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1. The concept of temperature. Heat and cold belong to the 
most fundamental of our sensations. From time immemorial the 
phenomena connected with them received a great deal of practical 
attention because of their vast importance in everyday life, but they 
were not made amenable to scientific treatment until a comparatively 
recent date. The possibility of doing this is due to the following 
empirical facts: (1) When a hot and a cold body are brought in con- 
tact, the former gets cooler, the latter warmer. After the lapse of a 
sufficiently long time a stationary state is reached in which no further 
changes occur. We say, then, that the two bodies are in thermal 
equilibrium. This equilibrium persists if the bodies are separated 
and then brought together a second time (provided they were not 
exposed, in the meantime, to the influence of other hotter or colder 
bodies). (2) Two bodies which are separately in thermal equi- 
librium with the same third are in equilibrium with each other. We 
say, then, that they have the same temperature. 

The second fact permits us to use one suitable body as a stand- 
ard or "thermometer" with which to compare the temperatures of 
others. In order to determine the degree of heat of the thermometer, 
one, usually, has recourse to the property of matter of changing its 
size or volume when heated. In this way a scale of temperatures is 
established. The current types of thermometers are well known and 
need not be described here. 

When brought into equilibrium with any body, the thermometer 
associates the degree of heat of this body with a definite reading of its 
scale. It replaces the vague notion of hot or cold by a precise numerical 
datum. Its invention was, therefore, the necessary antecedent of 
a scientific analysis of the phenomena of heat. (Compare 
section 39). 


2. Subject of thermodynamics. Definitions. Thermodynamics 
deals with systems which, in addition to mechanical and electromag- 
netic parameters, are described by a specifically thermal one, namely, 
the temperature or some equivalent of it. We have seen that for the 
very definition of temperature one must invoke the concept of thermal 
equilibrium. Thermodynamics is, therefore, of necessity, essentially 
a science about the conditions of equilibrium of systems and about 
the processes which can go on in states little different from the 
state of equilibrium. We introduce the following terms which, for 
the most part, are those in general usage: 

A homogeneous system is uniform in every part, with respect both to 
its chemical composition and to its physical condition. The fact men- 
tioned in the preceding section that two bodies set themselves into 
thermal equilibrium applies, of course, also to two adjacent parts of 
the same body. The statement that the temperature of a homogeneous 
system is uniform throughout is, therefore, equivalent to saying that 
it is in equilibrium. 

A simple homogeneous system or, for short, a simple system is one 
completely defined by the three parameters V (volume), p (pressure), 
and T (temperature). Such are gases and liquids in a static condition 
and solids in a state of isotropic stress removed from the action of 
gravitational and electromagnetic forces. The case of one or several 
simple systems is, by far, the most important. Simple systems offer 
the most interesting applications, and their treatment is sufficiently 
typical to bring out all the methods of thermodynamics so that a 
generalization is easily made when necessary. We shall, therefore, 
restrict ourselves in many of our discussions to simple systems, and 
shall take up the more general case only when this is required for 
special applications. 

A heterogeneous system is composed of a number of homogeneous 
ones separated either by surfaces of discontinuity or by wall partitions. 
Those homogeneous parts are called the phases of the system. 

We shall admit the existence and the use of partitions of different 
types of which the most important are the heat-conducting one which 
permits two bodies separated by it to set themselves into thermal 
equilibrium, and the insulating or adiabatic partition which prevents 
any exchange of heat. 

3. The thermal equation of state. We have mentioned in section 1 
that the temperature of a thermometer is usually inferred from its 
extension. The deeper reason why it is possible to do so lies in a very 
fundamental and general property of every simple system: there 
exists a functional relation between the temperature, the volume, and 


the pressure which is called the equation of state, or more accurately, 
the thermal equation of state. This relation between T, , and 7 
can be written in the form 

p-f(V,T). (1.01) 

Each of the three variables can, therefore, be regarded as a function 
of the other two. Subjecting the variables to small changes, we can 
form the following three partial derivatives: 

(1) We keep the pressure constant (A/> = 0) and give to 7 and T 
the small increments A 7 and AT. The limit of the ratio, for AT 1 = 0, 
is the partial derivative lim(A7/Ar) = (97/37%. (It is customary 
to indicate the parameter which is kept constant as a subscript.) 
Divided by the volume 7, this partial defines the coefficient of thermal 

< /^TA 


which represents the relative (i.e. referred to unit volume) increase of 
volume per unit increase of temperature, at constant pressure. 

(2) Keeping the temperature constant (AT" = 0), we obtain in a 
similar way the partial (QV/Qp)T connected with the coefficient of com- 

1 /^T/\ 


which gives the relative decrease of volume per unit increase of 
pressure, at constant temperature. 

(3) When the volume remains constant (A 7 = 0), the partial 
(dp/dT)v is obtained, related to the coefficient of tension 


The inverse partials are, simply, the reciprocals, (37/97% 
= l/(9r/97)p, etc., and do not give us anything new, and even the 
three partial derivatives which we have formed are not independent 
as there exists a relation between them. This becomes apparent 
when we write down the general expression for the total infinitesimal 
change A/> which results when both the volume and the temperature 
are changed by infinitesimal amounts. It is, according to the rules of 

'3P\ ,rr , (& 


In the special case, A = 0, the ratio AF/AT 1 is, precisely, the 
partial (dV/dT) p . Since the left side of the equation vanishes in 
this case, we can solve it with respect to this ratio and find 

This relation can be regarded as a differential form of the equation 
of state (1.01). 

In the general case of a homogeneous system described by other 
parameters (in addition to temperature and volume) there exist 
several equations of state as will become clear in section 7. 

Exercise 1. In nitrogen at t = 20 C and p = 0.134 atm, a = 0.00368 and 
ft = 0.00368. Calculate the compressibility 0i. 

Exercise 2. Suppose that one of the equations of state of a homogeneous 
system depending on many variables has the form 

The increment of y, analogous to the expression for Ap, is 

Show that by keeping all the variables y, Xi, . . . X n , T constant, except two, one 
can obtain n(n -f l)/2 new relations of the type (1.05) between triples of partials. 
(Remark: The set of relations so obtained includes all that are independent. From 
them there can be derived a great number of other (dependent) relations of a 
similar form.) 

4. The perfect gas. Absolute temperature. The so-called perma- 
nent gases of nature follow, more or less closely, a number of well- 
known laws. The first of these is the Boyle-Mariotte law according 
to which, for a constant temperature, the product of pressure and 
volume remains constant 

pV - const. (1.06) 

When we inquire about the behavior of a gas at varying tempera- 
tures, it is convenient to use a gas thermometer filled with this same 
gas as the thermometric fluid. The pressure of the thermometer is 
kept constant (o), and the zero of the temperature scale (t 0) 
is chosen arbitrarily, say, as that corresponding to a volume Fo. 


The temperature t is, then, defined as proportional to the relative 
increase of the volume over Fo, 


V - 7 (1 + <**)* (1.08) 

a being a constant. 

According to the law of Charles-Gay-Lussac, the formula (1.07) 
is universal for all permanent gases: Used as thermometric fluids in 
the same thermometer, they give, within a good approximation, the 
same readings, or, in other words, they have all pretty closely the same 
coefficient of expansion a. The uniformity can be best judged in the 
following way. Suppose we use a centigrade scale: that is, we choose 
the temperature of melting ice as / = and that of boiling water as 
t = 100. Equation (1.08) gives, then, for the volume at t = 100, 
Fioo = F (l + lOOa), whence a = (Fioo - Fo)/100F . The quan- 
tity defined in this way is called the mean coefficient of expansion, 
between and 100, and is denoted by o,ioo. In general this quan- 
tity depends on the pressure po kept constant during the expansion. 
The mean coefficients of expansion <*o, 100 were investigated by Henning 
and Heuse who measured them with considerable accuracy in helium, 
hydrogen and oxygen. These measurements are well represented by 
the expressions of Table I, 1 where po is expressed in atmospheres. 
Degrees of the centigrade scale will be indicated by the symbol C. 


Gas <>, too X 10 7 

He ......... 36604 - 25p 

H 2 ......... 36604 - 16p Q 

N 2 ......... 36604 + 167/> 

We arrive at a more rational choice of the zero point of our scale 
if we displace it by I/a, defining the new temperature as 

T = t + I/a. (1.09) 

The position of the old zero point / = on the new scale is then 
Jo = I/a, and eq. (1.08) takes the form V TtPo F r/7o. The 
temperature T is proportional to the total volume of the gas. We 
substitute this into eq. (1.06) whidi we refer to the temperature T 9 

* E. Kenning and W. Heuse, Zs. Physik 5, p. 285, 1921. 


writing pV = poV TtPo , and obtain a formula representing the com- 
bination of the Boyle-Mariotte and Charles-Gay-Lussac laws: 

pV CT, (1.10) 

where C is an abbreviation for the constant C 

A simple system in equilibrium is completely homogeneous and 
uniform in all its parts (compare section 2). The properties of our gas, 
in every small region of it, are, therefore, completely defined by the 
temperature T and pressure p and cannot depend on its total exten- 
sion or mass. If we compare two specimens of the same gas of dif- 
ferent masses, but at the same temperature and pressure, their respec- 
tive volumes V must be, obviously, proportional to their masses. It 
follows then from eq. (1.10) that the constant C must be also propor- 
tional to the mass of the gaseous system under consideration. 

