, never- 
theless, a somewhat full account of the subject will be given, 
for it has produced much experimental investigation, and 
further light is greatly to be desired. 

It is generally believed that the single potential difference 
at the common boundary of mercury and an electrolyte has 
been satisfactorily determined by experiments on capillary 
electrometers and by others in which mercury was allowed to 
drop into a solution. Nevertheless, uncertainties arise in the 


interpretation of the phenomena, and doubt may well be felt 
about the results deduced. Another method of investigation 
has been adopted by Exner and Tuma, and by Exner 1 . and 
although it is open to criticism we will first briefly consider it 
as it illustrates the difficulties inherent in the subject. 

The introduction of the dropping electrode is due to Lord 
Dropping Kelvin, and was applied by him to determine 

electrodes. fa e difference of potential between the earth 

and any point in the atmosphere. If a conductor is constantly 
giving off a stream of particles into the surrounding dielectric 
medium which is at a potential different from that of the 
conductor, each particle carries away an electric charge until 
the potential of the conductor is made equal to that of the 
conducting boundary of the insulator if the latter is itself 
unelectrified. By connecting the conductor with an electro- 
meter, the potential of the dielectric at the point of emission 
of the particles is determined. The particles may be drops 
of water or mercury, the products of combustion of a flame, 
or the smoke from a slow-burning match. For measuring 
the high voltages observed in meteorology, any of these arrange- 
ments are trustworthy, but when the differences of potential to 
be observed are one volt or less, the conductivity due to com- 
bustion restricts the method to the use of some kind of drop. 

In examining the junction of a metal with an electrolyte, 
Exner connected the metal to earth and to one quadrant of 
the electrometer, while the electrolyte was joined by means 
of a thread moistened with the same solution to a cylinder of 
filter paper also soaked in the liquid. The cylinder forms a 
virtually closed conductor, and the inside of it is therefore 
an equipotential region, and is assumed to have the potential 
of the surrounding electrolyte. A funnel having a capillary 
end is filled with mercury which falls in drops starting within 
the cylinder. The stock of mercury in the funnel is connected 
with the other quadrants of the electrometer. In this way 
the mercury gradually assumes the potential of the air inside 

1 Sitzungsber. Kais. Akad. Wien, xcvn. 917 (1889); c. 607 (1891); ci. 627 
and 1426 (1892). 


the moistened cylinder, except for any natural difference of 
potential between the air and the falling drops of mercury, 
or any electrification produced by friction with the funnel, etc. 
In order to correct for these effects, Exner arranged a null 
experiment in which the mercury drops formed inside a cylinder 
of carbon or platinum connected with the earth, and the reading 
of the electrometer then obtained was taken as zero. Now in 
doing this, any natural difference of potential between platinum 
and air is neglected, and thus all the results of the work are 
in error by a constant amount which should be added to or 
subtracted from them. The error may be small, but there is 
no means of estimating its magnitude. Moreover, if there is 
any natural potential difference between the electrolyte and the 
air, it also is included in the numbers obtained ; in fact, by the 
principle of the summation of electromotive forces, we see that, 
when electric transfer ceases, the total electromotive force of 
the circuit 

mercury / air / solution / metal / air / metal / mercury 

must vanish. To secure this result, the solution / metal inter- 
face is polarized by a double ionic layer; if the necessary 
double layer were too intense, there would presumably arise a 
continual leakage current, maintained by the energy of the 
falling drops. 

Another form of application of Lord Kelvin's dropping 
electrodes furnishes one of the methods in common use, to 
which reference has already been made, for the examination of 
the electric phenomena at the junction of mercury with an 
electrolyte. Von Helmholtz pointed out that a potential 
difference at such a junction would be produced by a double 
layer of electricity over the surface, the two opposing faces 
being oppositely charged on the side of the electrolyte by the 
congregation of ions as explained above. Such a system would 
take time to reach its final state, and he concluded that if 
mercury were allowed to drop rapidly from an orifice beneath 
the surface of a liquid electrolyte, the double layer would 
not be established, and the stock of falling mercury would 
be brought to the same potential as the electrolyte. The 


apparatus might therefore be used in connexion with an electro- 
meter as is the dropping electrode in meteorology, the other 
quadrants being joined to a quantity of mercury at rest in the 
same electrolyte. A difference of potential of about - 8 or 
0*9 volt is obtained between the dropping mercury and the 
mercury at rest in dilute sulphuric acid ; but it has been pointed 
out by Exner and by Brown 1 that the result is complicated by 
the electromotive force of the cell composed of the mercury with 
the clean surface newly exposed by the drop as it forms, the 
electrolyte, and the mercury, tarnished or affected in some way 
by the action of the solution, which is at rest in the bottom of 
the vessel. This part of the observed electromotive force may 
depend on potential differences of the type due to the regular 
orientation of bipolar molecules. It would be set up almost 
instantaneously at the interface, and thus would not be elimi- 
nated by the action of the falling drops. Again, Warburg 2 has 
suggested that owing to the formation of mercury salts in 
electrolytic solutions containing dissolved oxygen, an explana- 
tion of the phenomenon might be found in the electromotive 
force of the concentration cell, 

mercury / dilute mercury salt / concentrated mercury 

salt / mercury, 

since differences of concentration at the electrodes will be 
produced by the passage of a current. It seems likely at all 
events that both these effects are involved in the current 
which will flow through a connecting wire from the standing to 
the falling mercury. In examining the results of researches 
on mercury dropping electrodes, these inherent difficulties 
should be borne in mind. 

A third explanation of the dropping electrode is given by 
Nernst's theory of electrolytic solution pressure. The solution 
pressure of mercury is very low, and mercury ions tend to be 
deposited as metal on a mercury surface, even from a very 
dilute solution of one of its salts. As the drops form, mercury 
ions are absorbed by their newly exposed surfaces, and negative 

1 Phil. Mag. [5] xxvu. 384 (1889). 

2 Wied. Ann. xxxvm. 321 (1889). 


ions are attracted to the ionic layer of the electrolyte next the 
interface. These negative ions are carried down with the drops 
as they fall ; they enable mercury ions to redissolve in the 
lower parts of the solution when the drops coalesce with the 
standing mercury and the area of contact is diminished. This 
explanation involves a loss of mercury salt in the upper regions 
of the solution, and a corresponding gain below. Such changes 
of concentration have actually been observed by Palmaer by 
electrical and chemical methods in an unsaturated solution of 
calomel, through which mercury was allowed to drop 1 . 

Experiments have been made by different observers on 
electrodes dropping mercury directly into electrolytic solutions, 
with results that did not agree very well among themselves. 
If the orifice be within the electrolyte, the time of fall of the 
mercury as a continuous jet allows the ionic potential difference 
of the interface to partially establish itself, but Paschen 2 , who 
investigated the subject in 1890, came to the conclusion that 
concordant values could be obtained by making the mercury 
jet emerge from the orifice into air, but break into drops just 
as it reached the surface of the electrolyte. His experiments 
on liquid amalgams, however, seemed to indicate that even in 
this way the mercury or amalgam is not completely deprived 
of its electric charge on entering the solution. 

It is commonly assumed that in experiments on such 
mercury dropping electrodes the total potential difference be- 
tween the falling mercury and the solution is destroyed. There 
are observations, however, by G. Meyer' and S. W. J. Smith 4 
which make such an interpretation difficult to accept. We 
shall see later that, from a knowledge of the ionic mobilities, 
Nernst and Planck have calculated the rates of diffusion of 
electrolytes and hence the difference of potential between the 
solutions of two electrolytes in contact with each other. On 
their theory, and indeed on almost any possible view of the 
phenomena, there can be no potential difference between dilute 

1 Zeits.phys. Chem. xxv. 265 (1898); xxvm. 257 (1899); xxxvi. 664 (1901). 

2 Wied. Ann. XLI. 42 (1890). a Wied. Ann. LVI. 680 (1895). 
4 Phil. Trans, cxcm. A (1900). 



[CH. XI 

equivalent solutions of potassium chloride and iodide, which 
are ionized to the same extent, and contain ions possessing 
equal mobilities. The potential difference between either of 
these two solutions and a mercury dropping electrode should 
also vanish, and the electromotive force of the cell 

dropping mercury / KC1 / KI / dropping mercury 
should be zero. Its observed electromotive force is given by 
Meyer as 0'284 and by Smith as O262 and 0'256 volt. This 
result apparently indicates that part of the potential difference 
at a mercury- electrolyte interface depends on the nature of the 
anion ; it is not eliminated by the action of a dropping electrode, 
and is therefore probably established with much greater rapidity 
than the part of the potential difference which is so eliminated. 

The surface tension of the area of contact between the 
Eiectro-capiiiary mercury and a solution is affected by its elec- 
phenomena. trical state. If the surface be increased, an 

electric transfer is produced, and, conversely, if an external 
electromotive force be applied 
across the junction, the area 
tends to change, owing to an 
alteration in the effective surface 
tension. These phenomena have 
been applied by Lippmann 1 to 
the construction of capillary 
electrometers, of which several 
forms are in frequent use. In 
one variety, a vertical glass tube 
is drawn to a very fine capillary. 
The tube is partially filled with 
mercury, and the lower portion 
immersed in an electrolyte, usu- 
ally dilute sulphuric acid, in 
which is placed another quan- 

Fig. 60. 

tity of mercury. The capillary forces tend to raise the mercury 

surface in the little tube, and are balanced by the pressure of 

the long column. When the mercury in the vertical tube and 

1 Pogg. Ann. CXLI*. 561 (1873) ; Ann. de Chim. Phys. [5] v. 494 (1875). 


the mercury below the electrolyte are connected with two 
conductors at different potentials, such as the opposite poles 
of a galvanic cell, a change is produced in the level of the 
surface of contact in the capillary tube. A microscope with a 
micrometer eyepiece may be arranged to view the capillary, 
and, for small differences, the change in level is found to be 
proportional to the applied difference of potential. The move- 
ment is slow, and the final position of the meniscus is not 
reached for an appreciable time, probably owing to the high 
values of the electrical resistance of the column of electrolyte 
in the capillary, and of the frictional resistance to the move- 
ment of liquid through such a narrow tube 1 . Burch 2 has 
shown that, while the applied electromotive force is a small 
fraction of a volt, the electrometer behaves as a condenser of 
good insulation, retaining its charge for several hours when 
disconnected from the cell. On the other hand, when the 
applied voltage is greater, electrolysis at the surface seems to 
occur, and the charges leak away. If, during the process of 
charging, the cell be disconnected before the final position of 
the meniscus is reached, the movement at once stops, and, 
in any case, the instrument is quite dead beat, and never 
oscillates about its position of equilibrium except as the result 
of mechanical disturbance. 

An explanation of these phenomena, based on Lippmann's 
The theory of observations, has been given by von Helmholtz 3 , 

von Heimhoitz. on fa Q assumption that no electrolysis occurs. 
Any natural potential difference between two bodies implies 
an electrification over the boundary, in such a manner that 
an electric condenser of minute thickness is formed, with its 
parallel faces oppositely charged. This electric double layer 
will produce an electrostatic surface energy e, the value of 
which is ^CAU. 2 , where C is the capacity of the double layer 
per unit surface, A the area of contact, and II the potential 
difference across the layer. Now if II be kept constant, and 

1 Phil. Trans. CLXXXIII. A, 104 (1892). 2 Proc. R. S. LXX. 221 (1902). 

3 Wied. Ann. xvi. 35 (1882), Faraday Lecture, Ghent. Soc. Jour. (1882) ; see 
also Larmor, Phil. Mag. [5] xx. 422 (1885), and S. W. J. Smith, loc. cit. 


the area be increased, we have for defdA the value -^CTI 2 . This 
increase in available energy is obtained from the chemical 
energy which maintains the natural potential difference. The 
electric layer on either side of the interface tends to expand 
under the mutual repulsion of the different parts of its charge, 
and thus tends to increase the area of contact ; it therefore acts 
in a sense opposite to that of the ordinary surface tension $ . 
The total surface tension S will be 

On the assumption that the only effect of the potential differ- 
ence is to produce such an electrostatic effect, S will be 
independent of II, and the total observed surface tension will 
reach a maximum when II is zero. Nevertheless, it is possible 
that the potential difference may affect the nature of the 
surface chemically or otherwise, and thus change S the ordi- 
nary surface tension. The maximum value of S will, in this 
case, not necessarily imply that the potential difference of the 
double layer is zero. 

The usual methods, by which attempts have been made 
to determine the total natural potential difference between 
mercury and an electrolyte, really give, on the view advocated 
above, only the part of that total due to the ionic double 
layer. If we assume that the electrostatic effect is the only 
result of changing the ionic difference of potential by apply- 
ing an external electromotive force, and that the analogy 
with the condenser still holds good, the natural ionic potential 
difference will be equal and opposite to that which must 
be applied externally in order to reach the maximum value 
of the observed surface tension. Lippmann and others have 
in fact found that the curves drawn between the external 
electromotive force and the reading of a capillary electrometer, 
are roughly parabolic; with dilute sulphuric acid, the maximum 
of the curve is reached when about one volt is applied. Con- 
versely, when the surface of separation is stretched, a current 
flows to supply the charges for the increased area of the double 
layer of electricity, and Pellat 1 found that this current ceased 

1 Comp. Rend. civ. 1099 (1887). 


when an external electromotive force of 0*97 volt acted against 
the natural potential difference. 

By using liquid amalgams, the same electrolyte can be 
compared with mercury and what is effectively a different 
metal. Rothmund 1 and others have compared the difference 
of the values thus measured with the electromotive force of 

the cell 

amalgam / electrolyte / mercury, 

finding concordant results. 

Rothmund gives the following as the voltages required to 
give the maxima of surface-tension : 

Mercury in normal sulphuric acid solution + 0'926 volt 

Mercury hydrochloric +0'560 

Lead amalgam sulphuric +0'008 

Bismuth sulphuric + 0'478 

Tin hydrochloric +0-080 

Copper sulphuric +0'445 

Cadmium sulphuric -0*079 

Zinc sulphuric -0-587 

Thallium hydrochloric +0-089 

In the experiments with mercury, the acids were saturated 
with their mercurous salts, and when using amalgams a trace 
of the salt of the metal was added to the solution. Cells were 
then arranged with these amalgams in combination with the 
corresponding mercury electrodes. 

Observed Calculated 

E.M.F. E.M.F. 

Lead amalgam cell 0-923 0'918 

Bismuth 0'437 0'448 

Tin 0-534 0'480 

Copper 0-458 0-481 

Cadmium 1*090 1-005 

Zinc 1-472 1-513 

Thallium 0'652 0-471 

The results of these experiments, unlike those on cells with 
dropping electrodes above described, on the whole favour the 
view that the electromotive force of a cell is the sum of the 
1 Zeits. phys. Chem. xv. 1 (1894). 


single potential differences at its electrodes as determined by 
the capillary electrometer, and that any permanent part of the 
total potential differences due to the orientation of bipolar 
molecules is not involved. 

On the other hand, there is evidence to show that in 
general the result of an applied electromotive force on the 
surface tension is not merely the electrostatic effect con- 
templated by the Helmholtz theory, but depends also on the 
chemical nature of the electrolyte. 

The capillary electrometer may be imagined to consist of 
a very small condenser, composed of the mercury- electrolyte 
double layer in the capillary, arranged in series with a large 
condenser formed of the similar surface in the outer vessel. 
Such a small condenser will be charged by an electric transfer 
which does not appreciably affect the large one, and the varia- 
tion of the potential difference at the capillary electrode is 
the same as the variation of the external electromotive force. 

The Lippmann-Helmholtz theory rests on two hypotheses. 
It assumes that the electrometer circuit may in truth be treated 
as a system of condensers, and it assumes that, as explained 
above, the only effect of the potential difference, whether 
natural or applied, is the electrostatic one. 

To increase the surface tension, and thus to reduce the 
natural electrification, of the interface between mercury and an 
electrolyte, an external electromotive force has to be applied 
from the solution to the metal. The natural ionic double layer 
must therefore consist of negative anions, chlorine for example, 
in the electrolyte, and a corresponding positive charge, perhaps 
represented by positive electrons, on the surface of the mercury. 
On applying a gradually increasing reverse electromotive force, 
we may imagine that the chlorine ions diminish in number and 
finally disappear ; the surface tension then reaches a maximum. 
Beyond this point, positive metallic ions would be driven up to 
the interface, and a reverse double layer would arise. If this 
polarization exceeded a certain limit, a current would flow, and 
an amalgam might be formed. On the original Helmholtz 
theory, which took no account of differences between ions, and 
assumed that the reverse layer was similar to the first one, the 


experimental voltage surface-tension curves should be a single 
parabola. Observation shows, however, that there is usually a 
slight want of symmetry between the ascending and descending 
branches of the curves, possibly indicating the effect of the 
chemical nature of the ions. The result of this effect on 
the surface energy of the interface has been considered by 
van Laar 1 . When an electric transfer Sq occurs, the change 
in surface energy will be given by the expression 

66 = , 

dq da 

a denoting the electric charge per unit area. The surface 
tension will be altered by a term de/dA. Now 

<&>.-'& *' M* 

dA SA '' "da' 

The complete expression for the surface tension thus 

Whichever side of the electric double layer is positive, its 
effect is opposite to that of the natural surface energy, and the 
term crdS/do- is always negative, but there is no reason to suppose 
that its numerical value will be the same when it represents 
the effect of anions and cations on the electrolyte side of the 
double layer. The total electrocapillary curve therefore consists 
of parts of two parabolas which meet at the point for which 
a = and <y = $. Only the ascending branch has a maximum 
which is in general near, but not at, the point of intersection. 
Since the concentration of the ionic layer is always small, the 
variation of available surface energy can be formulated, and 
van Laar, by determining the constants of his detailed equa- 
tions, finds that they represent the experimental results of 
S. W. J. Smith with great accuracy. On this confirmation of 

1 Kon. Akad. Wetens. Amsterdam, March, 1902, p. 560. 



[CH. XI 

his theory, van Laar concludes that the capillary electrometer 
does not give a trustworthy means of measuring single potential 

In the work on electrocapillary phenomena to which we 
have referred, S. W. J. Smith 1 finds indications that, on 
changing the potential difference by external means, a leakage 
current will flow owing to the tendency of the electrode to 
revert to its original condition, so that the condenser analogy 
cannot be complete. He finds that a very high resistance in 
the potentiometer circuit changes the indications of the electro- 

As we said on p. 281, on almost any view of the pheno- 
mena there can be no difference of potential between dilute 
equimolecular solutions of potassium chloride and iodide, since 
they are equally ionized and contain ions of equal mobilities. 
The electromotive force of the cell 

mercury / potassium chloride / potassium iodide / mercury 

ought therefore to agree with the sum of the two potential 

mercury / potassium chloride + potassium iodide / mercury, 

Fig. 61. 

as determined by the capillary electrometer. If the latter 

values are estimated from the maxima of surface-tension, their 

1 Phil. Trans, ex cm. A, 47 (1900). 


sum for semi-normal solutions is 0*162 volt, while the cell 
gives 0'394 volt. Similar discrepancies occur in other cases. 
For these two solutions, Smith gives curves like those of 
Fig. 61, in which abscissae represent applied electromotive 
forces, and ordinates arbitrary scale readings of the electro- 
meter. While, in their ascending portions, the two curves have 
different slopes, they become parallel when they descend. It 
is probable that the effects of the ionic polarization are then 
the same for both. Let E be the external electromotive force 
required to give the same surface tension to the capillary 
surface of the potassium chloride solution as E' gives to the 
iodide solution. Then on the parallel parts of the curves, 
E E' is very nearly constant. Let II and II' be the natural 
potential differences between mercury and the chloride and 
iodide solutions respectively. On the first hypothesis of the 
ordinary electrometer theory (the condenser analogy), the 
potential differences between the solution and the capillary 
meniscus for two points of equal surface tension, one on each 
curve, are E II and E' IT respectively. On the second 
assumption, that the sole change is an electrostatic one, and 
the potential differences are the same in the two cases because 
the surface tensions are the same, we have 

or n-n' = ^- E' = a, 

where a is an observable quantity, measured by the horizontal 
distance between the parallel portions of the curves. If there 
is no potential difference between the two solutions when in 
contact, the electromotive force of the cell 

mercury / potassium chloride / potassium iodide / mercury 

is also II IT, and should thus be equal to a. The first four 
results in the following table give a comparison between the 
electromotive forces of such cells and the values calculated 
(1) by the method just described, (2) by estimating the 
maxima of the curves. 

w. s. 19 



[CH. XI 


Calculated E.M.F. 


E.M.F. of 

mercury cell 



J normal KC1 and KI 




To " " >' 




20 " " " 




1 KC1 and KCNS 



1 KC1 and Na 2 S 




"2 5) 3> 5) 




Here again we have results which suggest that the electro- 
static theory is insufficient. The maximum of surface-tension 
seems to depend on the nature of the anion, and, if that 
maximum be taken as a means of determining the natural 
potential difference, the electromotive force of a cell with two 
electrolytes having different ions apparently cannot be calcu- 
lated from the two single potential differences at its electrodes. 
The last two lines of the table indicate the same relations in 
solutions where both anion and cation are different, the greater 
discrepancies being explained by the uncertainty regarding the 
contact potentials of the two solutions. Electrocapillary curves 
for equivalent solutions of potassium, sodium and hydrogen 
chlorides, which contain the same anion, coincide within the 
limits of experimental error throughout both the ascending and 
descending portions. Hence it is concluded that the effect of the 
anion is considerable as long as the reverse applied electromotive 
force is less than the natural potential difference ; but the nature 
of the cation seems to have no appreciable influence on the 
potential difference throughout the whole range covered by the 
experiments. Assuming that the Nernst theory gives the true 
potential difference between two solutions, Smith, however, 
remarks that, although the slope of the lower portion of the 
descending curve varies little with the concentration of the 
solution, the absolute value of the surface tension for a given 
potential difference does show such variation. Thus the tension 
does not depend on the electrostatic effect alone even when 
the influence of the anion has presumably disappeared ; there 


is also a cation effect, which becomes evident as the solution 
grows increasingly positive with regard to the electrode, and 
the cation therefore tends to enter the mercury and form an 
amalgam. On the other hand, the anion effect increases as 
the electrode becomes more positive, and thus tends to dissolve. 

Warburg's theory of these phenomena can be extended to 
the capillary electrometer on the same lines as to the case of 
dropping electrodes 1 . When an external electromotive force is 
applied, Warburg traces the increase of surface-tension to the 
action of the polarizing current. This current removes from 
the neighbourhood of the meniscus the trace of mercury salt 
which always dissolves from the metal into solutions containing 
dissolved oxygen. The salt is slowly replaced by diffusion, and 
the actual change in concentration is the resultant of the two 
opposite effects. Owing to the minute quantity of salt in the 
capillary tube and the slowness of the compensating diffusion, 
the exhaustion may be very complete. The concentration cell 
which is formed may thus have a considerable electromotive 
force. The surface-tension will reach a maximum when the 
whole of the mercury ions are removed from the solution near 
the meniscus. In order to explain the descending branches of 
the surface-tension curves on Warburg's theory, it has been 
suggested that, as the electromotive force rises, an amalgam is 
formed with a surface-tension naturally lower than that of 

On Nernst's conception of electrolytic solution pressure, 
electrocapillary phenomena will be interpreted as follows. The 
low pressure of mercury causes positive ions to enter it even 
from a dilute solution. The mercury thus acquires a positive 
charge. An external electromotive force applied to an electro- 
meter from solution to metal causes a temporary current, which 
carries more mercury ions across the interface. In the capillary 
tube this process at once dilutes the solution, and therefore, in 
accordance with the logarithmic formulae of Chapter X., makes 
the mercury more anodic to the electrolyte and eventually stops 
the current. If the concentration of the mercury ions in the 
solution falls to the value corresponding to the solution pressure 
1 Wied. Ann. xxxvm. 321 (1889); XLI. 1 (1890). 



of the metal, the potential difference disappears; it' it falls 
below that value, the potential difference is reversed, the 
mercury becomes negative to the solution, and draws cations 
to the electrolyte side of the double layer. 

On the theories of both Warburg and Nernst, when the 
external electromotive force is removed, the processes of dif- 
fusion should gradually reduce the differences of concentration 
and the displacement of the meniscus of the electrometer. 
Burch has found, however, that a new and good electrometer 
will show the same deflection of the meniscus for many hours 
when charged to a small fraction of a volt and then left on 
open circuit with its electrodes insulated from each other. 
Such observations indicate that for small electromotive forces, 
the instrument acts as a condenser of good insulation. Never- 
theless, it seems certain that the changes of concentration 
contemplated by Warburg and Nernst must occur in some 
cases. It is possible that it is to the influence of such effects 
that are due some of the discrepancies which appear in the 
results of experiments on the potential differences at the 
surfaces of contact of mercury with electrolytes. 

Another set of electrocapillary phenomena, like those we 
Electric have been considering, probably depend on the 

endosmose. natural potential differences at the surface of 
separation of two unlike substances in this case an electrolyte 
and an insulator. If an electric current be passed through a 
vessel divided into two compartments by means of a porous 
partition and filled with some solution, we shall find that, in 
general, besides the changes in concentration at the electrodes 
which were described' on p. 207 under the head of migration, 
there is a bodily transfer of the liquid, usually in the direction 
of the current, through the porous plate. To this phenomenon 
the name of electric endosmose is given. It has been experi- 
mentally studied by Wiedernann 1 and Quincke 2 . 

If the pressure be kept constant on both sides of the 
partition, the volume of liquid which flows through, as measured 

1 Elektricitat, n. 166. 

2 Pogg. Ann. oxin. 513 (1861). 


by the overflow, is proportional to the total electric transfer, 
and is independent of the area and thickness of the plate ; it 
varies much with the nature of the solution, being greater with 
liquids of high specific resistance, and, in solutions of different 
concentrations of any one substance, is approximately propor- 
tional to the specific resistance. 

If the liquid is not allowed to overflow, the pressure on one 
side of the porous wall will increase. The final pressure is directly 
proportional to the electromotive force between the faces of the 
partition, and therefore to the current through it ; for a given 
current it varies inversely as the area of face of the porous 
wall and directly as its thickness. In this case, the flux of 
liquid due to the electric forces must be equal and in the 
opposite direction to that caused by the difference of hydro- 
static pressure. Considering the porous wall to consist of a 
collection of capillary tubes, we can apply Poiseuille's laws to 
the reverse flux under the hydrostatic forces, and this expla- 
nation has been supported by Quincke, who proved that the 
pressure produced by electric endosmose through a capillary 
glass tube was inversely proportional to the fourth power of 
the diameter of the tube. The pressures were considerable 
with distilled water, but ceased to be perceptible with liquids 
of high conductivity such as solutions of salts and acids. 

A detailed theory of the subject has been given by von 
Helmholtz 1 , on Quincke's hypothesis of a constant potential 
difference between the liquid and the walls of the capillary 
tubes. The electric charge which resides on the outermost 
layer of liquid and forms the inner face of the electric double 
layer, will be acted on by the external electromotive force and 
the skin of liquid will therefore be dragged through the tube. 
If a ditference of pressure is allowed to develop, one current of 
liquid is drawn forwards along the walls, and an opposite one 
flows down the centre of each tube under the action of the 
hydrostatic forces. The final pressure is reached when these 
two currents of liquid convey equal volumes per second in 
opposite directions. From these ideas von Helmholtz deduced 
the observed facts of electric endosmose, and calculated that 
1 Wied. Ann. viz. 337 (1879). 


the contact potential differences involved were of the order 
of one volt. A modification of the theory has been given by 
Lamb 1 , allowing for a slight slip between the liquid and the 
walls of the tube. 

In a similar manner is explained the motion through 
liquids of fine particles of clay or other material under the 
influence of an external electromotive force, a phenomenon 
which has been studied by Quincke and others 2 . 

It has been suggested by W. N. Shaw 3 that electric 
endosmose constitutes an essential part of the mechanism of 
electrolysis, the motion of the liquid being due to the drift 
of complex ions made up of an ion of the salt attached to a 
large number of solvent molecules. The inverse proportionality 
between the concentration of a solution and the endosmotic 
effect, shows that, in very dilute solutions, such complex ions 
must contain many hundred or thousand water molecules ; 
and it seems more likely that, in accordance with the usual 
view, electric endosmose is an independent phenomenon, not 
directly connected with the electrolytic process. 

t is possible that the results of experiments with capillary 
electrometers may be influenced by electric endosmose as soon 
as any current flows and a potential gradient established along 
the capillary tube. From Quincke's observations above de- 
scribed, however, it seems probable that the measurements 
would not appreciably be affected. 

It is evident from what has been said in the sections 
preceding the last, that there is some doubt 

Single potential 1,1,1 i i 

differences whether the experiments on dropping elec- 

trodes and on capillary electrometers really 
enable the natural potential difference, which is involved in 
the electromotive force of a galvanic cell containing a mercury- 
electrolyte surface, to be calculated even approximately. Never- 
theless, since many useful determinations of other single potential 

1 B. A. Report, 1887, 495. 

2 Wiedemann's Elektricitat, n. 181. 

3 B. A. Report, 1890, 202. 

CH. XI] 



differences, which, at all events, are relatively exact, rest on 
such measurements, in the present condition of the subject we 
must provisionally accept the value of about + 0*92 volt as the 
potential difference between mercury and dilute sulphuric acid, 
the mercury being positive to the acid. The step of potential 
as thus measured is in the opposite direction to that which 
occurs at the surface of a zinc plate. Results are obtained for 
other metal-electrolyte surfaces by subtracting this number, or 
another similarly estimated for mercury in contact with some 
other electrolyte, from the total electromotive force of galvanic 
cells arranged in the manner 

metal / electrolyte / mercury. 

Such indirect determinations will contain as a constant error any 
deviation of the primary measurement from the true value, but, 
as relative numbers, serving to compare the metals among 
themselves, they will retain their importance. 

In making such experiments, it is usual to employ what is 
known as a normal electrode, consisting of a quantity of pure 
mercury covered by a layer of mercurous chloride and a solution 
of potassium chloride of normal concentration, that is, a solution 
containing one gram-equivalent per litre. An indiarubber tube 
ending in a glass tube leads from the solution and is filled with 
it (Fig. 62). Contact can thus be made between the potassium 

Fig. 62. 

chloride and any other liquid. This electrode as measured by 
Lippmann's method gives a potential difference of 0*56 volt, 
the mercury tending to come out of solution and be deposited 


as metal. The chlorides can of course be replaced by other 
substances when their potential with respect to mercury is 
known. Thus a soluble sulphate, with mercurous sulphate as 
depolarizer, has been used. Assuming that we may neglect the 
small effects at the junction of the metals, and at the surfaces 
of contact of unlike solutions, if such surfaces are present, the 
measured electromotive force of the combination metal / electro- 
lyte/normal electrode enables the potential difference at the 
surface metal / electrolyte to be calculated by subtraction. 

In this manner, Neumann measured the single potential 
differences for many metals in contact with either normal or 
saturated solutions of their salts. The following are some of 
the most important results 1 . 

Metal Sulphate Chloride Nitrate Acetate 










In this table positive signs have been assigned to those 
metals which show a positive potential relatively to the liquids 
surrounding them. Assuming the accuracy of these results as 
absolute numbers, it follows that such metals tend to come out 
of solution, and the natural potential difference at their surfaces 
helps to drive a current in the direction to effect the deposition. 
Spontaneous separation of these metals, or solution of negative 
metals, however, will only occur if means are available for the 
simultaneous addition of opposite ions, or the removal of an 
equivalent quantity of similar ions. The numbers show that 
the electromotive forces of the cells used depend on the nature 
of the acid ion present, but Neumann also prepared centinormal 
solutions of many different thallium salts, and found sensibly 
equal values. In these solutions the ionization may be taken 

1 Zeits. phys. Chem. xiv. 229 (1894). 


- 0-503 











+ 0-095 

+ 0-115 


+ 0-238 


+ 0-150 

+ 0-515 

+ 0-615 

+ 0-580 


+ 1-055 


+ 0-980 

+ 1-028 


as complete, but it remains to be seen whether or not under 
such conditions the equality would extend to salts of all metals. 
There is some evidence to suggest that the variations of electro- 
motive force with the acid ion are to be traced to the presence 
of mercury, cells in which two other metals are used being 
usually free from such discrepancies 1 . 

The table on the last page gives a fair idea of the single 
potential differences calculated from the fundamental experi- 
ments on mercury, and, for slightly oxidizable metals such as 
silver, it will be seen that the method leads to numbers which 
have an opposite sign and an even larger numerical value than 
those obtained for very oxidizable substances such as zinc. The 
intimate connexion which exists between the electromotive force 
of a cell and the calorimetric heats of the resultant chemical 
actions, when allowance is made for the usually small reversible 
heat effects, has already been considered on p. 230. It seems 
reasonable to apply the same relations to each individual part 
of the circuit, and we should expect that metals which are only 
acted on with difficulty and have small heats of oxidation, 
would show a very much smaller potential difference than very 
oxidizable metals with large heats, though probably a difference 
of the same sign ; in fact the potential diagrams of the cell 
represented by the second parts of Figs. 58 and 59 seem 
a priori more likely to correspond to reality than those shown 
in the first parts. Moreover, in correlating these phenomena 
with those of the Volta contact effect between metals in air, it 
is probable that there will at all events be a general agree- 
ment between them. It is unlikely that metals would show a 
difference of sign in their potential differences with air, if that 
difference is due to actual oxidation or to an affinity which tends 
to oxidation. On the other hand, Nernst's theory of electro- 
lytic solution pressure offers a possible explanation of the 
difference in sign as usually accepted. Whatever be the final 
outcome of the problem, we may take Neumann's numbers and 
similar results as true relative values, though a constant error 
may eventually have to be added to or subtracted from them. 

1 Paschen, Wied. Ann. XLIII. 590 (1891) ; Taylor, Journ. Phys. Chem. i. 
1 and 81 (1896). 


Neumann's results enable the numerical value of Nernst's 
electrolytic solution pressure to be calculated. 