We shall use as the unit of mass of a chemically homogeneous or 
pure substance the mol, also called the gram-molecule. If /* denotes 
the molecular weight of this substance, its mol contains just n grams. 
To simplify our terminology we shall apply the term " molecular 
weight " also to monatomic substances where it has the same meaning 
as " atomic weight/' Let us consider a homogeneous system which 
consists of N mols of a pure substance. The volume occupied by one 
mol of it is called the specific molal volume or, simply, the molal volume 
and is denoted by v. 

We can apply this to our gas system if it is chemically homogeneous. 
The total volume V and the constant C can be represented as 

V**Nv, C = NR, (1.11) 

where R is an abbreviation for R = poVo/To or the constant C referred 
to one mol of the gas. The equation of state (1.10) takes then the form 

pV = NRT, (1.12) 


pv = RT. (1.13) 

The third empirical law which we have to invoke is that of Avo- 
gadro. It tells us that the molal volume v is, within a close approxima- 
tion, the same for all permanent gases, namely, under normal condi- 
tions (p = 1 normal atm = 1.013249 X 10 6 dyne cm- 2 , t = C) 
it is vo = 22414 cm 3 . Consequently, the gas constant R is also uni- 
versal. It is an immediate consequence of the definition of the mol, 
that 1 mol of any substance contains the same number n A of molecules, 
which is called the Avogadro number. It has been determined with 


fair accuracy and has been found to be n A =* (6.064 =fc 0.006) 
X 10 23 mol" 1 . A much-used quantity is the ratio 

* - R/n A9 (1.14) 

which represents the gas constant per molecule and is called the Boltz- 
mann constant. With its help eq. (1.13) can be written in an alterna- 
tive form. If we denote by Z the total number of molecules in N mols 
of a gas having the volume F, and by z = Z/F the number per unit 
volume, we can write Z = Nn A , NR = Nn A k = Zk = Vzk, and 
from (1.12) 

p = zkT. (1.15) 

If, instead of a single gas, we consider a mixture, we must again 
have recourse to experience. Observations on mixtures of permanent 
gases show that, with a good approximation, each of the constituent 
gases behaves as if the others were not present. In the case of <r 
gases contained in a vessel of the volume Fin the respective quantities 
of Ni, . . . N 9 mols, each of them exerts a partial pressure upon the 
walls of the vessel which can be computed in the same way as if the 
gas were single, i.e. by the formula 

p h V = N h RT. (1.16) 

The total pressure of the mixture is the sum of these partial 

P = pi + p2 + ...+p., (1.17) 

and its equation of state is obtained by summing the formulas (1.16) 
for h = 1, 2, ... <r 

(Ni + ... + N ff )RT. (1.18) 

None of the above laws is followed by any of the permanent gases 
quite rigorously, and the formula (1.18) which embodies all four of 
them is, therefore, only an approximate expression of the facts. If 
we compare two gas thermometers filled with different gases and 
having the same fixed points, the readings between the fixed points 
will not be strictly identical because the rate of expansion is, in reality, 
not constant but slightly dependent upon the temperature, in a dif- 
ferent way for the different gases. The same thing is true for non- 
gaseous thermometric fluids like mercury and alcohol. The scale of 
temperatures established by any particular thermometer construction 
is, therefore, special and arbitrary. In order to have a scale which 
does not depend on the individual properties of any special substance, 
we imagine the so-called perfect gas: an ideal fluid which strictly 


follows the equation of state given by formula (1.12). Let us, further, 
imagine a thermometer filled with a perfect gas and let us specify its 
scale in such a manner that between the points of freezing and of 
boiling water (both at 1 norm atm = 760 mm Hg) are 100 equal 
divisions. The temperature which we would measure with such a 
thermometer is called the absolute temperature. We shall, consistently, 
denote it by T (capital), reserving / (small) for the temperature mea- 
sured in an arbitrary scale. 

It might be thought, at first sight, that such a definition of the 
absolute temperature must be futile since the perfect gas, and the 
perfect thermometer filled with it, are ideals which do not exist in 
nature. We shall show, however, in section 29 that there is a simple 
procedure to determine the absolute temperature, in every case, by 
indirect measurements. Every thermometer can, therefore, be cali- 
brated in the absolute scale within the accuracy with which it can be 
read, and such calibrated thermometers are, in fact, readily available. 

Making use of the results of this calibration, we can state that the 
point of freezing water corresponds to TQ = 273.1 of the absolute 
scale. Combined with the value of the molal volume mentioned above 
(VQ 22414 cm 3 ), this leads to the following numerical values for the 
gas constant: R = (82.049 0.009) atm cm 3 deg" 1 mol^ 1 (if the 
pressure is measured in normal atmospheres) or R = (8.3136 0.001) 
X 10 7 erg deg-" 1 mol" 1 ) (if it is measured in dynes per cm 2 ). For 
Boltzmann's constant defined by (1.14) there follows the numerical 
value k = 1.371 X 10~ 16 erg deg" 1 . The question how closely the 
permanent gases approximate the behavior of a perfect gas will be 
more fully discussed in section 6. Here we shall only state that, 
according to their degree of perfection, they can be arranged in the 
following order: carbon dioxide, oxygen, air, nitrogen, argon, hydro- 
gen, helium. Carbon dioxide is the least and helium the most nearly 
perfect gas of this series. The deviations of helium from eq. (1.13) of 
perfect gases are, in fact, quite small under most conditions. Except 
at extremely low temperatures, helium can be used as the perfect 
thermometric fluid unless a very high accuracy is required. (Compare 
also section 29.) 

Exercise 3. Show that the coefficients of thermal expansion and of compres- 
sibility are for a perfect gas 


and check the validity of the relation (1.05). Calculate the coefficients numerically 
(per 1 deg, and 1 mm Hg as units) at normal conditions: r 273M, p 
760 mm Hg. 

5. The equation of Van der Waals. The next step in the approxi- 
mation to the properties of real matter is represented by an equation 
given by Van der Waals. In spite of its simplicity, it comprehends 
both the gaseous and the liquid state and brings out, in a most remark- 
able way, all the phenomena pertaining to the continuity of these two 
states. This equation has the form 

where a and b are two numerically small constants and v is the molal 
volume. It was derived by statistical considerations of so simple a 
nature that, even in a book devoted to thermodynamics, we can say a 
few words about the interpretation of the constants. It is shown in 
the kinetic theory that eq. (1.13) corresponds to a gas whose molecules 
are material points which do not collide or otherwise interact with one 
another. If we consider a gas with molecules of finite size, mutually 
exclusive as to their extension but not interacting in any other way, 
the difference is that the centers of the molecules cannot spread out 
in the whole volume of the gas but only in that part of it which is 
not occupied by other molecules and not immediately adjacent to 
them. In the first approximation, for the molal volume v there must 
be substituted the covolume, v b, where the constant b is proportional 
to the sum of the volumes of all molecules in one mol of the gas. The 
equation of state becomes, therefore, p = RT/(v b). On the other 
hand, if the molecules of the gas do interact at a distance, say, attract 
one another, there must be added to the external pressure p the 
internal pressure due to this attraction. Whatever its law is, it is 
safe to assume that two volume elements of the gas ATI and Ar2 
act upon each other with a force Fi2 proportional to the product of 
the masses in them, i.e. Fi% cc AwiAma. If we choose the mol as 
the unit of mass, we have Ami = ATI/IF, Aw2 = Ar2/v, so that 
F\2 cc ATiAT2/u 2 . This means that, when the density of the gas in a 
given vessel is changed by adding more gas or subtracting it, all the 
internal forces change in the ratio 1/v 2 . Since the pressure is defined 
as the force per unit area, this applies also to the internal pressure 
and we obtain for it the expression a/v 2 which is added to p in Van 
der Waals' equation. 1 

1 It must be assumed, also, that the force Fit decreases rapidly with the cfistance 
between the volume elements An and An. Otherwise the coefficient a would not 




Let us plot the pressure p against the molal volume v keeping the 
temperature T' in eq. (1.21) constant. Such a curve is called an 
" isothermal." As long as the molal volume v is very large, b is negli- 
gible compared with v, and a/v 2 compared with p. In the region of 
large volumes the curve is, therefore, identical with that given by 
the equation of perfect gases, namely, a hyperbola asymptotically 
approaching the horizontal axis p = for v = o . When the volume 
is very small and close to its lower limit 6, the second term on the 
right side is again negligible compared with the first and the curve 
becomes again hyperbolic. However, its asymptote is not the vertical 
V = (as in the case of the perfect gas) but the vertical v = b. The 
shape of the isothermals in the intermediate range is given in Fig. 1. 

They are curves with one minimum 
and one maximum which are real for 
low temperatures and complex for high 
so that, above a certain temperature 
T c (critical temperature), the pressure 
is, in them, a monotonically decreas- 
ing function of the volume (dp/dv <[0). 
Below T c the curves are not con- 
tinually decreasing but have a middle 
part (between the minimum and the 
maximum) in which p increases with 
v, (dp/dv > 0). It is obvious that this 
middle part, dotted in our curves, 
can have no physical reality. In fact, 
let us imagine the fluid in a state cor- 
responding to this part of the curve 
contained in a heat-conducting vertical cylindrical vessel whose top is 
formed by a piston. The piston can slide up and down in the cylinder, 
and we put on it a load exactly balancing the pressure of the gas. 
If we take a little weight off the piston, there will no longer be 
equilibrium and it will begin to move upward. However, as it moves 
the volume of the gas increases and with it its pressure. The resultant 
force upon the piston gets larger, retaining its upward direction. The 
piston will, therefore, continue to move and the gas to expand until it 
reaches the state represented by the maximum of the isothermal. Vice 
versa, if we add ever so little to the load of the balanced piston, the 
gas will collapse to the state corresponding to the minimum of the 

be an internal constant of the gas but would depend on the shape and size of the 
container. It does not follow from the above argument and is a separate assump- 
tion that a is independent of T and p. 


1. Isothermals according 
Van der Waals 1 theory 



isothermal. The relation dp/dv g is, therefore, the condition of 
dynamical stability, and in regions where it is not satisfied the fluid 
is in an absolutely unstable state. 

Leaving out the physically unreal dotted portion, the low tempera- 
ture curves break up into two disconnected branches. The one of 
small molal volume corresponds to the liquid state of matter while 
that of high volume represents its gaseous state. The important fact 
that the same equation applies to both states of aggregation was 
called by Van der Waals the continuity of the liquid and gaseous states. 
Experiments show that the following things actually happen when we 
gradually increase the volume (in a heat bath of constant temperature), 
starting from a point A of the liquid state (Fig. 1). The expansion 
goes along the liquid branch of the isothermal up to a certain point B. 
In this state the liquid begins to evaporate so that the system is no 
longer homogeneous, but heterogeneous, consisting of a liquid and a 
gaseous phase. As the volume is further increased, more and more 
liquid is evaporated while the specific properties of the two phases 
remain unchanged. In this part of the isothermal expansion the 
pressure (called the boiling pressure at the temperature 7') is constant, 
so that it is represented by a horizontal line, and remains so until all 
the liquid is evaporated (point E). From then on the system is again 
homogeneous and expands along the gas branch EF of the isothermal. 
Why the system behaves in such a way, and where the point B lies, 
are typical problems of thermodynamics which will be answered at 
length in section 43. The part BC of the liquid branch (like DE of 
the gas branch) represents only relatively unstable states: the vapori- 
zation of a liquid is greatly facilitated by the presence in it (or on its 
surface at the walls) of little bubbles of foreign gases. If these gases 
are driven out by a preliminary thorough boiling of the liquid and a 
suitable heat treatment of the vessel, it is possible to make the system 
expand along the liquid branch beyond the point B, bringing it into 
the so-called superheated state. In this state the pressure may become 
negative, in other words, the liquid is able to support a certain 
amount of tension; but at the slightest provocation it will jump 
over into the thermodynamically stable state on the straight 
line BE. 

Above the critical temperature there is no discontinuity in the iso- 
thermal, and it is arbitrary whether to call the fluid a liquid or a gas. 
Usually, it is considered a gas and, then, the statement applies: // is 
impossible to liquefy a substance at temperatures higher than the critical. 
If the substance follows eq. (1.21) of Van der Waals, its critical tem- 
perature can be determined in the following way. Below it, the iso- 


thermals have a maximum and a minimum which are analytically 
determined by the condition 

+ 2_* o. (1.22) 

As the temperature rises, the maximum and the minimum draw 
closer and closer together, and, in the critical isothermal, they coincide, 
forming a point of inflection. This point corresponds to the so-called 
critical state of the substance characterized by the values p c , v c , T c of 
pressure, molal volume, and temperature which are known as the 
critical constants. The analytical condition for a point of inflection is 

_ 6a 

^ ) 

In the critical point all three equations (1.21), (1.22), (1.23) must 
be satisfied. It is, therefore, easy to express the critical constants in 
terms of the Van der Waals constants a, b, R by solving these three 
equations with respect to the three variables p, v, T. The result is 


One can use the critical constants as the units in which to express, 
respectively, the pressure, specific volume, and temperature. The 
state of the substance is then described by the so-called reduced variables 

p v T . 

-ft * = :: T = V (L25) 

pc V c lc 

If we substitute these quantities for p, v, T and, at the same time, 
replace the Van der Waals constants by the critical, the equation of 
state (1.21) takes the form 

Referred to the reduced variables, the equation does not contain 
any specific constants and is, therefore, the same for all Van der Waals 
substances. States of two substances in which they have the same 
respective values of T, <f>, and T are called corresponding states, and 
eq. (1.26) expresses the law of corresponding states: If two of the 
reduced variables, IT, ^>, T, are the same, respectively, for different sub- 
stances the third is also the same and their states are corresponding. The 
mathematical reason for the validity of the law of corresponding 


states in connection with Van der Waals' equation is that it contains 
just as many constants as variables, namely three. However, the 
law of corresponding states is not restricted to Van der Waals sub- 
stances. It is often satisfied with a fair degree of precision in groups 
of substances which show marked deviations from eq. (1.21), as will 
be shown more fully in the next section. 

It is well to add here a few words about gas mixtures. The additiv- 
ity of partial pressures which exists in perfect gases, according to 
eq. (1.17) does not hold in mixtures of Van der Waals gases. The 
physical reason for that additivity is that each perfect gas behaves as 
if the others were not present, and this means that there is no inter- 
action between their molecules. On the contrary, the aim of Van der 
Waals' theory is, precisely, to take into account the interaction. 
For a binary mixture (i.e. consisting of two gases) the kinetic theory 
has been carried through by Lorentz. 1 The equation of state obtained 
by him for such a mixture is formally identical with the ordinary 
Van der Waals eq. (1.21), but the symbols v t a, b have a different 
meaning. If the mixture contains NI mols of the first gas and N 2 of 
the second, v = V/(N\ + N 2 ), while the quantities a and b depend 
on the mol fractions Xi = N\/(N\ + N 2 ) and x 2 = N 2 /(Ni + N 2 ) 
and have the form 

a = #n xi 2 + 2ai2 Xix 2 -f- 022 x$ 

b = bn xi 2 + 2bi2 xix 2 + b 22 


The coefficients an #22 are those of the pure gases, while #12 
characterizes the mutual action ; the same applies to the coefficients b. 

Exercise 4. Calculate by means of eqs. (1.24) the Van der Waals constants 
a, b and the ratio RT c /p c v c for the following substances from the critical data 
(R 82.07): 

Methyl acetate 506.9 46.33 227.6 

Methyl formate 487.2 59.25 172.0 

Exercise 5. Derive the coefficients of thermal expansion (1.02) and of compres- 
sibility (1.03) for the equation of Van der Waals. (Remark: as this equation is 
readily resolved with respect to />, it is convenient first to express a and pi in terms 
of partials of p t by means of (1.05) or in any other way.) 

6. Behavior of real substances. As far as gases are concerned, it 
was pointed out in the preceding section that eq. (1.13) of a perfect 

1 H. A. Lorentz, Ann. Physik 12, p. 127, 1881. For a discussion of this equation, 
in relation to experimental data, see: J. D. Van der Waals, Continuity of the 
Gaseous and Liquid States. Vol. II. German edition of 1900. 



I 6 

gas is already a good approximation as long as the molal volume v is 
large. Although the equation of Van der Waals has its limitations 
(see below), the numerical values of its constants a and b are apt to give 
an idea how much a given substance deviates from the law of perfect 
gases and indicate the order of accuracy here involved. We give in 
Table 2 these constants and the critical data of a number of gases. 




t c *C 

PC (atm) 

b = \v c 

a X 10- 
(atm cm 6 ) 














H 2 












N 2 






O 2 





Carbon monoxide 
Carbon dioxide 

CO 2 





Nitrous oxide 

N 2 O 





Water vapor 

H 2 O 






C1 2 





Sulfur dioxide 

SO 2 





The constant ft, depending on the molecular volume, is not greatly 
different for the different gases. The relative correction in the magni- 
tude of the pressure, due to the presence of this constant in eq. (1.21), 
is roughly b/v. In the neighborhood of the normal conditions 
(/ = C, p = 1 atm), the molal volume is about 22 400 cm 3 /mol 
and the correction has the numerical value of 0.001 to 0.002. On the 
other hand, there is a considerable variation, from gas to gas, in the 
constant a, and the correction a/v 2 depending on it varies widely. 
For helium under normal conditions it is only 0.00007, for nitrogen 
and oxygen 0.0027. For the latter two gases, it is of the same order 
of magnitude as the first correction (0.0017 and 0.0015), and, since 
its sign is opposite, the total correction is reduced. For water steam 
at the boiling point (t = 100 C, p = 1 atm), it becomes -0.006. 

It was known already to Van der Waals himself that his equation 
does not give an accurate representation of the properties of a fluid 
in all ranges of temperatures. In fact, it is possible to determine the 
coefficients a and b from the data about pressures and volumes relating 
to a single isothermal. When the coefficients, obtained in this way 
for widely different isothermals, are compared, they turn out not to 




be the same, especially, a is found to increase as the temperature is 
lowered. Another indication that eq. (1.21) is not accurate in the 
vicinity of the critical state is as follows. Theoretically there exists 
the following relation between the critical data p e , v c , T if the expres- 
sions (1.24) are valid: 


RT C 8 

- - 2.67. 


In reality, the ratio of the empirically measured critical data 
RT c /pcV c is nearly the same within large groups of substances but it 
has not the value 2.67 required by the theory. The constancy of the 
ratio R is an indication that the law of corresponding states still 
approximately holds when the Van der Waals equation fails. In 
Table 3 we give the critical data of a few organic substances. 






v e 

RT c /p c v 

Ethyl ether 

C 4 Hi O 












CflH 6 Br 






C 6 H 6 C1 





Ethyl formate 






Ethvl acetate 

C 8 H 8 O2 






C*5l~l 1J 






v^5i~i 12 












C 7 Hi 





Hexane .... 

(-.gH 14 





Carbon tetrachloride .... 