Electrolytic r 

solution in the last chapter it was shown that, on the 

analogy of the junctions between two liquids, 

the potential difference between a metal and a solution might 

D/TT ~p 

be expressed in the form - log -^ , E being the gas 

j. u 

constant for the gram-molecule of the metal, P m the solution 
pressure of the rnetal, P the osmotic pressure of its ions in the 
electrolyte, q the charge on the monovalent gram-equivalent, 
and y the valency of the ions. The potential difference can be 
observed, and the osmotic pressure is approximately known from 
the concentration of the solution. Thus the electrolytic solution 
pressure can be calculated ; the following are some of Neumann's 
values in atmospheres recalculated by Le Blanc. 

Zinc 9-9 xlO 18 Hydrogen 9'9 x 10~ 4 

Iron 1-2 xlO 4 Mercury 1-1 x 1(T 16 

Lead 1-1 x 10~ 3 Silver 2-3x10-" 

In the logarithmic expression, P m denotes the osmotic 
pressure of the solution at which it would show no potential 
difference with the metal. Nernst extends this idea, and 
identifies P m with a characteristic property of the metal itself, 
which, on the analogy of the vapour pressure of a liquid, is 
taken to measure the tendency of the metal to diffuse in the 
form of electrolytic ions in the liquid surrounding it. The 
legitimacy of this extension is still a matter of discussion, and, 
as indicated on p. 257, by writing the formula as 

RT\ogP m /qy-RT\ogP/qy, or M-RTlogP/qy, 
we may treat the part of the expression referring to the 
metal simply as a function of its properties of unknown form. 
Accepting provisionally, however, the solution pressure hypo- 
thesis, the absolute values given above are still open to ob- 
jection, not only as based on the mercury-electrolyte difference 
of potential, but in another way. The formula from which 
they are calculated is transferred from that deduced on the 
assumption of the ideal gaseous laws for the junction between 
two liquids, and the extension of these laws to the very high 
pressures here dealt with is clearly unjustified. 


In the derivation of the theory for liquid junctions on 
p. 246, it is shown that the electromotive force is measured by 
the integral fvdP. Lehfeldt has calculated the value of this 
integral on the assumption that the deviation from the gaseous 
laws in solutions is represented by an expression of the form of 
van der Waals' equation for gases. Putting 

~ v-b 
we have, since in concentrated solutions i is nearly unity, 

+ b (P m - P). 

P f 

Applying this to the case of zinc in normal zinc chloride solution, 
we may put 22 atmospheres for P, and 0'5 volt for E\ b is 
assumed to be the volume of a gram-molecule of the salt in 
the solid state, about 46 cubic centimetres. The value of the 
solution pressure P m can then be calculated, and comes out 
about 2 x 10 4 atmospheres, instead of about 10 19 , as deduced 
from the simple logarithmic formula. It is clear that such 
considerations as these enable the deviations of concentrated 
solutions from the ideal gaseous laws to be estimated, and 
Lehfeldt has calculated the osmotic pressures of such solutions 
from measurements of the electromotive forces of concentration 

We shall reconsider the hypothesis of solution pressure 
under the head of electrolytic diffusion. 

It was one of the objects of the early experimenters to 

arrange the metals in order in an electro- 

chemical chemical series. The two tables of potential 

differences, set forth on pp. 268, 296, are 

quantitative solutions of this problem under given conditions. 

Whereas it was formerly thought that the metals occupied 

the same relative positions in all circumstances, it is now obvious 

that the potential differences which they yield will depend on 

the nature of the surrounding medium, and, if that medium is a 

solution, on the concentration of the dissolved substance, though 

if a table of electrolytic solution pressures could be calculated, 

it would enable the effects of concentration to be eliminated. 


The general accuracy of the theories explained above 
indicates that a metal immersed in the solution of one of its 
salts should be the less electropositive as the concentration of 
its ions in the solution increases. If the metal dissolves as a 
compound salt, as do gold and silver in cyanide solutions, it 
may be that the metal can exist in the solution in the form of 
simple ions in very small quantity only 1 . 

In accordance with this result, the electromotive force of 
such metals as gold and silver in solutions of cyanides is very 
high, and places them in a position in the electrochemical 
series different from that which they occupy when the metals 
are studied in contact with solutions of their own salts or 
the corresponding acids. As stated in the last chapter, Hittorf 
found that in the cell Cu / KCN / K 2 SO 4 / ZnS0 4 / Zn, copper is 
dissolved when a current flows. 

The contact potentials of different metals with cyanide solu- 
tions have also been studied by von Oettingen 2 and by Christy 3 . 
The latter observer traces the influence of the concentration of 
a solution of potassium cyanide on its potential difference 
against gold, and shows that the rate at which gold dissolves 
when shaken with the liquid is a function of this potential 
difference and also of the amount of dissolved oxygen. As a 
combination of these two effects, the rate of solution of the 
gold reaches a maximum at a concentration of ten to twenty 
per cent, of potassium cyanide, and then again decreases as 
that proportion is exceeded. 

Another aspect of the subject now under consideration is 
given by an examination of the phenomena of 

Polarization. . J . * rn. TTTTT 

polarization. As we said in Chapter VIJLJL., 
it requires a certain minimum electromotive force to drive a 
permanent current through an electrolyte between electrodes 
which are not dissolved. If a single Daniel 1's cell be connected 
through a galvanometer with two platinum plates immersed 

1 See footnote p. 255. 

2 Journ. Chem. and Metallurgical Soc. South Africa, 1899. 

3 Amer. Imt. Mining Engineers, Trans, xxx. (1899), reprinted as Bulletin oj 
Depart. Mining, etc., Univ. of California. 


in dilute sulphuric acid, the galvanometer is at first deflected. 
The current, however, rapidly falls off, and soon sinks nearly to 
zero. If the platinum plates are now disconnected from the 
cell, and joined with each other through the galvanometer, 
they will send a current through it in the reverse direction. 
The plates are said to be polarized. The electromotive force 
of polarization, in the case we have chosen, soon diminishes, so 
that in order to measure its maximum value, the connexions 
must be rapidly reversed. Raoult 1 found that a speed of 
reversal of one hundred alternations a second was enough to 
secure this result. The best method of experimenting is to 
use a tuning-fork commutator which vibrates very rapidly. 

If the electromotive force is gradually raised from a very 
small value, the reverse force of polarization is also found to 
rise, keeping equal to that applied, until a nearly constant limit 
is reached. A further rise in the applied electromotive force 
causes little or no more increase in the polarization, and the 
current through the solution can then be calculated from 
Ohm's law by taking as the effective electromotive force the 
value found by subtracting that of polarization from the force 
externally applied. 

The phenomena of polarization have been very fully studied 
by Le Blanc 2 . There is a certain decomposition 
1 value for the applied electromotive force, beyond 
which a permanent current flows. Le Blanc 
found that the decomposition voltage can be easily and exactly 
determined for salts from which a metal is precipitated, the 
current starting from that point to rise proportionally to the 
electromotive force ; but for other salts, as well as for acids and 
alkalies, the measurements are more uncertain. 

The following decomposition values were found with 
platinum electrodes for salts from which the metal is pre- 
cipitated ; the salts were mostly in normal solutions. 

1 Ann. Chim. Phys. iv. 2. 326 (1864). 

2 Zeits. phys. Ghem. vin. 299 (1891) ; or Le Blanc's Elektrochemie, Eng. 
Trans, p. 247. 



[CH. XI 


Zinc sulphate 

2-35 volts 

Zinc bromide 


Nickel sulphate 


Nickel chloride 


Lead nitrate 


Silver nitrate 


Cadmium nitrate 
Cadmium sulphate 
Cadmium chloride 
Cobalt sulphate 
Cobalt chloride 

1-98 volts 
1-78 , 

Whereas the values given in the above table for metallic 
salts vary from metal to metal, the values for acids and 
alkalies show a maximum decomposition point, which is 
approached by most of these compounds and exceeded by 




1-67 volts 
















1-57 volts 














Sodium hydrate 
Potassium hydrate 
Ammonium hydrate 

1-69 volts 
1-74 , 

Diethylamine (| normal) 1/68 volts 
Tetramethyl ammonium 

hydrate (| normal) 1-74 

Methylamine ( normal) 1/75 

Acids and alkalies which evolve hydrogen and oxygen on 
electrolysis, show the maximum decomposition voltage nearly 
independently of the concentration of the solution. For acids 
which are more easily decomposed, the numbers increase on dilu- 
tion with a simultaneous change in the nature of the products. 

The use of other non-oxidizable electrodes such as gold or 
carbon instead of platinum, leads to different numerical results, 
though the relations between them remain unaltered. The 
differences may be explained by remembering that, although 
the resultant chemical process is in each case the liberation of 



hydrogen and oxygen, the production of bubbles of gas at the 
surface of a metal, which does not occlude the gas, is an 
essentially irreversible operation depending on conditions which 
may well vary from metal to metal 1 . 

The electromotive force of polarization evidently consists 
of two parts, one depending on the electrical 

^'"electrode! WOrk d De at the anode > and tne tner n 

that at the cathode. In order to examine 
these separately, an arrangement due to Fuchs was used 
by Le Blanc 2 . The tuning-fork commutator is adapted to a 
double U-tube apparatus shown in Fig. 63. The primary 

Fig. 63. 

or polarizing current is passed from Q between the electrodes 
a and b. If the electrode b is to be examined, the bent glass 
tube of the normal electrode described on p. 295 is inserted at 
c, and the effect of the cell so formed is balanced by an 
adjustable electromotive force at M, an electrometer being used 
as indicator. The potential difference between the plate b and 
the liquid can then be found by subtracting that of the normal 
electrode and that at the contact of the two solutions at c. As 
the primary current from Q is increased from zero, it is found 
that the electromotive force of polarization is at first nearly 
equal to that of the primary current, but it gradually comes to 

1 In this connexion reference may be made to a paper by Nernst and 
Dolezalek [Zeits. Elektrochem. May 10, 1900] ; and another, containing a 
criticism on it, by C. J. Heed [Journ. Phys. Chem. v. 1 (1901)], on the "Gas 
Polarization of Lead Accumulators." 

2 Zeits. phys. Chem. xn. 333 (1893); xin. 163 (1894)*; also Electrochemistry, 
p. 244. 


a nearly constant value, though Le Blanc states that no exact 
final limit is ever reached. 

When the solutions which deposit metals are examined in 
this way, Le Blanc finds that at the decomposition point the 
polarization potential difference at the cathode is equal to the 
potential difference which a plate of the metal itself gives if 
placed in contact with the solution, both, of course, depending 
on the value taken for the fundamental mercury -electrolyte 
difference of potential. The polarization at a junction is thus 
exactly correlated with the single potential difference, which 
can be measured by experiments on capillary electrometers or 
dropping electrodes. If, as previously explained, we refer the 
total potential difference to a combination of molecular and 
ionic effects, Le Blanc's results indicate that electrolytic polari- 
zation is an ionic phenomenon a natural result to anticipate. 
As the external electromotive force is gradually increased from 
zero, the measured potential difference at the cathode, like the 
total electromotive force of polarization, rises also, approaching a 
limit, though the electromotive force necessary to reach this limit 
is often less than that required to give the maximum polarization 
of the whole apparatus which includes the anode also. The 
limit seems to be reached when the deposit of metal on the 
electrode is enough to cover its surface with a continuous layer. 
The converse phenomenon has been studied by Oberbeck 1 . who 
deposited small quantities of the metal of a salt solution on a 
platinum plate, and then measured the potential difference 
between the plate and a solution of the same salt placed in 
contact with it. As the amount of deposit was increased, 
this potential difference rose, and finally reached the value 
found for a solid plate of the metal. As a final result of all 
these investigations, it is concluded that the deposition and 
solution of metals from solutions of their salts are reversible 
processes. The single potential differences exhibited in 
Neumann's table on p. 296, may therefore also be taken as 
measuring the polarization when the metal is electrolytically 
deposited from its solution in the salts there indicated. 

1 Wied. Ann. xxxi. 336 (1887). 


In considering the total effects of polarization the anode also 
has to be taken into account. When the anode 
!af gases? * s ^ platinum or a similar metal, gas is usually 
evolved there, and it thus becomes of great 
importance to determine how far the conditions of reversibility 
hold good in the evolution of gas at an electrode. As we have 
seen, a minimum electromotive force is required to continually 
electrolyse a dilute solution of sulphuric acid in water ; when 
gold or bright platinum electrodes are used, about 1*7 volts are 
necessary. The reverse electromotive force of polarization is, 
however, only T07 volts, and as is well known, if the hydrogen 
and oxygen are collected in tubes and kept in contact with 
platinum electrodes, an arrangement called Grove's gas battery 
is obtained, which furnishes a secondary electromotive force of 
1*07 volts, and will yield a current as long as any gas remains. 
Thus the development of gas at a bright platinum surface is 
an irreversible process. When, however, the electrodes are 
coated with platinum black by previously passing a current 
backwards and forwards between them through a solution of 
platinum chloride, Le Blanc proved that the decomposition 
point was 1*07 volts ; so that, with platinized electrodes, the 
process is reversible. The difference is explicable when we 
remember that platinum occludes a large amount of gas. The 
platinized electrodes absorb the gases when slowly developed, 
and when the plates become 'saturated, if parts of them are 
outside the liquid, they can gradually give up the gases by 
diffusion without the formation of bubbles. Thus, if an external 
electromotive force of 1'07 volts be applied, the system is in 
equilibrium, while, if the applied electromotive force exceeds or 
falls short of that value by an infinitesimal amount, an in- 
definitely small current will flow one way or the other, and the 
gases are slowly set free or dissolved. The arrangement is 
therefore reversible, and the thermodynamic treatment of the 
effects of pressure, etc., on the electromotive force of the 
oxy-hydrogen gas battery, which was given on p. 240, applies 
equally to their effects on the reverse electromotive force of 
polarization in the decomposition of water between platinized 
w. s. 20 


electrodes. We may therefore in this case also write the 
equation then deduced, 

W P* 

where R is the usual gas constant for one gram-molecule, T the 
absolute temperature, q the charge of electricity passing when 
one univalent gram-ion is liberated, y the valency of the ions, 
and p l and p 2 the pressures in the two cases considered. 

Now if pz be gradually reduced, the value of this expression 
can be made as great as we please, and thus, at a certain very 
low pressure, the reverse electromotive force must vanish, and 
below this pressure actually be reversed, so that water would 
decompose spontaneously. This critical pressure will be so low 
that it is quite out of reach of experimental confirmation ; in 
fact the vapour pressure of the water itself would prevent its 
ever being reached. 

The information that the decomposition of water could 
theoretically be effected at a low pressure by a very small 
electromotive force is exceedingly striking, for the heat 
developed by the direct chemical combination of oxygen and 
hydrogen at constant pressure is nearly independent of the 
absolute value of that pressure. It furnishes a good illustration 
of the want of proportionality between the heat of chemical 
union and the electromotive force when other transformations of 
energy are involved, and shows the need of the second term in 
von Helmholtz's equation, p. 236, 

Let us now return to the case when gold or bright platinum 
electrodes are used instead of platinized ones. As we have said, 
the decomposition point is then 17 volts, while the reverse 
electromotive force is still only 1'07 volts, showing that the 
process is irreversible. Bright electrodes have very little power 
of absorbing gas; consequently if an electromotive force be- 
tween TOY and T7 volts be applied, the gases cannot be 
removed from the electrodes nearly fast enough by diffusion, 
and, when the solution in the neighbourhood of the electrodes 
becomes saturated with dissolved gas, the evolution will cease. 


Slow diffusion from the liquid into the air and back through 
the liquid will however go on, and this process allows more 
gas to be evolved, while a slight leakage current continually 
flows, as indicated by the galvanometer. In order to produce 
a permanent large current and a constant evolution of gas in 
appreciable quantities, it is necessary to raise the electro- 
motive force till it is able to cause the formation of bubbles 
at the surface of the electrodes, a process which involves an 
amount of work depending on the surface-tension, the state of 
the electrodes and other uncertain and irreversible conditions. 
That these conditions vary with different kinds of electrode is 
shown by the unequal potential differences needed to liberate 
hydrogen at cathodes of platinum, gold, lead, copper, etc. In 
such cases, when bubbles of gas are formed, part of the available 
energy of the chemical action is not expended on electrical 
separation ; thus the reverse electromotive force, which depends 
on the free energy of this separation, is less, and the process is 
not reversible. 

It will be noticed that the 1*7 volts needed to evolve oxygen 
and hydrogen at bright platinum electrodes is 
the maximum value of the decomposition point 
of solutions of acids and alkalies (p. 302). This 
fact is explicable if we consider in detail the process of electro- 
lysis in such cases. All the ions in the solution, of whatever 
nature, are acted on by the electric forces, and must therefore 
all carry the current by moving through the solution; as, indeed, 
was shown by the experiments of Hittorf. At the electrode, 
however, if more than one kind of ion is present, that kind will 
first be deposited which has the lowest deposition value. Now 
we shall find later that in water, even when pure, a certain 
number of hydrogen and hydroxyl ions are always present, and 
unless they are removed in some way, these ions will cause 
hydrogen and oxygen to be evolved before any substances in 
the solution which possess higher deposition voltages can appear 
at the electrodes. 

Now for acids and alkalies, the electrolytic processes allow 
this preferential action to occur. The hydrogen ions derived 



from the electrolyte in one case, and its hydroxyl ions in the 
other, travel to the electrode at which they can respectively be 
converted into neutral hydrogen and oxygen. Thus while in 
the interior of the solution the current is almost entirely carried 
by the ions of the acid or the base, the transmission from the 
solution to the electrode is effected primarily by the ions of the 
water. From solutions of some salts also, hydrogen and oxygen 
are evolved; but here the conditions are different. Alkali is 
developed at the cathode, and its hydroxyl ions, combining with 
some of the hydrogen ions of the water, enormously reduce 
the number available. Thus the potential difference required to 
liberate the hydrogen at the electrode is increased, in accordance 
with a relation to be afterwards deduced and already used on 
p. 253 to explain the high electromotive forces of certain 
concentration cells. 

Returning to the consideration of acids and alkalies, we see 
that the decomposition voltage of such of them as contain ions 
of higher values than hydrogen and hydroxyl cannot rise above 
the potential difference which liberates hydrogen and oxygen. 
Those acids on the other hand, which, like hydrochloric, contain 
an anion of low deposition point, show a smaller decomposi- 
tion value when present in fairly concentrated solutions. As 
the concentration falls, it becomes difficult for the diffusion of 
the acid in solution to replace fast enough the chlorine ions 
which are removed from the layer of liquid in contact with the 
electrode. Increasing numbers of hydroxyl ions are therefore 
used to convey the current into the electrode, and this causes 
a rise in the polarization, which in dilute solutions reaches the 
maximum 1*7 volts. From strong solutions of hydrochloric acid 
the gases evolved are hydrogen and chlorine, but as dilution 
proceeds, the chlorine is gradually replaced by oxygen from 
the hydroxyl. This rise in the polarization is well seen in the 
following table, due to Le Blanc. 

2 normal hydrochloric acid, decomposition point 1'26 volts 

U 55 5) 55 55 5) 1*34 

~6 It 55 55 55 5) 1*41 ,. 

1(5" J) 55 55 55 55 162 

TT2 55 55 55 55 5 -*- "** 55 


The products of the continuous electrolysis of any mixed 
solution, containing two metals, depend on conditions more 
complicated than those which control the initial decomposition 
voltage of the solution or the polarization at one electrode. It 
is evident that the conditions determining the appearance of 
a second kind of ion of higher deposition point depend on such 
things as the current density, the transport numbers for the 
different ions present, the rate of diffusion of the dissolved 
substances, the existence and intensity of convection currents 
in the liquid and any mechanical mixing or stirring. The 
initial decomposition voltage of a solution, however, does not 
involve these dynamical problems, and solely depends on the 
potential differences required for the liberation of the ions first 
appearing at the two electrodes. If we accept the logarithmic 
expression for the electrolytic solution pressure, it is easy to 
see that two ions can only be simultaneously liberated by an 
electromotive force E when, with the usual notation, 

RT P mi RT P m , 

- log 1 -f^ = -- - log pr-' , 

qyi 5 A qy* * P* 
or, for two monovalent ions, when 

that is, when the partial osmotic pressures of the two ions in 
the liquid are in the same ratio as their solution pressures 1 . 
Even if this result be only a rough indication of the conditions 
of the problem, it serves to show that enormous differences in 
concentration would be necessary in order that the two metals 
of different deposition-voltages should be deposited together 
from a well-stirred solution, by a current of small intensity. 
Experiments confirming these conclusions have been made 
by Sand on mixed solutions of copper sulphate and sulphuric 
acid, in which convection was prevented. He finds that copper, 
which, at a copper electrode, has a deposition value lower than 
that of hydrogen by about 0*507 volt, is first liberated at the 

1 Nernst, Zeits. phys. Chem. xxn. 541 (1897) ; Sand, Proc. Phys. Soc. Lond. 
xvn. 496 (1901). 


cathode. As the current is increased, hydrogen also appears ; 
but this is due to the exhaustion of copper from the layers of 
solution in contact with the electrode which proceeds more 
rapidly than the replacement effected by the diffusion of the 
salt. By efficient stirring it is possible to prevent any evolu- 
tion of gas in cases where, without stirring, over sixty per 
cent, of the electro-chemical equivalents liberated would be 

Electrolysis has long been used to separate metals from 
each other. The theory of this process will now be clear. 
Let us suppose that we have a mixed solution of zinc and 
copper sulphates. The deposition point of copper is ~0'5 15 
volt, and that of zinc -I- 0'524 volt. Thus if the total electro- 
motive force applied be enough to give a potential difference 
at the cathode greater than 0*515 volt but less than +0*524 
volt, copper only will be deposited, for although its deposition 
point rises as the amount of copper gets less, this change is 
very small, and all traces of copper which could be detected 
by chemical analysis will be removed from the solution before 
the deposition point rises to that of zinc. If the -electromotive 
force at the cathode be now increased above -I- 0'524 volt, the 
zinc likewise can be separately removed from solution. 

Even without this adjustment of electromotive force, if the 
solution be kept well stirred to prevent the local exhaustion of 
one metal at the electrodes, complete separation can be nearly 
effected. For, as we have seen, as long as there is any of the 
metal of lower deposition point present, none of the other is 
liberated. This principle is used in a process of copper refining. 
A plate of pure copper forms the cathode in a bath of copper 
sulphate. The anode is a thick plate of impure copper, pro- 
bably containing metals both less and more easily deposited 
than copper. The bath is stirred, and when the current flows, 
copper and all more oxydizable metals are dissolved, while the 
less oxydizable metals, such as gold and silver, fall to the bottom 
of the vessel, for while copper is present in excess the current 
will dissolve it rather than more resisting metals. In the 
neighbourhood of the cathode, however, there will be a large 
excess of copper together with other metals, such as zinc, more 


easily oxydizable and therefore of higher deposition points. As 
long as any copper is near, therefore, none of the other metals 
are deposited, and pure copper is obtained at the cathode. 

On the other hand, by increasing the current density, it 
is usually possible to exhaust the one metal from the layers 
of solution next the electrode faster than either stirring or 
diffusion will replace it. The other metal must then also be 
used by the current, and, by proper adjustment of conditions, 
it is possible to deposit alloys, the percentage composition of 
which can be altered by varying the current density. 



Introduction. Osmotic pressure of electrolytes. Additive properties of 
electrolytic solutions. Dissociation and chemical activity. The mass 
law. Equilibrium between electrolytes. Thermal properties of elec- 
trolytes. Heat of ionization. Dissociation of water. The function of 
the solvent. Hydrolysis. Conclusion. 

THROUGHOUT our investigation of the electrical properties 
of solutions we have constantly been led to 

Introduction. . . 

infer that the ions of electrolytes are to a 
certain extent independent of each other. The flow of the 
current is in accordance with Ohm's law, and as we have 
already pointed out, that law implies freedom of interchange 
between the parts of the dissolved molecules. The existence 
of specific coefficients of mobility as characteristic properties of 
certain ions in very dilute solutions, involves the idea of inde- 
pendent migration, and suggests that the freedom of the ions 
from each other persists during the greater part of the time, 
and is not merely a power of interchange at the moments of 
molecular collision. If it were only a momentary freedom, the 
convective passage of the ions in opposite directions through 
the liquid, indicated by Faraday's law, would be explained by 
a continual handing on of the ions from molecule to molecule. 
The ions would work their way along by taking advantage of 
the intermolecular collisions, and the ionic velocities would 
depend on the frequency of these collisions ; a frequency, which, 
as indicated by the kinetic theory, depends on the square of the 
concentration. Now, as we saw on page 213, the conductivity 


of a solution varies as the product of the concentration and the 
relative ionic velocity ; on this view, then, the conductivity will 
be proportional to the cube of the concentration. The facts 
described on page 203 do not bear out this result. In dilute 
solutions, the conductivity is proportional to the concentration, 
and, as the concentration rises, the conductivity increases at a 
slower rate. It is difficult to see how these relations could hold 
except as a consequence of an almost complete migratory free- 
dom of the ions of dilute solutions, and very strong evidence 
is thus obtained in favour of a theory of ionic dissociation. 

Preconceived ideas would not, perhaps, lead us to expect 
that substances, which, like the mineral salts and acids, show 
great chemical stability when solid, should almost completely be 
dissociated into their ions when dissolved in water. It must, 
however, be remembered that it is precisely these bodies which 
possess the greatest chemical activity, that is to say, most 
readily exchange their parts with those of other substances. 
That a solution of hydrochloric acid, for example, does not ex- 
hibit the properties of dissolved hydrogen and chlorine, though 
it has been urged as an objection, is not a valid argument 
against the theory of dissociation, for the ions are certainly in 
conditions differing from those in which the atoms of the same 
elements exist in their usual state. Whether or not there is 
combination between the ions and the solvent, and whatever 
be the exact relation between the ions and the charges they 
carry, we are at least certain that a definite quantity of elec- 
tricity has to pass between an ion and the electrode before 
the substance can be liberated in a normal chemical state, say 
as gaseous hydrogen or chlorine. The energy associated with 
a substance when ionized must therefore be very different in 
quantity and character from that associated with it when in 
its normal chemical condition, and there is no reason to assume 
identity of properties in the two states. 

It has been suggested that, if really dissociated from each 
other, the two ions of a dissolved salt would generally diffuse at 
different rates, and ought therefore to be separable. If such 
separation occurred, however, electrostatic forces between the 
ions would at once arise and increase till further division was 


prevented. Nevertheless, some separation should undoubtedly 
occur, and, as a matter of fact, a volume of water in contact 
with the solution of an electrolyte is found to take, relatively to 
the solution, a potential of the same sign as the charge on the 
ion which has the greater mobility and therefore the quicker 
rate of diffusion. The phenomena involved will be studied in 
Chapter XIII. 

An experiment described by Ostwald 1 is instructive in 
connexion with this subject. A membrane of copper ferro- 
cyanide can be prepared which will allow potassium chloride in 
solution to pass through it, but is quite impermeable to barium 
chloride. Now, according to the theory, the chlorine ions of 
this salt will again pass, since they could do so in the first case, 
but the electric forces will prevent any considerable separation 
from taking place. If, however, we place some substance like 
copper nitrate on the other side of the membrane, the chlorine 
ions, which diffuse in one direction, are replaced by nitric acid 
ions, which diffuse in the other. In this way electrostatic 
charge is prevented, and the process will continue till we soon 
find nitrate mixed with the barium chloride, and chloride 
mixed with the copper nitrate. The salts cannot have directly 
reacted, for neither alone can pass through the membrane, but 
the exchange is readily intelligible on the hypothesis that the 
ions possess migratory independence. 

The dissociation required by the theory is a separation of 
the ions from each other, securing complete migratory indepen- 
dence. There is nothing to suggest that the ions are free from 
all chemical combination. As pointed out in the chapter on 
theories of solution, the hypothesis of electrolytic dissociation is 
entirely independent of any particular view as to the nature of 
solution or the physical mode of action of the osmotic pressure. 
All that is required to interpret the electrical phenomena is the 
freedom of the migrating ions from each other ; they may quite 
possibly be combined in some way with the solvent. If we take 
a chemical view of the nature of solution, it is in fact necessary, 
as shown on page 174, to imagine such combinations between 

1 B. A. Report, 1890, p. 332. 


the ions and the solvent in order to explain the abnormal 
osmotic pressures of electrolytes. We may perhaps represent 
what occurs by supposing that double molecules, such as 

NaCl . 2H 2 0, are formed, and dissociated into Na . H 2 0, and 

Cl . H 2 O, complex ions analogous to those described on pages 
226 229, for the existence of which there is definite evidence. 
On the other hand it may even be that a double decompo- 
sition goes on, as suggested by Reychler 1 , a molecule of sodium 
chloride, for example, decomposing with water thus : 

Na Cl + HOH = Na OH + HCL 

The water here is separated into parts which are non-electrical; 

a positive molecule of the composition NaOH, and a negative 

molecule HC1 are consequently formed. These molecules are 
not the same as ordinary soda and hydrochloric acid, which 
themselves are imagined to react with water in accordance with 
the equations 

Na OH + HOH = Na OH + HOH 

and H Cl + HOH = HOH + HCL 

The acid, by the interchange of its hydrogen, becomes nega- 
tively electrified, and produces a positive molecule of water 
which acts as the cation ; the alkali itself becomes a positive 
ion, and produces a negative molecule of water to form the 
anion. This hypothesis may not accurately represent the facts 
the suggested decomposition of water under the action of 
dissolved electrolytes into non-electrical hydrogen and hydroxyl 
cannot readily be accepted but it does not seem to be contra- 
dicted by any of the electrical relations ; and it is from the 
consideration of this and other similar ideas that we may hope 
to ascertain the essential features of the dissociation theory. 

1 Outlines of Physical Chemistry, Eng. trans., London and New York, 1899, 
p. 216. 


In solutions of electrolytes the osmotic pressure and the 
Osmotic correlated effects, the depression of the freezing- 

pressure of eiec- point and the lowering of the vapour pressure, 
are abnormally great. When an organic body 
such as cane sugar is dissolved in water, the osmotic pressure 
effects of dilute solutions are found, approximately at any rate, 
to agree with the values deduced by Van 't Hoff's theory. The 
osmotic pressure of dilute dissolved gases can be deduced by 
the principles of energetics either from the observed solubility 
relations, or from general molecular theory, and the reasonable 
extension of the results to solutions of other substances is justi- 
fied by experimental measurements in many different solvents, 
examples of which, due to Raoult, are given on pages 156 and 
157. Such theoretical and experimental considerations prove 
that, when water is used as solvent, the lower series of 
values, obtained with organic solutes, are normal ; it is the 
higher values characteristic of electrolytic solutions, which 
need further explanation. 

As long as a solution is dilute enough for the particles 
of solute to be outside each other's sphere of action, theory 
indicates that the osmotic pressure of a number of dissolved 
molecules has a value identical with that of the pressure the 
same number of gaseous molecules would exert at the same 
temperature when confined in a volume equal to that of 
the solution. Thus the osmotic pressure effects of dilute 
substances must depend on the number and not on the 
nature of the dissolved molecules. When experiments yield 
abnormally small values, it follows that the number of the 
solute molecules is less than that indicated by the chemical 
formula weight ; it is then natural to conclude that aggre- 
gation of molecules to form complexes has occurred. When, 
on the other hand, abnormally great values are obtained for 
solutions of electrolytes, it is necessary to infer that the number 
of solute particles is increased, and that some of their molecules 
have dissociated. Attempts have been made to explain the 
phenomena by an association of solvent molecules instead of 
a dissociation of those of the solute, but the general theory 
indicates that it is the volume and not the state of molecular 


aggregation of the solvent that is involved ; moreover, in the 
particular case of vapour pressure which led to the idea of the 
association of the solvent, the equations of page 128 clearly 
show that such an association would not affect the osmotic 

As soon as Van 't HofT 1 made known his investigations on 
osmotic pressure they were applied to the theory of electrolytic 
dissociation by Planck 2 and by Arrhenius 3 . Planck showed that 
the abnormally great osmotic pressure effects of electrolytes, 
when considered from the point of view of thermodynamic 
theory, required the hypothesis of some form of electrolytic 
dissociation. Arrhenius pointed out that the amount of dis- 
sociation thus existing in a solution might be estimated by two 
independent methods ; it might be determined by the com- 
parison of the actual equivalent conductivity with its value at 
infinite dilution, or by the measurement of the osmotic pressure 
effects, of which the importance had been recognized by Van' t 

The depression of the freezing-point has been more 
thoroughly investigated than the other correlated properties, 
and as the experimental error is probably less, a comparison 
may rightly be instituted between the results of this method 
and the electrical ones. Following Arrhenius, let us suppose 
that every electrolytically active molecule produces an abnor- 
mally great osmotic pressure, and that its effect is proportional 
to the number of ions into which it can be resolved. Thus the 
effect of an active molecule of potassium chloride should be 
twice that of an inactive one, and the effect of a molecule of 
potassium sulphate, which in dilute solutions yields two K 
ions and one S0 4 ion, should be three times that of an undisso- 
ciated molecule. 

If then, in a certain solution, we have ra inactive and n 
active molecules, each of the latter giving k ions, the total 

1 Kongl. Svenska Akad. Handl xxi. 38 (1885) ; Zeits. phyK. Chem. i. 481 

2 Wied. Ann. xxxn. 499 (1887). 

3 Zeits. phys. Chem. i. 631 (1887) ; Eng. trans. Harper's Science Series, iv. 47. 


osmotic pressure produced will be proportional to m -f kn, 
whereas the normal osmotic pressure would be proportional to 
m + n. By measuring the conductivity we can, for the dilute 
solutions of simple salts (see p. 225), find the fraction of the 
number of molecules which is at any moment active. Let us 
call it . Then, on Arrhenius' theory 

a = 

m + n ' 

so that, if the ratio of the actual osmotic pressure to the normal 

is called i, 

in -\- kn _, , , n _, N 

i= - - = l+(k-I)a (61). 

m + n 

This same ratio can also be found by direct experiment on the 
depression of the freezing-point, for by Van 't Hoff s equation we 
know the normal value, and if t be the observed depression for 
a solution of one gram-equivalent per litre, 

._ t 
= T8Q' 

We can thus compare the value of i as directly determined 
by observations on the freezing-point, with its value as calcu- 
lated from the conductivity. The table on the opposite page is 
part of that given by Arrhenius 1 for aqueous solutions. 