C C1 4 





Tin tetrachloride 

SnCl 4 





In spite of the shortcomings just mentioned, the Van der Waals 
equation is extremely useful. It gives an excellent quantitative account 
of the behavior of gases whose density is not too high. Beyond this 
it offers a qualitative picture of the phenomena of condensation, 
critical state, etc. However, it must be always remembered that it 
cannot be used for quantitative predictions of phenomena involving 
high densities. Many other equations of state were proposed but, 
in part, they apply only to a narrow range of the variables, in part, 
they offer a gain in accuracy insufficient to compensate for the much 
more complicated form. Therefore, we shall mention here only two 
of them. 


(A) The mean empirical, reduced equation of Kamerlingh Onnes 1 
is expressed in terms of the reduced variables (1.25), 

K is the ratio (1.28) of the critical data, while the coefficients 
depend on r 

iii #2, . . being constants. The other coefficients have the same 

This equation grew out of the studies of its author on the validity 
of the law of corresponding states. He found that there exist large 
groups of substances which obey the law of correspondence to the 
following extent. The substances of a group can be arranged in a 
series so that each of them has a close correspondence with the pre- 
ceding substance while the series, as a whole, shows progressive and 
considerable changes in the constants. The above equation applies 
to a fictitious substance representing the mean type of the series. The 
results of Kamerlingh Onnes show us two things: (1) how extremely 
complicated the conditions become when a high accuracy is aimed at; 
(2) that the law of corresponding states has an approximate validity 
beyond that of any of the simpler equations of state. 

(B) The equation of state of Beattie and Bridgeman 2 


* v 2 ^ ' ~ l v 2 ' 

[ e = c/vT*, 

AQ, Bo, a, i, c being constants. This equation is one of the most recent 
that have been proposed so that its authors had the benefit of the 
experience of their predecessors and of more accurate experimental 
data. 3 It gives good results over a wide range of the variables. 

1 H. Kamerlingh Onnes and W. H. Keesom, Die Zeistandsgleichung. Encyklo- 
paedie math. Wiss. Vol. VI, Article 10. 

2 J. A. Beattie and O. C. Bridgeman, Proc. Am. Acad. 63, p. 229, 1928. 

1 Another of the more recent equations is that due to A. Wohl (Zs. physik. 
Chemie 87, 1, 1914; 99, p. 207, 1921). It is often quoted, but the writer has diffi- 
culty in understanding it, as he is not convinced that it satisfies the requirements of 
an equation of state. 

tl - e) ,. . A B _ ^ _ 




The above equations of state apply, primarily, to gases, secondarily, 
to liquids at temperatures not far from the critical. It is not necessary 
to say much about matter in the condensed state. The thermal 
expansion and compressibility 
of condensed substances are 
so small (compared with 
those of gases) that they are 
not important in most ther- 
modynamical applications. 
In the first approximation, 
they are altogether neglected ; 
in the second, they are taken 
as constant over finite ranges. 
The process of transition from 
the liquid to the gaseous state 
is in real substances the 
same as that described in 
connection with the Van der 
Waals equation and illus- 
trated by Fig. 1. As an 
example we give in Fig. 2 the 
actually observed isothermals 

in carbon dioxide. (The mp- FlG 2 .-Empirical isothermals in carbon 
lal volume is given here in 
fractions of the normal vol- 
ume at C and 1 atm, v, 
marked by the symbol X). 


D .005 ".01. 


FIG. 2. Empirical isothermals 

= 22 256 cm 3 ; the critical point is 

Exercise 6. Calculate the critical data in terms of the constants of the equation 

of state 

/ n \ 

- b) RT (Berthelot) 

by way of the formulas (1.22), (1.23), (1.24). 
Exercise 7. Do the same for the equation 

p(v -b) = RT exp (-c/RTv). (Dieterici) 

(Remark: when the first and second partials of p vanish, those of log p also vanish.) 


7. The element of work. The concept of work is defined in 
mechanics as follows. When a material point moves under the action 
of a force F, through the infinitesimal distance dl in the direction of F, 
the elementary work done by this force is Fdl. When the directions 
of the path dl and of the force do not coincide but include an angle a 
only the projection of F on the direction of motion Fi = F cos a con- 
tributes to the work, and the element of work has the expression 

DW = Fidl = Fdl cos a = F x dx + F y dy + F,dz, (2.01) 

if we denote, respectively, by F x , F v , F, and dx, dy, dz the projections 
of F and of dl upon cartesian axes of coordinates. In the case of the 
general dynamical system of n degrees of freedom, the element of 
work is expressed in a similar way 

DW - yi dXi + y 2 dX 2 + . . . + y. dX n (2.02) 

where X\ 9 . . . X n are the geometrical coordinates and yi, . . . y n the 
generalized forces. If the work is due in part to non-mechanical 
forces (electric, magnetic, capillary, etc.), the form of this expression 
is not changed : there only appear additional terms of the same type. 
Of particular importance to us is the form which the element of 
work takes in simple systems (section 2). The only force with which 
such a system can act upon the outer world is the pressure upon its 
boundaries. The pressure is the force per unit surface of the wall and 
has the direction of the normal to it. Consequently, the force acting 
upon an element d2 (Fig. 3) of the surface is F = pd2 in the direction 
n. Let us suppose that, because of the pressure in the system, its 
surface gets slightly displaced from the position AB into A'B' 
so that the element dS moves into the new position d2' through 
the very small distance A/. The work of the force upon this 
element is then, according to the expression (2.01), equal to 
dSA/ cos(n,A/), and the work of the forces of pressure upon all 
the surface to DW = p f dSA/ cos(n,A/). In the limit, when A/ is 
infinitesimal, every element, in being displaced from dl, todS', sweeps 





over an oblique cylinder whose volume is, precisely, dA/ cos (n,A/). 
The integral f dSA/ cos(n,A/) is, therefore, the total increase of the 

volume of the system dV: 


We shall, in general, denote by capital letters quantities referred to 
the whole system (N mols) and by small letters quantities referred to 1 mol 
of it. w represents, therefore, the molal work (W = Nw, N being the 
mol number), or 

Dw - pdv. (2.04) 

It is work done by the system : We define the sign in such a way that 
work of this kind is counted as positive, while work done by outer 

FIG. 3. Element of work 
in simple system. 

FIG. 4. Graphical repre- 
sentation of work. 

o v 

FIG. 5. Work in cyclic 

forces against the system is considered as negative. The expression 
(2.04) receives a simple graphical representation in the (p, v)- diagram 
(Fig. 4). Let the points A, A' correspond respectively to the initial 
state of the system and to its state after an infinitesimal expansion dv. 
The product pdv = Dw is, then, represented by the infinitesimal area 
under the element AA' y the shaded strip of our diagram. If a finite 
process of expansion is such that it can be represented by a curve 
of the (p, fl)-diagram, for instance, by the segment AB, the total 
work (per mol) done in it by the system is equal to the integral of the 
expression (2.04) 



* A 



Its graphical representation is, obviously, the total area under the 
segment AB. Since the work w is, in this case, positive, Fig. 4 permits 
us to formulate the following rule. An area represents positive work 
when it lies to the right of the curve giving the process (for the observer 
looking in the direction of the process, i.e. from A to B). This rule is 


borne out also in the case of a cyclic process where the system returns 
to its initial state (Fig. 5). While expanding (ACB), the system does 
positive work; while contracting (BDA), negative. The negative 
area (double shaded) is to the left of the curve BDA and must be sub- 
tracted from the positive area under A CB. The difference is the net 
(positive) work done by the system; it is represented by the area 
enclosed by the circuit ACBD and lying to the right of it. 

Exercise 8. (If the force is measured in dyne /cm 2 and the volume in cm 3 , the 
work is expressed in erg). Calculate the work of vaporization of water. When 
1 mol of water (18 grams) is vaporized at 100 C, the increase of volume is VB VA = 
30186 cm 8 mol" 1 . The pressure in the process is constant and equal to 1 atm: 
p = 1.013 X 10 6 dyne/cm 2 . 

Exercise 9. Calculate the work necessary to compress 1 mol of a gas from the 
normal state (VA = v = 22414cm 8 , PA = 1 atm) to half the normal volume 
(VB = fo), assuming that the process follows the law of Boyle-Mariotte: pv = 

8. Linear differential expressions and exact differentials. It will 
be well to recall to the reader a few mathematical facts which, although 
simple, are very fundamental in thermodynamics since the whole of 
this science may be said to be built up on them. Mathematically 
speaking, formulas (2.01) to (2.04) are examples of linear differential 
expressions, whose general form is 

k(xi ' Xn)dXl " 


For instance, in the particular case of only two variables x,y 
it reduces to 

L(x,y) = M(x,y)dx + N( Xj y)dy. (2.07) 

The most interesting question which arises with respect to these 
expressions is whether they are " exact differentials 1 ', meaning whether 
they can be obtained by differentiating some function f(xi, . . . x n ) of 
the same variables. According to the rules of differentiation the total 
differential of / is 

-dx k . (2.08) 

We see, therefore, that the linear differential expression (2.06) is 
an exact differential when 

Jf*-j, (*-l,2...n); (2.09) 


II 8 



in words, when the coefficients Mk are the partial derivatives of a 
function/(#i, . . . x n ). If we single out the two independent variables 
Xk, xi the second partial does not depend on the order of differentiation 

Fropl'the UJfmulas (2.09) there follows, therefore, 

; , (ft,/ - 1, 2 ...), 


the so-called reciprocity relations. It is clear that the validity of the 
reciprocity relations is a necessary condition for the expression (2.06) 
being an exact differential. Moreover, it is shown in calculus that 
this condition is also sufficient: Whenever the relations (2.10) are 
satisfied, a function can be found such that the coefficients Mk are its 
partial derivatives. This function is called 
the potential of the expression. 

We conclude from this that a linear 
differential expression of the type (2.06) is 
not, in general, an exact differential. If the 
functions Mk(xi, . . . x n ) are selected at ran- 
dom, they will, in general, not satisfy the 
reciprocity relations. The exact differential 
is, therefore, a rare exception. 

Of great importance is the concept of the p IG> 
line integral of a linear differential expression. 
To fix our ideas, we shall explain this notion 

for the case of expression (2.07) with only two variables x, y, which we 
shall interpret as the cartesian coordinates of a plane (Fig. 6). The 
line integral along the curve from A to B 

[M(x,y)dx + N( X ,y)dy} 

is constructed in the following way. The first element is obtained by 
substituting, for the arguments of the functions M and N, the values 
of x and y in the point A and, for dx and dy, the increments which lead 
from A to an infinitely close point A 1 on the curve. For the second 
element one takes the values of M and N at the point -4' and the 
increments leading to the point -4", and so one moves from point to 
point along the curve, until its end B is reached. The integral is the 
sum of all the elements so constructed. In the case of n variables they 

6. Notion of line 


must be interpreted as the coordinates of an w-dimensional hyper- 
space, and the generalization is obvious. 

When the linear differential expression is an exact differential (2.08), 
its line integral can be given explicitly. For instance, 


J A 

The line integral depends, then, only on the coordinates of the initial 
point #i A) , . . . x (A) and of the final x{ B \ . . . x ( * } and not on the path 
between them. It has the same value for all the curves going from 
the same initial to the same final point. In particular, when the curve 
is closed, the initial and the final points coincide and the expression 
(2.11) is equal to zero: 

= 0. (2.11) 

If V^ Mkdxk is an exact differential, its line integral over any closed 

circuit vanishes. It is shown in mathematics that the inverse theorem- 
is also true : If the line integral of a linear differential expression vanisjte$ 
for every arbitrarily chosen closed circuit, it is an exact differential. 

To summarize, we can say that the following three possible prop 
erties of the linear differential expression (2.06) are completely equiva- 
lent: (1) It is an exact differential; (2) its coefficients satisfy the 
reciprocity relations (2.10) ; (3) its line integral vanishes for any closed 
curve. If the expression has one of these three properties, it has also 
the other two. 

Supposing that the linear differential expression (2.06) is not an 
exact differential, the further question arises whether it can be trans- 
formed into one by multiplying it by a suitable function X(*i, . . . x n ). 
If this is the case, we say that it admits an integrating multiplier or, 
simply, that it is integrable. After multiplication by X the coefficients 
of the expression become XM*, and, if it is integrable, there must be 
satisfied, instead of (2.10), the reciprocity relations 

In the case of the differential expression (2.07) with two variables, 
there is only one reciprocity relation, 3(XAf)/3y = 9(XW)/9#, which 
can be regarded as a partial differential equation with X as the depend- 
ent variable. Since such an equation has always a solution, a linear 


differential expression with two variables is always integrable. This is 
no longer the case if n is larger than 2. For instance, when n = 3, 
there exist the three relations 

9*3 9*2 ' 9*i 9*3 9*2 9*i 

of which two are independent. In general, it is impossible to satisfy 
two equations by one function. This is brought out by the fact that 
X can be eliminated from the three relations which leave as the result 
of this elimination 

,, ,_ / 

Mi [ ---- ]+M 2 [ --- f+Mz I ~ -- 1 = 0. (2.13) 
9*2 / \9*i 9*3 / \9*2 9*i 

(The simplest way of checking this is to substitute the form of the 
coefficients M k = (9//9**)A following from (2.12). This equation is 
known as the condition of integr ability. Unless it is satisfied the dif- 
ferential expression does not admit an integrating multiplier. In the 
general case of n variables, there must be satisfied a condition of the 
type (2.13) for each triple of variables. 

It should be noted that the integrating multiplier, when it exists, 
is not unique. In fact, suppose that the linear differential expression 
(2.06) is transformed by multiplication by X into the exact differential 
df(xi 9 . . . x n ). The further multiplication by any function of/, say 
<p(f), changes it into <p(f)df, which is also an exact differential. There- 
fore, X^>(/) is another integrating multiplier. 

Exercise 10. Check by means of eq. (2.12) that L(p, v) = pdv vdp admits 
an integrating multiplier, namely X = 1/t; 2 . 

Exercise 11. Decide by means of the criteria (2.10) and (2.13) whether the 
following linear differential expressions are exact differentials, integrable, or non- 

(a) (2xydx + x*dy 

(b) zdx + xdy +]ydz, 

(c) yzdx + zxdy -f xydz, 

(d) (y + z)dx + (z+ x)dy + dz 

9. Application to the element of work. In order to apply these 
mathematical theorems to the physical concepts of section 7, we have 
to find out whether in a simple system the element of work, Dw = pdv, 
is an exact differential. As pointed out in section 3, the state of a 
simple system is completely determined by two of the three variables 


p, v, T. Let us use for its description T and v: The fact that dT does 
not appear in the linear differential expression Dw means, then, that 
the coefficient of dT is zero. The expression should be written 
Dw SB pdv + Q.dT, and it is easy to see that it does not satisfy the 
reciprocity relation (2.10) or dp/$T = 30/90, since the left side is 
equal to the coefficient /3i (1.04) and is, generally, finite and the right 
is 0. We conclude, therefore, that the element of work is not an exact 
differential. This is confirmed by considering the line integral of Dw 
over a closed circuit: we found in section 7 that it is represented by 
the enclosed area while, in the case of an exact differential, it would 
be equal to nothing. The same applies to the general case of a system 
with n degrees of freedom. The difference between the element of 
work (2.02) in mechanics and in thermodynamics is that, in mechanics, 
the generalized forces y k are supposed to depend only on the coordi- 
nates of position Xi, . . . X n , whereas, in thermodynamics, they are, 
in addition, functions of the temperature T. 

y k = y k (Xi, X 2 ,... X n , T), k = 1, 2, . . . n. (2.14) 

The term with the differential dT has the coefficient zero, and, 
therefore, it is impossible to satisfy some of the reciprocity relations, 
just as in the case of the simple system. This is the reason why we 
use for the element of work the symbol Dw: we wish to reserve the 
symbol d for exact differentials. 

A system can be considered as defined only then when the forces it 
exerts on its environment are known. In other words, eqs. (2.14) 
must be considered as given. As they are completely analogous to 
the equation of state, p = p(v,T), of the simple system, we conclude 
that a homogeneous system has one less equation of state than it has 
independent variables. (Compare also section 27). 

10. Heat and heat capacity. As the invention of the thermometer 
led to the concept of temperature (section 1), so another physical 
instrument, the calorimeter, helped to develop the important concept 
of heat. We have already seen that, when two systems of different 
temperatures are brought in contact, the warmer cools down and the 
colder heats up until equilibrium is reached. A calorimeter is a 
standard system with which different bodies are brought in contact 
in order to study their respective losses of temperature in relation to 
its gain. To fix our ideas, let us suppose that the calorimeter consists 
of one kilogram of water (initial temperature T\) into which the bodies 
experimented upon (initial temperature Ti > 7i) can be dropped, 
after which the combined system assumes a final temperature T inter- 


mediate between the two (T% > T> 7\). The notion of heat was 
evolved from such measurements in the eighteenth century when the 
calorimeter technique was not yet very precise. Within this limited 
accuracy the following result was obtained: In experiments with 
the same body, the final temperature T adjusts itself for all variations 
of the initial temperatures T\ and T% in such a way that the ratio 

rip rr* 

C = 2 _ - constant. (2.15) 


C is, therefore, a characteristic constant of the particular body or 
system and is called its heat capacity. Comparing different bodies of 
the same material, it was, further, found that their heat capacities are 
proportional to their masses. For a body of the mass Af, one can 
write C = Me', where c f is the heat capacity per unit mass or the 
specific heat of the material. Pouring into the calorimeter an additional 
kilogram of water, we can determine the heat capacity C w of this 
water like that of any other system. In the units implied in 
formula (2.15) it turns out to be 1, and this is also the heat capacity 
of the calorimeter itself, which was taken to consist of the same 
amount of water (C w = 1). Equation (2.15) can, therefore, be 
rewritten as follows 

C (T - r 2 ) + C w (T - Ti) = 0. (2.16) 

This relation has the form of an equation of conservation: It sug- 
gests that there is something that does not change in the process of 
calorimetric conduction. We call this invariable function or quantity 
heat. In fact, both terms of the left side have the form 

Q = C(T - T), (2.17) 

where T' is the initial and T the final temperature. The first term 
relates to the body experimented upon and is negative; it represents 
the heat lost by that body. The second, oppositely equal to the first, 
is the heat imparted to the calorimeter. 