It will now be seen that there are two relations involved, in 
the dissociation theory. Firstly, the number of ions into which 
a molecule must be resolved in order to explain its electrical 
behaviour when completely dissociated in a very dilute solution 
should be the same as the number required to give its observed 
osmotic pressure ; secondly, in dilute solutions of simple salts, 
where the phenomena are not obscured by complex ions or 
changing ionic viscosity, as the concentration rises, the abnor- 
mally great osmotic pressure should diminish with the coefficient 
of electric ionization. Since the publication of Arrhenius' 
original paper, the results of which were accepted as a rough 
proof of the approximate accuracy of both these relations, 
a great quantity of experimental work has been undertaken ; 
some of it must now be passed in review. 

1 Zeits. phys. Chem. n. 491 (1887). 


Substance dissolved 

No. of gram 
per litre 

i observed 
from freez- 

i calcu- 
lated from 

a coeffi- 
cient of 

A. Non-Conductors. 

Methyl alcohol 
CH 3 OH .1 






Ethyl alcohol j 
C 2 H 5 OH \ 



Phenol ( 
C fl H 6 .OH \ 




? 1-00 

> o 




Cane Sugar 
C H 



^12 rL 22 v -'ll 







B. Electrolytes. 

Lithium hydrate j 










Acetic acid f 
CH 3 COOH | 






Phosphoric acid 
H 3 P0 4 1 











Sodium chloride 















Silver nitrate J 
AgN0 3 { 









Potassium sulphate 





K 2 S0 4 

- 0-227 













Calcium chloride 





CaCl 2 


















Copper sulphate 













To investigate the first relation it is necessary to measure 
the electrical conductivity and the freezing-point of solutions 


so dilute that the ionization may be taken as complete, or, at 
all events, that its value at infinite dilution may be estimated. 
The freezing-point experiments are complicated by sources of 
error pointed out in Chapter VI. In Raoult's book 1 on 
"Cryoscopie" are given the results obtained by Loomis at a 
concentration of O'Ol gram-molecule of salt to 1000 grams of 
water, i.e. O'Ol normal, as in themselves trustworthy and in 
accordance with the best of other results known at the time. 
The following are the molecular depressions of the freezing- 
point for certain substances in aqueous solution, the concentra- 
tion being expressed as gram-molecules of solute per thousand 
grams of water. 

Potassium hydrate 3'71 Nitric acid 3'73 

Hydrochloric acid 3'61 Potassium nitrate 3'46 

Potassium chloride 3*60 Sodium nitrate 3'55 

Sodium chloride 3'67 Ammonium nitrate 3-58 

Sulphuric acid 4'49 Calcium chloride 5'04 

Sodium sulphate 5*09 Magnesium chloride 5 '08 

Magnesium sulphate 2 -66 Zinc sulphate 2 - 90 

In the first group are substances which are shown by the 
electrical properties to yield in solution two monovalent ions. 
On the dissociation theory, therefore, the osmotic pressure 
effects should, at high dilutions, have double their normal value. 
The normal value for the molecular depression of the freezing- 
point is 1'857, as calculated from the osmotic pressure theory, 
and confirmed by experiments on dilute aqueous solutions of 
non-electrolytes (see page 147). Twice this value is 3*714, 
a number to which all the observed molecular depressions of 
substances in group 1 closely approximate. The electrical 
behaviour of bodies in the second group similarly indicates 
dissociation into three ions, producing a theoretical molecular 
depression of 5 '57. The experimental numbers differ from this 
value by perhaps 10 per cent., but the error is in the right 
direction since the electrical conductivities at the concentration 

1 Paris, Oct. 1901. 


used by Loomis in these freezing-point experiments, namely, 
about 0*01 normal, show that the ionization is still far from 
complete in the salts containing divalent ions. The corre- 
sponding error is still greater in the salts of the third group, 
which yield two ions, both divalent ; the molecular depression 
should be again 3 '7 14, greater by about a third than the 
observed values. All discrepancies are thus of the kind to be 
expected from a consideration of the electrical phenomena ; and 
the first group, the salts of which are about 95 per cent, ionized 
at the concentration used in the cryoscopic experiments, yield 
very concordant results. 

In the investigation of the freezing-points of very dilute 
solutions carried out by E. H. Griffiths, to which reference was 
made on pages 147 and 158, results for salts with divalent ions 
have not yet been published. The molecular depression of 
potassium chloride at a concentration of about 0*0003 normal 
was found to be 3720, exactly double, within the limits of 
experimental error, the number given by an equivalent solution 
of cane sugar. At this concentration the conductivity indi- 
cates that 99*7 per cent, of the salt is dissociated. 

Thus the evidence at present available goes to support the 
accuracy of the first relation of Arrhenius' theory in the case of 
aqueous solutions. The observed depressions never appreciably 
exceed the theoretical values, and the discrepancies in the other 
direction are readily explicable by incomplete ionization. In 
fact consideration shows that the relation can only be exact for 
those solutions which reach a definite limit of equivalent con- 
ductivity as the dilution is increased ; it is only these solutions 
that are fully ionized. 

Passing to solutions in solvents other than water, we find 
that sufficient data are not available to decide whether the 
same relation between the electrical and the osmotic pheno- 
mena holds good. The difficulties of experiment are much 
increased, and no observations on osmotic pressure effects seem 
to have been made on solutions in which the dilution was 
carried far enough to secure a constant value for the equivalent 
conductivity and so justify the assumption of complete ioniza- 
tion. In many aqueous solutions, such as those of acetic acid 
w. s. 21 


and ammonia in particular, complete ionization cannot be ex- 
perimentally reached; and, without definite evidence, we cannot 
assume that it is ever obtained in another solvent. Measure- 
ments on stronger solutions are of little use, for as soon as the 
dissolved particles come within each other's sphere of influence 
their change of available energy by dilution will not be inde- 
pendent of the nature of the solvent, and the thermodynamic 
deduction of the gas-value for the osmotic pressure ceases to be 
valid. Moreover, for non-aqueous solutions, we have little know- 
ledge of such electrical constants as the transport numbers, and 
it is not safe to conclude that the ions are of the same nature as 
in water. In alcoholic solutions, at any rate, what little evidence 
is forthcoming indicates that complex ions are very numerous, 
even at moderate dilutions, (see pages 195 and 218), and any 
such complexity must diminish the number of dissolved par- 
ticles, and consequently the osmotic pressure effects. Kahlen- 
berg, however, states that solutions of diphenylamine in methyl 
cyanide show abnormally low molecular weights, and yet are 
not conductors of electricity 1 . Such a result perhaps indicates 
a dissociation yielding products which are not electrically 
charged, or a non-electrical double decomposition with the 
solvent. Until further observations have been made it is im- 
possible to say whether or not the first relation suggested by 
the dissociation theory holds for non-aqueous solutions. 

The second relation enunciated by Arrhenius indicates that 
the coefficient of ionization measured electrically should agree 
with its value calculated from the osmotic pressure effects ; but 
this relation can only hold within very narrow limits of con- 
centration. The thermodynamic theory of osmotic pressure is 
valid only when the solute particles are beyond each other's 
sphere of influence, and any further addition of solvent can 
consequently not affect that part of their available energy which 
is due to their connexion with the solvent. For greater con- 
centrations, the osmotic pressure will depend on the nature of 
the solvent and of the solute, and on the interaction between 
them. Again, if complex ions are present, and, in any case, as 

1 Jour. Phys. Chem. v. 344 (1901) ; vi. 48 (1902). 


soon as the concentration becomes great enough to affect the 
ionic fluidity, the ratio of the actual to the limiting equivalent 
conductivity, as shown on p. 225, ceases to measure the fraction 
of the number of molecules which are resolved into indepen- 
dent ions. Moreover, except in solutions of such simple salts 
as potassium chloride, etc., there is some doubt whether the 
limiting value of the equivalent conductivity has ever been 
reached, and if not, even in dilute solution, the electrical 
measurements do not give the true coefficient of ionization. 
Nevertheless, experiments on these lines are of great interest ; 
confirmation of the relation for dilute aqueous solutions of 
simple salts would be valuable evidence that the Arrhenius 
relation gives, in such cases, a complete explanation of the 
phenomena, and the amount of divergence in other cases would 
supply useful indications of the nature and amount of the dis- 
turbing influences. It must be clearly recognized that, while 
the dissociation theory requires the agreement in the numbers 
of the ions indicated by the electrical and osmotic methods in 
the few cases in which the conductivity phenomena show the 
ionization to be complete, it is far from suggesting that the 
first relation holds in other cases, or that the second relation 
exists except for a limited range of concentration of salts which 
in dilute solution are fully ionized. As soon as the concentra- 
tion begins to increase, the complications we have indicated are 
appreciable, and the relation between the electrical and the 
osmotic values of the ionization coefficient must become more 
and more inexact. Stress is laid on these restrictions because 
the relations under consideration have played a great part in 
the history of the dissociation theory, and the want of quanti- 
tative agreement in the results of the two lines of research has 
often been adduced to deny the claims of the theory to give a 
true explanation of the difference between electrolytic and 
non-electrolytic solutions. 

It may here be pointed out that relations, which are true 
when the system possesses the properties of dilute matter, 
must be expected to begin to fail at much smaller concentra- 
tions in the case of electrolytes than of non-electrolytes. On 
the supposition that the forces between atoms and molecules 



are electrical, the force between' two electrically bipolar mole- 
cules will quickly diminish as the distance between them 
increases probably as the fourth power. The force thus 
rapidly becomes insensible beyond a certain small range, the 
sphere of molecular action ; but the force between two dis- 
sociated ions is of a different order. Here we have positive 
and negative charges which are not permanently connected to 
opposite charges to form molecular doublets. The forces will 
be more of the nature of those between small isolated electrified 
bodies, and their variation with distance will approximate to 
the law of inverse squares; their range is greatty increased, and 
ionic influence will not rapidly vanish beyond a definite limit in 
the same way as do the intermolecular forces. We might expect, 
for example, that, for cane sugar, the molecular depression of the 
freezing point should keep constant throughout a much greater 
variation of concentration than for the solution of a metallic 
salt, even when the latter is corrected for ionization. It is 
possible that, as the concentration increases, the electric forces 
between the dissociated ions become sensible before any re -com- 
bination can occur ; if so, the ionic velocities, and therefore the 
conductivity, would be reduced before the ionization ceases 
to be complete, and the coefficient a of p. 224 would not 
represent the ionization even at great dilution. On the other 
hand, the osmotic pressure effects might be affected also at 
about the same concentration, though not necessarily to the 
same extent, and perhaps any such inter-ionic electric forces as 
are here contemplated may, for the purposes of the properties 
we are considering, be truly reckoned as combination. At all 
events, the forces would interfere with that complete migratory 
independence of the opposite ions as regards each other which 
may be taken as the meaning of complete ionization. 

References to cryoscopic determinations on solutions of 
electrolytes have been made in the chapter on freezing points, 
pp. 153 to 158. Many of the results have been compared 
with the ionizations as measured by Kohlrausch at 18 or 
Ostwald at 25 ; but to obtain a satisfactory basis of com- 
parison, the electrical data also must be determined at the 
freezing point. 


Experiments at 0, on solutions of moderate dilution, have 
been made by R. W. Wood 1 , Archibald 2 , and Barnes 3 , the 
limiting values of the equivalent conductivity being estimated 
by reference to Kohlrausch's data. Other experiments were 
made by Kahlenberg and Hall 4 and by the present writer 5 , 
who carried the dilution far enough to reach the limiting 
equivalent conductivity of simple salts. The experiments 
were planned in connexion with those of Griffiths on freezing 
points, and were made in a platinum cell similar to his 
apparatus. The results showed an appreciable difference 
when the ionizations were determined at and at 18. The 
following values were obtained from smoothed curves drawn 
between the cube root of the concentration and the equivalent 
conductivity, and represent the most probable numbers for the 
ratio /ji/fM^ of the actual to the limiting equivalent conductivity 
throughout the range of concentration employed. The values 
for magnesium sulphate were obtained later from experiments 
in a glass cell. 

It will be seen that definite limits were found for the 
equivalent conductivities in the cases of potassium chloride 
and permanganate and of barium chloride. In these solutions, 
therefore, complete ionization was reached at the concentrations 
indicated. In the cases of the other salts used, potassium 
bichromate and ferricyanide and copper sulphate, no exact 
limit was found, and the value of the equivalent conductivity 
corresponding to complete ionization had to be estimated by 
exterpolating the curves. For sulphuric acid which, as at 
18, reaches a maximum equivalent conductivity at a certain 
dilution, and then falls off again as the concentration is still 
further diminished, the maximum was taken as the limit, a 
mode of procedure which, however, almost certainly leads to 
too high values for the ionization coefficients. 

Zeits. phys. Chem. xvm. 3 (1895) ; Phil. Mag. [5] XLI. 117 (1896). 

Trans. Nov. Sco. Inst. Sci. x. 33 (1898). 

Ibid. x. 139 (1899) ; Trans. Roy. Soc. Canada, n. 6 (1900). 

Jour. Phys. Chem. v. 339 (1901). 

Phil. Trans. A. cxciv. 321 (1900) ; Zeits. phys. Chem. xxxm. 344 (1900). 





1 a' 

i o 

^ # 

<* $ 



r & 

a s, 

" a 

^3 -M 

i J 

*- % 

b | 

8 -S 


^ *3 

-i ? 


H 8 


M 1 









O5 O5 00 I>- CO Tt< i I J>- CO 

1>- CD iO CO C 

O5 C5 O5 Oi 00 00 00 

Tt< O O5 00 J>- *O 

t i CM o i t oo -^ 

O CD >0 ip 

Oi 00 CO "^ ^^ CO ^> ^H CO O5 ^O 


CO G^ C^O O CO i^ CO ^^ GO G*^ 


co o 
O^ O5 00 00 00 



GO CO (M 1^* CD G<1 C^ ^H ^cj 

O5 O5 O5 O5 O5 O5 O5 O5 O5 


o o o i i r- 1 CM co 
p o a o p o p 



The investigation has lately been extended in glass appa- 
ratus to greater concentrations, and the following smoothed 
values have been obtained : 




JEBaCl 2 

^CuS0 4 

iMgS0 4 



































































The first column under KC1, like those under all the other 
salts, gives the value of p / fi x when the concentration is ex- 
pressed in terms of gram-equivalents of salt per thousand grams 
of solution. In the second column under KC1, the concentra- 
tion is expressed per thousand grams of solvent. This method, 
if applied to the other salts would, in a similar way, reduce the 
calculated results. 

The corresponding freezing point experiments are not yet 
completed, and an exact comparison of the two lines of research 
is not possible. If, however, we accept Griffiths' result for 
potassium chloride as establishing the first relation, namely, 
the agreement between the numbers of the ions at infinite 
dilution as estimated in the two ways, we can take 1*858 k, 
where k is the number of the ions, as the limiting value of the 
molecular depression, and calculate the ionization a from cryo- 
scopic determinations on stronger solutions. The following 
comparisons with the electrical measurements described above 
may be given, n being expressed in gram-molecules of salt per 
thousand grams of solvent. 




Potassium chloride. KC1 = 74-59. 



a (Baoult) 


























a (Jones 


a (Loomis) 



























In the annexed diagram, the smooth curve W denotes the 
electrical values of *&, and the dotted lines indicate the 


-*^ \ 


ionizations as deduced from the cryoscopic observations of 
Raoult (R), Loomis (L), Jones (J), and Ponsot (P). The cross 


shows the concentration at which Griffiths found complete 
ionization. Accepting his result, it follows that Jones' 
numbers, at great dilution at any rate, are too high, while 
the complete difference of Ponsot's curve from those of other 
observers makes it unsafe to lay stress on his experiments. 
We are thus left with the results of Raoult and Loomis, and 
the lower part of Jones' curve as a basis of comparison with 
the electrical measurements. As a general conclusion, we may 
perhaps say that there are indications that the electrical and 
cryoscopic curves approach each other at greater dilutions, and 
perhaps also at greater concentrations. Throughout the range 
at which a direct comparison can be made, however, there is a 
considerable difference between the two sets of results. This 
difference is greater than has been usually supposed. The 
electrical curve, deduced from conductivity measurements made 
at 18, which has generally been adopted as the basis of com- 
parison, lies below the curve for given in the diagram, and 
happens nearly to coincide with the cryoscopic results. The 
question cannot be regarded as settled, but the evidence at 
present available indicates that the ionizations as measured in 
the two ways become consistent only at extreme dilution, even 
in simple salts such as potassium chloride. 

The divergencies between the cryoscopic results of different 
observers, considerable in the case of potassium chloride, be- 
come even more conspicuous for more complicated salts, and 
in the present position of the subject no useful purpose would 
be served by a detailed examination. It seems that, in some 
cases at any rate, the curves cross each other at moderate 
concentration, the electrical becoming lower than the freezing 
point results. In barium chloride solutions, the change occurs 
within the range of comparison given by Loomis' experiments ; 
it is possible that some of the agreements that have been 
obtained may depend on accidental conjunctions of this nature, 
and would fail at lower and higher concentrations. 

More accurate determinations of the freezing points are 
needed before such comparisons as these can be taken as a 
satisfactory basis for theoretical generalizations. Collections of 


Barium chloride. 



n = 2M 



. /*/A* 





















the present data have been made by MacGregor 1 and Kah- 
lenberg 2 . 

Passing as before to solutions in solvents other than water, 
we again find the difficulties of experiment much increased. 
Many observations have been made, but very seldom has the 
dilution been pushed to such extremes as are necessary to 
produce complete ionization in aqueous solutions. The phe- 
nomena seem to be more complicated, and sometimes the 
equivalent conductivity increases with concentration even in 
fairly dilute solutions 3 . Limiting values of the equivalent 
conductivity for salts of the alkali metals dissolved in methyl 
and ethyl alcohol have however been obtained by increasing 
the dilution by Fitzpatrick 4 , Vollmer 5 , and Zelinsky and 
Krapiwin 6 . 

For solutions in these solvents, then, it is possible to 
calculate the dissociations at moderate dilutions by deter- 
mining the ratio of the equivalent conductivities. To compare 
these results with the osmotic values, it is necessary to use 
the vapour pressure or the boiling point method, as the 
freezing points of these solvents are very low. A collection 

1 Trans. Nova Scotia Inst. Sci. x. 211 (1899 1900) ; Phil. Mag. [5] L. 505 

2 Journ. Phys. Chem. v. 339 (1901). 

3 Kahlenberg, Journ. Phys. Chem. v. 342 (1901). 

4 B. A. Report, 1886, p. 328 ; Phil. Mag. xxiv. 378 (1887). 

5 Wied. Ann. LII. 328 (1894). 

6 Zeits. phys. Chem. xxi. 35 (1896). 


of results of such comparisons has been made by A. T. Lincoln 1 , 
who gives the following tables from boiling point experiments 
by Woelfer, and conductivity experiments by Vollmer. 




in per cents. 



boiling point 



tions in meth^ 

tl alcohol. 

Lithium chloride 




Potassium iodide 




Sodium iodide 




Potassium acetate 




Sodium acetate 




Solutions in ethyl alcohol. 

Lithium chloride 




Potassium acetate 




Potassium iodide 




Silver nitrate 




Sodium iodide 




55 55 




Sodium acetate 




In alcoholic solutions, what little evidence is available 
indicates that complex ions are very frequent even at moderate 
dilutions (see pp. 195 and 218), and the above results show 
that, as we should expect, the disturbing factors have greater 
influence than in aqueous solutions. The results for sodium 
iodide suggest that better agreement would be obtained at 
smaller concentrations. 

Experiments, summarized in Lincoln's paper, on acetone by 
Carrara, Laszozynski, and Dutoit and Aston, and on pyridine 
by Laszozynski and Gorski, indicate limiting values for the 
equivalent conductivities of the iodides of the alkali metals 
when dissolved in these solvents 2 . Lincoln's measurements on 

1 Journ. Phys. Chem. m. 457 (1899). 

2 Carrara, Gazz. Chim. Ital. xxvn. 1. 207 (1897) ; Laszozynski, Zeit. 
Elektrochem. n. 55 (1895); Dutoit and Aston, Compt. rend. cxxv. 240 (1897); 
Laszozynski and Gorski, Zeit. Elektrochem. iv. 290 (1897). 


silver nitrate in pyridine show no sign of reaching a limit, but 
his greatest dilution was only 784 litres per gram-molecule 1 . 
Many other solvents give conducting solutions, but, in them, 
limiting values of the equivalent conductivity have seldom or 
never been obtained. A general account of the work that has 
been done on non-aqueous solutions will be found in Lincoln's 
paper. Cases in which the boiling or freezing points of con- 
ducting solutions indicate molecular weights equal to or greater 
than the normal have been pointed out by Kahlenberg 2 . Such 
phenomena probably indicate association of the non-ionized 
solute molecules. 

Many attempts have been made by chemists to trace con- 
Additive nexions between the physical properties of 
eT P troi ieS i f compounds and their chemical constitution, 
solutions. Details can be found in any book on physical 
chemistry. The general result may be summarized by saying 
that, while some properties, such as the atomic volumes, the 
atomic heats, and the power of magnetic rotation of the plane 
of polarization of light, seem to be permanent characteristics of 
the elements and keep nearly the same values even when those 
elements change their state of combination, such additive 
relations are limited to a few properties, and never seem to 
be more than approximations. 

In solutions of electrolytes, additive relations are applicable 
to many more properties, and are much more accurately true. 
The explanation of these results by means of the hypothesis of 
the practical independence of the constituents of the solutes 

1 A detailed study of this solution would be useful, for in it Faraday's 
law has been confirmed by Skinner (B. A. Report, 1901, p. 32) ; from boiling 
point experiments Werner concluded that the molecular weight is nearly normal 
(Zeits. Anorg. Chem. xv. 23 (1897)) ; while, between dilutions of one and forty 
litres, the transport numbers for the cation have been found by Schlundt to 
be : 1 litre, 0-326 ; 2 litres, 0-342 ; 10 litres, 0-390 ; 40 litres, 0*440 (Jour. Phys. 
Chem. vi. 168 (1902)). This rapid change indicates the existence of complex 
ions. Schlundt remarks that, as a rule, the experiments of Hittorf and others 
show that the transport number for the cation increases rapidly with dilution 
in solutions of salts which show a marked affinity for the solvent. 

2 Journ. Phys. Chem. v. 342 (1901). 


was suggested by several observers, even before the develop- 
ment of the electrolytic dissociation theory. That theory 
indicates that when the ionization is complete, the difference 
between any physical property of a solution and the cor- 
responding property of its solvent should be compounded 
additively of the differences produced by the two ions. When 
the ionization is not complete, the differences referred to must 
be similarly compounded of those produced by the undis- 
sociated molecules and the dissociated ions. It should thus 
be possible to express the numerical values of the various 
properties in terms of the state of ionization, by means of an 
expression of the form 

P == P w + K ( 1 - a) n + L an 

where P is the numerical value of any property, such as the 
density, etc., P w the value of the same property for the solvent 
under the same conditions, n the molecular concentration of 
the solution, a the ionization coefficient, and K and L two 
constants independent of the concentration. MacGregor has 
supported this equation for many properties of dilute solutions 
by tabulating known data 1 . An extended account of the 
additive relations of salt solutions will be found in Ostwald's 
Lehrbuch der Ckemie. A short summary only is here attempted. 
Valson 2 found that the specific gravities of salt solutions 
could be calculated from a table of moduli of the elements of 
the substance dissolved, the modulus for each element being 
experimentally determined. The relation is better investigated, 
however, by considering the specific volume instead of its 
reciprocal the specific gravity, and Groshaus 3 found that the 
molecular volume of the dissolved salt was, in dilute solution, 
the sum of two constants, one determined only by the acid and 
the other only by the base. The densities and thermal expan- 
sions of solutions have since been redetermined by Bender 4 , 
who confirmed Valson's conclusions. The thermal expansion of 

1 Phil. Mag. [5] XLIII. 46 and 99 (1897). 

2 Compt. rend. LXXIII. p. 441 (1874). 

3 Wied. Ann. xx. p. 492 (1883). 

4 Wied. Ann. xxii. 184 (1884) ; xxxix. 89 (1890). 


salt solutions is more uniform the more the concentration is 
increased, the curved temperature-volume diagram for water 
becoming more straight as salt is added. Ostwald 1 has measured 
the volume-changes accompanying the neutralization of bases 
by acids, and shown that, here again, additive relations appear. 
The subject has been fully discussed by Nicol 2 . 

Similar phenomena appear when we study the colour of a 
salt solution 3 , which is found to be produced by the super- 
position of the colours of the ions and the colour of the undis- 
sociated salt. If the absorption spectra of a series of coloured 
salt solutions containing a common ion are examined, the additive 
character of the colour is well seen, the absorption bands due 
to the common constituent being unaffected by the presence 
of the other part of the salt. The light transmitted through a 
solution is composed of all those rays which have been absorbed 
by neither constituent. Anhydrous cobalt chloride is blue, 
while in cold aqueous solution all cobalt salts are red. Red, 
then, is the colour of the cobalt ion, and only appears when the 
salt is more or less dissociated. When cobalt chloride is dissolved 
in alcohol, the conductivity is very low, showing very incomplete 
ionization. The colour is 1 , accordingly, the blue of the undis- 
sociated salt. If we slowly add water to this solution, the 
ionization gradually increases, and the colour changes to purple 
and then red. An aqueous solution, boiled with potassium 
cyanide, is decolorised, for a cobalti-cyanide, K 3 Co(CN) 6 , has 
been formed; the ions of this compound are 3K and Co(CN) 6 ; 
the free cobalt ions no longer exist, and the solution ceases to 
respond to the usual tests for cobalt. That the red colour is 
really due to the ionization, and not to a hydrate formed 
between the cobalt salt and the solvent, is indicated by the 
additive nature of the phenomena ; for, like many other pro- 
perties, the colour of non-electrolytes depends on the consti- 
tution and is not additive. The use of indicators, which show 
the presence of acids or bases by a change in colour, depends 
upon similar phenomena. Thus para-nitrophenol is a weak 

1 Lehrbuch, or Solutions, p. 257. 

2 Phil. Mag. xvi. 121 ; xvm. 179 (18834). 

3 Ostwald's Lehrbuch; Zeits.phys. Chem. ix. 584 (1892). 


acid, very little dissociated. The addition of an alkali, soda for 
example, causes the corresponding salt to be formed. This is 
largely dissociated, and the intensely yellow colour of the ion 
C C H 4 NO 2 . is at once seen. 

A rise of temperature generally reduces the dissociation of a 
salt in solution, and increases the number of combined molecules 
the accompanying increase of conductivity being brought about 
by a still greater reduction in the viscosity which the solution 
opposes to the motion of the ions. We should expect, therefore, 
on heating a coloured solution in which this temperature 
relation exists, that the colour would become more like that of 
the undissociated salt. Thus anhydrous copper chloride is a 
yellow solid, and the combination of this with the blue of the 
copper ion produces the green colour of the strong solution. 
On adding water the colour gets more blue, but on heating 
it goes back to green. Other cases have been described by 
J. H. Gladstone 1 . 

Similar additive relations have been traced in the refraction 
coefficients, which were found by Gladstone to be additive 
properties in solutions of active i.e. dissociated salts, in the 
optical rotatory powers, in the surface tensions, and in the 
viscosities of salt solutions ; while Perkin, from the phenomena 
of magnetic rotation, concluded, without reference to the disso- 
ciation theory, that salts were dissociated into acid and base. 
The thermal capacities are complicated in that a change of 
temperature usually causes a change in the state of dissociation 
to an amount dependent on the nature of the substance ; but, 
in completely dissociated solutions, the thermal capacity is also 
an additive property 2 . 

The rapidity and ease with which reactions occur between 
solutions of electrolytes are in sharp contrast 

Dissociation * i i ' i 111 n i 

and chemical with the difficulty and delay usually experienced 
in producing chemical changes in organic sub- 
stances. The close connexion between chemical activity and 

1 Phil. Mag., 1857, [4], xiv. p. 423. 

2 Marignac, Ann. Chim. et Phys., 1876, [5], vm. p. 410. 


electrolytic conductivity was noticed by Hittorf, and Arrhenius, 
who afterwards investigated the subject, was able to establish 
definite numerical relations. 

The existence of specific coefficients of affinity, which are 
characteristic properties of individual acids and bases whatever 
the reaction in which they are engaged, is clearly recognized in 
modern chemistry. The relative strengths of these affinities 
may be measured in different ways with consistent results. If 
one acid acts on the sodium salt of another, some of the sodium 
salt is decomposed, and, unless its acid is removed from the 
sphere of action by evaporation or precipitation as fast as it is 
formed, eventually certain quantities of both sodium salts and 
both acids will be left in solution. The relative amounts will 
finally be the same whichever possible pair of components we 
use as reagents. The final composition of the solution cannot, 
in general, be ascertained by chemical means, for the addition 
of a new substance would alter the equilibrium. Physical 
methods of investigation have therefore been employed. 

Thomsen determined how much of the sodium salt of one 
acid was decomposed by another, by measuring the heat evolved 
during the action. He thus measured the ratio in which the base 
is shared by the acids a ratio which may be said to express 
their relative avidities. Ostwald 1 also investigated the relative 
avidities of acids for potash, soda, and ammonia, and proved 
them to be independent of the base. The method employed 
was to measure the changes in volume caused by the action. 
The results are given in column I. of the table which follows, 
the avidity of hydrochloric acid being taken as one hundred. 

Another method is to allow some acid to act on an insoluble 
salt, and to measure the quantity of substance which goes into 
solution. Determinations have been made with calcium oxalate, 
which is easily decomposed by acids, oxalic acid and a soluble 
calcium salt being formed. The avidities of acids relative to 
that of oxalic acid are thus found, so that the acids can be 
compared among themselves. Their relative avidities as thus 
measured are given in column II. of the table. 

A property of acids, at first sight unconnected with the 
1 Lehrbuch der Ally. Chemie. 


avidity, is their accelerating influence on such actions as the 
" inversion " of cane sugar, which consists in its transformation 
into dextrose and laevulose. It has long been known that strong 
acids produce much greater accelerating effects than weak acids, 
the acid itself being in all cases unchanged. The relative 
strengths of acids as thus determined agree with their avidities 
for decomposing the salts of other acids. Another instance 
of accelerating action is seen in aqueous solutions of methyl 
acetate, which, if allowed to stand, undergo a very slow decom- 
position into alcohol and acid. This process is much quickened 
by the presence of a little dilute foreign acid, though the 
accelerator remains unchanged. It is again found that the 
influences of different acids on this action may be taken as 
specific coefficients of affinity. The results of this method are 
given in column III. 

Finally in column IV. the electrical conductivities of normal 
solutions of the acids have been tabulated. A better basis of 
comparison would be the ratio of the actual to the limiting 
conductivity; but, since the conductivity of acids is chiefly 
due to the hydrogen, the limiting value is nearly the same 
for all, and the general result of the comparison would be 

As we have already noticed, the electrolytic conductivities 
of solutions of different mineral acids attain approximately 
equal values and their ionizations are nearly complete. Similar 
phenomena are observed in the case of their chemical affini- 
ties. The values of the affinity for hydrochloric, nitric and 
other strong acids are practically the same, and cannot by any 
means be increased. Ostwald has found that the introduction 
of oxygen, sulphur or a halogen, which increases the affinity of 
a weak acid (compare acetic acid with the three chloracetic 
acids), has no effect on the affinity of strong acids. The 
limit has evidently been reached, and the whole substance 
obtained in a state of activity. In each column of the following 
table the number for hydrochloric acid has therefore been made 
equal to 100. 

w. s. 22 

































Monochloraceti c 






























Similar methods can be used for determining the relative 
strengths of bases. The avidities can be compared by sharing 
an acid between two bases competing for it ; and their in- 
fluence on the rate of saponification of methyl acetate gives 
the accelerating power 1 . Since the velocity of the hydroxyl ion 
is less than that of the hydrogen ion, the conductivities yield 
a less accurate method of comparison than in the case of acids, 
and the ionizations have therefore been calculated for the 
concentration of one-fortieth normal, at which the accelerating 
power was measured. 





Lithium hydroxide 















The difficulties and the experimental errors of some of these 
chemical measurements are very considerable, and, in many 
cases, the solutions of the acids given in the table are not 
of comparable concentrations. Nevertheless, the remarkable 
general agreement of the results is quite enough to show the 
intimate relation which exists between the chemical activity of 
1 J. Walker, Physical Chemistry, London, 1899, p. 277. 


electrolytes in aqueous solution and their electrical conductivity. 
For solutions in other solvents no such numerical data are 
available. Kahlenberg has shown that chemical reactions which 
are practically instantaneous occur in non -electrolytic solutions 
in benzene 1 . As an example, dry hydrochloric acid gas passed 
into a solution of copper oleate in benzene produced at once a 
heavy brown precipitate of copper chloride, though the solution, 
even at the instant of reaction, showed no more conducting 
power than did the pure benzene. Again, an insulating solution 
of stannic chloride in benzene mixed with the solution of copper 
oleate, gave instantly a copious precipitate. It seems, then, 
that electrolytic ionization is not in all cases the mode of 
operation of rapid chemical action, and that the encounters 
between two molecules must sometimes be accompanied by 
chemical interchanges. 