The modern improved measurements have completely borne out 
the fact that there is conservation of heat in all processes of conduction. 
However, its definition in terms of the temperature had to be refined 
somewhat, since it was found that the heat capacity is not strictly 
constant but depends on the temperature interval. If Q is the heat 
imparted to a system when its temperature is raised from T' to T, 
we define 

C Q/(T - n (2-18) 


as the mean heat capacity for this interval. When the interval is 
small, T = T - Ar, this can be written C = A^/Ar, and in the 
limiting case of Ar = it becomes 



the heat capacity at the temperature T. (We prefer the symbol D be- 
cause DQ is not an exact differential, as we shall see in section 13). 
Multiplying this by dT and integrating, we find the more accurate 
expression for the heat imparted to the system 


CdT, (2.20) 

which takes the place of the approximation (2.17) and must be, also, 
substituted into the equation of conservation (2.16): 


T s*T 

CdT+ I C w dT = 0. (2.21) 

J Tl 

The accepted unit of heat is the (gram-} calorie, the heat which 
must be imparted to 1 g of water in order to raise its temperature by 
1C. There are several different definitions of it in use, the most 
important being: (a) The 15 calorie, the heat necessary to warm 1 g 
of water from 14.5 to 15.5 C. (b) The mean calorie, one one- 
hundredth of the heat which raises the temperature of the same mass 
of water from to 100 C. The latter is larger than the former in 
the ratio 1.00024. One thousand gram-calories form one kilogram- 

As to specific heats, we shall refer them, in the case of chemically 
pure substances, to 1 mol of matter and denote by c the molal heat, i.e. 
the heat which raises the temperature of 1 mol of the material by 1 C. 
This definition is incomplete because the specific heat depends also 
on the pressure conditions during the measurement; we shall have to 
say more about this in section 14. Some authors use two terms: atomic 
heat (heat capacity of 1 mol of a monatomic substance) and molecular 
heat (1 mol of a chemical compound), but we shall apply the single 
term, molal heat, to both cases. 

Joseph Black, who established the " doctrine of specific heats," 
discovered also another phenomenon, the latent heat of transformations. 
He observed (1761) that during the process of melting of ice (while 
the temperature of the water-ice mixture remains constant) a definite 
amount of heat must be imparted to it, to transform it from the solid 


state into the liquid, which he called the latent heat of fusion. Later 
he also proved the existence of a latent heat of vaporization and approx- 
imately measured it. We shall denote the total latent heat by L and 
the molal latent heat by /. 

Chemical reactions are also accompanied by a positive or negative 
heat development called the heat of reaction. If heat must be imparted 
to the system in order to make the reaction proceed, it is called endo- 
thermic; in the opposite case, exothermic. We shall adhere to the con- 
vention that Q is positive when it is imparted to the system. Therefore, 
the heat of an exothermic reaction will be counted as negative, of an 
endothermic as positive. 

In modern usage, the phenomena of change of volume and pressure 
with temperature (treated in Chapter I) are spoken of as the thermal 
properties of matter, while heat capacities, latent heats, etc., belong to 
the caloric properties of matter. 

Exercise 12. Into a calorimeter consisting of 200 g of water (initial temperature 
/i =s 20 C) are thrown pieces of metal weighing 300 g and having the initial tempera- 
ture /2 = 100 C. The temperature of equilibrium / is for (a) aluminum (ju = 27.0) 
39.5C, (b) iron (/z = 55.8) 31.9C, (c) zinc (ju - 65.4) 29.8 C, (d) silver ( - 
107.9) 26.2 C. Find the specific heats per 1 g and the molal heats of these four 
metals, (/z denotes the atomic weight.) 

Exercise 13. The molal latent heat of fusion of water (/z 18) is / 1440 cal 
mol" 1 . We drop into a mixture of 1000 g ice and 1000 g water 300 g of copper 
(/u = 63.6, molal heat c = 5.73 cal mol"" 1 ). How much ice will melt before the 
system comes to equilibrium at C? 

11. History of the first law. The notion of energy and of its con- 
servation was, first, established as a theorem restricted to the science 
of mechanics. Leibnitz showed in 1693 that, in an isolated mechanical 
system, there remains constant the sum of the kinetic and the potential 
energies, called by him the live and the dead forces (vis viva and vis 
mortua). If the system is not isolated, these energies are convertible 
into work done against outer forces. About seventy years later was 
formulated the law of the invariability of heat in all processes of con- 
duction, as stated in the preceding section. The first law of thermo- 
dynamics asserts that mechanical energy and heat, the two quantities 
subject to laws of conservation, are not generically different, but 
equivalent to each other and interconvertible. 

A vague suspicion that mechanical action can have heating effects 
seems to have existed since early times, but the first precise statement 
that mechanical work can be converted into heat and the first attempts 
to put this theory upon an experimental foundation were due to 
Benjamin Thompson, a native of Connecticut, better known under 


his Bavarian title of Count Rumford (1798). Supervising the boring 
of cannons in the arsenal of Munich, Rumford was struck by the high 
temperature which the brass shavings attained and by the general 
large development of heat in the process. He could think only of two 
causes, other than the conversion of work into heat, which, conceivably, 
could have produced this effect: On the one hand, there may exist, 
he thought, a latent heat of breaking up solid brass into chips,; on the 
other, the borer may induce some chemical action of the air upon 
brass attended by heat development. The first possibility was dis- 
proved by special measurements of two different kinds. One should 
expect that the surmised latent heat is due, largely, to a difference in 
the heat capacities of the solid metal and of the shavings. However, 
no such difference could be detected by calorimetric determinations 
especially undertaken for the purpose. Still more conclusive was a 
second line of reasoning: When a blunt borer rotating with much 
friction was substituted for the sharp one, the production of chips 
decreased many times while the yield of heat remained about the 
same. As to the possible action of air, Rumford tried to meet this 
objection by immersing the whole apparatus in water so that the air 
had no access to any of the heated parts. This change of conditions 
had no appreciable effect on the rate of the heat development. 
Rumford gives very explicit data how much heat the power of a live 
horse produces per hour. Translated into modern units, they are 
estimated to mean that 1 (gram)-calorie is equivalent to 5.45 X 10 7 
erg, a result which is 31% above the value at present accepted. A 
sequel to Rumford's work was a paper by his associate in the Royal 
Institution of London, Sir Humphry Davy (1799). In order to refute 
the objection that the heat, in the above experiments, could have 
come from the water (which might have exercised upon brass a chem- 
ical influence similar to that of air), Davy studied the conversion of 
work into heat in vacuo. By means of a clockwork he rubbed two 
pieces of ice against each other under the jar of an air pump. The 
effect was unmistakable, inasmuch as the ice melted faster with 
rubbing than without, but no quantitative determinations were 

The brilliant lead of Rumford and Davy was not followed up by 
other investigators. Forty years passed as a period of quiescence dur- 
ing which these problems were, practically, never mentioned as far as 
the literature of physics and chemistry is concerned. Not until the 
decade from 1840 to 1850 was the principle of conservation of energy 
established and accepted, owing to the independent work of several 
men belonging to different nations. The French engineer S6guin 


(1839) formulated the theory of the equivalence of work and heat with 
great clarity and carried out much experimental work trying to prove 
it. The idea of these experiments was excellent and amounted to a 
reinterpretation of the formula of adiabatic expansion from the point 
of view of conservation of energy (compare section 18). Their execu- 
tion and analysis, however, were imperfect, and Seguin himself 
admitted that his results succeeded only in making his contention 
probable and did not constitute a convincing proof. Quite independ- 
ently and with equal lucidity the principle was stated by the German 
physician Robert Julius Mayer (1842), who also computed the ratio 
between the units of heat and work (mechanical equivalent of heat) 
from data on the specific heat of gases then available (compare section 
15). This method, although not quite free from objection, disclosed 
an unusual insight into the nature of the phenomena involved in it. 
The result at which he arrived is 1 calorie = 3.57 X 10 7 ergs, or 15% 
below the correct value. However, Mayer seems to have overlooked 
that even the most ingenious hypothesis needs an experimental 
demonstration in order to become a scientific law: no experiments 
were either planned or suggested by him. The credit of having set 
the principle of conservation of energy upon a firm experimental 
foundation is, entirely, due to James Prescott Joule of Manchester 
(England). Owing to delicate health, Joule did not receive a regular 
course of instruction but was tutored at home; among his teachers 
was the chemist, John Dalton. Joule conceived the idea of the 
equivalence of heat and work at an early date and began his experi- 
ments upon it in 1840 at the age of 20. They were carried through 
in his private laboratory built for him by his father, a retired brewer. 
He was led to the problems of conservation of energy from a study of 
electric currents, having ascertained (in 1840) that there is a constant 
relation between the work necessary to maintain a current and the 
heat generated by it (Joule's heat). In his first determination of the 
mechanical equivalent of heat (1843) he utilized just this fact: The 
current was supplied by an electromagnetic generator and passed 
through a resistance immersed in a calorimeter. While this circuit 
operated at a stationary rate, he determined, on one hand, the work put 
into the generator, on the other, the heat developed in the calorimeter 
(result: 1 cal = 4.6 X 10 7 erg). Best known is his direct determina- 
tion of 1849 in which he produced heat by churning water and other 
liquids with paddle-wheels, and which gave 1 cal = 4.154 X 10 7 erg 
(for a long time, the accepted standard value, unsurpassed for several 
decades), but his work included the study of a great variety of processes 
relating to the interconvertibility of mechanical, thermal, electrical, 


and chemical energies. As one of the founders of the principle of 
conservation of energy must be also regarded Hermann Helmholtz, 
at that time a young surgeon in the Prussian army. He at once recog- 
nized its importance and scope and was instrumental in bringing it 
to the attention and acceptance of scientists by publishing a brilliant 
paper (1848) in which he followed its workings through numerous 
applications to the sciences of physics, chemistry, and physiology. 1 

It is very remarkable that none of these four men was by training 
a physicist. Sguin was brought up as an engineer, Mayer and 
Helmholtz as physicians, while Joule having received an irregular edu- 
cation cannot be classified with any professional group. The significant 
fact that the contemporary physicists had no part in the development 
of a most important phase of their science cannot be denied, even if 
Helmholtz and Joule later became ornaments of their profession. 
The reason for this has been pointed out by Ostwald and others: It 
was the time of reaction after the collapse of Schelling's " Natur- 
philosophie". All the physicists had been, for a longer or shorter 
period, under the influence of that movement. Now they were tired 
of fruitless generalities and anxious to make amends for the wasted 
time by conscientious factual research. They became suspicious of 
anything that was reminiscent of a speculative anticipation of nature. 
This frame of mind accounts for the hostility with which the mechan- 
ical theory of heat was received in physical circles. On the other hand, 
the active interest developed by engineers on its behalf hardly needs 
any explanation : The production of power from fuel is the engineer's 
everyday pursuit which compels him to think continuously about 
the connection between heat and work. 

These facts help us to understand the aloofness of the physicists 
and the active participation of an engineer in investigating the equiva- 
lence of mechanical and caloric energy. It remains, however, unex- 
plained how this principle could enlist the interest of two distinguished 
medical men. The writer began to feel that there might lie a historical 
problem in this and that it would be worth his while to inquire into the 
interests and the scientific background of physicians and physiologists 
of the period between 1820 and 1840. These suspicions were confirmed 

1 Posthumous notes of S. Cannot disclose that this author, who wrote his " Reflex- 
ions sur la puissance motrice du feu " strictly from the point of view of the indestruc- 
tible caloric fluid, in later years (before his death in 1832) became more friendly to 
the idea of the equivalence of heat and work. However, it is hardly possible to 
assign him a share in the discovery of the first law, on the strength of this, because 
he did not possess either a rigorous formulation of it or any new empirical evidence 
to support it. Besides, his notes were published for the first time in 1878, a quarter 
of a century after the law had ceased to be controversial. 


by the following passage in one of Joule's papers: " On conversing a 
few days ago with my friend Mr. John Davies, he told me that he had 
himself, a few years ago, attempted to account for that part of animal 
heat which Crawford's theory had left unexplained, by the friction 
of the blood in the veins and arteries, but that finding a similar 
hypothesis in Haller's 'Physiology 1 , he had not pursued the subject 
farther " (Postscript to the paper in Phil. Mag. 23, 1843). These 
lines tell us about a third physician who was familiar with the idea of 
the equivalence of heat and work. Moreover, they give us a hint 
that the source of his interest in the subject lay in the theory of animal 
heat, with a reference to Haller's physiology as a clue. 

At present, the name of Albert Haller may not be generally 
known, but in the eighteenth century he was considered a scientific 
colossus and enjoyed an international prestige, as an authority in 
medicine and botany, which is equaled only by that of Newton, in 
his sphere and at his time. A native of Berne (Switzerland), he became 
the main scientific figure at the university of Gottingen. In German- 
speaking countries his reputation was further enhanced by important 
poetical writings. His " Outline of Physiology " appeared in 1747 and 
was immediately translated into French and English. It is not too 
much to say that it marked a new epoch in that branch of science 
and became the main influence in the education of several generations 
of physicians. The last German edition was published in 1822, and 
we have seen that Dr. Davies consulted this treatise even about the 
year 1840. The passage which Dr. Davies must have had in mind is 
contained in section 303 of the first volume. 1 On the preceding pages 
Haller analyzes the uses of the lungs and comes to the conclusion that 
the animal heat is produced in this organ and imparted to the blood 
passing through it. Of considerable interest to us is his hypothesis as 
to how the heat originates: In his own words, it arises " from the 
alternate extension and contraction, relaxation, and compression of 
the pulmonary vessels, by which the solid parts of the blood are per- 
petually rubbed together and closely compressed in the attrition that 
is made during expiration, as it is more rapidly moved and ground 
together during expiration". Very important is, also, the following 
sentence: " Nor is it any objection to this that water cannot be made 
hot by any friction. Nor in reality is this assertion true; for water by 
violent winds and motion, as well as milk, acquires some degree of 
warmth". These quotations contain Haller's theory of respiration 
which, for a long time, dominated the science of medicine. It is quite 
a surprise to find that this theory was essentially based on the idea 
1 Not section 304 of Vol. II, as Joule gives the reference. 


that heat can be generated by mechanical work, although it was 
enunciated 50 years prior to the work of Rumford. 1 

We have found a convincing proof of the fact that, at a certain 
epoch, the physiologists were much concerned with the problem of 
the relation between work and heat, and we shall see below that they 
were keenly aware of the work of Rumford and Davies, seeing in it a 
confirmation of Mailer's hypothesis. It is surprising, at first sight, 
that this interest could have persisted until the middle of the nineteenth 
century, since the modern combustion theory of respiration, suggested 
already by Priestley, received strong support from the measurements 
of Lavoisier and Laplace (about 1781) and was rediscovered indepen- 
dently by the English physician Crawford (referred to by Joule as 
Crawford's theory). A closer investigation shows, however, that there 
were excellent reasons which prevented the acceptance of the combus- 
tion theory as a complete explanation of animal heat. Lavoisier and 
Laplace had found that the oxidation of carbon contained in the car- 
bohydrates of the blood is insufficient to account for the developed 
heat, and they expressed the conjecture that the balance is produced 
by the oxidation of hydrogen. An experimental test of this hypothesis 
required measurements extremely delicate and difficult for the prim- 
itive technique of those times. A long time passed, therefore, before 
they were even attempted. At length (in 1821), the Academic des 
Sciences announced a prize for the investigation of this problem, and 
this induced Dulong and Despretz to start work on it. Despretz 's 
essay was submitted to the academy in due time and received the prize 
on June 1, 1823 (published in the Ann. Chem. Phys. (2) 26, p. 337, 
1824). He concluded from measurements on 200 animals that the 
complete heat of oxidation of both carbon and hydrogen accounts 
for the larger part of the animal heat but not for the whole of it. 
There remained a defect of about 20% for the explanation of which 
Despretz fell back upon Haller's theory, saying that it could be 
produced " by the motion of the blood and the friction of its different 
parts". The results obtained by Dulong were quite in agreement with 
those of Despretz's. He read his paper before the academy in 1822, 
but he did not contend with it for the prize and he resisted the urging 
of his friends to have it published (appeared after his death: Ann. 
Chim. Phys. (3) 1, p. 440, 1841). The reasons for this strange reluc- 
tance became clear when his posthumous papers were examined: 
Dulong had no faith in the thermochemical data on which his own 
and Despretz's conclusions were based. He intended to redetermine 

1 Even Haller cannot claim priority in this matter, since similar views were held 
in the seventeenth century by the latro-Physical School. 


the heats of formation of carbon dioxide and of water in order to 
revise his essay. This rede termination, finally carried out, was the 
last scientific work done by Dulong: he obtained new and more 
accurate data but died before bringing his results into shape for 
publication (posthumously edited by Cobart, Ann. Chim. Phys. (3) 8, 
p. 183, 1843). While his work confirmed the previously accepted 
heat of formation of carbon dioxide as fairly accurate, it established 
an entirely new value for the heat of formation of water, showing 
that the old data were much too low. This correction was more than 
sufficient to remove the discrepancy between the animal heat and the 
chemical heat development of respiration and constituted a conclusive 
proof of the oxidation theory. However, it must be remembered that 
this knowledge came rather late: between 1823 and 1843 the last 
word of physiological science was the existence of a defect of animal 
heat which, apparently, was left unaccounted for by the combustion 

There remains the question how widely this knowledge was diffused 
among physiologists and physicians. The writer happened to come 
across a very comprehensive German " Handbuch " of physiology of 
that time written in six volumes by leading authorities under the 
editorship of Karl Friedrich Burdach. The account of the theory of 
respiration, due to Ernst Burdach, is contained in the sixth volume, 
which appeared in the year 1840. Many pages are devoted to the 
history of the two theories, and both sides of the question are discussed 
at considerable length. After a lucid presentation of Haller's views 
(p. 544), Burdach turns to " the doctrine of the resemblance of respira- 
tion and combustion". It appears from his account that in the period 
between 1820 and 1840 the physicians interested in the theory of 
respiration were preoccupied, like Joule's friend Dr. Davies, by 
" that part of animal heat which Crawford's theory left unexplained". 
Most of them pointed to Haller's theory as to the most likely explana- 
tion and maintained that the origin of the unaccounted heat lay in 
the mechanical action of friction in the arteries. These mechanical 
effects were, in their turn, supposed to be produced by the " vital 
principle " (puissance vitale) invented by Barthez. Burdach gives 
many references to authors who advocated this view. Among them 
were some of the leading men of science, like Despretz and Treviranus, 
but also some of the rank and file. For instance, the writer had the 
opportunity of reading in the originals a German pamphlet by F. Lau 
(Bonn 1830) and an English paper by J. M. Winn (Phil. Mag. 14, 
p. 174, 1839). They are of great interest as an indication into how 
wide a circle the problem had spread. Lau and Winn were, apparently, 


ordinary practitioners without much knowledge of the literature of 
the question or understanding of its deeper implications. Yet, their 
theories were only modifications of Haller's views. It is interesting 
to note that Burdach himself, who was not in sympathy with these 
explanations, did not question the fact that heat can be produced by 
friction, but regarded it as a matter of common knowledge, citing, in 
this respect, Rumford and Davy. He only doubted whether this 
cause would be sufficient to account for the desired effect quantitatively. 
The above facts throw a new light on the role of the medical 
profession in the history of the principle of conservation of energy. 
There was, actually, no break of tradition between the work of the 
eighteenth and the nineteenth centuries. However, this tradition 
was handed down through the unexpected channel of the science of 
physiology, whose representatives acted, for half a century, as the 
custodians of the idea of equivalence of heat and mechanical work. 
If this statement implies that the historical role of Mayer and 
Helmholtz consisted in turning back this principle to physics, the 
branch of science where it naturally belongs, this is in no way a 
detraction from the merits of these two great men: They were not 
merely passive transmitters of a concept, because there is an enormous 
gap between the vague knowledge current among physicians, that heat 
is generated by mechanical action, and the scientific formulation of 
the principle of equivalence, which could be bridged only by men of 
unusual ability. Helmholtz, eventually, developed into one of the 
greatest scientists of all times, and Mayer showed in his later physio- 
logical writings (1845) a clearer understanding of the nature of animal 
metabolism than any of his contemporaries. Nevertheless Mayer was 
entirely unsuccessful in his endeavors to persuade the physicists and 
chemists of the validity of his conception. The reason for this was, 
without doubt, that he could not offer them much in the way of an 
experimental proof at a period when they were particularly reluctant 
to accept anything on insufficient empirical evidence. A few years 
latter Helmholtz had far better success because, in the meantime, 
Joule's work had put the principle of conservation of energy on a 
secure experimental foundation. 