The phenomena of reversible chemical action have already 
been considered from a kinetic standpoint on 

The mass-law. L 

pp. 205 and 206. The results can also be 
obtained by an application of the principles of energetics. The 
most direct way to treat the problem is to consider the increase 
of available energy due to the appearance of new molecules or 
atoms of given species during the process of chemical change 2 . 
This increase in free energy will involve two terms ; one 
expressing the work a done in forming the particle at the 
temperature chosen and at a standard pressure, and another 
giving the work required to bring by isothermal operation the 
new substance when formed to the actual pressure at which the 
system exists. If the system is a gas or dilute solution, the 
second term will be of the form RT\ogp/p , where p is the 
actual and p the standard pressure. Thus the change in the 
available energy of the system is a + RT log bC, an expression 
already used on p. 26, where a is a function of the temperature 
T, R is the gas constant per gram-molecule and has the same 
value for all kinds of dilute matter, (7 is the final number of 
molecules of the given species per unit volume, and b is a 
constant expressing the dilution of the molecules at the standard 

1 Jour, Phys. Chem. vi. 1 (1902). 

2 Larmor, Phil. Trans. A. cxc. 276 (1897). 



pressure and at the existing temperature. A reaction in the 
system involves the disappearance of molecules of some of the 
species present, and the appearance of others to an equivalent 
amount. When equilibrium is reached, the change of available 
energy arising from a further slight transformation of the kind 
considered must vanish ; thus 

n, (a, + RT log kCi) + n, (a z + RT log & 2 C 2 ) + . . . = 0, 
or, jRT(log &!*'&,* ... + log C^Cf*...) = - (n^ + n z a 2 +...), 
where n a , n 2) ... are the numbers of the molecules of the different 
types which are involved in the reaction, reckoned positive 
when they appear, negative when they disappear. We then 
see that 

Cf*Cp...=K (62), 

K being a function of the temperature alone. This expression 
is the mass-law of chemical equilibrium, originally derived by 
Guldberg and Waage from statistical considerations. 

This law of mass action has been applied to reversible 
chemical actions such as the dissociation of gaseous nitrogen 
peroxide and the like processes, and has been found to lead to 
results in accordance with the observed facts. It has been 
extended to electrolytic dissociation by Ostwald. For a binary 
electrolyte such as potassium chloride, it is natural to suppose 
that the change consists in the dissociation of one molecule 
into two ions ; in this case in the equation of equilibrium n^ will 
be 1, and n 2 and n 3 will each be +1, so that the equation 


Cr l C z C 3 = K (63), 

or, since the concentration of the two ions must be the same, 

O^Cf = K (64), 

where Cj denotes the molecular, and (7 2 the ionic concentration. 
This result is explained on kinetic principles by assuming 
that the rate of dissociation is proportional to the active mass 
(7j of the remaining molecules ; and that the rate of recombi- 
nation varies as the frequency of collision between the ions, 
a frequency which is proportional to the product of the active 
masses of the ions, that is to (7 2 2 . For equilibrium, the two rates of 
transformation must be equal, and we regain the mass equation. 


Considering one gram-molecule of electrolyte dissolved in a 
volume V of solution, the ionization being a, we have 

v ) (v) ~ K> 




This equation is called Ostwald's dilution law. It should 
represent the effect of dilution on the ionization of binary 
electrolytes ; and, for small concentrations, when the ioniza- 
tion may be measured by the ratio of the actual equivalent 
conductivity to its value at infinite dilution, an experimental 
confirmation of its accuracy should be possible. Many observa- 
tions show, however, that the law fails to express the ionization 
of strong acids and salts, though Ostwald has confirmed it with 
considerable accuracy in the case of weak acids with small 
coefficients of ionization. For such bodies 1 a is nearly equal 
to unity, and only varies slowly with dilution. The equation 
then becomes 

or * = >JVK (66), 

so that the molecular conductivity should be proportional to 
the square root of the dilution. If we determine a for a 
number of solutions of different strengths, and use our results 
to calculate K, we may expect the values obtained to be con- 
stant. The following table is given by Ostwald: 

Acetic acid. 








































V is the number of litres containing one gram -molecule ; 
//. the molecular conductivity (in mercury units), and //.^ its 
maximum value which is calculated as 364 from the velocities 
of the acetic acid ion and of hydrogen, determined by Kohlrausch 
from the conductivity of sodium acetate and mineral acids. 

The following are further examples of Ostwald's experi- 

Cyanacetic acid. 


































Formic acid #='0000214 
Acetic -0000180 

Monochloracetic acid '00155 

Dichloracetic -051 

Trichloracetic 1-21 

Propionic acid -0000134 

Butyric -0000149 

Isobutyric -0000144 

Iso valeric -0000161 



If we have once determined the constant K for any elec- 
trolyte, we can, by the help of the equation, calculate the 
conductivity for any dilution. Ostwald considers that this 
constant, K, gives the "long sought numerical value of the 
chemical affinity." 

If we choose states of dilution V L and F 2 for two different 
substances, such that the products V l K l and V 2 K 2 are equal, 

a 2 
then , and therefore a, must be the same for both. If we 

alter both dilutions in the same ratio, the products 
and V 2 K 2 are still equal, so that the dilutions at which two 
substances are dissociated to the same extent are always pro- 
portional, whatever the absolute values of the dilution. This 
was experimentally discovered by Ostwald before he had applied 
the theory of dissociation to electrolytes. 


As already stated, the dissociation of highly ionized electro- 
lytes does not conform to Ostwald's dilution law. The failure 
occurs not only in the case of acids and alkalies, when the con- 
ductivity curves are abnormal, but also in solutions of normal 
salts. Thus Ostwald gives the following numbers calculated 
from Kohlrausch's measurements for potassium chloride. 











Rudolphi 1 has given an empirical relation which seems to 
hold for such cases, though no physical meaning has been 
attached to it. The equation is 

2 v 4 v 

VF(l-a) = r (T^aFF = 

The values of the first constant for potassium chloride are : 











Van 't Hoff 2 shows that equally good results are obtained 
from the equation 

or T 

Thus for silver nitrate at 25 the comparable constants are : 



K! (Rudolphi) 

K' (Van 't Hoff) 

























1 Zeits. phys. Chem. xvn. 385 (1895). 

2 Zeits. phys. Chem. xvm. 300 (1895). 


Van 't Hoff's equation can be deduced by the kinetic method 
on the assumptions that the number of molecules dissociating is 
proportional to the square of the whole number of undissociated 
molecules, and that the number of ions recombining is propor- 
tional to the cube of the whole number of ions, the equation of 
equilibrium being 

Kohlrausch points out that Van 't Hoff's formula, if written 
in the form 

C 3 G 

,- = constant, or ~ = constant, 

G! Oj 3 

and divided on each side by C^, becomes 
C 2 _ constant 

c. = ~W~ 

C denotes the average nearness of the molecules, so that, 
if T be the average distance between them, we get the very 
simple relation 

C 2 

rx constant, 


the ratio between the ionic and the molecular concentrations 
being proportional to the average distance between the undis- 
sociated molecules. 

Turning from these empirical relations, let us consider once 
more Ostwald's original dilution equation, which extends the 
chemical law of mass action to the dissociation of electrolytes. 
In deducing the law by the application of thermodynamics, 
the restriction to dilute systems is necessary, in order that 
the reacting particles may be beyond each other's sphere of 
influence, and the change with dilution of available energy be 
thus independent of the nature of the solvent. Now, as we 
pointed out on p. 324, on the hypothesis that chemical forces 
are of electrical origin, the influence of a dissociated ion will 
extend far beyond the range at which the forces between the 
non-dissociated molecules cease to be sensible. A solution 
containing dissociated ions will therefore fail to show the 
properties of dilute matter at a much less concentration than 


will the solution of a non-electrolyte. It is not surprising, there- 
fore, that the dilution law does not hold at the concentrations 
at which it is tested. It is possible that it would only be 
applicable at dilutions so great that most solutions of strong 
electrolytes would be almost completely dissociated ; in fact, as 
already stated, it is possible that, as the concentration increases, 
the electric forces between the dissociated ions would become 
sensible sooner than any combination could occur. In the case 
of weakly dissociated bodies, like acetic acid, the number of 
ions at moderate concentrations is enormously smaller than for 
strong acids and salts ; it is possible, also, that the presence of 
a large quantity of undissociated solute affects the properties 
of the medium and diminishes the electric forces between the 
separated ions. Such solutions, therefore, show the phenomena 
of dilute systems at comparatively high concentrations, and 
conform to the dilution law. 

The application of the mass law as hitherto considered is 
concerned only with substances which dissociate into two ions. 
For salts or acids, which, like barium chloride or sulphuric acid, 
may be expected to give three ions, equation (62) on p. 340 

and we get 

for the dilution law. If a. is small we may write 

a = \/~FZ ........................ (67). 

In the case of weak polybasic acids, succinic for example, the 
ionization at high concentrations conforms to the law for mono- 
basic acids, and varies with the square root of the dilution in 
accord with equation (66) on p. 341. This behaviour indicates 
that the ions are H' and HA', where A denotes the acid group, 
and the dashes the valency of the ion. When about half the 
molecules are dissociated, some begin to produce three ions, 
and, at greater dilutions, the dissociation becomes normal in 
agreement with equation (67) above, indicating H', H', and 
A" as the ions. From strongly ionized bodies three ions are 


usually formed at more moderate dilutions, as shown by the 
conductivity and the depression of the freezing point, though, 
as we have seen, some of them may be linked with solute or 
solvent molecules to form complex ions. For these bodies, as 
for the corresponding binary compounds, the theoretical dilution 
law fails. 

When solutions of two electrolytes are mixed, there will, 

Equilibrium * n g enera ^ De a change in the amount of ion- 

between ization in both. For particular concentrations 

of bodies which conform to the dilution law, 

however, we can show that no such change occurs, and the 

solutions can be mixed without affecting the number or nature 

of the ions, or the mean conductivity. Any two solutions which 

fulfil these conditions were called by Arrhenius isohydric. Let 

us, as the simplest case, consider two simple electrolytes which 

possess one ion in common, such as two acids HA l and HA 2 . 

Let the coefficients of ionization be a-^ and 2 , and the dilutions 

V 1 and F 2 respectively. Then, for the two solutions 

1 x-rr = KI and 7^- 

If the solutions be isohydric, we can mix them without changing 
the ionizations ; the total volume becomes Fi + F 2 , and the 
number of hydrogen ions a x + a 2 . For the acid HA in the 
mixed solution, since the number of A ions remains unchanged 
at !, we have by equation (63) on page 340, 

(! + 2 ) ttj _ 

Dividing this equation by the first, we get 

_(ai + OFi _ ,' ai + <*2 _ F^F, 
(F.+ F.X" , F, 

Thus - ( or - ............... (68). 

Now I/F! and a. 2 /F 2 are the respective concentrations of the 
hydrogen ions in the two isohydric solutions HA l and HA 2 . It 
follows, then, that solutions of electrolytes containing a common 


ion are isohydric when the concentration of the common ion in 
the different solutions is the same. 

Two solutions with a common ion will so act on each other 
when mixed that they become isohydric, for then alone will the 
undissociated part of each be in equilibrium with the dissociated 
ion common to both. 

In deducing this result, the dilution law has been used ; the 
investigation therefore only applies in the case of electrolytes 
which conform to that law. Nevertheless, similar principles 
probably hold in other cases, and we may use our conclusion 
to qualitatively elucidate the interaction of solutions of any two 
acids. If the solution of a strong acid like hydrochloric be 
mixed with that of a weak acid like acetic, equilibrium can 
only occur when the two acids are isohydric and the concen- 
tration of the hydrogen ions the same for both. In order to 
secure this condition, it is necessary that a large amount of 
the feebly dissociated acetic acid, and a small amount of the 
highly dissociated hydrochloric, should exist. Now when 
dilute hydrochloric acid is mixed with dilute sodium acetate, 
acetic acid is formed, and this process continues till the two 
acids are isohydric, and the dissociated hydrogen ions in 
equilibrium with both. A large quantity of undissociated 
acetic acid must therefore be formed, and consequently most 
of the acetate be decomposed. This replacement of a weak 
acid by a strong one is a matter of common observation in the 
chemical laboratory. As indicated on p. 336 however, it must 
be noticed that the relative strengths of two acids can only be 
determined when both remain within the sphere of action ; if 
one of them is removed by precipitation or evaporation, it will 
be completely replaced, irrespective of the relative strengths. 

The theory of isohydric solutions can also be applied to 
investigate the effect on the solubility of one salt of adding to 
its solution a quantity of another salt containing an ion common 
to both. Nernst has pointed out 1 that in all likelihood the 
equilibrium between a solid salt and its solution is primarily 
an equilibrium between the crystals and the undissociated 
dissolved molecules, which on the other hand, are themselves 
1 Zeits. phys. Chem. iv. 372 (1888). 


in equilibrium with the dissociated ions. Under constant 
external conditions, therefore, we may conclude that the 
amount of the undissociated solute present in the liquid is 
not changed by adding more of either of its ions. In order 
to simplify the theory as much as possible, let us consider the 
case of a sparingly soluble salt, silver bromate for example 1 . 
The concentration of the silver ions can be increased by adding 
a soluble silver salt, and that of the bromate ions by adding 
a soluble bromate. Let us add a quantity of silver nitrate. 
The quantity of undissociated silver bromate is unchanged, and 
must still be in equilibrium with the silver and bromate ions. 
According to the mass law, the product of the concentrations 
of these two ions must be equal to the concentration of the 
undissociated salt multiplied by a constant ; in this case the 
product must itself be constant. By increasing the number 
of the silver ions, then, the concentration of the bromate ions 
must be diminished in the same ratio. A bromate ion can only 
be precipitated in company with some positive ion ; thus silver 
bromate, formed by the combination of its ions, is deposited to 
restore equilibrium. The effect of adding the silver nitrate 
therefore is to reduce the solubility of the silver bromate 
proportionally to the quantity of nitrate added. The effect 
of a soluble bromate is exactly similar, and the solubility of the 
silver bromate is lowered to the same extent as by an equivalent 
quantity of silver nitrate. 

This account of the subject has been quantitatively confirmed 1 . 
A solution of silver bromate, saturated at 24'5, contains 0*0081 
gram-equivalents per litre. Assuming that the salt is practically 
completely dissociated, the product of the concentrations of the 
two ions is 0'0081 2 , or 0*0000656. To such a solution, a quantity 
of silver nitrate was added sufficient to give a 0'0085 normal 
solution of the nitrate when dissolved in the volume of water 
which contained the bromate. Assuming complete dissociation 
for the silver nitrate also, let us calculate #, the total quantity 
of silver bromate which remains dissolved. The concentration 

1 The details of the calculations which follow are taken from Walker's 
Physical Chemistry (1899), p. 294. 


of the bromate ions is x, and that of the silver ions 0*0085 4- x. 

(0-0085 + as) x = 0*0000656, 

whence x = 0'0049. 

An experimental measurement showed that the solubility was 
actually reduced from 0*0081 to 0*0051 normal. If the cal- 
culation be corrected for the changes in the ionization of the 
salts indicated by conductivity determinations, the theoretical 
number becomes 0*00506, even nearer to the observed result. 

Two sparingly soluble salts, shaken up with the same 
quantity of water, each reduce the solubility of the other. 
The saturated solutions of thallium chloride and thallium 
thiocyanate have concentrations of 0*0161 and 0'0149 normal 
respectively. If x and y denote the solubility of the two salts 
respectively each in presence of the other, the concentration of 
the Cl ions is x and that of the SON ions is y, while the sum 
x + y gives the concentration of the thallium ions. We thus 


whence we calculate that x is 0*0118, and y is 0*0101 ; direct 
experiment gives 0*0119 and 0*0107 for the same quantities. 
These results justify the use of the principles here involved in 
such cases as the electromotive force of concentration cells, etc., 
examples of which we have already considered (pp. 253, 254). 
Owing to the failure of the mass law for solutions of strong 
electrolytes, we should expect these results, which depend on 
the same principles, to be limited in their application. The 
concordance between theory and experiment in the cases given, 
indicates that, for very slightly soluble salts, the theory is 
justified. This result is of great interest, for it shows that 
at great dilution, when the ions are beyond each other's 
spheres of influence, the mass law holds for strong electrolytes. 
Attempts have been made to use this solubility method to 
determine the ionization of the salt added, but consistent 
results are obtained only when the salt precipitated is very 
slightly soluble 1 . 

1 Arrhenius, Zeits. phys. Chem. xi. 391 (1893). 


These principles are often used in the chemical laboratory 
to precipitate salts from solution in a state of purity. Thus 
sodium chloride can be separated from a strong solution by 
the addition of hydrochloric acid, a very soluble substance also 
containing chlorine ions. The product of the concentrations of 
the ions of sodium and chlorine exceeds the equilibrium value, 
and salt is precipitated. 

The problem of the equilibrium of two electrolytes which 
have no common ion is much more complicated. When, for 
example, the solutions of two salts M^A^ and M 2 A% are mixed, 

the final system will contain the ions M ls M 9 , A 1 and A z , 

together with undissociated molecules of M 1 A lt M 2 A 2 , M^A^ 
and MzA-i; the equilibrium to be investigated is that which 
holds between all these bodies under the conditions of the 
experiment. Of the four salts, any two which contain a common 
ion can be treated by the methods already adopted. Thus a 
solution of M 1 A 1 can be made isohydric with one of M^A^. 
But the same solution of M^A^ can be made isohydric with 
M^A^ with regard to the other ion, and the solution of M^A^ 
can be made isohydric with that of M 2 A 2 . The conditions 
which must hold between the volumes of the four isohydric 
solutions to secure that their equilibrium is not disturbed when 
they are mixed, can be investigated on the assumption that 
they all conform to Ostwald's dilution law. For one of them, 
say M l A l , we have with the usual notation, 

If all the solutions be now mixed without change of equili- 
brium, the number of the M ions will increase in the ratio of 
(Fj + F 2 )/Fj, where F 2 is the volume of the isohydric solution of 
M^A Z in which the concentration of the M l ions must be the 
same. The volume in which the M l ions are now contained is, 
however, V l + F 2 4- F 3 + F 4 , so that the concentration of the 
M- ion is 



Similarly the number of A l ions is increased in the ratio 
(Fj + F 8 )/F 1} and its concentration becomes 

V^F.+ F. + F.+ F,)' 

where F 3 and F 4 are the volumes of the isohydric solutions of 
and M 2 A 2 respectively. The new equilibrium of the salt 
.j will thus be given by the equation 

(F,+ F 2 ) a(F 1+ F 3 ) 

F.CF.+ Fs+Fs+F,)- F,(F 1 +F 2 +F 8 +F 4 ) 
- ~ - = K - 

F.+ F.+ F.-f F< 

Hence by the dilution law 

Thus (F.+ r.)(F 1+ F.) _ 

F 1 (F 1 +F 2 +F 3 +F 1 )- 1 ' 
which reduces to 

F,F 4 =F 2 F 3 ..................... (69), 

an equation giving the relation which must hold between the 
volumes of the four isohydric solutions, in order that there 
should be no disturbance in equilibrium when mixture occurs. 
In words, we may say that the products of the volumes of such 
pairs of solutions as contain no common ion must be equal to 
each other. The solutions were all isohydric; that is, they 
had the same ionic concentration. The equilibrium condition, 
therefore, means that the total number of ions in each of the 
four solutions must be the same. In the chemical equilibrium 

M 1 A 1 + M Z A Z : M^ + M^, 

let us call the total masses of the four substances ra^ w 2 , ra 3 
and w 4 respectively, and their coefficients of ionization a lt c^, 
3 and 4 . For equilibrium on mixing, and therefore when 
equilibrium is reached in any case, our relation becomes 

This result is evidently an expression of the mass law, but 
it shows that the active mass of one of the substances we are 
considering is not its total mass, but only the fractional quantity 
which is dissociated into electrolytic ions. 


We have seen that the solubility of a salt may be reduced 
by the addition of an electrolyte containing one of the same 
ions. On the other hand, under certain conditions, the addition 
of an electrolyte may increase the solubility of a precipitate or 
other sparingly soluble body. For example, the small quantity 
of calcium tartrate which dissolves in water is highly dis- 
sociated. If hydrochloric acid be added, tartaric acid is 
produced, which, in presence of the ions of calcium chloride, 
etc., is only slightly dissociated. The concentration of the 
tartrate ions is therefore much reduced, the ionic product falls 
below the saturation value, and more calcium tartrate is dis- 
solved. Further applications of this theory of chemical 
equilibrium in electrolytes will be found in Arrhenius' paper, 
and in most books on Physical Chemistry. 

When two neutral salt solutions are mixed, there is, in 

general, neither evolution nor absorption of heat. 

properties of This experimental result has been formulated 

in what is known as Hess' law of thermo- 

neutrality. The dissociation theory indicates that, when all 

four possible salts are fully ionized, and consequently exist 

in solution in a dissociated condition, no appreciable change 

can occur on mixing, and no thermal phenomena can appear. 

If one of the reagents or one of the products of the action 

is only slightly dissociated, a separation of molecules or a 

combination of ions will take place when the solutions are 

mixed, and heat will be developed or absorbed. 

In the same way, the remarkable conclusion at which 
Thomsen arrived from his experiments on the heat of neutral- 
ization of acids and bases may be explained. He found that 
when a strong acid reacted with an equivalent quantity of a 
strong base in dilute solution, the heat evolved was always 
about 13,700 or 13,800 calories per gram-equivalent, whatever 
the acid and base used. The dissociation theory considers the 
reaction, for example, between hydrochloric acid and potash to 
be represented by the equation 


The ions K and Cl suffer no change, but the H of the acid 
and the OH of the alkali unite to form water, which, being 
present in a relatively enormous quantity, is only dissociated 
to an exceedingly small amount. An exactly similar process 
occurs when any strongly ionized acid acts on any strongly 
ionized base, and it is thus evident why, in such cases, the 
heat evolution should remain about constant. 

The law of thermo-neutrality, and the constancy of the heat 
of neutralization of strong acids and bases have been explained 
in ways which do not involve the dissociation hypothesis. 
Crompton 1 , for example, points out that the experiments of 
Thomsen prove that, when hydrogen is replaced in a mono- 
molecular organic compound by any radicle, the heat evolved 
is independent of the group with which the hydrogen was 
originally united. Extending this result to inorganic bodies, 
it might be that the heat of replacement of hydrogen in the 
acid is equal and of opposite sign to that of the replacement of 
hydroxyl in the base. The total heat of neutralization would 
then be zero, except for the thermal change accompanying the 
alteration in state of the elements of water, which before the 
action exist as parts of diluted foreign molecules, and after 
the action are added to the liquid solvent as part of itself. 
The heat evolution, then, would be a thermal value analogous 
to the heat of condensation of water from vapour to liquid, and 
the law of thermo-neutrality of reacting salts would hold good 
when no .water is formed. Such a theory of course expresses 
the isolated facts to explain which it was framed, but it 
cannot connect the thermal and electric properties of solutions. 

In reactions with the weaker acids and bases, the ionization 
of which is less complete, the heat evolved will diverge from 
the normal value, for the salts produced are usually more 
dissociated, and ions will be formed during the process. For 
instance, in the solutions used by Thomsen, sulphuric acid 
is only about half dissociated, and shows a heat of neutralization 
higher than the normal, so that heat must be evolved when it 
is resolved into its ions. 

1 Chem. Soc. Journ. Trans. LXXI. 951 (1897). 
w. s. 23 


Since the energy associated with a quantity of substance 
Heat of when ionized is different from that associated 

ionization. wUh it when in the normal chemical state, the 
heat of formation in aqueous solution of the molecule of an 
electrolyte from its ions will generally be different from that 
evolved when it is produced from its non-ionized elements. 

The heat of ionization of an electrolytic substance can be 
calculated by an application of the principles of thermodynamics. 
In the deduction of the mass law of chemical equilibrium on 
p. 339, it was shown that the change of available energy of a 
system per molecule of isothermal reaction could be expressed 
in the form 

S^ = w 1 a 1 + w a a,+ ...+RT(log b^b^... C^Cf* ...). 
Thus ty = S^ + RTlogK', 

where B^ is a standard of reference, and K' is b^ 6 2 ?i2 . . . K. 
As the condition of equilibrium at each temperature, SA/T must 
vanish, so that ty JZZMogZ'. 

By partial differentiation, for unaltered materials, 



a result which is independent of the unknown term 
In the free energy equation (p. 29) 

^ and e refer to changes in the free and internal energy 
respectively, so that T/T is equivalent to 8^/r above. Now the 
heat X absorbed by the system is 

X- -^--7^- -T 2 ( 
* 2 dT~ l dT( 

Hence from (70) we get 

\-Rpl\ogK ..................... (71) 

as the heat absorbed by the system per molecule of isothermal 
reaction. This expression, due to Van 't Hoff, is a more general 
form of equation (17) on p. 115, which gives the heat of 


solution of a substance in terms of the temperature coefficient 
of solubility. Equation (17) can of course be obtained from (71). 
If we make the assumption that the heat of reaction is 
independent of the temperature, which will restrict our result 
to somewhat small temperature ranges, we may integrate this 
new equation and obtain 


*? - * ( I _ i 

ZK^R^T! T Z 

These results apply to chemical changes in any dilute system, 
and may therefore be used to calculate the heat of ionization 
per gram-molecule when the constants K l and K a are known 
for two neighbouring temperatures. 

The coefficient of dissociation of aqueous solutions is generally 
found to decrease as the temperature rises ; and by experiment- 
ally determining its value for different temperatures and 
calculating the rate of variation, Arrhenius 1 has measured the 
heats of formation of various molecules from their ions by 
means of this equation. It is important to observe that his 
results only apply to solutions in water, and that, for the 
strongly dissociated bodies, for which Ostwald's dilution law 
fails, it is not to be expected that this theory, depending on 
similar principles, should lead to true results for the heats of 
ionization. The numbers for strongly dissociated bodies in the 
following table are calculated from observations on decinormal 


X at 21 -5 

X at 35 

Acetic acid CH 3 COOH 
Propionic acid C 2 H 6 COOH 

+ 28 
- 183 

- 386 
- 557 

Butyric acid C 3 H 7 COOH 
Phosphoric acid H 3 P0 4 
Hydrofluoric acid HF 

- 427 

- 935 


Hydrochloric acid HC1 


Nitric acid HN0 3 


Soda NaOH 


Potassium chloride KC1 

- 362 

Barium chloride BaCl 2 
Sodium butyrate C 3 H 7 COONa 

- 307 

+ 547 

1 Zeits.phys. Chem. iv. 96 (1889); ix. 339 (1892). 





From this table, by adding to the heat of formation of water 
from its ions that evolved by the completion of the dissociation 
of the acid, Arrhenius has calculated the total heats of neutral- 
ization of soda by different acids. 




Hydrochloric acid HC1 



Hydrobromic HBr 



Nitric HN0 3 
Acetic CH 3 COOH 



Phosphoric H 3 P0 4 
Hydrofluoric HF 

14959 14830 
16320 16270 

In the case of strongly dissociated substances, the number 
of molecules undissociated is so small that the variation from the 
normal value of the heat of neutralization is too slight to test 
the equation by experiment. For the weak acids, phosphoric, 
acetic, and hydrofluoric, which conform to the dilution law, the 
concordance is seen by the table to be satisfactory. 

From equation (71) on p. 354, it follows that, if the heat 
of formation is negative, that is, the heat of dissociation positive, 
the value of d (log K)/dT is also negative, and the dissociation 
must become less with increasing temperature. The con- 
ductivity is dependent on two factors, (1) the dissociation, 
and (2) the frictional resistance offered by the solution to 
the passage of the ions through it. If we call the reciprocal 
of this resistance the ionic fluidity of the solution, the mole- 
cular conductivity will be proportional to the dissociation and 
to the ionic fluidity. At great dilution the dissociation is 
complete, and the ions are so far apart that no change in 
temperature can affect the state of dissociation. Any alter- 
ation in conductivity with change of temperature must then be 
due to an alteration in fluidity, and the temperature coefficient 
of fluidity can be determined by measuring the temperature 
coefficient of conductivity at a dilution so great that the 
molecular conductivity has reached its limiting value. Now 
the table on p. 355 shows that the heats of formation from 
their ions of the substances examined have a greater negative 


value at the higher temperature. From equation (71) it follows 
that the rate of decrease of dissociation with increase of tem- 
perature must therefore increase as the temperature rises. If 
the temperature coefficient of fluidity either decreases with 
rise of temperature, keeps constant, or increases more slowly 
than the negative coefficient of dissociation, it is clear that 
a maximum conductivity must be reached at a certain tem- 
perature, beyond which any further heating will decrease the 
dissociation more than it increases the fluidity, and so, on 
the whole, diminish the conductivity. 

Arrhenius calculated, by deductions from the equation, that 
solutions of the two slightly dissociated bodies, hypophosphoric 
and phosphoric acids, should have maximum values for the con- 
ductivity at 57 and 78 respectively. He then experimentally 
determined their conductivities at different temperatures, and 
actually found maxima at 55 and 75. Sack 1 , by measuring 
the conductivity of copper sulphate solutions in closed vessels, 
found a maximum at 96 for a 0'64 per cent, solution ; calcu- 
lation by Arrhenius' method gives 99 for a solution of this 

The heat of ionization hitherto considered is the heat 
evolved when the molecule of a salt or acid is dissociated into 
its ions in aqueous solution. The determination of the heat 
change associated with the formation of an equivalent weight 
of ions during the process of solution of a metal is a different 
problem, and Ostwald 2 has attacked it on the assumptions that 
the single potential difference at the interface between a 
metal and a solution is known, and that the Gibbs-Helmholtz 

is applicable not only to the whole cell, but also to each 
individual surface of contact within it. E is then the single 
potential difference, and X the total heat effect at the electrode, 
which measures the heat of ionization generated by the passage 
of the metal into the ionic state. From this point of view, 

1 Wied. Ann. XLIII. 212 (1891). 

2 Lehrbuch, p. 955. 




such thermo-chemical data as the heat of precipitation of 
copper from its solution by zinc, are always the sums or 
differences of two heats of ionization, and if one heat of 
ionization is known, others may be calculated from thermo- 
chemical values. The following are some of the heats of 
ionization given by Ostwald : 

Per gram- Per gram- 
atom equivalent 

Potassium +612 +612 
Zinc +331 +166 

Cadmium +165 + 83 
Thallium +8 +8 
Iron (ferrous) + 202 + 101 

Per gram- 

Iron (change from 

ferrous to ferric ions) 1 2 1 

Lead - 14 

Copper - 177 

Silver - 264 

Mercury - 207 

Per gram- 



- 89 

The thermal unit is 100 calories. The results depend on 
the presumed correct determination of single potential dif- 
ferences by the use of capillary electrometers and dropping 

The conductivity of carefully distilled water is very small, 
and it can therefore only be dissociated to a 

Sr water! 10 " ver y s % nt extent - Tne Desfc water which can 
be prepared by distillation in presence of air 
has a conductivity of about 07 x 10~ 6 measured in reciprocal 
ohms across a centimetre cube. Kohlrausch and Heydweiler 1 
have distilled water in a vacuum and collected it directly in 
a resistance cell, which had been ^kept for ten years full of 
distilled water in order to dissolve all the soluble constituents 
of the glass ; in this manner they obtained water with a 
conductivity of 0'015 x 10~ 6 at 0, and 0'043 x 10~ 6 at 18. 
From the experimental results alone it is impossible to tell 
whether the slight trace of conductivity which remains is due 
to residual impurities or to ionization of the water itself, but 
an examination of the question may be made from the thermo- 
dynamic standpoint. The constant heat of neutralization of 

1 Wied. Ann. LIII. 209 (1894). In the paper the conductivity is expressed in 
terms of that of mercury ; the numbers have here been reduced to reciprocal 
ohms across a centimetre cube. 


strong acids and alkalies is due on the dissociation theory 
to the combination of the ions of water, and therefore gives for 
the heat of ionization per gram-molecule a value of 13,700 
calories at 18. Van 't Hoff's equation (71) 

9 , ^ _ X 

log K - 

in combination with the dilution law for weak binary electro- 
lytes which shows, as on p. 341, that K is a?/V, leads to the 

id_ x 


from which the temperature coefficient of the dissociation can 
be calculated. The conductivity, in accordance with Kohlrausch's 
theory, depends on the product of the dissociation and the sum 
of the ionic mobilities, which varies with the ionic fluidity. The 
ionic mobilities of the hydrogen and hydroxyl were found by 
experiments on 0*001 normal solutions of potash, hydrochloric 
acid, and potassium chloride, to vary with temperature in 
accordance with the equation 

t being the temperature on the Centigrade scale. At this 
great dilution the ionization of the solutes may be taken as 
complete, so that the influence of temperature on conductivity 
is due to its effect on fluidity alone. The total temperature 
coefficient of conductivity of pure water calculated from 
these data is 0*0581. The experiments showed that, as con- 
tinual purification of the water lowered its conductivity from 
0-29 x 10~ 6 to 0*043 x 10~ 6 , the temperature coefficient of its 
conductivity increased from 0*027 to 0*0532. It was hence 
estimated that the temperature coefficient would rise to the 
thermodynamic value when the conductivity had sunk to 
0*0386 x 10~ 6 at 18. This result, then, was taken to be the 
conductivity of pure water. It will be seen that about ten per 
cent, of the conducting power of Kohlrausch's best water is due 
to impurities. From the conductivity of pure water, its dis- 
sociation can be calculated ; and Kohlrausch's values indicate 
that the number of gram-equivalents dissociated per litre is 


0-35 x 10- 7 at 0, 0-80 x 10~ 7 at 18, T09 x 10~ 7 at 26, and 
2*48 x 10~ 7 at 50. Thus a cubic metre of water at 18 contains 
about 1*4 milligrams of dissociated molecules, or O08 milligrams 
of hydrogen ions. It may be observed that the ionic mobilities 
assumed in this investigation are the maximum values. Now 
at extreme dilution the equivalent conductivity of acids and 
alkalies diminishes, and it is possible that this phenomenon 
may somewhat affect the result of the calculation. 

Another value for the dissociation of water has been 
obtained by examining its influence on chemical reaction 
velocities. Methyl acetate and water form methyl alcohol and 
acetic acid at a rate proportional to the number of hydroxyl 
ions present in the solution. Wijs 1 used this reaction to 
measure the dissociation of water; he prepared an aqueous 
solution of methyl acetate carefully freed from acid or other 
impurity, and titrated it at intervals with standard alkali to 
measure the amount of acetic acid produced. The acid, as it is 
formed, retards the action, so that it is necessary to estimate 
the rate of transformation at the beginning of the process. 
The concentration of the dissociated ions appeared to be about 
1'2 x 10~ 7 gram-equivalents per litre at 25. 

A third method of estimating the dissociation of pure water 
lias been used by Ostwald 2 . A plate of spongy platinum in 
contact with hydrogen and an electrolyte acts as a hydrogen 
electrode, and if two such electrodes are arranged, one in an 
acid and the other in an alkali, the system may be treated as a 
concentration cell with regard to the hydrogen ions. In a 
normal acid solution, owing to the incomplete ionization, the 
concentration of the hydrogen ions is about 0'8, so that the 
concentration of the same ions round the alkali electrode can 
be calculated from the logarithmic formula [(48) p. 248] 

riRT, P, 

E = - -logW 

qy *Pi 

with the notation there indicated. The electromotive force of 
the cell is complicated by the potential difference of contact 

1 Zeits. phys. Chem. xi. 492; xn. 514 (1893). 

2 Zeits. phys. Chem. xi. 521 (1893). 


between the two solutions, as was pointed out by Nernst 1 ; and 
at 18 the corrected value is given as 0'81 volt. Putting in this 
number, we may consider the effect of the liquid contact to be 
eliminated and assume the transport ratio r to be 0*5, and Van 't 
Hoff's factor i to be 2. The gram ionic charge q is 9644 C.G.S. 
units, and y the valency of. the ions is 1. Transforming to 
common logarithms, the equation then gives 

0-81=0-0575 Iog 10 , 

or = 10 14 . 

* i 

Since P 2 is 0'8, P 1 the concentration of the hydrogen ions in 
the alkali solution is 0'8 x 10~ 14 . By the mass law we know 
that the product of the two ionic concentrations divided by the 
concentration of the undissociated water should be a constant. 
The water is present in large excess and its quantity may be 
taken as unalterable, so that the ionic product itself is constant, 
and will have the same value in pure water as in the solution 
of alkali, namely 0'64 x 10~ 14 . In pure water the concentrations 
of the hydrogen and hydroxyl ions must be equal, and the 
dissociated fraction is therefore 0'8 x 10~ 7 gram-equivalents per 
litre. This exact agreement with Kohlrausch's result may not 
be justified by the approximate nature of the calculations, but 
it shows that values of the same order are obtained in the two 

The differences which exist between the conductivities of 

the same substance when dissolved in different 

Jf h the U soivent. solvents show that the power of conducting a 

current depends on the nature of the solvent as 

well as on that of the solute. The conductivity depends on two 

factors, the ionization and the ionic fluidity of the liquid, and, 

to secure ready conduction, both these properties must have 

high values. 

A suggestion made independently by J. J. Thomson 2 and 
Nernst 3 may possibly explain the property possessed by certain 

1 Zeits. phys. Chem. xiv. 155 (1894). 

2 Phil. Mag. [5] xxxvi. 320 (1893). 

3 Zeits. phys. Chem. xi. 220 (1893). 


solvents of ionizing substances dissolved in them. If the forces 

holding the ions together in a molecule are electrical in their 

nature, they will be much weakened by immersing the molecule 

in a medium of high specific inductive capacity. 

The effect can be illustrated by considering the 

influence of a mass of conducting material placed 

near two little particles charged with opposite 

kinds of electricity. The result of the presence 

of the conductor can be represented by imagining 

that electrical images of opposite sign are formed 

near the charges just inside the conductor. The 

external forces due to the charged particles are 

reduced, and thus their attraction for each other 

may be so much diminished that separation may 

occur. The effect of an insulator of high dielectric constant 

is similar in kind, though rather less in magnitude ; and, other 

things being equal, the relative ionization powers of solvents 

should be proportional to their specific inductive capacities. 

Some results, which, as far as they go, support this con- 
clusion for solutions in water, methyl alcohol, and ethyl alcohol, 
have been given by the present writer 1 . The specific inductive 
capacities of the three solvents are, according to Tereschin : 
water, 83'7 : methyl alcohol, 32'65 : ethyl alcohol, 25'8. If we 
suppose provisionally that the resistances offered by these 
solvents to the motion of the ions are in about the same ratios 
as their viscosities, we must divide these numbers by 100, 63 
and 120, respectively. We then get for the theoretical ratio of 
the conductivities, 

Water 100 Methyl Alcohol 63 Ethyl Alcohol 26. 
An investigation by Vollmer showed that, for many salts, the 
ratio of the conductivities in the three solvents was 

Water 100 Methyl Alcohol 73 Ethyl Alcohol 34. 
It seems probable, then, that the specific inductive capacity 
and the viscosity are important factors in determining the 
relative ionization powers of solvents. More recently an at- 
tempt was made to ionize water by dissolving it in different 

1 Phil. Mag. xxxvin. 392 (1894). 


solvents. The object of the work was not attained, but it was 
shown by Novak and by the writer that, for mixtures of water 
with excess of formic acid, of which the dielectric constant is 
about 62, the conductivity curve is more like that of an 
electrolyte in water than it is when substances of lower 
dielectric constant, such as acetic acid, are used as solvent 1 . 
Slight dissociation of water dissolved in methyl alcohol has 
been found by Carrara, who shows that extremely dilute solu- 
tions conform to the mass law 2 . 

On the other hand there seem to be many exceptions to 
this rule of concordance between ionizing power and dielectric 
constant. Liquefied ammonia and sulphur dioxide dissolve salts 
to form solutions which conduct well, but both solvents have 
low specific inductive capacities. Other exceptions have been 
given by Kahienberg and Schlundt 3 . In view of the well- 
known fact that many aqueous solutions, such as those of 
ammonia and acetic acid, are only ionized to a very small 
extent, it is evident that no such rule as that under discussion 
can be universally true. Influences due to the specific nature 
both of solvent and solute must prevent any complete generali- 
zation. The fundamental idea of the Thomson-Nernst theory 
is, however, a valuable advance towards the explanation of the 
ionizing power of solvents. 

It is worthy of remark that, as well as reducing the forces 
between ions, the conducting body in Figure 65 will attract 
each ion to itself. The same thing would occur in a solvent 
of high specific inductive capacity. When the forces between 
two ions have been loosened, a slight collision with other mole- 
cules, or with molecules of the solvent, may suffice to cause 
dissociation ; the liberated ions may be annexed by the solvent, 
and loose compounds formed. The ions, being dissociated from 
each other and readily passed on from one particle of the 
solvent to the next, would then be able to work their way 
through the liquid under the action of the external electric 

1 Phil. Mag. [5] XLIV. 1 and 9 (1897). 

2 Gazz. Chim. Ital. xxvn. 1. 422 (1897). 

3 Journ. Phys. Chem. v. 382 and 503 (1901). 


Briihl 1 has pointed out that since oxygen can act as a 
quadrivalent as well as a divalent element, water and other 
substances containing it must be looked on as unsaturated 
compounds. Hence arise their high dielectric constants, great 
powers of ionization and readiness of combination. The ions 
in such substances may be supposed to be loosely and distantly 
connected, so that the electric moment of a molecule is great. 
Such a molecule when in solution will come under a powerful 
influence from any ion it encounters, and is therefore easily 
dissociated. Alcohols, ketones, ethereal salts, and acids also 
contain oxygen, and their dissociating power decreases as their 
molecular weight rises and their content of oxygen diminishes. 
The valency of nitrogen, like that of oxygen, can vary, and 
nitrogen compounds, nitriles, etc., give conducting solutions. 
Hydrocarbons and other saturated substances have little or no 
dissociating power. Dutoit and Aston 2 have further remarked 
that the liquid solvents considered above, when examined by 
the capillary methods of Ramsay and Shields and otherwise, 
are found to consist of polymerized molecules. It is probable 
that all these properties are connected, though again excep- 
tions to any law of exact correlation have been indicated by 

Solutions of salts which are strong electrolytes give a 

neutral reaction; but a salt such as chloride 

MssoclaSon. or nitrate of copper or zinc, containing a 

strong acid and a weak base, is found to give 

an acid solution, or if the base is strong and the acid weak, 

as in sodium carbonate or potassium cyanide, the reaction of 

the solution is alkaline. These results are explicable if we 

remember that water is to a slight extent dissociated, and may 

thus act either as a weak acid in virtue of its hydrogen ions 

or as a weak base because it contains hydroxyl. When a salt 

MA is dissolved in water, the reversible decomposition known 

as hydrolysis 


1 Ber. xxvni. 2866 (1895) ; Zeits. phys. Chem. xvm. 514 (1895) ; xxvu. 319 
(1898). 2 Compt. rend. cxxv. 240 (1897). 


may produce a non-electrolytic dissociation of the salt. This 
process is always possible, and must therefore occur to some degree 
in every case. The condition of equilibrium, equation (69) on 
p. 351, shows that the product of the ionic concentrations must 
be the same on each side of the equation. The ionic concentra- 
tion of water is excessively small, so that when both the acid 
and base are strongly dissociated, they must be present in very 
small quantities, and there is practically no hydrolysis. On the 
other hand, if the salt contains a weak acid or base, having 
an ionic concentration comparable with that of water, the 
conditions of equilibrium will require an appreciable amount 
of acid or alkali, and a considerable fraction of the salt will be 
found to be hydrolytically dissociated. If the acid is strong 
and the base weak, there will now be an excess of hydrogen 
ions, and the solution will have an acid reaction, while if 
the base is strong and the acid weak, that reaction will be 

In determining experimentally the amount of hydrolysis in 
any given case, it is impossible to estimate the acid or base 
produced by the usual chemical methods, for, by them, the 
equilibrium would be disturbed, and progressive hydrolysis 
would eventually decompose all the original salt. Measurements 
of optical or other physical properties of the solutions can, 
however, be employed, and the accelerating influence on certain 
reactions of the free hydrogen or hydroxyl ions has also been 
used to investigate the subject. The velocity constant for the 
catalysis of methyl acetate, or the inversion of cane sugar 
(p. 337), is approximately proportional to the number of free 
hydrogen ions present in the solution, while the hydroxyl can 
be estimated by observing the initial rate of saponification of 
ethyl acetate. The numbers in the first table which follows are 
given by Walker 1 as the percentage hydrolysis of the hydrochlo- 
rides of weak bases, at a temperature of 25, and a dilution of 
32 litres per gram-molecule. 

Aniline 2*6 

Parat oluidine 1 5 

Orthotoluidine 3-1 
Urea 76 

1 Physical Chemistry, p. 281. 


The next table is due to Shields 1 , and expresses the per- 
centage hydrolysis of salts of weak acids and strong bases at 
24, and at the given dilutions per gram-molecule. 

Dilution in litres Hydrolysis 

Potassium cyanide 1 0'31 

4 0-72 

10 1-12 

40 2-34 

Sodium carbonate 5 2'12 

10 3-17 

20 4-87 

40 7-10 

Potassium phenate 10 3'05 

50 6-65 

Borax 32 0'92 

Sodium acetate 10 0-008 

The amount of hydrolytic dissociation being small in all 
these cases, the mass law simplifies to a proportionality between 
the percentage ionic concentration and the square root of the 
dilution (see p. 341), and this result is borne out by the values 
for the more dilute solutions given above. In all these cases 
the hydrolysis is slight; but Shields found that trisodium 
phosphate was about 98 per cent, dissociated into free caustic 
soda and phosphoric acid at a dilution of 50 litres, and Walker 
states that salts of the very weak base diphenylamine are 
almost completely hydrolysed by water. 

Another case of considerable hydrolytic dissociation is found 
in salts of the weak base ferric oxide ; in fact, ferric hydrate can 
be obtained in a soluble form by placing ferric chloride in 
a vessel separated from a large volume of water by a sheet 
of parchment paper. After some days, owing. to progressive 
hydrolysis, nearly all the hydrochloric acid will be found to have 
passed into the water, leaving the iron behind as a brown 
solution of ferric hydrate. 

Since the equivalent conductivity of hydrochloric acid is 
much greater than that of a normal salt, it is possible to roughly 
estimate the amount of hydrolysis in a solution of ferric chloride 

1 Phil. Mag. (5) xxxv. 365 (1893). 


from the conductivity data. Taking the figures for the acid 
and for the ferric chloride given in the Appendix, and assuming 
that the equivalent conductivity of a normal salt is about 100 
in reciprocal ohms, it is easy to calculate that a solution of ferric 
chloride at a dilution of 1000 litres is hydrolysed to about 56 
per cent. This result neglects the influence of the residual 
ferric chloride on the dissociation of the acid and is therefore 
probably too low. Ferric acetate, which has both a weak acid 
and a weak base, seems to be more completely hydrolysed, the 
conductivity being of the same order as that of pure acetic 
acid 1 . 

The theory of electrolysis described in this chapter has 
Conciusi n proved one of the most stimulating hypotheses 

in the recent history of physical science. At 
the outset it met with much opposition, chiefly from chemists 
who held that its fundamental demands were inconsistent with 
well-established chemical conclusions. At present, criticism 
comes mainly from another side, and seeks to show that the 
relations which the theory suggests between the electrical, 
osmotic, and chemical properties of solutions, aqueous and other, 
fail when examined experimentally. The reasons for such 
failure have been pointed out in this chapter ; the theory only 
indicates the relations in question under certain simple con- 
ditions, which can seldom be secured in practice. As experi- 
mental arrangements approximate to ideal conditions, the 
correspondence between theory and observation increases, and 
the variations in other cases are explicable by causes suggested 
in the development of the theory itself. We must again em- 
phasize the complete mutual independence of the theory of ionic 
dissociation and any particular view of the nature of solution 
or the mode of action of osmotic pressure. It is quite poss- 
ible that solution is a process of chemical combination ; the 
dissociation required by the electrolytic theory is a separation 
of the opposite ions from each other, and would not in the 
least prevent a connexion of those ions with molecules of the 
solvent. Some form of dissociation theory seems to be clearly 
1 Whetham, Phil. Trans. A. CLXXXVI. 516 (1895). 


indicated by the electrical properties of solutions, and, until 
these properties are otherwise explained, the theory as at 
present formulated will be a guide in further investigation. 
The extended study of more concentrated solutions will throw 
light on the nature of the interactions between the different 
solute molecules, and between the solute and the solvent ; the 
effects of these interactions on many of the properties of any 
given solution are eliminated by working at such extreme 
dilution that the dissolved substance conforms to the laws 
of dilute matter. The complete theory of electrolysis needs 
further experimental data upon which to build, but the 
fundamental conception of ionic dissociation seems to secure 
a foundation for further development. 


Theory of diffusion. Experiments on diffusion. Diffusion and osmotic 
pressure. Diffusion of electrolytes. Potential differences between 
electrolytes. Liquid cells. Complete theory of ionic migration. 
Electrolytic solution pressure. Diffusion through membranes. 

IT is well known that a solution, left to itself, gradually 

becomes of equal concentration throughout. 

diffusion / This process implies an automatic drift of the 

dissolved substance through the liquid, and 

has received the name of diffusion. 

The diffusion of matter is analogous to the conduction of 
heat, and Fick 1 applied Fourier's treatment of the latter 
phenomenon to the elucidation of the former. The quantity 
of substance which diffuses through unit area in one second 
may be taken as proportional to the difference in concentration 
between the fluids at that area and at another parallel area 
indefinitely near it. This difference in concentration is propor- 
tional to the rate of variation of the concentration c with the 
distance cc, so that the number of gram-molecules of solute 
which, in a time &t, cross an area A of a long cylinder of 
constant cross section is 

St .................. (73), 

where D is called the diffusion constant or the diffusivity. 

1 Pogg. Ann. xciv. 59 (1855). 
W. S. 24 


At another area at a position x + Sac near the first, the 


concentration will be c ,- &e and the transfer across it is 



Hence in unit time the element of volume comprised between 
the two areas will on the whole gain in contents by 

But the volume of this element is Abx, and the rate of increase 
of concentration is dc/dt. We thus obtain the equation of 

n A d z c . A dc . 
DA ~j- n ox = A -=- Sx, 
dx 2 - dt 

^ d~c dc 

D ^=di ........................ (74) - 

This differential equation represents the general nature of 
diffusion. It can be integrated for definite cases, when the 
process is simplified by the geometrical and other conditions 
of the system. 

A systematic investigation of diffusion without any separat- 

ing membrane was first made by Graham 1 , who 

fn^iff^ion 8 immersed in a large volume of water a wide- 

mouthed bottle containing a solution, and after 

some time measured the quantity of substance in the water. 

By this method he found that acids diffused about twice as 

quickly as neutral salts, and that the rate of diffusion of 

these salts varied much according to their composition. Two 

dissolved substances diffused independently of each other, so 

that it was possible to separate the constituents of some double 

salts, the alums for example, which are decomposed by water. 

The quantity which diffused was found to be nearly propor- 

tional to the concentration of the original solution, and to 

depend largely on the temperature. Substances like tannin, 

1 Phil Trans. 1850, pp. 1, 805; 1851, p. 483. 


albumen and gums, diffused very much more slowly than the 
other bodies examined, and only at about one-fiftieth the rate 
of hydrochloric acid. These less diffusible bodies are non- 
crvstalline, and Graham called them colloids in distinction to 
the more diffusible crystalloids. 

Weber 1 was the first to work out a satisfactory method of 
determining the absolute value of the diffusion constant in 
Fick's equation. When two plates of amalgamated zinc are 
placed in two solutions of zinc sulphate of different concen- 
trations, the solutions being in contact with each other, a 
difference of electrical potential is produced between the plates 
which is proportional to the difference in concentration, pro- 
vided that difference is small. A concentrated solution of 
zinc sulphate was placed in the lower part of a cylindrical 
vessel, the bottom of which was made of an amalgamated zinc 
plate, and a dilute solution gently poured in on the top of the 
first. The electromotive force between the lower zinc plate 
and a similar plate placed in the topmost layer of liquid was 
measured, and found to decrease as the difference in concen- 
tration became less. If we apply Fick's law to this case we 
get an infinite series in the expression for the electromotive 
force, but when the time is long, the first term only is impor- 
tant, and we get, if H is the height of the vessel, and t the 
time, ^2 

E = be-JT* D * ...(75). 

The following table gives the observed values of TFiA 
which should be constant if Fick's law holds good. 

Days ^D 

45 -2032 

56 -2066 

67 -2045 

78 -2027 

89 -2027 

910 -2049 

1011 -2049 

Mean *2042 

1 Wied. Ann. vn. 469 and 536 (1879). 





Stefan 1 showed that in the case of a very long cylinder, in 
which the concentration at one end remains constant, the 
quantity diffusing through an area A should be, according to 
Fick's law, 

To apply this to a finite cylinder we must imagine that the 
amount which would have passed beyond the limiting layer is 
reflected, and, travelling backwards in accordance with the 
same laws, is added to the quantity present in the lower layers. 
Experimentally realizing these conditions, Scheffer 2 placed a 
solution underneath a volume of pure water and measured the 
quantity of substance which diffused upwards. The following 
are some of his results, n being the number of molecules of 
water in which one molecule of substance is dissolved. 





Hydrochloric acid 








Nitric acid 




) 5) 




Sulphuric acid 




Acetic acid 




















In general the diffusion constant was found to be independent 
of the dilution, but, in the case of hydrochloric acid, it appeared 
to increase somewhat with the concentration. 

Graham* and Voigtlander 4 found that the rate of diffusion 
in solid agar-agar jelly solutions was nearly the same as in 

1 Wien. ATcad. Ber. LXXIX. 161 (1879). 

2 Ber. xv. 788, xvi. 1903 (18823), and Zeits. phys. Chem. 390 (1888). 

3 Phil. Trans. 1861, p. 183. 

4 Zeits. phys. Chemie, ILL 316 (1889). 

CH. XIIl] 



water, and, as these jellies obviate all disturbing effects due 
to shaking or convection currents, they have been extensively 
employed. For a 0'72 per cent, solution of sulphuric acid, 
diffusing into a cylinder of agar jelly, Voigtlander gives the 
following numbers; N represents the number of milligrams 
of sulphuric acid diffusing through a given area. The results 
confirm Stefan's formula. 

Time in 
















The distance to which a determinate concentration reaches 
is proportional to the square root of the time of diffusion. 
Thus the formula can be tested and the constants determined 
by tracing the decolorization of a dilute alkaline solution, 
coloured red by phenolphthallein, as the acid diffuses upward. 
The following table, due to Voigtlander, gives the value of the 
diffusivity at 0, 20 and 40, and, in the last two columns, the 
mean temperature coefficients from to 20 and 20 to 40. 




D 40 


a 2 

Formic acid 












Propionic acid 












Hydrochloric acid 
Nitric acid 
















Potassium chloride 






Sodium chloride 
















When the concentration of different parts of a solution is 
not uniform, the osmotic pressure must also 
o D smot S ic n pr a e n ssure. vai T- B J imagining the parts of the solution 
separated by ideal semi-permeable membranes, 
we see that the osmotic pressure is the force per unit area, or 
the partial pressure, which must be applied, by the diaphragm 
or otherwise, to the dissolved molecules in bulk in order to 
prevent their diffusion. By the principle of reaction, it follows 
that, in a solution of varying concentration, the force which 
causes diffusive translation of the molecules in a thin slice of 
the liquid is the reversed difference of osmotic pressures on the 
two faces of the slice 1 . The phenomena of diffusion have been 
investigated on these lines by W. Nernst 2 and M. Planck 3 . 
If we have a vertical cylinder with a solution of some non- 
electrolyte in its lower part, and pure water at the top, the 
dissolved substance gradually makes its way upwards through 
the water, and, neglecting the small disturbing effect of 
gravity, a uniform solution will finally result. At a height x 
in the cylinder let the osmotic pressure be P, so that if A be 
the area of cross section, the substance in the layer whose 
volume is ASx finds itself under the action of a force equal 
to ASP, the negative sign being taken because the force acts 
in the direction in which the pressure decreases. If c be the 
concentration in gram-molecules per cubic centimetre, the force 
which in this layer acts on each gram-molecule is 

^dP = _ldP 

cA dx c dx ' 

Let F denote the force required to drive one gram-molecule 
through the solution with the velocity of one centimetre per 
second. Since the velocity of drift is constant, F must also 
denote the resistance offered by the viscous medium. The 
velocity attained is 



1 Larmor, Aether and Matter, p. 293. 

2 Zeits. phys. Chem. n. 615 (1888); iv. 129 (1889). 

3 Wied. Ann. XL. 561 (1890). 


and if BN be the number of gram-molecules which cross each 
layer in a time &, since the number crossing unit area per 
second is proportional to the concentration and to the average 
velocity of the individual molecules, we get 

When the solution is dilute, and there is no polymerization or 
dissociation of molecules with change of concentration, we may 
apply the gas equation for the osmotic pressure, and write 
P = cRT, the value of the constant jR corresponding to one 
gram-molecule of any substance being taken as usual This 

SA^-^^fc .................. (76). 

By comparison with Fick's equation (73) 


D, the diffusion constant, is seen to correspond to the factor 

The slow rate of diffusion has led to the adoption of the day 
instead of the second as the unit of time for experimental work, 
so that the observed diffusivity D is given by the expression 

D dc . 


From equation (76) we see that the force required to 
drive one gram-molecule through the solution with a velocity 
of one centimetre per second is 

RT dc ., 
F^-^^A -j-ot 
8N dx 

86400 RT 

Thus if we know the diffusion constant, we can calculate 
the force required to produce unit velocity. Voigtlander 
gives 0*472 as the diffusivity of formic acid at C., and from 
this we can calculate that the force required to drive one 


gram-molecule (46 grams) of formic acid through water with 
a velocity of one centimetre per second is equal to the weight 
of 4340 million kilograms. The necessity for such an enormous 
force is at once realized if we remember the minute size of the 
molecules and the consequent great influence of the resistance 
of the medium. 

A solution of uniform temperature will in the end become 
homogeneous ; but if the upper layers be kept hotter, the con- 
centration in the lower layers must be greater, in order that 
the osmotic pressure should be the same throughout. This 
result was experimentally established by Soret 1 and explained 
as above by Van 't Hoff 2 . The experiments supply a method 
of determining the influence of temperature on osmotic pressure, 
and the results are in accordance with the gas law for dilute 

If the osmotic pressure-gradient were the only driving force, 
the different mobility of the two ions of an 
Sectrdiytes! electrolyte, such as hydrochloric acid, would 

cause separation between them. 

In a solution of hydrochloric acid at the bottom of a tall 
glass cylinder, with pure water lying above it, the hydrogen ions 
travel faster than the chlorine, and carry their positive charges 
with them, leaving the lower layers negatively charged. An 
electrostatic force thus arises, which opposes the process of 
separation, and keeps the number of opposite ions in each 
part of the system very nearly the same. Nevertheless some 
separation does occur, and this explains the fact that water, in 
contact with an aqueous solution of an electrolyte, takes, with 
regard to it, a positive or negative potential as the positive or 
negative ion travels the faster. 

When solutions of two different electrolytes are placed in 
contact, a similar state arises. Let us suppose that we have 
a solution of hydrochloric acid in contact with one of lithium 
bromide. On the one hand more hydrogen ions than chlorine 
ions will diffuse out of the acid solution, and therefore the 

1 Ann. Chim. Phys. xxn. 293 (1881). 

2 Zeits. phys. Chem. i. 487 (1887). 


salt solution will receive a positive charge. On the other hand, 
more bromine ions than lithium ions will diffuse from the 
salt solution into the acid, and thus the potential difference 
will be increased. 

Let us return to the consideration of the solution of a 
single electrolyte containing two monovalent ions, placed 
beneath pure water. From the velocities of the two ions 
under unit potential gradient, as found by Kohlrausch's theory, 
it is easy to deduce the velocity with which they will travel 
when unit force acts on them. Let us call these velocities 
V and V for the cation and ariion respectively. The actual 

... . , , UdP VdP 

velocities in our case will therefore be ---- 7- and -- -= , 

c dx c dx 

so that the amounts passing any cross section of the cylinder 
in a time &t are 

-UA.$t and --. 
dx dx 

When U is different from V, a difference of potential is set 
up; with the effect, on reaching a steady state of electric 
separation, of making the ions travel together. If the poten- 
tial gradient is dEjdx the force on a gram-equivalent of an 
ion carrying a charge q is qdEjdx, and numbers of the two ions 
which would cross, under the action of this force alone, are 

- UAcq~^t and + VAcq^St. 

Under the influence of both the osmotic and the electric 
forces the number of gram-equivalents which diffuse in a given 
time must be equal, so that we get 

dE \ -rrAtofdP dE \ 
+ cq^}^- K4&(-T--egf-T-j; 

* dx ) \dx * dx ) 

or eliminating dEjdx, 


U +V dx 

For dilute solutions we may assume that the gaseous laws hold 
good, so that 


c, the concentration, being the reciprocal of the volume in 
which one gram-molecule is dissolved. 

Therefore SN = 

We shall need the intermediate steps of this investigation 
when we consider Nernst's account of contact differences of 
potential ; this last equation merely states that the resistance 
offered by the liquid to the passage of an electrolyte is the sum 
of the resistances offered to the passage of its ions, and can be 
directly deduced on that assumption without further electric 
hypotheses. Thus the osmotic pressure of a binary electrolyte 
has double the normal value, so that the number of gram- 
molecules of hydrochloric acid diffusing across any section of 
the vessel in a time 8t is, by equation (76), 

s , r 2RT . dc .. 
B N = -- ^- A -r~ St. 
F dx 

The resistances to hydrogen and chlorine moving with unit 
velocity are I/ 7 and I/ V respectively, so that the resistance to 
hydrochloric acid is 

F I I U+V 
" U"*' V~ UV 

and we recover Nernst's equation 


oN = 


From the general theory of diffusion we have already 
deduced equation (73) 



By comparing this with Nernst's equation, we see that, for 
electrolytes, the diffusion constant is given by the expression 

T is the absolute temperature, R the gas constant corresponding 
to one gram-equivalent of substance, 1*980 calories per degree 
or 8*284 x 10 7 ergs per degree, so that it only remains to 
calculate U and V, the velocities with which the ions move 




under the action of unit force. The quantity of electricity 
associated with one gram-equivalent of any ion is + 9644 
electromagnetic units. If the potential gradient is one volt 
(10 8 c.G.s. units) per centimetre, the force acting on this gram- 
equivalent will be 9644 x 10 8 dynes. This, in dilute solution, 
gives the ion its specific velocity, say u. Thus the force P^r 
required to give the ion unit velocity is 9*644 x 10" /u dynes or 
9'83 x ~LQ 5 /u kilograms weight. If the ion have an equivalent 
weight W y the force P x producing unit velocity when acting on 
one gram is 9'83 x IQ*/Wu kilograms weight. Thus, in order 
to drive one gram of potassium ions with a velocity of one 
centimetre per second through a very dilute water solution, 
a force is required equal to the weight of 38,000,000 kilograms. 
The table gives other examples 1 . 

Kilograms weight 

Kilograms weight 



Pr . 



15 x 10 8 

38 x 10 6 


14 x 10 8 

40 x 10 6 










N0 3 



NH 4 









C 2 H 3 2 






C 3 H 6 2 



Since the ions move with uniform velocity, the frictional 
forces brought into play must be equal and opposite to the 
driving forces acting, and therefore these numbers also represent 
the ionic friction coefficients in very dilute solution at 18 C. 

Let us now return to the consideration of the velocity. We 
have seen that the force acting on one gram-equivalent of an 
ion, when the potential gradient is one volt per centimetre, is 
9644 x 10 8 dynes, and that, in dilute solution, this gives to the 
ion its specific velocity u. The velocity it would attain under 
unit force will therefore be 

U = TT x 10~ 8 cms. per second. 

1 Kohlrausch, Wied. Ann. L. 385 (1893). 




In the case of hydrochloric acid, for example, the specific 
mobility of the hydrogen is 0'0032, and that of the chlorine 
0-00069; thus 

U= 3-32 x 10~ 15 , and F= 715 x 10~ 16 
and, for the diffusion coefficient, we have 

the velocities, for convenience, being reckoned in centimetres 
per day. 

The agreement between theory and Scheffer's observations 
on diffusion is shown by the table. 


D observed 

D calculated 

Hydrochloric acid, HC1 



Nitric acid, HN0 3 



Potash, KOH 



Soda, NaOH 



Sodium chloride, NaCl 



Sodium nitrate. NaNO 3 



Sodium formate, NaCOOH 



Sodium acetate, NaC0 2 CH 3 
Ammonium chloride, NH 4 C1 



Potassium nitrate, KN0 3 



The theoretical numbers are slightly increased by the assumption 
that the ionization of the solutions is complete, which is not 
accurately the case. This correction, then, would improve the 
agreement. The possibility of thus correctly calculating the 
diffusion constant must be regarded as very strong evidence in 
favour of the methods of the investigation. 

Further developments for the cases of other solvents and of 
mixed electrolytes have been traced by Arrhenius 1 , who shows, 
for example, that the rate at which hydrochloric acid diffuses 
will be increased by the presence of one of its salts. This is 
confirmed experimentally; when 1'04 normal HC1 diffuses into 
01 NaCl, D is calculated as 2'43 and observed as 2'50, and 
when the NaCl solution is 0'67 normal, calculation gives 3'58 
and observation 3*51. 

1 Zeits.phys. Chem. x. 51 (1892). 


As we have seen above, when a solution is placed in contact 
with water, the water, which becomes a dilute 

Potential differ- . . . 

ences between solution, will take a positive or negative potential 
with regard to the stronger solution, in accord- 
ance with the greater specific mobility of the cation or the 
anion. Taking the equation which expresses the relation that, 
when a steady state is reached, the ions migrate at equal 
rates, viz. 

we get 

dE_ V-UdP 

dx~ cq V+U dx'' 
or, since for dilute solutions P = cRT, 

dE _RTV- UdP 
~dx ~ Pq V+U dx> 

which gives on integration 

RTV-U, P, 
ET^U 1 ^' 

where P^ and P 2 denote the osmotic pressures of the ions in the 
dilute and concentrated solutions respectively, and E denotes 
the difference of potential, i.e. the electromotive force between 
the two liquids. Now U and V, the ionic velocities under unit 
forces can, by multiplying by q, the quantity of electricity asso- 
ciated with one gram-equivalent of an electrolyte, be transformed 
into u and v, the velocities under unit potential gradient. We 
have already restricted the investigation to the case of dilute 
solutions, so that we can also replace the ratio between the two 
osmotic pressures by the corresponding ratio between the two 
concentrations. The equation now becomes 


q v + u d 

Thus the potential difference between two solutions with 
different concentrations of the same electrolyte, containing 
only univalent ions, is proportional to v u, the difference 
between the mobilities of the anion and the cation. 




If the valency of the cation be y l and that of the anion 
a similar investigation shows that 

q v + u 




Liquid cells. 

In order to compare these equations with observation, Nernst 1 
devised a form of concentration cell in which 
the electromotive force depends only on the 
two solutions. Such arrangements are sometimes known as 
liquid cells. We may take as an example, the following series : 
Hg/HgCl/0-1 normal 

KCl/0-01 KCl/0-01 HCl/0-1 HC1/01 KCl/HgCl/Hg. 
Two things are here to be observed ; the first, that the ends 
of the chain are identical, and the potential differences there 
neutralize each other; the second, that, in dilute solutions, 
it is only the ratio and not the absolute values of the osmotic 
pressures or the concentrations that are involved. Thus the 
effect at the junction O'Ol KCl/0'01 HC1 is equal and opposite 
to that at the junction O'l HCl/0'1 KC1, and the only effective 
junctions are those between 0*1 KCl/0'01 KC1 and between 
O'Ol HCl/0'1 HC1. From the ionic mobilities of potassium, 
chlorine and hydrogen, the difference of potential at each of 
these junctions can be calculated, and the sum of the two 
results compared with the experimental value of the electro- 
motive force of the arrangement. The following table gives 
the results of Nernst's comparison of the calculated and observed 
values for this and other similar liquid cells. 


E.M.F. calculated 

E.M.F. observed 

KC1, NaCl 



KC1, LiCl 



KC1, NH 4 C1 
NH 4 C1, NaCl 







KC1, HNO, 



KC1, C 9 H n S0 3 H 


- 0-0469 

1 Zeits. phys. Chem. iv. 129 (1889). 




The more general case of any two electrolytes in contact 
with each other has been considered by Planck 1 . The equations 
are somewhat complicated, but, when the total concentration 
of the ions in the two solutions is the same, and all the ions 
have the same valency y, the expressions reduce to the simple 

_ RT, Ui + v, 
E = - - log - . 
qy * 

Nernst has determined the electromotive force of cells which 
can be used to verify this equation ; the following are the results 
of the comparison. 


E (calculated) 

E (observed) 

HC1, KC1 



HC1, NaCl 



HC1, LiCl 



KC1, NaCl 



KC1, LiCl 



NaCl, LiCl 



Hittorf's account of the phenomena of ionic migration deals 
only with the initial changes of concentration 
on. which appear at the two electrodes on the 
passage of a current, before the diffusion that 
supervenes produces a sensible effect 2 . As long as the middle 
part of the solution retains its original concentration, Hittorf s 
investigation holds good, and this condition must be maintained 
in experimental measurements of transport numbers. 

When the current flows for a long time, the electrode 
regions of densities modified by the current extend and meet 
each other, and the results of backward diffusion become im- 
portant. The general problem of electrolytic conduction which 
then arises has been investigated by the use of Fourier's 

1 Wied. Ann. xxxix. 161; XL. 561 (1890); account in Ostwald's Lehrbuch, 
n. 848. 

2 See above, pp. 208212. 


diffusion analysis by Planck, Larmor 1 , and others, on the 
assumptions that the ions both migrate and diffuse indepen- 
dently of each other and that the ionization is complete. 
Ultimately a steady state will be reached ; with a constant 
current and a non-dissolvable anode, the concentration dimi- 
nishes uniformly with the time as the electrolysis proceeds, 
and its gradient has a definite value irrespective of the value 
of the concentration itself, changing uniformly from I/2RTqu 
at the anode to I/2RTqv at the cathode, where / denotes 
the current, R the usual gas constant per gram-molecule, T 
the absolute temperature, q the electric charge on one gram- 
equivalent of a monovalent ion, u the mobility of the cation, 
and v the mobility of the anion. The difference in the 
concentrations at the anode and cathode in the steady state 

is found to be 

II u v 

+ v 

which is equal to Hittorf's difference produced initially per 
unit time divided by Djl, D being the diffusion constant and 
I the length between the electrodes. As the ions diffuse at 
different speeds, whether electrolysis is going on or not, any 
changes of concentration at once give rise to internal electro- 
motive forces. Even when the steady state is reached, the 
gradient of electromotive force is of complex character. When 
the applied electromotive force is kept constant, and the 
current allowed to change, the quantities will vary exponentially 
with the time. 

A special case of Larmor's equations, in which the circuit is 
imagined to be broken, so that the current is zero, gives Nernst's 
expressions for the potential differences at the interface of two 
solutions of an electrolyte of different concentrations. Here 
the state of concentration is not steady, the only possible steady 
state being one of uniform density. 

Another application of the principles of the investigation 

enables the effect of a transverse magnetic field to be examined, 

and the coefficient of the resultant Hall effect to be calculated ; 

for, by the laws of electrodynamics, a transverse magnetic field 

1 Aether and Matter, p. 291. 


must produce a sideways force on the moving ions which con- 
stitute the current. Larmor shows 1 that a magnetic field H is 
equivalent to a transverse uniform electric force F which, if c 
denotes the concentration of the solution, has the value 

v + u 2cq ' 

An investigation of the more general case which arises when 
the electrolyte is only partially ionized, has been given by 
F. G. Donnan 2 . 

Nernst's hypothesis of a solution pressure of metals in 

contact with electrolytic solvents may also be 

Electrolytic approach ed from the point of view of ionic dif- 

sure. fusion. To each metal is ascribed a definite 

solution pressure, depending only on the nature 

of the solvent and the temperature ; this pressure tends to 

carry the metal into solution in the form of positively charged 

ions. The process will electrify the solution positively, and 

leave the metal with a negative charge. In this manner, 

according to Nernst, is set up the potential difference at the 

surface of the metal, the phenomena of which we have pre- 

viously studied. The electric forces will oppose the further 

solution of the metal, tending to drive back again the ions 

already in the liquid. The electrostatic charges on the ions 

are very great, and the potential difference of equilibrium may 

be reached long before a weighable quantity of metal has been 


On any view, the process of solution can only continue if 
negative ions can simultaneously dissolve, or other positive ions 
be removed from solution. The latter condition is illustrated 
by the replacement of hydrogen in acids, or the precipitation 
of one metal by another. When hydrogen is evolved, it is 
probable that it is first dissolved by the metal, from which it 
separates when its vapour pressure exceeds that of the atmo- 
sphere. The action can be stopped by a sufficient external 

1 Phil. Trans. CLXXXV. A. 815 (1894), or loc. cit. 

2 Phil. Mag. Nov. 1898. 

w. s. 25 


pressure, the value of which can be determined by thermo- 
dynamic considerations, and, on Nernst's ideas, depends on 
the solution pressure of the metal. Thus Beketoff 1 and 
Brunner 2 have shown that hydrogen at a high pressure can 
precipitate silver, platinum and palladium ; Cailletet arrested 
its evolution from zinc and sulphuric acid ; while Nernst 3 and 
Tammann 4 have examined the action of other metals. 

From Nernst's standpoint, this process of metallic solution 
is analogous to the diffusion of ions across the interface 
between a concentrated and a dilute solution of the same 
electrolyte. Such a metal as zinc is looked on as a solvent in 
which the concentration and the osmotic pressure of its own 
positive ions are very high. Some of these ions diffuse into a 
liquid in contact with the metal, till the characteristic ionic 
potential difference is set up. The electric forces then prevent 
further change, and, since the metal maintains the constancy 
of its ionic concentration, the osmotic pressures on the two 
sides of the interface are never equalized, unless the metal and 
solvent happen to show no difference of potential. 

We have seen that on Nernst's theory of electrolytic diffu- 
sion, the potential difference between two solutions of an 
electrolyte of different concentrations can be expressed as 

RTv-u, P 2 

E = - - log ~ . 

q v + u & P! 

If, ignoring the essential difference in the two cases, we extend 
this equation to the interface between a metal m and a solution, 
we are concerned with the cation only, for no anion is trans- 
ferred across the boundary. Thus v is zero and we get 

where P m now denotes the osmotic pressure of the cations in 
the substance of the metal itself, that is, its solution pressure. 
We thus regain the equation which was deduced in a former 

1 Compt. Rend. XLVIII. 422 (1889). 

2 Pogg. Ann. cxxn. 153 (1864). 

3 Compt. Rend. LXVIII. 395 (1869). 

4 Zeits. Phys. Chem. ix. 1 (1892). 


chapter (p. 257) from a consideration of the osmotic work 
equivalent to unit electric transfer. We have already con- 
sidered the limitations of this equation and pointed out that, 
following Helmholtz, the term involving P m may be treated as 
an affinity constant, characteristic of the metal and the solvent. 

In applying these considerations to common chemical gal- 
vanic cells, such as Daniell's, we neglected the electromotive 
force at the junction of the liquids ; as will now be clear, the 
theory of ionic diffusion enables us, in simple cases, to supply 
the term previously missing from the equation. 

In concentration cells the metal is the same at each elec- 
trode, so that P m can be eliminated ; it is therefore possible to 
develop the theory of such cells from the study of ionic diffusion. 
For a monovalent metal, such as silver, we have 

7?T / P 4i 11 P P 

, JLtJ. /, J. Af. V U , JT i , 

A . P Ag \ 
-jf - loo- -=p 
P 2 ^ P l J 

PT " ^u 

RT 2v P 2 

= "log TT- 

q u + v ' P 1 

Since v/(u + v) is for dilute solutions equal to r, the transport 
ratio for the anion, and n, the number of ions given by a 
molecule of the electrolyte, is here 2, this result is identical 
with equation (50) on p. 24$. By similar methods we can 
regain the equations already given in Chapter X. for other 
kinds of concentration cell. 

The exact significance of the physical constant named 'solu- 
tion pressure ' is uncertain. Following Nernst, Ostwald considers 
that, in a given solvent, it is a function of the metal and 
temperature only, and consequently that the single potential 
difference at the interface is independent of the nature of the 
negative ion. Measurements, in part described in Chapter XL, 
of the potential differences at single reversible junctions, 
when the cation is of the same metal as the electrode, 
have often been made from this point of view. We may here 
again refer to those of Le Blanc 1 and Neumann 2 . These 
observers measured the electromotive forces of cells made up 

1 Zeits. phys. Chem. xn. 345 (1893). 

2 Zeits. phys. Chem. xiv. 225 (1894). 



with the junction in question at one electrode, and mercury in 
the usual normal potassium chloride solution with an excess of 
calomel at the other. Assuming the potential difference be- 
tween the electrolytes to be small, Neumann found that at great 
dilution the electromotive force of the cell was in general 
independent of the anion ; but Paschen, Bancroft, and other 
observers, working with metals in solutions not of their own 
salts (arrangements which possibly form limiting cases of 
reversible electrodes and are subject to the same laws), have 
found that the potential difference does depend on the anion 
when the metal is copper, platinum, or mercury. Many ex- 
periments on cells containing non-reversible electrodes have 
been made to determine the influence of the nature of the ions 
and of concentration. Among these experiments we may 
mention those of Paschen 1 , Ostwald 2 , Oberbeck and Edler 3 , 
Bancroft 4 , and A. E. Taylor 5 . Taylor suggests that the differ- 
ences found by some of the observers on changing the anion 
may be due to large potential differences of non-osmotic type 
at the surface of contact of the two liquids in the cells, for he 
finds that such large differences often arise in cases where there 
is a tendency to form complex salts. 

Before Graham's experiments on free diffusion through 
water, many observations had been made on 

Diffusion J 

through mem- the passage of dissolved matter through various 
animal and vegetable membranes. Such mem- 
branes, made of bladder, parchment paper, and similar materials, 
are of the nature of colloids, and appear to be quite imperme- 
able to other colloidal substances. Solutions of colloids may be 
freed from crystalloids by placing the mixture in a vessel closed 
by a membrane, which, on its other side, is in contact with a 
large volume of the pure solvent. The crystalloids pass through, 
and, after a considerable time, are completely separated from 
the dissolved colloids. The process is known as dialysis. 

1 Wied. Ann. XLIII. 590 (1891). 

2 Zeits. phys. Ghent, i. 583 (1887). 

3 Wied. Ann. XLII. 209 (1891). 

4 Zeits. phys. Chem. xn. 289 (1893); Physical Review, in. 250 (1896). 

5 Jour. Phys. Chem. i. 1 and 81 (1896). 


The rate at which crystalloids pass through one of these 
membranes depends on the nature both of the diffusing sub- 
stance and of the membrane. Water will usually pass more 
freely than salts dissolved in it, and thus a temporary osmotic 
pressure can be obtained by using a membrane of bladder or 
parchment paper. The septum is not a perfect semi-permeable 
wall, however, and the limiting value of the osmotic pressure is 
never reached ; gradually the salt diffuses outwards, and the 
concentration of the liquid becomes identical on both sides. 
The relative rates of passage of solute and solvent were shown 
by Eckard 1 to depend on the nature of the membrane, which 
must therefore also control the temporary pressure observed. 
The true maximum value of the osmotic pressure, which we 
have studied in Chapter V., can only be obtained by aid of a 
perfect semi-permeable wall, and must clearly be independent 
of the nature of the partition ; for if not, a perpetual motion 
arrangement could at once be devised. With the membranes we 
are now considering, irreversible processes are involved, and the 
conditions are entirely different from those which theoretically 
hold when an ideal perfect semi-permeable wall is used. Such 
ideal partitions are theoretically possible, and, by making use 
of this idea, we can simplify the application of the principles 
of thermodynamics to the elucidation of the phenomena of 
solution. It is however very difficult to construct perfect 
semipermeable membranes, and a considerable number of those 
prepared in accordance with the directions given on p. 96 will 
always be found to show some leakage of salt. The greatest 
pressure actually reached will then vary with different mem- 
branes, but this variation is a consequence of the imperfection 
of the partition, which allows some of the available energy of 
mixture to pass directly into heat, and, in accordance with 
theory, vanishes if membranes are obtained which show no 
leakage. The theories which we have developed cannot 
be applied to the phenomena shown by leaking membranes, 
whether natural or artificial, for such leakage must render 
the system essentially irreversible. Nevertheless, the study 

1 Pogg. Ann. cxxvm. 61 (1866). 


of the temporary pressures which can be obtained by the 
help of organic membranes, the rate of diffusion of different 
substances through them, and their thermodynamic or osmotic 
efficiency, are problems of fundamental importance for the 

The mode of action of the membrane is at present little 
understood. The various possible views are described on 
p. 97, under the head of inorganic semi-permeable membranes, 
and it is as yet impossible to say if the separating process 
depends (1) on a mechanical sieve-like action, (2) on the satura- 
tion of the membrane or the formation of loose chemical 
compounds with it on one side and their decomposition on the 
other, or (3) on the filtering action of capillarity explained on 
p. 98. As there suggested, it is possible that the three 
modes of explanation may run into each other, different 
aspects being more prominent in different cases. 



The colloidal state. Process of gelation and structure of gels. Coagulative 
power of electrolytes. The nature of colloidal solutions. 

THERE is a marked difference in physical and chemical 
The colloidal properties between bodies of definite crystalline 
state ' form, such as most inorganic salts and minerals, 

and soft or amorphous substances, such as albumen and the 
various kinds of jelly, Graham distinguished the two groups 
as crystalloids and colloids respectively, and particularly ex- 
amined them with regard to their relative diffusive powers. 
Many different kinds of chemical compounds show colloidal 
properties. Besides a vast number of animal and vegetable 
substances, some of which seem to play a great part in the 
phenomena distinctive of living matter, many of the precipitates 
which are formed in the course of inorganic chemical reactions 
appear in an amorphous or colloidal state. The sulphides of 
slightly oxidizable metals such as antimony and arsenic are 
good examples. Thus if a solution of arsenious acid is allowed to 
flow into water saturated with sulphuretted hydrogen by means 
of a continuous current of the gas, a colloidal hydrosulphide is 
formed, which can be freed from excess of sulphuretted hydro- 
gen by passing a current of hydrogen, and from salts by 
dialysis. Many hydrates, too, are colloids, ferric hydrate, for 
instance, which can readily be prepared from the corresponding 
salts of iron. By treating dilute solutions of gold chloride with 


reducing agents, such as a few drops of a solution of phosphorus 
in ether, the gold is set free in the colloidal condition, forming 
a ruby-coloured solution which can be purified by dialysis. 
Silver, bismuth and mercury can also be obtained in colloidal 
solution. Colloid solutions seem to be non-conductors of 
electricity, the dissolved colloids moving as a whole up or down 
the potential gradient in the same way as non-conducting solid 

The classification of substances into colloids and crystalloids 
again brings us to the study of that part of our subject which 
is concerned with the phenomena of allotropy, amorphous 
modifications, and crystallization, and has been referred to on 
pp. 45 to 47 in the chapter on the Phase Rule. If, as there 
indicated, a true solid is always crystalline, colloids which 
possess some of the properties of solids must really be under- 
cooled liquids ; in fact, such a view of their nature was sug- 
gested by Graham 1 . Fluid colloids seem to be capable of 
existing in a coagulated or insoluble condition, which they 
readily assume under a slight disturbing influence. The solu- 
tion of hydrated silica, for instance, may remain liquid for days 
or weeks in a sealed tube, but is sure to coagulate at last. 
The existence in nature of mineral and crystalline forms of 
silica, which have been deposited from water, suggests that, 
even in its coagulated condition, the colloidal substance is 
unstable, passing eventually into a crystalline variety. Glass, 
too, usually a typical colloid, may become crystalline with lapse 
of time. We may conclude, then, that colloids are essentially 
unstable bodies, never in true equilibrium, though the forces, 
viscous or other, opposing a change of state, may be so large 
that the condition will persist for a very long time. 

When examined chemically, colloids show very little 
activity, and chemical changes are produced in them slowly and 
with difficulty. They freely form addition products, however, 
with such bodies as water and alcohol, such combined water or 
alcohol being readily interchangeable with other similar sub- 
stances. The process of absorption of water is often accompanied 

1 Phil. Trans. CLI. 183 (1861) ; Collected Papers, p. 553. 


with considerable increase of volume. The corresponding con- 
traction when the water is removed by evaporation sometimes 
gives rise to considerable forces ; a solution of isinglass, drying 
in a glass vessel over sulphuric acid in vacuo, may tear away 
strips from the surface of the glass owing to its strength of 
adhesion. Many solid colloids, and solid solutions of colloids 
and water, can be used as solvents for mineral salts and acids ; 
as we have stated (pp. 217, 372), the ionic mobility and the 
diffusivity are then very little less than in liquid aqueous solu- 
tions of equivalent strength. On the other hand, the power 
of separating colloids and crystalloids by dialysis shows that 
colloidal membranes are almost impermeable to other colloids. 

Solutions of colloids in crystalloid solvents, such as water 
or alcohol, seem to be divisible into two classes. Both classes 
appear to mix with warm water in all proportions, and the 
mixture will solidify under certain conditions to form a mass 
which may be called a gel ; but one class, represented by 
gelatine and agar jelly, will, when solidified, redissolve on 
warming or dilution, while the other class, containing such 
substances as hydrated silica, albumen, and metallic hydro- 
sulphides, will, under the influence of heat or on the addition 
of electrolytes, form gels which cannot be redissolved. The 
solidification of members of the first class into redissolvable 
substances is termed setting, that of substances in the second 
class, which form insoluble precipitates, is termed coagulation 1 . 
Liquid solutions of colloids in water have been called by 
Graham hydrosols, and the solids, formed by setting or coagu- 
lation, hvdrogels. Hardy has distinguished the two kinds of 
systems forming soluble and insoluble precipitates as reversible 
and non -reversible 2 . The names are convenient, but as there 
appears to be a considerable difference between the melting 
and solidifying points of jellies, etc., it must be understood that 
such systems are not necessarily reversible in the thermody- 
namic sense of the word. 

1 W. B. Hardy, Proc. R. S. LXVI. 95 (1900). 

2 loc. cit. 


The mechanism of gelation in reversible colloidal systems 
has been studied experimentally by van 
gdaTn "and Bemmelen > and by Hardy. 
structure of Van Bemmelen measured the vapour pres- 

sures of gels which had been formed by the 
coagulation of hydrated silica and contained varying proportions 
of absorbed water. When water is removed slowly, a regular con- 
tinuous curve is obtained ; when the removal is rapid, the diagram 
shows changes of curvature. Van Bemmelen considers that the 
system does not consist of two definite phases in the sense of 
the Phase Rule, for the two parts into which it separates on 
coagulation are not divided by a definite interface. Still, 
two parts can be distinguished, one of which is colloidal, viscous, 
and possesses a net-like structure in which the other more 
fluid liquid is partly absorbed and partly retained mechanically. 
The colloidal liquid passes into the solid state by the lapse of 
time or by the influence of foreign bodies. 

The emission and absorption of water vapour by colloidal 
matter has been investigated theoretically by Duhem 2 , who 
deduces the results observed by van Bemmelen from the ther- 
modynamic properties of a system of which two of the controlling 
variables are subject to hysteresis. 

Hardy examined mixtures of agar and water, agar water 
and alcohol, and gelatine water and alcohol. The last-named 
ternary system gives a homogeneous liquid when warm, but 
divides on cooling into two phases possessing different refractive 
indices, and is therefore suitable for microscopic investigation. 
When the proportion of gelatine is small, from 6 to 14 per 
cent., fluid droplets are seen to form as the liquid cools ; they 
solidify and eventually join together into a loose framework. The 
mass has then become a more or less solid gel. With a higher 
proportion of gelatine, 36'5 per cent., this arrangement was 
inverted, and the drops formed contained less gelatine than the 
residual substance, which now forms a solid solution, interrupted 
by spherical spaces filled with liquid. The temperature at 
which the separation into two phases occurs is raised by an 

1 Zeits. anorgan. Chem. xm. 233 (1896) ; xvm. 14 (1898). 

2 Jour. Phys. Chem. iv. 65 (1900). 


increase in the proportions of gelatine or alcohol, and lowered 
by the addition of the common solvent water. No binary 
system was found in which the changes could be followed by 
the microscope ; but with a mixture of agar and water, the gel 
could be separated into two phases consisting of a solid and a 
liquid solution respectively, by expressing the latter through 
canvas. The constitution is therefore probably similar to 
that investigated for the three-component system described 

Graham observed that the addition of salts, sometimes in 
minute quantities, often caused colloidal solu- 


power of electro- tions to coagulate 1 . Hydrated alumina, for 
instance, prepared by dialysing a solution of 
the chloride containing excess of the hydrate, was so unstable 
that a few drops of well-water at once produced coagulation, 
and the same change was brought about by pouring the 
solution into a new glass vessel, unless the vessel had repeatedly 
been washed with distilled water. This action of salts was 
further investigated by Schulze 2 , who found that hydrosols of 
sulphide of arsenic were coagulated by salts at a rate depending 
largely on the valency of the metal. Denning the coagulative 
power as the reciprocal of the concentration in gram-molecules 
per litre necessary to coagulate a given solution, Schulze found 
that the relative coagulative powers of univalent, divalent, and 
trivalent metals were in the ratios 1 : 30 : 1650. These results 
were verified by Prost 3 , who used sulphide of cadmium as the 
colloid, and by Linder and Picton 4 , working with sulphide of 
antimony. Linder and Picton found that a slight trace of the 
metal is entangled in the coagulum, the salt apparently being 
decomposed to a corresponding extent. Their measurements 
showed that, for different salts of a given metal, the coagulative 
powers are proportional to the equivalent electrical conductiv- 
ities, and that the relative coagulative powers of various 

1 Collected Papers, p. 580. 

2 Jour, prakt. Chem. xxv. 431 (1882). 

3 Bull. Acad. Eoy. Sci. de Belg. [3] xiv. 312 (1887). 

4 Jour. Chem. Soc. Trans. LXVII. 63 (1895). 


sulphates of univalent, divalent, and trivalent metals ranged 
round the mean values 1 : 35 : 1023. The effect of adding a 
small quantity of the salt of one metal was to reduce the amount 
of the salt of another metal with the same valency which was 
required for coagulation ; but if the metal of the second salt had 
a different valency, the amount of salt needed was actually 
increased by the presence of the first salt : more strontium 
chloride, for instance, was necessary when a little potassium 
chloride was previously dissolved. It is probable that the 
molecular changes which accompany coagulation are not sudden 
discontinuous processes, for Linder and Picton 1 found that, as 
the point of coagulation is approached, the size of the colloid 
particles increases even though actual coagulation does not 
occur, and, under similar conditions, Graham 2 observed a gradual 
increase in the viscosity of the solution. 

An explanation of some of these remarkable valency rela- 
tions has been offered by the present writer 3 . The connexion 
with electrical conductivity discovered by Linder and Picton 
shows that the coagulative power of a salt depends on its 
electrical properties. Let us suppose that, in order to produce 
the aggregation of colloidal particles which constitutes coagula- 
tion, a certain minimum electrostatic charge has to be brought 
within reach of a colloidal group, and that such conjunctions 
must occur with a certain minimum frequency throughout the 
solution. Since the electrical charge on an ion is proportional 
to its valency, we shall get equal charges by the conjunction of 
2n triads, Sn diads, or Qn monads, where n is any whole 

In a solution where ions are moving freely, the probability 
that an ion is at any instant within reach of a fixed point is, 
putting certainty equal to unity, approximately represented by 
a fraction proportional to the ratio between the volume occupied 
by the spheres of influence of the ions and the whole volume of 
the solution, and may be written as Ac, where A is a constant 

1 Jour. Chem..Soc. Trans. LXVIII. 73 (1895). 

2 Collected Papers, p. 619. 

3 Phil. Mag. [5] XLVIII. 474 (1899) ; also Hardy and Whetham, Jour. Physio- 
logy, xxiv. 288 (1899). 


and c represents the concentration of the solution. The chance 
that two such ions should be present together is the product of 
their separate chances, that is, (Ac) 2 . Similarly, the chance for 
the conjunction of three ions is (Ac) s , and for the conjunction 
of n ions is (Ac) 11 . 

In order that three solutions, containing trivalent, divalent, 
and univalent ions respectively, should have equal coagulative 
powers, the frequency with which the necessary conjunctions 
should occur must be the same in each solution. We should 
then have, the constant being assumed equal in each case, 

Ac 3 zn = A sn c 2 3n = A 6n c, 6n = a constant = B. 


B 3n B Qn 

d, C 2 , C 3 representing the concentrations of monads, diads, and 
triads in their respective solutions. Thus we get for the ratios of 
the concentrations of equi- coagulative solutions 

j_ i_ i_ _L 1 

d : C 2 : c 3 = B Gn : B* n : B 2n = l:B Qn : B Sn . 

1 i 

Let us put B Qn = - ; the ratios can then be written 



The reciprocals of the numbers expressing the relative concen- 
trations of equi-coagulative solutions give values proportional 
to the coagulative powers of solutions of equal concentration ; 
so that, calling the coagulative powers of equivalent solutions 
containing monovalent, divalent, and trivalent ions respectively 
Pi>p2,p 3 , we get 

Pi-.pz'.p^l \x\x\ 

Let us now take some numerical examples. Putting x = 32, 
we get the series 

1 : 32 : 1024, 

which agrees very well with Linder and Picton's results for 
colloidal solutions of antimony sulphide, 



and putting x = 40, we get 

numbers comparable to Schulze's values for sulphide of arsenic, 

1 : 30 : 1650. 

When we consider the difficulty of the experiments, and 
remember that the coagulative powers of solutions containing 
different ions of equal valency are not equal, but vary as the 
equivalent conductivities, these results show a better agree- 
ment between the calculated and observed values than might 
have been expected. 

The particles in solutions of colloids in water generally move 
when in an electric field, the direction of motion depending on 
the nature of the colloid and of the solvent. Thus Hardy 1 
found that proteid, modified by heating to the boiling point 
when dissolved in water, reverses the direction of its motion 
under the influence of electric forces, when the reaction of the 
fluid holding it is changed from slightly acid to slightly alkaline. 
A minute quantity of free alkali causes the proteid particles to 
move against the current, while the addition of an equally minute 
quantity of acid is followed by movement in the same direction 
as the current. Movement in an electric field shows that the 
particles must be charged electrically, a double layer probably 
being formed at their surfaces, in accordance with Quincke's 
theory of electric endosmose. The reversal of direction implies 
a reversal of sign in the charges on the particles, and therefore, 
by slowly adding acid or alkali to the liquid, it is possible to 
obtain an iso-electric point at which there is no potential 
difference between the liquid and the particles. Hardy finds 
that as this point is approached, the stability of the system 
diminishes, and at the iso-electric point it is probable that 
coagulation spontaneously occurs. The same observer has also 
discovered that, in the case of colloids travelling with the 
electric current, it is the anion which is active in causing 
coagulation, and not the metallic ion as in the experiments of 

1 Jour. Physlol. xxiv. 288 (1899) ; Proc. R. S. LXVI. 110 (1900). 


Schulze, Prost, and Linder and Picton, who all used colloids 
which move against the current. Thus it is always the ion 
possessing a charge of opposite kind to that on the colloid 
particles that is effective in producing coagulation. 

These observations suggest that the coagulative power of 
electrolytes may depend on a modification, under the electro- 
static influence of the ionic charges, of the surface energy of the 
interface between the two phases of the colloidal system. As 
shown in the Chapter on contact electricity, the natural surface 
energy of such an interface is diminished by the presence of 
certain kinds of electric double layers. The tendency of the 
surface tension to condense many small particles into a few 
larger ones (p. 43) is thus reduced, and the system of small 
particles may become stable. Approach to the iso-electric 
point will, by decreasing the intensity of the double layer, 
increase the surface tension, and diminish the stability of 
the system, while the absorption of ions possessing charges of 
opposite sign to those on the particles will reduce the charges 
on the particles, and again act in a similar manner. The average 
size of the particles will be increased, and, if the influence at 
work is sufficient, the particles may be precipitated, or if enough 
colloid matter is present, the whole solution may coagulate into 
a more or less solid mass. As another mode of stating the 
explanation, we may say that a high potential difference implies 
a great mutual affinity between the two phases, tending to 
expand the interface. As the opposite charges of the electric 
.double layer are annulled, the affinity diminishes; it vanishes 
at the iso-electric point, and the solution becomes unstable. 

A different explanation of the coagulative properties of 
certain substances has been offered by Quincke 1 , on the basis 
of changes of surface tension only. These changes are sup- 
posed to be produced by the spreading of the electrolytic 
solutions over the surfaces of the particles, forming a new inter- 
face with the surrounding liquid. This hypothesis makes no 
attempt to explain the relations of coagulative power with 
electrical conductivity and valency. 

1 B. A. Report, 1901, p. 60. 


Liquid solutions of colloids may be regarded either as ordinary 
solutions, of which the solutes possess enor- 

Nature of . 

colloidal soiu- mously high molecular weights, or as systems 
of two phases, composed of suspensions of par- 
ticles in the liquid, different from it and of greater than molecular 
dimensions. Much discussion has taken place on the relative 
merits of these two hypotheses. It is certain that some colloid 
solutions may be kept almost indefinitely without precipitation ; 
but, if the foreign particles are small enough, the viscosity of 
the water is enough to make the settling process almost inde- 
finitely slow. The properties of hydrosols differ considerably from 
those of true solutions; the rate of diffusion is very much less, 
the heat of solution is usually inappreciable or at all events 
very small, while no certain indications have been obtained of 
measurable osmotic pressures or depressions of the freezing 
point of the solvent. 

In some colloid solutions, the presence of suspended particles 
can readily be detected. Picton and Linder 1 , who have made 
extensive investigations on this subject, have observed a con- 
tinuous gradation in size from particles large enough to be 
visible under a microscope. Such particles exist in solutions of 
mercuric sulphide and of arsenious sulphide prepared from the 
tartrate ; under a magnification of 1000 diameters they appear 
as minute solid particles in rapid Brownian movement, crowded 
together so closely that very little free space is left. Other 
solutions of colloidal sulphides, together with those of ferric 
hydrate, chromic hydrate, aluminium hydrate, silicic acid, 
cellulose, starch, and acid and neutral Congo-red, while non- 
resolvable under the microscope, contain particles large enough 
to scatter and polarize a beam of lime-light. These optical 
methods fail to show the presence of particles in the colloidal 
solutions of molybdic acid, and of silicic acid in presence of 
hydrochloric acid. On the other hand, certain crystalloids 
possessing very complicated molecules, oxyhaemoglobin, car- 
bonic oxide haemoglobin, and a compound of ferric hydrate with 
ferric chloride, which is said to crystallize as 9Fe 2 O 3 . FeCl 3 , 
yield solutions which contain particles large enough to scatter 
1 Jour. Chem. Soc. Trans. LXL 148 (1892). 


and polarize light 1 . A colloidal solution of arsenic sulphide, 
prepared from the aqueous solution of pure arsenious acid, 
showed a diffusive power comparable with that of crystalloids, 
though the same solution polarized light. Picton and Linder 
conclude that there is no distinction in kind between colloid 
and crystalloid solution, but that a continuous gradation exists 
between solutions containing colloid particles visible under a 
microscope and electrolytic solutions of common crystalloid salts 
and acids. The absence of measurable osmotic-pressure pro- 
perties appears to be merely an affair of arithmetic. Particles 
of a size to scatter light in accordance with the observations to 
which we have referred, must be comparable in size with the 
wave-length of light, about 5 x 10~ 5 centimetre. If the parti- 
cles are close together we may conclude there are about 10 4 
in a linear centimetre, or 10 12 in a cubic centimetre. In a 
normal solution of a crystalloid which gives an osmotic pressure 
of about 22 atmospheres, the number of solute molecules is 
approximately 5 x 10 20 . The osmotic pressure of the colloid 
solution in question will therefore be about 2 x 10~ 9 of that of 
a normal solution of a crystalloid, a value much too small to be 

It is worthy of note that turbid suspensions of clay, kaoline, 
etc., in water are rapidly cleared by the addition of small 
quantities of metallic salts 2 . This action, which is almost 
certainly of the same nature as the coagulation studied above, 
probably helps in the formation of sand-banks at the mouths of 
rivers, the salts of the sea water clearing the suspensions of 
clay brought down by the fresh water ; precipitation then occurs 
owing to the diminished velocity. 

The conditions which determine the colloid or crystalloid 
nature of a substance are at present little understood. The 
persistence of the colloid properties when a substance passes 
from the dissolved to the non-dissolved state, shows that the 

1 Gamgee has shown in other ways that oxyhaemoglobin has a mixture 
of colloidal and crystalloidal properties. Proc. R. S. LXX. 79 (1902). 

2 Schulze, Pogg. Ann. cxxix. 366 (1866) ; Schloesing, Compt. Rend. LXX. 1345 
(1870) ; Bodlander, Gott. Nachr. 1893, p. 267 ; Spring, Rec. Trav. Chim. Poys- 
Bas, 1900, pp. 222, 294. 

w. s. 26 


determining conditions must be of fundamental importance. 
The molecular forces seem to be much less active in colloids, 
but the freedom with which some of them disintegrate and 
dissolve in presence of water and other liquids indicates that some 
interaction between them and their solvent must occur. On 
these lines, making certain assumptions as to the nature of the 
forces at work and their variation with the distance, Donnan 1 
has offered an investigation of the conditions which would secure 
the disintegration and solution of a colloid. 

It seems likely that the forces and interactions which are 
involved in crystalloid solution are of the nature of those which 
are classed as chemical or molecular, while, when colloids dis- 
solve, the actions between solvent and solute are conditioned 
also by the phenomena which are studied under the names of 
surface tension and capillarity. As the size of the dissolved 
aggregates or particles increases, the importance of the chemical 
forces diminishes and that of the capillary forces grows. In 
studying the properties of colloidal solutions by the light of the 
Phase Rule, we must remember that the surface of separation 
between the phases is enormously extended owing to the minute 
size of the particles, and the surface energy therefore becomes 
of great, perhaps of preponderating, importance. An investiga- 
tion of the influence of capillarity on the theory of equilibrium 
will be found in Gibbs' work 2 ; he shows that an interfacial 
transition layer provides in a sense a new phase coexistent with 
those on each side of it and having its own characteristic equa- 
tion. Again, colloids may be regarded as undercooled liquids, 
and in a condition of unstable equilibrium. Their condition 
may therefore depend on the time, which introduces a new 
variable beyond those contemplated by the ordinary application 
of the Phase Rule 3 . 

In conclusion we may point out that, if colloid and crystal- 
loid solution are but the extreme limits of a continuous series 
of phenomena, the study of dissolved colloids of varying degrees 
of aggregation promises to throw light on the general problem 
of the fundamental nature of solution. 

1 Phil. Mag. [6] i. 647 (1901). 3 Trans. Connect. Acad. in. 380 (1877). 

8 Bancroft, The Phase Eule, p. 234. 


BRUNI and Padoa 1 have prepared solid solutions by the sublima- 
tion of a mixture of two crystalline substances, 
^oHd'sohiions? 1 ' sucn as mercuric iodide and bromide. The 
mixture was placed at the bottom of a glass bulb, 
which was then partially exhausted and kept in a bath of alloy 
at a temperature a few degrees below the fusing point of the 
mixture. After a day or so, homogeneous crystals of a solid 
solution were found in the upper part of the bulb. 

Lord Rayleigh 2 has measured the compositions of liquid 
and vapour in the distillation of mixtures, 

Chapter III. p. 75. . 

Distillation of drawing curves with the two compositions as 
axes. With 96 per cent, alcohol the compo- 
sitions were identical, in agreement with the minimum boiling 
point found by Noyes and Warfel 3 . By proper arrangements, 
separation of the mixture can be effected by a single continuous 

Since the year 1897 a series of researches has been carried 
out by Van 't Hoff and his pupils on the solu- 

Chapter IV. p. 93. * 

solubility of bility of simple salts, double salts, and mixtures, 
with the immediate object of elucidating the 
geological process involved in the formation of oceanic salt 
deposits, such as those of Stassfurt 4 . 

1 Atti Accad. dei Lincei, Roma, xi. 565 (1902). 

2 Phil. Mag. [6] iv. 521 (Nov. 1902). 

8 Amer. Chem. Soc. Jour. xxm. 463 (1901). 

4 A report on the work was communicated by E. F. Armstrong to the 
British Association (1901, p. 262) ; the original papers will be found in the 
Abhand. Kon. Akad. Wissensch., Berlin, 18971902. 




When several salts are simultaneously present, the possible 
number of solids in equilibrium with the solution may be 
predicted by the Phase Rule, but the nature of any double 
salts, the possibility of their co-existence, and the order in 
which the various solids appear, must be determined by experi- 

As an example we may take a solution containing potassium 
chloride KC1 and magnesium chloride MgCl 2 . 6H 2 O, substances 
which form a double salt of composition KC1 . MgCl 2 . 6H 2 O 
called Carnallite. Here we have three components, namely, 
water and the two simple 
salts ; the double salt, not 
being an independent vari- 
able, is merely a possible 
solid phase. To obtain a 
monovariant system, which 
is in equilibrium at one 
given temperature when the 
pressure is fixed, we must 
assemble four phases, two 
being solids. Working at 
the constant pressure of the 
atmosphere, and at a con- 
stant temperature of 25, the 
conditions of such a system 
are completely determined, 
and the liquid phase can 
have only one composition. 
When, at the beginning of 
the deposition, the only solid 
phase is composed of the 
crystals of the less soluble 
salt, the system is not de- 
termined, and the composi- 
tion of the liquid will vary 

as evaporation proceeds till a new solid phase appears. The 
composition of the liquid phase then remains constant as long 
as both solids are deposited. The phenomena may be repre- 


10 20 30 40 

Molecules K, C1 2 

Fig. 66. 


sented on a diagram (Fig. 66), for which the necessary data 
are given in the following table, the amount of potassium 
chloride being reckoned in double molecules as K 2 C1 2 , for the 
sake of comparison with MgCl 2 . 

Number of molecules per 

Substances saturating Points on thousand molecules 

the solution the diagram of water 

K 2 C1 2 MgCl 2 

KC1 A 44 

KC1 and Carnallite B 5-5 72-5 

CarnaUite and MgCl 2 .6H 2 C 1 105 

MgCl 2 .6H 2 D 108 

Points lying inside the lines ABCD represent the composition 
of unsaturated solutions ; points on the line AB indicate the 
number of double molecules of potassium chloride which 
saturate the liquid as the amount of magnesium chloride 
is increased ; similarly the lines BC and CD represent 
the conditions of saturation with Carnallite and magnesium 
chloride respectively. The points B and C, representing the 
composition of solutions simultaneously saturated with two 
solid phases, are determined by the experimental results given 
in the table. 

In more complicated systems the number of variables can 
often be reduced by remembering that in what are known as 
reciprocal salt pairs, for example, 

MgCl 2 + K 2 S0 4 = K 2 C1 2 + MgSO 4 , 

the amount of the second pair can be expressed in terms of 
the amount of the first. This consideration makes it often 
possible to represent on a solid model the phenomena observed 
when several salts are present. 

By studying the evaporation of sea water on these lines, it 
has been found that the order in which salts should be depo- 
sited is probably as follows. (1) Sodium Chloride ; (2) Sodium 
Chloride and Magnesium Sulphate ; (3) Sodium Chloride and 
Leonite ; (4) Sodium Chloride, Leonite, and Potassium Chloride, 
or Sodium Chloride and Kainite ; (5) Sodium Chloride, Kiese- 
rite and Carnallite ; (6) Sodium Chloride, Kieserite, Carnallite, 


and Magnesium Chloride, the solution then drying up without 
further change of composition. 

This succession agrees with that found on actually evapo- 
rating sea water at 25, and approximately conforms to that 
observed in the geological deposits at Stassfurt. 

Morse and Horn 1 precipitated a membrane of copper 
ferrocyanide in the walls of a porous cell by 

Chapter V. p. 96. J ? J 

semi-permeabie placing solutions of copper sulphate and 
potassium ferrocyanide on each side of the 
wall, and passing an electric current till the resistance became 
1500 to 3000 ohms. Membranes thus obtained easily with- 
stood pressures of four or five atmospheres. 

It has been shown by Lyle and Hosking 2 that, when plotted 
with the temperature, the equivalent conduc- 
tivity and the fluidity of solutions of sodium 
chloride give similar but not identical curves, 

which indicate that both conductivity and fluidity would vanish 

at a temperature of 35'5 Centigrade. 

The conductivity of solutions of salts etc. dissolved in liquid 
hydrocyanic acid has been investigated by 
Kahlenberg and Schlundt 3 . The potassium 
sa l ts Dave conductivities about three times 
tnose of tne corres P on d m g aqueous solutions. 
These aqueous solutions are themselves highly 
ionized, so that the increased conductivity in hydrocyanic acid 
must be due to the smaller ionic viscosity of that solvent. In 
light of the high dielectric constant (about 95) of the liquid, it 
is interesting to observe that some other salts showed less con- 
ductivity than in water, and that both water and alcohol were 
found to dissolve to form non-conducting solutions. (See p. 362.) 

1 Amer. Chem. Jour. xxvi. 80 (1901). 

2 Phil. Mag. [6] in. 487 (1902). 

8 Jour. Phys. Chem. vi. 447 (1902). 





and Reprinted, by permission, from the Report of the British 
Association for the Advancement of Science, 1893. 

THE comparison of the numerical results of electrolytic observa- 
tions is rendered difficult by the fact that the data are scattered in 
various periodicals and expressed by different observers in units 
that are not comparable without considerable labour. The following 
table has been compiled with the object of facilitating the com- 

In the table are included all observations known to the com- 
piler, with the exception that a selection only is made in the case of 
organic bodies. Observations for a number of additional substances 
will be found in Ostwald's papers in the Journal fur Chemie, vols. 
xxxi., xxxii., and xxxiii., and in the Zeitschrift fur physikalische 
Chemie, vol. i. With this restriction it is hoped that no important 
observations published before the year 1893 have been omitted, and 
that, in the reduction of results, expressed in such varied units, the 
table is sufficiently free from mistakes for it to be of service. The 
data included refer to the strength and specific gravity of solutions, 
with the corresponding conductivities, migration constants, and 
fluidities. The several columns are as follows : 

I. Percentage composition, i.e. the number of parts by weight 
of the salt (as represented by the chemical formula) in 100 parts of 
the solution. 


II. The number of gramme equivalents per litre, i.e. the number 
of grammes of the salt per litre divided by the chemical equivalent 
in grammes, as given for each salt. 

III. The specific gravities of the solutions : in most cases the 
specific gravities of the solutions are not given by the observers, 
and the numbers given have been deduced from Gerlach's tables in 
the Zeitschrift fur analytische Chemie, vol. viii. p. 243, <fec. 

IV. The temperatures at which the solutions have the specific 
gravities given in the previous column for the given strength of 

Y. The conductivity, as expressed by the observer. In the cases 
in which the observer has expressed his results for specific equivalent 
conductivity no numbers are given in this column. 

VI. The temperature at which the conductivities of the solu- 
tions have been determined. 

VII. The temperature coefficient referred to the conductivity 

VIII. The specific equivalent conductivity of the solutions at 18 
in terms of the conductivity of mercury at ; the specific equivalent 
conductivity is the conductivity of a column of the liquid 1 centi- 
metre long and 1 square centimetre in section, divided by the 
number of gramme equivalents per litre. 

In some few cases, in which no temperature coefficients have 
been determined, the results have been given for the temperature 
at which the observations were made. 

The numbers given in the column are the values for the specific 
equivalent conductivity x 10 9 . 

IX. This column contains the values for specific equivalent 
conductivity at 18 in C.G.S. units : they are obtained from those in 
the previous column by being multiplied by the value of the conduc- 
tivity of mercury at in C.G.S. units. The factor is 1-063 x 10~ 5 . 
The values in Kohlrausch's units, reciprocal ohms e J centimetre 
cube per gram equivalent of salt per cubic centimetre, can be 
obtained by dividing by 10 the numbers actually given in the 
column (i.e. 1293 becomes 129-3 for the first solution of potassium 
chloride, &c.). The results may in some cases differ by a few parts 
in a thousand from Kohlrausch's latest values given in his book das 
Leitvermogen der Elektrolyte (Leipzig, 1898). 


X. The migration constant for the anion ; for instance, in the 
case of copper sulphate (CuS0 4 ), for (SO 4 ). Some more recent 
results are given on p. 212. 

XI. The temperatures at which the migration constants have 
been determined. 

XII. The number of gramme equivalents per litre, as defined for 
column II., for which the fluidity data are given in the following 

XIII. The fluidity of the solutions of the strength given in the 
previous column. 

Most of the results given for the fluidity of solutions are 
expressed in terms of the fluidity of water at the same temperature ; 
to obtain the absolute values for the solutions they have been 
multiplied by the value for the fluidity of water at the given 
temperature. The values used for this purpose have been taken 
from Sprung' s observations for the viscosity of water given in 
Poggendorff's Annalen, vol. clix. p. 1. 

To obtain the values for fluidity in c.G.s. units, the numbers in 
this column must be multiplied by the factor '1019. 

XIY. The temperature at which the solutions have the fluidity 
given in- the previous column. 

XV. The temperature coefficient of fluidity at 18, that is 

XVI. In the last column are given the references to the 
various papers from which the data are taken : against each refer- 
ence will be found a number, which appears also against the first of 
the data which have been taken from the paper in question. 

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The numbers refer to the pages. Authors' names are printed in small capitals. 
The electrochemical properties of aqueous solutions of the substances of which the 
names appear in italics will be found tabulated in the Appendix. 

ABEGG and NERNST, theory of freezing 

point determinations, 153 
Absolute electric charge on ions, 189 
Absolute ionic velocities, 214 ; table of, 223 
Absolute scale of temperature, 15. 
Absorption coefficients, 85, 87 
Absorptiometer (Fig. 33), 84, 85 
Accelerating influence of acids, 337 
Accumulators or secondary cells, 263 ; 

electromotive force, 265; origin, 181 
Acetic add, 453 ; abnormal vapour pres- 
sure, 129; freezing point of solutions 
in, 156 ; ionic mobilities, 215, 218 ; 
ionization, 321; ionization constant, 

Acetate of silver, solubility of, 93 
Acetone, conductivity of solutions in, 331 
Acids, accelerating influence, 337 ; elec- 
trical conductivities, 204, 337, 357 
Action, secondary electrolytic, 194 
Additive properties of solutions, 332 
Adiabatic relations of ideal gas, 7 
ADIE, absolute value of osmotic pressure, 


Affinity or avidity, coefficient, 205, 336, 
342 ; measurement, 336 ; residual, 173 
AITKEN, supersaturation, 43 
Alcohol, ionization of solutions in, 322, 

331 ; vapour pressures, 72, 74, 403 

ALEXEJEFF, mutual solubility of liquids, 8<S 

Alkalis, electrolytic decomposition of, 178 

ALKEMADE, VAN KIJN, VAN, f curves, 26, 63 

Allotropic solids (Fig. 12), 45 

Alloys, 59 ; eutectic, 60 ; mixed crystals, 

69 ; freezing point, 60, 160 ; microscopic 

study, 60, 62, 68 ; rate of cooling, 68 ; 

structure (Figs. 43, 44), 144; copper 

and tin, 71 ; gold aluminium, 61, 62 ; 

fusion curve (Fig. 20), 62; silver 
copper, 59, 60 ; fusion curve (Fig. 19), 

Alternating currents, used in measure- 
ment of electrolytic conductivity, 199 

Aluminium chloride, 420; sulphate, 445 

Amalgams, in capillary electrometers and 
galvanic cells, 285 ; in concentration 
cells, 244 ; in dropping electrodes, 281 ; 
vapour pressure of, 141 

Ammonia, 471 ; ionization, 216, 322 ; 
solubility, 86 

Ammonium bichromate, 450; chloride, 412; 
chromate, 450 ; iodide, 424 ; nitrate, 
428 ; sulphate, 438 

Amylamine, 474 

Anhydrous solutes (Fig. 13), 50, 51 

Animal electricity, 176 

Anion, definition of, 180 

Anode, definition of, 180 

ARCHIBALD, equivalent conductivities at 0, 
325 ; freezing point, 153 

ARMSTRONG, H. E., hydrate theory of 
solution, 170 

ARMSTRONG, E. F., Keport by, 403 

ARRHENIUS, chemical activity, 336; co- 
efficient of ionization, 322 ; electrolytic 
dissociation, 317 ; diffusion of electro- 
lytes, 380; freezing points, 153; heat 
of neutralization, 356 ; heat of ioniza- 
tion, 355 ; ionic fluidity, 224 ; maxima 
conductivities, 357 ; osmotic pressure 
of electrolytes, 317 

ASTON and DUTOIT, equivalent conduc- 
tivities, 331 

Available energy, see Energy 

Avidity, measurement, 336 ; table of rela- 
tive, 337 



AYRTON and PERRY, potential differences, 

BANCROFT, "the Phase Rule," 77, 402; 

potential differences, 388 
Barium chloride, 415 ; hydrate, 471 ; 

nitrate, 432 
BARNES, freezing points, 153 ; equivalent 

conductivity at 0, 325 
Battery, see Cell 

BECKMANN, boiling points, 137; freezing 
point determinations, apparatus for 
(Fig. 45), 155 ; freezing points of con- 
centrated solutions, 161; molecular 
weights in solution, 140 ; vapour pres- 
sures, 132, 133 

BEETZ, electrolytes, resistance of, 198 ; 
conductivity of supersaturated solutions, 

BEIN, electrolytic transport numbers, 210 
BEKETOFF, precipitation of metals by 

hydrogen, 386 

BEMMELEN, VAN, gelation, 394 
BENDER, properties of solutions, 333 
Benzine, freezing points of, solutions in, 


Benzoic acid, 465 
BERKELEY, Earl of, growth of crystals, 81; 

time required for saturation, 90 
BEKTHELOT, occlusion, 82 

of succinic acid, 93 

BERZELIUS, electrochemical theory, 191 
BERZELIUS and HISINGER, electrolytic de- 
composition of salt solutions, 178 
BINDEL, supersaturated solutions, 80 
BLAGDEN, freezing points of solutions, 153 
Boiling points, 135, 331 ; measurement of, 
by Beckmann, 137 ; Buchanan, 138 ; 
Regnault, 138 ; Wade, 140 
BOLTZMANN, laws of osmotic pressure, 169 
BOTTOMLEY, potential differences, 271 
Boundary of two solutions, 219, 376, 382 
BOUTY, measurement of electrolytic con- 
ductivity, 198 
Boyle's Law, 7 

BREDIG, specific ionic mobility, 229, 230 
BREDIG, COHEN and VAN 'T HOFF, transi- 
tion cells, 261 
BREDIG, NOYES and OSTWALD, concentrated 

solutions, 162 
Brompropionic acid, 459 
BROWN, J., temperature coefficient of 
fused salt cells, 237; mercury drop- 
ping electrodes, 280 ; potential differ- 
ences, 271 

BRUHL, ionizing power of water, 364 
BRUNI and PADOA, solid solutions, 403 
BRUNNER, precipitation of metals by 
hydrogen, 386 

BUCHANAN, JL Y., boiling point of solu- 
tions, 138, 139 ; cryohydric point, 
lowering of, 146 ; freezing of sea water, 
145 ; freezing point of sodium chloride 
solutions, 142 ; structure of ice, 145 

BUNSEN, absorption coefficient, 85 ; con- 
firmation of Henry's law, 86; deter- 
mination of solubility, 84 

BURCH, action of capillary electrometer, 
283 ; electrometer as condenser, 282 

BUSCEMI, temperature coefficient of fused 
salt cells, 237 

Butyric acid, 460 

Cadmium bromide, 422 ; chloride, 419 ; 
iodide, 424 ; nitrate, 433 ; sulphate, 443 

CAILLETET, evolution of hydrogen by 
metals, 386 

Calcium chloride, 416 ; hydrate, 471 ; 
nitrate, 432 

Calorie, definition of, 1 

Cane sugar, inversion of, 337 ; freezing 
point of solutions, 147 

Capacity for heat of solutions, 171, 335 

Capacity, specific inductive, of solvents, 

Capillary action, electro-, 282 ; electro- 
meter, 282 

CARLISLE and NICHOLSON, early experi- 
ments on 'electrolysis, 177 

Carnallite, deposition of, 404 

Carnot's engine, 12 

CARRARA, dissociation of water, 363 ; equi- 
valent conductivities in pyridine, 331 

Cathode, definition of, 180 

Cation, definition of, 180 

Cells, galvanic: amalgam, 285; bichro- 
mate, 182 ; chemical, 255 ; Clark's stan- 
dard, 184, 235, 261; Daniell's, 182, 233 ; 
effect of pressure, 239 ; electromotive 
force, 232, 235, 236, 285, 288, 382, 383 ; 
fused salt, 237; Grove's, 235, 237; 
irreversible, 235, 262; liquid, 382; 
mercury, reversible heat of, 238; re- 
duction and oxidation, 258; reversible 
234, 235, 237 ; secondary, 263 ; Smee's 
182; temperature coefficient of, 237; 
transition, 258; Weston's, 261 

Cells, concentration : calomel, 250 ; von 
Helmholtz's theory, 242 ; hydrogen, 242 ; 
silver chloride, 249 ; silver nitrate, 245, 
248; different electrodes, 242; differ- 
ent solutions, 245 ; double concentra- 
tion, 250; effect of low concentration, 
253 ; ionization in, 255 ; migration in, 
250 ; table of electromotive forces, 250 

Cells, osmotic (Fig. 36), 96, 118, 168 

Cells, resistance (Figs. 48 to 51), 201 

Change of volume and osmotic pressure, 



CHAKPY, alloys, 59 ; microscopic study of, 

Chemical affinity, sec Affinity 

Chemical combination, theory of solution, 

Chemical constitution and mobility of 
ions, 229 

Chemical potential, 25, 34 

Chloric acid, 434 

CHRISTY, contact potentials, 300 

CLARK, LATIMER, standard cell, 184, 235, 

CLAUSIUS, on electrolysis, 205; latent 
heat equation, 38 

Coagulation of colloidal solutions, 45, 395 

Cobalt sulphate, 445 

COHEN, hydrated solids, 53 ; transition 
point, 259 

sition cells, 261 

chemical equivalent of silver, 185 

Colloid, definition of, 371 

Colloidal solutions, coagulation of, 45, 
395; nature of, 400; separation from 
crystalloids, 388 

Colour of salt solutions, 334 

Combination, chemical, theory of osmotic 
pressure, 169 

COMEY, A. M., Dictionary of Chemical 
Solubilities, 93 

Commutator, revolving, 201 ; tuning-fork, 

Complete cycles, 6 

Complex cycles, 18 

Complex ions, 226, 322, 331 

Complex, molecular, theory of solutions, 

Components, definition of, 35; one com- 
ponent, 39; two components, 48; 
consolute, 48; two liquid, 58; two 
volatile, 71; three, 76 

Compound, definition of, 48 

Concentrated solutions, equation for, 162; 
freezing point of, 160; freezing point 
and osmotic pressure, 150; vapour 
pressure of, 126 

Concentration cells, 241; see Cells, 

Concentration curves, 55 ; see Curves 

Concentration, influence of, on equivalent 
conductivity, 223 

Concentration, ionic, 340, 346, 351, 365 

Conductivity, electrolytic, 197, 408; 
additive nature of, 207; equivalent, 
202, 408; connexion with osmotic 
pressures, freezing points and vapour 
pressures, 160, 316; influence of con- 
centration (Fig. 12), 202, 203, 223, 
322; of liquid films, 230: in various 

solvents, 322, 330, 362; measurement 
of, 198, 325; of mixed solutions, 88; 
temperature coefficient of, 202, 408; 
of supersaturated solutions, 80; of 
water, 193, 358 
Consolute components, 48 
Consolute solid solutions, 63 
Contact electricity, 267 et seq. 
Contact potentials, 300; table of, 285 
Contact of two solutions, 219, 376, 382 
Contraction on formation, 171 
Convergence temperature, 154 
Co-ordinates, generalized, 18 
Copper chloride, 419; sulphate, 443 
Copper refining, 310 
Corpuscles, or negative ions, 191, 274 
CROMPTON, heat of neutralization, 352 
CRUIKSHANK, experiments on electrolytes, 


Cryohydrates, 49 

Cryohydric point, 142; lowering of, 146 
Crystalline structure of alloys, 68, 71, 

144; of ice, 47, 144 
Crystallization, 44, 81, 171, 392 
Crystalloid, definition of, 371 ; separation 

of, 380 

Crystals, mixed, 62, 69, 403 
Crystals, surface energy of, 45, 81 
Cupric chloride, 419; sulphate, 443 
Current, alternating, used in measuring 

electrolytic conductivity, 199 
Curves, concentration or solubility (Figs. 
16 to 32, 34, 35, 52, 64, 66), 55 to 79, 
89, 91, 328, 404; conductivity (Figs. 
52, 64), 203, 328; electrocapillary 
(Fig. 61), 288; fusion and solidification 
(Figs. 16 to 25, 40, 64), 55 to 70, 125, 
328; ionization (Fig. 64), 328; Phase 
Bule (Figs. 8 to 15, 31, 40), 39 to 54, 
76, 125; vapour pressure (Figs. 26 to 
30, 40, 41), 72 to 75, 125, 136; f (Figs. 
7, 21 to 24), 26, 64 to 67 
Cyanacetic acid, ionization constant of, 342 
Cycles, complete, 6; complex, 18 
CZAPSKI, temperature coefficient of cells, 

Dalton's law, 48 

DANIELL, cell, 182, 233; nature of ions, 

DAVY, Sir HUMPHRY, electrolysis, 178; 

electrolytic decomposition of alkalis, 

178; polarization, 181 
Decomposition, electrolytic. 178 ; of water 

178, 306, 307 
Decomposition voltages, 301 ; tables, 302 

DE COPPET, cryohydric temperatures, 

143; freezing point of solutions, 153; 

of supersaturated solutions, 80 



Density of solutions, 170, 333, 408; 
of supersaturated solutions, 80 

Depression of the freezing point, 126; 
see also Freezing Points 

DBS COUDRES, mercury concentration 
cell, 244 

DEVILLE and TROOST, occlusion, 82 

DE VRIES, isotonic solutions, 119 

DEWAB, occlusion, 82 

Diagrams of apparatus, absorptiometer 
(Fig. 33), 85; boiling point (Fig. 42), 
139; capillary electrometer (Fig. 60), 
282; electrolytic conductivity (Figs. 
47 to 51), 200 to 202; freezing point 
(Fig. 45), 155; ionic migration (Figs. 
54, 55, 56), 210, 217, 221; normal 
electrode (Fig. 62), 295; osmotic cell 
(Fig. 36), 96; polarization (Fig. 63), 303 

Dialysis, 388 

Dicldoracetic acid, 456 

DIETERICI, vapour pressure of sulphuric 
acid, 265 

Diethylamine, 475 

Diffusion, 369; constant of, 369 ; absolute 
value, 371; tables, 372, 373; of electro- 
lytes, 376; experiments on, 370; 
through membranes, 388; and osmotic 
pressure, 374; theory, 369, 374, 376 

Diffusivity or diffusion constant, 369 

Dilatometer, 46 

Dilution, effect of, 169; heat of, and 
osmotic pressure, 111; law of, 341, 
343, 344, 350 

Dimethylamine, 474 

Dissipation of energy, principle of, 110 

Dissociation, electrolytic, theory of, 206, 
312; and chemical activity, 335; heat 
of, 354 ; and osmotic pressure, 159, 316 ; 
of water, 358, 362 

Dissociation, hydrolytic, 364 

Dissolution, heat of, 112, 171 ; table, 117 

Divariant systems, 36 

DOLEZALEK, theory of accumulators, 264, 

DONDERS and HAMBURGER, temperature 
and osmotic pressure, 118, 119 

DONNAN, colloid solutions, 402; Hall 
effect in electrolytes, 385 

Double concentration cells, 250 

Double salts, 92; electrolysis, 195; 
deposition, 403 

Dropping electrodes, 278, 281 

DUHEM, theory of colloids, 394 

DUTOIT and ASTON, conductivities of 
solutions in acetone, 331; properties 
of solvents, 364 

ECKARD, dialysis, 389 
EDLER and OBEBBECK, potential differ- 
ences, 388 

Efflorescence of crystals, 54, 58 
Electric charge of ions, 188, 189 
Electric endosmose, 292 
Electricity, animal, 176; contact, 267 
Electro-capillary action, 282, 294 
Electro-chemical equivalents, 184; table 

of, 187 

Electro-chemical properties, table of, 407 
Electro-chemical series, 177, 296, 298, 


Electrodes, definition of, 180; of different 
concentration, 242; dropping, 278, 281 ; 
platinum, preparation of, 199; tapping, 

Electrolysis, 176, et seq.; of gases, 187 
Electrolytes, additive properties, 332^ 
coagulative properties, 395; conduc- 
tivity of, 197; conductivity of and depres- 
sion of freezing point, 160 ; diffusion of, 
376 ; equilibrium between, 346; measure- 
ment of conductivity of, 198; potential 
differences between, 381; solution 
pressure of, 256; thermal properties 
of, 352 

Electrolytic conductivity, 197, 408 
Electrolytic separations, 301 
Electrolytic solution pressure, 274, 280, 

297, 385 

Electrometer, capillary, 282 
Electromotive force, of galvanic cells, 

236, 238, 240, 243, 247, 257, 259, 381 
Electromotive series of metals, 177, 296, 

298, 299 

Electrons, theory of, 191, 274 

Electroplating, 178, 194, 307 

EMDEN, vapour pressure of solutions, 132 

Endosmose, electric, 292 

Energy, available or free, 28, 29; appli- 
cations to chemical change, 339; 
coordination of physical science, 165; 
dilution of solutions, 169; electro- 
capillary action, 284; electromotive 
force, 235; heat of ionization, 354; 
latent heat, 29; osmotic pressure, 103, 
113, 165 

Energy, conservation of, 1 ; internal, 19 ; 
surface, 43, 81, 402 

ENGEL and ETARD, influence of tem- 
perature on solubility, 91 

Engine, Carnot's reversible, 12 

Entropy, 20 

Equilibrium, 10, 32, 205, 225, 339 j 
conditions of, 25; electrolytic, 205, 
225, 339, 346; false, 11, 37; labile, 42; 
of phases, 33, 68, 78; in saturated 
solutions, 78 

Equivalent conductivity, tables of, 408 

Equivalent conductivity, curves showing 
(Fig. 52), 202, 203; influence of con- 
centration, 223; limiting value, 332; 



measurement at 0, 325; in various 

solvents, 321, 330, 362 
Equivalent, electro-chemical, 184; table 

of, 187 

ERMAN, voltaic pile, 178 
EBSKINE MURRAY, contact electricity, 272 
ETARD and ENGEL, influence of tempera- 
ture on solubility, 91 
Ethereal solutions, lowering of vapour 

pressure in, 132 

Ethyl alcohol, ionization in, 322, 330 
Ethylamine, 473 
Ethyl-sulphuric acid, 446 
EWAN, T., freezing, point of concentrated 

solutions, 150 
EWING and ROSENHAIN, structure of alloys 

(Fig. 44), 144 
Eutectic alloys, 60, 144 
Evolution of gases in polarization, 305 
EXNER, mercury-dropping electrodes, 280 ; 

single potential differences, 278 
EXNER and TUMA, single potential 

differences, 278 
Expansion, thermal, 171 ; of salt solutions, 

EYK VAN, equilibrium of solid and liquid 

phases, 68 
EYKMAN, freezing points, 150, 161 

False equilibrium, 11, 37 
FARADAY, early experiments on electro- 
lysis, 179; laws of electrolysis, 182; 

polarization, 181 ; vapour pressures, 132 
Faraday's laws, 182, 184, 332; in fused 

salts, 187; in gases, 187 
PAVRE, occlusion, 82 
Ferric chloride, 421; equilibrium of 

phases (Fig. 17), 56; efflorescence of 

crystals, 54; hydrolysis, 366 
Ferrous chloride, 420 
Ferrous sulphate, 444 
FICK, diffusion, 369, 371 
Films, gaseous, 82, 269, 271; liquid, 

conductivity of, 230 
FITZGERALD, osmotic pressure and surface 

tension, 100, 103 
FITZGERALD and TROUTON, conductivity 

of electrolytes, 204 
FITZPATRICK, electrolytic conductivity, 

201, 330; tables of electro-chemical 

properties, 211, 407 
Fluidity, ionic, 224, 356 
Fluidity of solutions, 409; temperature 

coefficient of, 409 
Forces, generalized, 19 
Formic acid, 452; freezing points of 

solutions in, 156 

Formic acid series, mobility of ions, 230 
FOURIER, conduction of heat, 369 
Practionation, 75, 403 

Free energy, see Energy 

Free surface of volatile liquid, 98 

Freezing points, 40, 126, 141, 153, 317; 

of alloys, 59, 62, 69, 144, 160; 

depression of the, 123, 126, 142, 147, 

149, 150, 156, 160, 162, 317, 320, 322, 

328, 330, 400; diagrams (Figs. 16 to 

25, 40, 64), 55 to 70, 125, 328; 

experimental methods (Fig. 45), 153, 

155, 158 ; non-aqueous solutions, 156, 

321, 330; connexion with electrolytic 

conductivity, 159, 316 to 332; with 

osmotic pressure, 126, 147, 152; with 

vapour pressure, 122, 125 

Friction coefficients, ionic, tables of, 379 

FUCHS, electromotive force of polarization, 

Fused salt cells, temperature coefficient 

of, 237 
Fusion curves, see Freezing Point Diagrams 

GALVANI, origin of electrolysis, 176 ; animal 

electricity, 176 
Galvanic circuit, distribution of potential 

in (Figs. 58, 59), 275 
Galvanism, identity with electricity, 178 
GAMGEE, colloids and crystalloids, 401 
Gas, adiabatic relations of an ideal, 17 ; 
electrolysis, 187, 189 ; polarization, 
305 ; solubility in liquids, 84, 130 ; 
solubility in mixed solutions, 87 ; 
solubility in salt solutions, 87; solu- 
bility in solids, 81 
Gas, battery, Grove's, 305 
Gaseous film, 82, 269, 271 
Gaseous pressure, identity with osmotic 

pressure, 104, 120, 166 
Gelation, 394 

Generalized co-ordinates, 18; forces, 19 
GIBBS, chemical potential, 25, 34 ; elec- 
tromotive force of reversible cells, 235 ; 
growth of crystals, 45, 81 ; latent heat 
equation, 38 ; phase rule, 35 ; theory 
of osrnotics, 109 ; transition cells, 258; 
surface energy, 81, 402 
GILBAULT, effect of pressure on electro- 
motive force, 240 
GILL, reversible heat of cell, 238 
GLADSTONE, colour of salt solutions, 335 
GOCKEL, temperature coefficient of electro- 
motive force, 237 

GOODWIN, double concentration cells, 252 
GORE, surface of contact of two solutions 


GORSKI and LASZOZYNSKI, equivalent con- 
ductivities, 331 
GRAHAM, colloids, 391, 396; diffusion, 

370, 372; occlusion, 82 
Gram-molecule, definition of, 4 
GRIFFITHS, freezing points of dilute solu- 



tions, 158, 321 ; mechanical equivalent 
of heat, 1 ; molecular lowering of freez- 
ing point, 147 

GKOSHAUS, properties of solutions, 333 
GROTTHUS, electrolytic chain, 180 ; electro- 
lytic decomposition, 179 
GROVE, cell, 235; gas cell, 305 
GULDBERG and WAAGE, the mass law, 

205, 340 
GUTHE and PATTERSON, electro-chemical 

equivalent of silver, 185 
GUTHRIE, alloys, 59; cryohydrates, 49; 
equilibria of mixtures of salts, 76 

Hall effect in electrolytes, 384 

HALL and KAHLENBERG, equivalent con- 
ductivity at 0, 325 

HAMBURGER and BONDERS, influence of 
temperature on osmotic pressure, 118 

HARDY, coagulation, 398 ; gelation, 394 ; 
gels, 393 

HAUTEFEUILLE and TROOST, occlusion, 82 

Heat, latent, equation, 29, 30, 37, 125 

Heat of dilution, calculation of freezing 
point from, 170 ; and osmotic pressure, 
111 ; of sulphuric acid, 266 

Heat of formation, determination, 238 ; 
table, 117 

Heat of ionization, 354 

Heat of precipitation, 117 

Heat of reaction, 237 

Heat of solution and solubility, 90, 115; 
and osmotic pressure, 112 ; of super- 
saturated solutions, 80 ; table of, 117 

Heat, reversible, of cell, 237 

Heat, specific, of supersaturated solu- 
tions, 80 

HEIDENHAIN and MEYER, absbrptiometer 
(Fig. 32), 84 

HEIM, electrical conductivity of super- 
saturated solutions, 80 

chemical equivalent of silver, 185 

HELMHOLTZ, VON, electro-capillary action, 
283 ; electric endosmose, 293 ; electro- 
motive force, 235 ; free energy, 28 ; 
migration in concentration cells, 250 ; 
osmotics, 109 ; potential differences, 

HENDERSON and STROUD, measurement of 
electrolytic conductivity, 197 

Henry's law, 85 ; confirmed by Bunsen, 86 

HESS, law of thermo-neutrality, 352 

HEYCOCK and NEVILLE, on alloys, 59 et 
seq. ; copper and tin (Fig. 19), 71 ; 
depression of freezing point, 160 ; gold 
and aluminium (Fig. 30), 62, and (Fig. 
43), 144 ; microscopic investigations, 
59, 61, 62, 68, 71, 144 (Fig. 43); 
osmotic pressure, 244 

w. s. 

tivity of pure water, 193, 358 

HISINGER and BERZELIUS, electrolysis of 
salt solutions, 178 

HISSINK, equilibrium of solid and liquid 
phases, 68 

HITTORF, chemical activity and conduc- 
tivity, 336 ; complex ions, 226 ; electro- 
chemical series, 300; electrodes of 
concentration cells, 254 ; electrolysis of 
double salts, 195 ; migration of ions, 
208, 210, 383; secondary action in 
electroplating, 194 

HOFF, VAN 'T, diffusion and osmotic pres- 
sure, 376 ; dilution law, 343 ; influence 
of pressure on solubility, 90 ; latent 
heat equation, 38 ; molecular depression 
of freezing point, 149 ; osmotic pressure, 
absolute value of, 103, 107; osmotic 
theory, 172 ; solubility of mixtures, 403 ; 
table "of heats of solution or precipita- 
tion, 117 

HOFF, VAN 'T, COHEN and BREDIG, transi- 
tion cells, 261 

HOITSEMA, solid solutions, 83 

HOITSEMA and KOOZEBOOM, occlusion, 82 

HOLBORN and KOHLRAUSCH, equivalent 
and electrochemical weights, 186 ; trans- 
port numbers, 211 

HOPFGARTNER, transport numbers, 210 

HORN and MORSE, semi-permeable mem- 
branes, 406 

HORSFORD, resistance of electrolytes, 198 

HOSKING and LYLE, ionic viscosity, 406 

HOULLEVIGNE and OSMOND, electrolysis of 
salts of iron, 196 

Hydrated solids, 53 

Hydrates, crystallization of, 171 ; forma- 
tion of, 57 ; isolation of, 171 

Hydrate theory of solution, 170 

Hydriodic acid, 422 

Hydrobromic acid, 421 

Hydrochloric acid, 410 ; solubility, 75, 
84, 87 

Hydrocyanic acid, 451 ; solutions in, 406 

Hydroferrocyanic acid, 451 

Hydrofluoric acid, 425 

Hydrogels, 393 

Hydrogen, concentration cell, 242 

Hydrolysis, 364 

Hydrolytic dissociation, 364 

Hydrosols, 393 

Ice, arctic, 145 ; crystalline varieties of, 

47 ; structure of, 47, 144 ; 
Index, refractive, 171 ; and boundary of 

solutions, 121 
Indicator diagram, 14 
Internal energy, 19 
Inversion of cane sugar, 337 




Inversion point, 41, 55, 259, 262 

Iodides, mixture of, freezing point curve 
(Fig. 25), 69, 70 

lodopropionic acid, 460 

Ionic concentration, 245, 253, 300, 308, 
317, 326, 340, 348, 365, 397 

Ionic fluidity, 224, 356, 379, 406 

Ionic migration, theory of, 208, 213, 383 

Ionic viscosity, 224, 356, 379, 406 

Ions, charge on, 189 ; complex, 226, 322, 
331 ; as condensation nuclei, 43 ; dis- 
sociation, in electrolysis, 206 ; fluidity, 
224; migration, 207, 210; mobility, 
211 to 226, 229, 230; nature of, 191 

lonization, 225, 316, 325, 328, 337, 341, 
354, 358, 362, 406; heat of, 354; in 
various solvents, 330, 331, 362, 406 

lonization of dilute solutions at 0, 321, 
325, 328 

Irreversible cells, 262 

Isobutylamine, 473 

Isobutyric acid, 461 

Isohydric solutions, 346 

Isomorphous salts, 92 

Isotonic coefficients, 120 

Isotonic solutions, 119 

JAHN, heats of formation, 238 ; reactions 
in electrolysis, 194 ; reversible heat in 
cells, 237, 238 ; temperature coefficient 
of cells, 237 

JONES, H. C., freezing points, 153 ; ioni- 
zation at 0, 328 

JOULE, measurement of thermal equiva- 
lent of work, 1 

of succinic acid, 93 

KAHLENBEBG, abnormal molecular weights 
in solution, 322, 332; chemical re- 
actions, 339; freezing point data, 330 

KAHLENBEKG and HALL, equivalent con- 
ductivity at 0, 325 

power of solvents, 363, 406 

KELVIN, Lord (Sir Wm Thomson), 
capillary action, 100; dropping elec- 
trodes, 278; electromotive force and 
heat of reaction, 237; latent heat 
equation, 38 ; principle of classification, 
110; similarity of laws for gases and 
solutions, 169; thermo-electricity, 272 

KISTIAKOWSKY, electrolytic transport 
numbers, 210 

KOHLRAUSCH, F., alternating currents, 
199 ; boundaries of solutions, 220 ; 
conductivity of solutions, 80, 199, 202; 
conductivity of water, 193; electrolysis 
of platinum chloride, 196; equivalent 
conductivity of solutions, 202; ionic 

friction coefficients, 379 ; ionic mobility 
or velocity, 211 ; Ohm's law in electro- 
lysis, 177, 204; use of telephone, 199 

KOHLRAUSCH, F. and W., electro-chemical 
equivalent of silver, 185 

tivity of pure water, 193, 358 

chemical and equivalent weights, 186; 
tables of electrolytic transport numbers, 

KONOWALOFF, vapour pressure of miscible 
liquids (Figs. 26 to 30), 72 to 75 

KRAPIWIN and ZELINSKY, conductivity of 
solutions in alcohol, 330 

KUMMEL, electrolytic transport numbers, 

KUSCHEL, electrolytic transport numbers, 

LAAR VAN, electrocapillary phenomena, 287 

Labile equilibrium, 42 

Lactic acid, 462 

LAMB, theory of electric endosmose, 294 

LARMOR, diffusion, 374, 384; electro- 
capillary action, 283 ; migration of 
ions, 209; osmotic theory, 105, 109; 
thermo-electricity, 272 

LASZOZYNSKI, conductivity of solutions in 
acetone, 331 

LASZOZYNSKI and GORSKI, conductivity of 
solutions in pyridine, 331 

Latent heat, and available energy, 29, 
30; le Chatelier's theorem, 37, 38; 
and boiling point, 136; and freezing 
point, 142; osmotic pressure and heat 
of solution, 114; freezing point and 
vapour pressure, 125 

Law of available or free energy, 28, 29 

Law of diffusion, Fick's, 371 

Law, dilution, 341, 344 

Law of thermoneutrality, 352 

Law, Henry's, 85 

Law, the mass, 339 

Law, Ohm's, 197, 204 

Law, Volta's, 276 

Laws, Faraday's, of electrolysis, 182 

Laws of osmotic pressure, 103, 120, 166 

Laws of thermodynamics, 1, 2 

Laws of vapour pressure for mixed 
vapours, 71, 406 

LE BLANC, decomposition point, 308; 
evolution of gases, 305; polarization, 
301, 303, 304; single potential differ- 
ences, 387 

Lead nitrate, 434 

LE CHATELIER, alloys, 59; latent heat, 38 

LEHFELDT, electrolytic solution pressure, 
299 ; electromotive force of concentration 
cell, 247 



LEHMANN, nature of amorphous bodies, 47 
LENZ, electrolytic transport numbers, 210 
LINCOLN, ionization in various solvents, 

LINDER and PICTON, coagulative power 

of electrolytes, 395, 396, 399; nature 

of colloidal solution, 400 
LIPPMANN, capillary electrometer, 282, 284 
Liquid cells, 382 

Liquid, free surface of a volatile, 98 
Liquids, miscibility of, 88 
Liquids, mixed, laws of vapour pressure 

for, 71, 403; mixed, solubility in, 93; 

separation of, by fractionation, 74, 403 ; 

solubility of, in liquids, 88; under- 
cooled, 42, 392, 402 
Liquidus curve, 66, 69 
Lithium butyrate, 461 ; carbonate, 449 ; 

chlorate, 435 ; chloride, 414 ; formate, 

453; hydrate, 470; iodide, 424; 

isobutyrate, 462; nitrate, 430; per- 

chlorate, 436; propionate,45$; sulphate, 

440; trichloracetate, 458 
LODGE, Sir OLIVER, measurement of ionic 

velocity, 216; potential differences, 270 
LOEB and NERNST, electrolytic transport 

numbers, 210 
LONGDEN, conductivity of metallic films, 

LOOMIS, freezing points of solutions, 

153, 321, 328 
LYLE and HOSKING, conductivity and 

fluidity of solutions, 406 

MACGREGOR, freezing point data, 330; 
measurement of electrolytic conduc- 
tivity, 201 ; properties of dilute solutions, 

Magnesium chloride, 417; nitrate, 433; 
sulphate, 440 

Magnetic rotation of solutions, 171, 335 

Manganese chloride, 420 

MARIGNAC, thermal capacity of salt 
solutions, 335 

Mass law of chemical action, 205, 339 

MASSON, ORME, ionic velocities, 220; 
table, 221 

Membranes, diffusion through, 388; 
perfect semipermeable, 102; semi- 
permeable, 95, 406 

MENDELEEFF, hydrates in solutions, 170 

Mercury concentration cells, 243 

Metachlorbenzoic acid, 468 

Metals, electromotive series, 177, 296, 
298, 299 

Meta-nitrobenzoic acid, 467 

Meta-oxybenzoic acid, 466 

Methyl alcohol, ionization in, 331 

Methylamine, 472 

Methyl sulphuric acid, 445 

MEYER, G., electromotive force of con- 
centration cell, 246; mercury dropping 
electrodes, 281 

MEYER and HEIDENHAIN, absorptiometer 
(Fig. 33), 84 

Microscopic study of alloys, 68, 71, 144 

Migration of ions, 207, 383 

Migration constants, 210, 212, 222, 408 

Migration, elimination of, in concentra- 
tion cells, 250 

Migration, ionic, theory of, 208, 383 

Miscibility of liquids, 88 

Mixture, definition, 48; solubility, 92 
(Fig. 66), 403, 404 

Mobility, ionic, 214 to 226 

Molecular bombardment, theory of, 167 

Molecular complexes, 170 

Molecular weight in solution, 140, 158, 
316, 324 

MONCKMAN and J. J. THOMSON, filtration 
of permanganate, 98 

MOND, EAMSAY and SHIELDS, occlusion, 
82, 83 

Monobromacetic acid, 457 

Monochloracetic acid, 455 

Monovariant systems, 36 

MORGAN, ionization of double cyanides, 

MORSE and HORN, semipermeable mem- 
branes, 406 

MOSER, theory of concentration cell, 242 

NERNST, chemical cells, 255; concentra- 
tion cells, 248, 250, 382; diffusion, 
281, 374, 376; electrolytic solution 
pressure, 274, 280, 385; equilibrium in 
solutions, 347; galvanic cells, 240; 
ionizing power of solvents, 361 ; liquid 
cells, 252, 382; metallic and electro- 
lytic conductivity, 184; reversible heat 
of mercury cells, 238; solubility in 
mixed liquids, 93; solubility of silver 
acetate, 93 

NERNST and ABEGG, theory of freezing 
point determination, 153 

NERNST and LOEB, electrolytic transport 
numbers, 210 

NEUMANN, single potential differences, 
296, 387 

NEVILLE, alloys forming mixed crystals, 
69; freezing and melting point curves, 


NICHOLSON and CARLISLE, early experi- 
ments on electrolysis, 177 

Nickel chloride, 420; sulphate, 444 

NICOL, additive properties of salt solutions, 

Nitric acid, 426 



Nitrobenzene, solutions in, freezing points 

of, 157 
Non-electrolytes, freezing points, 153, 

319; osmotic pressures, 104, 120; 

vapour pressures, 129 
Non-variant systems, 35 
NOVAK, ionization of water, 362 
NOYES, electrolytic transport numbers, 

NOYES, BBEDIG and OSWALD, freezing 

points of concentrated solutions, 162 

OBEBBECK, polarization, 304 

Occlusion, 82 

OETTINGEN, VON, contact potentials, 300 

Ohm's law in electrolytes, 197, 204 

Organic cells and osmotic pressure, 118, 

OSMOND, alloys, 59, 61 

OSMOND and HOULLEVIGNE, electrolysis of 
salts of iron, 196 

Osmotic pressure, 95, et seq., absolute 
value of, 103, 106, 109, 120, 166, 316 
to 332 ; and boiling point, 122 ; cor- 
puscular, 274 ; and diffusion, 374 ; 
effect of concentration on, 117 ; effect 
of temperature on, 118 ; of electrolytes, 
116, 120, 157, 159, 175, 316 to 332; 
experimental measurement, 95, 117, 
406; and freezing point, 126, 152; 
and gaseous pressure, 103, 106, 109, 
120, 166; and heat of solution, 112; 
of metallic solutions, 141, 160, 244; 
and organic cells, 118, 389 ; and surface 
tension, 97, 100, 390 ; theoretical laws 
of, 103, 120, 166, 316 to 332; and 
vapour pressure, 91, 98, 123, 127 

Osmotics, theory of, 103 to 112, 120, 
166, 175, 316 to 332 

OSTWALD, additive properties of solutions, 
333 ; acidity, measurement of, 336 ; 
colour of solutions, 334 ; conditions for 
production of current, 234 ; concentra- 
tion cells, 253 ; dilution law, 341, 344 ; 
dissociation of water, 360 ; heat of 
ionization, 357; ionic mobility, 229, 
230 ; mass law, 340 ; solubility, 84, 85, 
92; volume change of salt solutions, 

trated solutions, 162 

OSTWALD and WALKEB, measurement of 
vapour pressure, 133 

Oxidation and reduction cells, 258 

Oxygen, valency and ionizing power, 364 

PADOA and BBUNI, solid solutions, 403 
PALMAEB, electrolytic solution pressure, 

PASCHEN, mercury dropping electrodes, 281 

PATTEBSON and GUTHE, electro-chemical 
equivalent of silver, 185 

PELLAT, electro-capillary action, 284 

PELLAT and POTIEB, electro-chemical equi- 
valent of silver, 185 

Peltier effect, 238, 273 

PEBKIN, magnetic rotation in solutions, 

PEBEY and AYETON, metallic potential 
differences, 268 

PFEFFEE, osmotic pressure, 95, 117, 120 

Phase Rule, 32 et seq., 394, 402, 433 

Phase, definition of, 33 

Phases, equilibrium of, 33 

Phenol and water, concentration curve 
(Fig. 18), 58 

PICKEBING, concentrated solutions, 163 ; 
densities of solutions, 170; freezing 
points, 153 ; hydrate theory of solution, 
170, 171 ; permeability of membranes, 
96, 172 

Pile, Volta's, 176 

PLANCK, diffusion, 281, 374, 384 ; electro- 
lytic dissociation, 317 ; galvanic cells, 
240; liquid cells, 383 

Platinum thermometry, 59, 147, 158 

POINCABE, L., temperature coefficient of 
fused salt cells, 237 

POISEUILLE, laws of, 293 

Polarization, 181, 300, 303, 305 ; elimina- 
tion of 182, 198, 199; and contact 
electricity, 267 

Polymerisation, gaseous, 161 ; at high 
concentrations, 161 ; in solutions, 364 ; 
in solvents of benzene series, 159 

PONSOT, convergence temperature, 154; 
freezing points of solutions, 161, 162. 

Potassium acetate, 454 ; bichromate, 450 ; 
bromide, 422; butyrate, 460; carbonate, 
448 ; chlorate, 434 ; chloride, 411 ; 
chromate, 450 ; cyanide, 451 ; fluoride, 
426 ; formate, 452 ; hydrate, 468 ; iodide, 
423; isobutyrate, 462; lactate, 463; 
nitrate, 427 ; oxalate, 464 ; perchlorate, 
436 ; propionate, 458 ; sulphate, 437 ; 
trichloracetate, 457 

Potassium chloride, ionization, 321, 327, 

Potential, chemical, 25, 34; graphical 
method of representing (Figs. 7, 21 to 
24), 26, 64 to 67 

Potential differences, 267 et seq., 381; 
between electrolytes, 242 et seq., 381; 
in galvanic circuit (Figs. 58, 59), 275; 
single, 275, 294; tables of, 285, 296; 
summation of, 268, 276 

Potential, thermodynamic, 23, 25, 64 

POTIEB and PELLAT, electro-chemical 
equivalent of silver, 185 



POINTING, depression of freezing point, 
126 ; osmotic and gaseous pressure, 
173 ; theory of osmotic pressure, 174 

Pressure, osmotic, see osmotic pressure 

Pressure, effect on electromotive force of 
cells, 239; effect 011 metals, 41 ; effect 
on solubility, 83, 85, 90 ; vapour, see 
vapour pressure 

Pyridine, equivalent conductivity of solu- 
tions in, 331, 332 

QUINCKE, coagulation, 399 ; electric en- 
dosmose, 292 

RAMSAY, vapour pressure of amalgams, 

141, 244 
RAMSAY, MOND and SHIELDS, occlusion, 

82, 83 

RANKINE, latent heat equation, 38 
RAOULT, freezing points, 147, 153, 156, 

161, 162, 328; polarization, 301; 

vapour pressures, 129, 132 
RAOULT and RECOURA, vapour pressure of 

acetic acid, 129 
RAYLEIGH, Lord, osmotic pressure, 106, 

110; distillation, 403 

electro-chemical equivalent of silver, 

Rays, Rontgen, charge on ions produced 

by, 189 
RECOURA and RAOULT, vapour pressure of 

acetic acid, 129 
Reduction cells, 258 
REED, C. J., accumulators, 263 
Refining, copper, 310 
Refractive index of solutions, 171, 335 ; 

used to determine boundary of solutions, 

REGNAULT, measurement of boiling point, 


REICHER, transition point of sulphur, 46 
REINDERS, equilibrium of solid and liquid 

phases, 68 
REINOLD and RUCKER, conductivity of 

liquid films. 230 
Residual affinity, 173 
Resistance of electrolytes, 197 et seq.; 

see Conductivity 
Reversible engines, 12 
Reversible processes, 8 
chemical equivalent of silver, 185 
RUN, VAN ALKEMADE, VAN, f curves, 26, 63 
RITTER, action of accumulator, 181; 

electromotive series of metals, 177 
minium alloy, 61 

59, 68 

Rontgen rays, charge on ions produced 
by, 189 

ROOZEBOOM, allotropic solid (Fig. 12), 
46, 47; alloys, 59; equilibrium of 
hydrates, 57 ; liquidus and solidus 
curves, 66; mixtures of iodides (Fig. 25), 
69, 70; theory of solid solutions, 63, 68 

ROOZEBOOM and HOITSEMA, occlusion, 82 

ROSCOE, distillation of nitric and hydro- 
chloric acids, 75 

of Chemistry, 93 

ROSENHAIN and EWING, structure of alloys 
(Fig. 44), 145 

Rotation, magnetic, of solutions, 171, 335 

ROTHMUND, electrocapillary action, 285 

ROWLAND, mechanical equivalent of heat, 1 

RUCKER and REINOLD, conductivity of 
liquid films, 230 

RUDOLPHI, dilution law, 343 

RUDORFF, freezing points of solutions, 
153; solubility of mixtures, 92 

Rule, the Phase, 32 et seq., 394, 402, 

RUTHERFORD, velocity of gaseous ions, 189 

SACK, conductivity of copper sulphate, 357 

Salicylic acid, 466 

Salt deposits, oceanic, 403 

Salts, double, 92; electrolysis of, 195 

Salts, isomorphous, 92 

Salts and water, equilibrium of, 50 et seq., 


SAND, electrolysis of mixed solutions, 399 
Sandbanks, formation of, 401 
Saturated solutions, 49 et seq., 78 et seq., 

112, 143, 403 

Saturation, time required for, 90 
SCHEFFER, diffusion, 372, 380 
SCHLUNDT, complex ions, 226 ; transport 

numbers, 332 

power of solvents, 363, 406 
SCHORLEMMER and ROSCOE, Text-book of 

Chemistry, 93 

SCHULZE, coagulative power of electro- 
lytes, 395, 399 

Sea-water, deposition of salts from, 405 
Secondary action, 192, 194 
Secondary cells, 263 
Selenic acid, 446 
Semi-permeable membranes, 95, 102, 168, 

172, 406; perfect, 102 
Series of metals, electromotive, 177, 268, 

296, 298, 299 

SETSCHENOFF, absorption coefficient, 87 
SHAW, W. N., electric endosmose, 195, 

294 ; electro-chemical equivalent of 

copper, 186 
SHELDON and DOWNING, ionic velocity, 217 



SHIELDS, hydrolysis, 366 

SHIELDS, KAMSAY, and MOND, occlusion, 

82, 83 

electro-chemical equivalent of silver, 185 
Silver, electro- chemical equivalent of, 184, 


Silver acetate, solubility of, 93 
Silver nitrate, 430; sulphate, 440 
Single potential differences, 267 to 292, 294 
SKINNER, electrolysis of solutions in 

pyridine, 332 
SMALE, temperature coefficient of Groves' 

gas cell, 237 
SMEE, galvanic cell, 182 
SMITH, S. W. J., electro-capillary action, 

283 (Fig. 61), 288; mercury-dropping 

electrodes, 281 
Sodium acetate, 455 ; butyrate, 461 ; 

carbonate, 449; chloride, 413; ethyl 

sulphate, 446; formate, 452; hydrate, 

469; iodide, 424; isobutyrate, 462; 

lactate, 463 ; methyl sulphate, 445 ; 

nitrate, 429 ; perchlorate, 436 ; pro- 

pionate, 459 ; selenate, 447 ; sulphate, 

439; trichloracetate, 457 
Sodium sulphate, Phase Kule diagram 

(Fig. 15), 54; solubility curves (Figs. 

16, 32), 55, 79 

Solidifying point, see Freezing point 
Solid solutions, 62 to 71, 83, 403 
Solids, allotropic, 45; amorphous, 47, 

392; hydrated, 53 
Solids, solubility in liquids, 89, 90 
Solidus curve, 66, 69 
Solubility, 27, 48, 55, 78; curves (Figs. 

16 to 25, 32, 34, 35, 66), 55 to 70, 79, 

89, 91, 404; of gases, 81, 84, 85, 87; 

of liquids, 88, 92; of solids, 89, 90; 

tables of, 93, 94 
Solute, definition of, 49; anhydrous, 50; 

hydrated, 53 
Solution, 48, 77, 165; heat of, 112; 

table of heats of, 117; theories of, 165 
Solvent, 49, 361, 406 ; specific inductive 

capacity, 362, 406 
SORET, temperature, diffusion and osmotic 

pressure, 376 

SPIERS, contact electricity, 271 
SPRING, effect of .pressure on metals, 41 

59, 68 

Stassfurt, salt deposits, 403 
STEAD, alloys, 59, 68 
STEELE, complex ions, 226, 228; ionic 

velocities, 221 

STEFAN, theory of diffusion, 372 
STREINTZ, accumulators, 264 
Strontium chloride, 415; hydrate, 471; 

nitrate, 431 

STROUD and HENDERSON, measurement of 

electrolytic resistance, 199 
Structure of ice, alloys, etc., 143 
Succinic acid, solubility of, 93 
Sulphur, allotropic forms (Fig. 12), 45 
Sulphur dioxide, solubility of, 86 
Sulphuric acid, 437; densities of, 170; 

heat of dilution, 266 
Sulphurous acid, 446 
Supersaturated solutions, properties of, 80 
Supersaturation, 43, 44, 80 
Surface energy or surface tension, 43, 44, 

80, 287; and potential difference, 28S; 

osmotic pressure, 97, 100, 390 
Surfusion, 42, 45, 80, 155, 392 
Systems, divariant, 36; monovariant, 36; 

non variant, 35 ; one component, 39 ; 

two component, 49 

Tables, accelerating powers, 338 ; avidities, 
337, 338; boiling points, 137, 331; 
conductivities of acids, 338 ; contact 
potentials, 285; cryohydric tempera- 
tures, 143; decomposition voltage, 302, 
308; diffusion constants, 372, 373; 
electro-chemical properties, 407 ; electro- 
lytic solution pressure, 298; electro- 
motive force of accumulators, 266 ; 
amalgam cells, 285, concentration cells, 
250, 252, liquid cells, 382, 383, mercury 
cells, 290; equivalent conductivities at 
0, 326, 327; equivalent weights and 
electro-chemical equivalents, 187; freez- 
ing points, 149, 156, aqueous solutions 
of alcohol, 163, cane sugar, 163, electro- 
lytes, 319, 320, 328, 330 ; non-aqueous 
solutions in acetic acid, benzene, formic 
acid, 156, nitrobenzene 157; heat of 
ionization, 355, 358; heat of neutrali- 
zation, 356 ; heat of precipitation or 
solution, 117; hydrolytic dissociation, 
366 ; ionic friction coefficients, 379 ; 
ionic mobilities, 211, 212, 215, 218, 221, 
222, 223 ; ionization constants, 242 ; 
ionization of barium chloride, 330, 
potassium chloride, 328, of solutions in 
alcohols, 331; migration constants or 
transport numbers, 212, 222, 408 ; 
potential differences, 296; reversible 
heat of cells, 238; solubility, 93, 
94 ; transport numbers or migration 
constants, 212, 222, 408; vapour pres- 
sures, 132, 133, 134, 137, 331 

TAMMANN, alloys, freezing point of, 160.; 
amorphous solids, 47 ; crystalline varie- 
ties of ice, 47; crystallization, 44; 
osmotic pressure, 119, 120; pressure 
and evolution of hydrogen, 386; vapour 
pressures, 132, 134 

Tartaric acid, 465 



TAYLOR, A. E., potential differences, 388 

Telephone, used as indicator, 199 

Temperature, absolute scale of, 15; co- 
efficient of cells, 237; coefficient of 
conductivity, 408 ; coefficient of fluidity, 
409 ; convergence or equilibrium, 154 

TERESCHIN, specific inductive capacity of 
solvents, 362 

Theory of chemical combination, 169; 
of direct molecular bombardment, 167 ; 
dissociation, hydrate, of solution, 170; 
of osmotics, 109 

Theories of solution, 165 

Thermal properties of electrolytes, 333, 
335, 352 

Thermodynamics, elements of, 1 to 31 ; 
laws of, 1, 2 

Thermodynamic potential, 23, 25 

Thermo-electricity, 272 

Thermometry, platinum, 59, 147, 158 

Thermo-neutrality, law of, 352 

THOMSEN, affinity, 336; heat of dilution 
of sulphuric acid, 266; heat of neu- 
tralization, 352 

THOMSON, JAMES, latent heat equation, 38 

THOMSON, J. J., charge on ions, 189; 
corpuscles, 191; effect of pressure on 
cells, 240; electrolysis of gases, 187; 
ionizing power of solvents, 361 

THOMSON, J. J., and MONCKMAN, filtration 
of potassium permanganate, 98 


Three component systems, 76 

TILDEN, solution and chemical action, 172 

Tin, solutions in, lowering of freezing 
point, 160 

TOWNSEND, J. S., charge on ions, 190 

Transition cells, 258 

Transition points 41, 55, 259 (Fig. 57), 
261, 262 

Transport numbers, 207, 210, 212, 408 

TBAUBE, preparation of semi-permeable 
membranes, 95 

Triangular diagram (Fig. 31), 76 

Triclilor acetic acid, 456 

Trietliylamine, 475 

Trimethylamine, 475 

TROOST and DEVILLE, occlusion, 82 

TROOST and HAUTEFEUILLE, occlusion, 82 

TROUTON and FITZGERALD, conductivity 
of electrolytes, 204 

TUMA and EXNER, single potential 
differences, 278 

VALSON, specific gravity of salt solutions, 

Vapour pressure, abnormal, 129, 134; 

of amalgams, 141; calculation of, 130; 

of concentrated solutions, 126; curves 

for, water and alcohol (Fig. 27), 74, 403, 
water and formic acid (Fig. 30), 75, 
water and isobutyl alcohol (Fig. 25), 
72, water and methyl alcohol (Fig. 28), 
74, water and propyl alcohol (Fig. 26), 
74; measurement, 132; ethereal solu- 
tions, 132; and freezing points, 122; 
mixed solutions, 71, 403; and osmotic 
pressure, 98, 127; tables, 132, 133, 
134, 137, 331 

Vapour, supersaturated, 42 

Velocity of the ions, 179, 188, 189, 208 
to 230, 312, 376, 379; absolute, 214; 
measurement of, 216, 217; tables of, 
215, 218, 221, 222, 223 

Viscosity of solutions, 335, 409; ionic, 
224, 356 

VOLLMER, conductivities in alcohol, 330, 

VOIGHTLANDER, diffusion, 372, 375 

Volatile components, 88 

Volatile liquid, free surface of, 98 

VOLTA, contact electricity, 267; early 
experiments on electrolysis, 176 

Volta's law, summation of potential 
differences, 268, 276 

Volta's pile, 176, 178 

Voltage decomposition, 181, 301, 308 to 

Voltameter, silver, 184 

Volume, change of, and osmotic pressure, 
111; of solutions, 333, 336 

WAAGE and GULDBERG, the mass law, 

205, 340 
WAALS, VAN DER, equation of state for 

gases, 161 
WADE, E. B. H., measurement of boiling 

points, 140 
WALKER, J., vapour pressures, 132; 

hydrolysis, 365 
WALKER and OSTWALD, measurement of 

vapour pressures, 133 
WARBURG, mercury-dropping electrodes, 

280; surface tension and electromotive 

force, 291 
Water, conductivity of, 193, 203, 358, 

362; decomposition of, in electrolysis, 

178, 193, 306; ionization, 193, 203, 

358, 362; preparation of pure, 193, 

358; sea, freezing of, 143 
WATTS, Dictionary of Chemistry, 93 
WEBER, C. L., specific ionic velocity of 

copper ions, 216 

WEBER, H., boundaries of solutions, 220 
WEBER, H. F., diffusion, 371 
Weights, equivalent and electro-chemical, 

187; molecular, determination of, in 

solution, 140, 158; abnormal, 108, 159, 

322, 332 



WEKNEB, molecular weight of pyridine, 332 
Weston cell, transition point, 261 
WEYPBECHT, freezing of sea-water, 145 
WHETHAM, W. C. D., coagulative power 
of electrolytes, 396; complex ions, 226; 
conductivity, 195, 201, 204, 325; 
hydrolysis, 367; ionization power of 
solvents, 362; mobile equilibrium in 
electrolytes, 225; preparation of plati- 
num electrodes, 200; specific ionic 
velocities, 217 
WIEDEMANN, electrolytic conductivity, 

198; endosmose, 292 
Wus, dissociation of water, 360 
WILDEBMANN, freezing points, 153 
WILLIAMSON, theory of chemical change, 

WILSON, C. T. R., charge on ions, 189; 

supersaturation and condensation of 

water vapour, 43 
WOELFEB, boiling points of solutions in 

alcohol, 331 
WOLLASTON, galvanism and electricity, 

WOOD, B. W., equivalent conductivity, 


Work and energy, 3 
WULLNEB, vapour pressures, 132 

f curves (Figs. 7, 21, 22, 23, 24), 26, 
64, 67 

ZELINSKY and KEAPIWIN, equivalent con- 
ductivities in alcohol, 330 

ZENGELIS, electrodes of concentration 
cells, 254 

Zinc chloride, 418 ; sulphate, 441 





OCT 17 IW5 

LD 21-100m-12,'43(879Gs) 

J 4 I 3