12. Reversible and irreversible processes. It was pointed out in 
section 2 that the thermodynamical description of a system, by such 
variables as its temperature and pressure, implies that it is in equi- 
librium. In general, this equilibrium is disturbed if the system is 
subjected to any physical process. For instance, if we exert a mechan- 
ical action upon it or bring it under the influence of a source of heat, 
the result is a complicated transient state, of longer or shorter dura- 
tion, during which all the physical characteristics of the system are 
different in every point of it. The quantitative description of such an 
occurrence is, therefore, entirely outside the scope of thermodynamics. 
It is possible to follow through by thermodynamical methods only 
processes which take place so slowly that they can be regarded as a 
succession of states of equilibrium. Strictly speaking, the change of 
the system in a finite time must be even infinitesimal and the rate of 
change infinitely slow. 

Such infinitely slow processes have an important theoretical 
advantage in that they are reversible, that is, they can be gone through 
in both directions. In fact, let us take the example of a simple system: 
If we change two of the three variables, say v and T, so slowly that 
the system can be regarded as in equilibrium at any moment, the 
equation of state applies all the time and the third variable p is also 
completely determined. When we go through the same values of 
v and T in the opposite direction, the succession of states of the system 
is, simply, reversed. For every infinitesimal step, the work done by 
the system is expressed by Dw = pdv. When the process-is reversed, 
dv changes its sign while p has the same values so that the work retains 
the same absolute value as in the direct process,, but has the opposite 
sign, i.e. is done by outer forces against those of the system. We shall 
see in the next section that the heat received by the system in every 
infinitesimal part of the process is also a linear differential expression 
so that the same conclusions hold with respect to it: If a certain 
amount of heat is imparted to a system in an infinitely slow process, 



the same amount is withdrawn from it when the process is reversed. 
We have restricted ourselves to simple systems, only, to fix our ideas : 
All our results apply also to the most general thermodynamical system 
because, for every phase of it, there exist equations of state of the type 
(2.14) and the elements of work and heat have always the form of 
linear differential expressions. 

When we turn to processes of a finite speed, we can no longer assert 
that they are reversible. As an example, we consider a gas in a 
vertical cylindrical vessel with a piston as its top. The reversible way 
of compressing such a gas would be to increase the load on the piston 
infinitely slowly by adding infinitesimal weights. When a finite 
weight is added at a time, the compression goes at a finite rate, but 
owing to the lag in the propagation of stresses, the compression and 
the pressure immediately under the piston will be higher than in the 
remainder of the gas and higher than in the reversible process. To 
produce the same reduction of volume, the outer forces have, there- 
fore, to do more work than in the infinitely slow case : A part of the 
work is consumed by elastic waves, currents, and other wasteful 
processes in the gas, the so-called phenomena of internal friction. On 
the other hand, if we expand the gas back to the original volume, 
proceeding at a finite speed, the same lag will cause the pressure at 
the piston to be lower than in an infinitely slow expansion. The gas 
will do, therefore, less work than in the latter process. It follows 
from this that the compression and expansion, at a rate which is not 
infinitely slow, are irreversible processes since the work which must 
be applied in the compression is larger than that received from the 
system in the expansion. 

The extreme case of rapid expansion is that when there is no outer 
force 1 all, for instance, when the piston is suddenly removed. Such 
an expansion into the void takes place without the system doing any 
work. It is quite obviously irreversible, since it is necessary to apply 
work of outer forces to compress the system back to the initial volume. 
Another example of an irreversible process is the phenomenon of 
thermal conduction of which we have made use in defining the con- 
cepts of temperature (section 2) and of heat (section 10) : If we bring 
in contact two bodies having a finite difference of temperatures, heat 
flows, spontaneously, from the warmer to the cooler, but never in the 
opposite direction. All processes of nature proceed at a finite speed, 
and experience tells us that they are all accompanied by effects of the 
nature of friction or conduction which make them irreversible. The 
engineer would call these effects " wasteful' ', since they do not produce 
any work, and the irreversible process " less efficient " than the 


reversible. Strictly speaking, the reversible processes are an ideal 
which exists only in our imagination. However, this is enough to 
make its discussion very useful; besides, many real processes are con- 
ducted at so slow a rate that they can be regarded as reversible within 
the limits of experimental error. Only such processes can be mathe- 
matically described in thermodynamics, or represented by means of 
curves in diagrams. If the system is not in equilibrium no unique 
values of temperature and pressure can be assigned to it. 

13. Analytical formulation of the first law. Internal energy. 
The purport of the first law has been made sufficiently clear in Chapter 
II. Its gist is that heat is convertible into mechanical work and other 
forms of energy, and vice versa, in the ratio of the mechanical equiva- 
lent of heat: 1 15-gram-calorie = 4.185 X 10 7 erg. By means of 
this conversion factor we shall always express heat and work in the 
same units. What is, then, the fate of an infinitesimal amount of 
heat DQ, imparted to a system by bringing it in contact with a suit- 
able heat reservoir? Since thermodynamics deals only with states 
which are infinitely close to equilibrium we need not consider its con- 
version into kinetic energy: The heat DQ will, in part, cause the 
system to do the work DW against outer forces; in part, it will be 
used to raise the temperature and, perhaps, to overcome the resistance 
of inner forces attending a change in volume and in other inner prop- 
erties. We denote the portion utilized for the latter purpose by DU 
and obtain the equation : 

= DU + DW. (3.61) 

Wejmow that the element of work DW is not an exact differential, 
but we do not yet know what the elements DQ and DU a^e, in a 
mathematical sense, and it will be our next objective to find this out. 
This is the reason why we use the symbol D: According to the con- 
vention made in section 8, the symbol d is reserved for expressions 
which, we are sure, are exact differentials. 

Let us now consider a reversible process in which the heat Q is 
imparted to our system by bringing it, successively, in contact with 
an infinity of heat reservoirs differing in temperature by infinitesimal 
amounts. For every step of this process eq. (3.01) holds, and we have, 
simply, to integrate it from the initial state of the system (1) to its 
final state (2) : Q = f* DQ, and, according to section 7, W =/f DW. 
We find, therefore, 

DU = Q - W. (3.02) 



Of particular interest is the case when the process is cyclic so that 
at the end of it the system is, precisely, in the same state as in the 
beginning. The system itself does not, then, contribute anything to 
the energy balance. If we compare the conditions before and after 
the cycle is carried through, the only changes are outside the system. 
Namely, the heat Q is withdrawn from the reservoirs and the work W 
done against outer forces. The law of conservation of energy leads, 
therefore, to Q W 0, instead of (3.02), whence 

DU=Q, (3.03) 

for any reversible cyclic process. As we have seen in section 8, 
eq. (2.11), it follows from the property (3.03) of the integral that the 
expression DU is an exact differential. This means that there exists 
a certain function U of the variables defining our system whose total 
differential d U is identical with the differential expression D U. This 
function U is known as the internal energy, Every function which is 
completely determined by the variables of the system we shall call a 
characteristic function of it. The internal energy is, therefore, one of 
the " characteristic functions " of thermodynamical systems. 
Instead of eq. (3.01) we have to write, therefore, 

DQ = dU 4- BW. (3.04) 

The element DQ being the sum of an exact and a non-exact dif- 
ferential is itself not an exact differential according to the criteria of 
section 8. The integral (3.02) over a reversible process takes now 
the form 

Q - U 2 - Ui + W. (3.05) 

In words: the heat imparted to the system is used, partly, to 
increase its internal energy from the initial value U\ to the final Z/2, 
and partly to do outer work, Though derived for a reversible process, 
this equation has a wider range of application. In fact, suppose that 
in an initial state of equilibrium the system has the internal energy Ui. 
We subject it, then, to an irreversible process and, after it is over, 
wait until the system and its environment are again in equilibrium. 
^Suppose it is now found that the internal energy has become C/2, that 
outer sources have lost the heat Q, and that the work W has been 
done against outer forces. The law of conservation of energy leads 
again to the relation (3.05). In this sense it can be applied also to an 


irreversible process. In the particular case when the process is cyclic, 
the initiaT and the final state are identical, J7 2 Ui = 0, and 

Q - W. (3.06) 

The gist of the first law of thermodynamics is that work can be 
produced only at the expense of heat or some other form of energy. 
Therefore, it can be formulated as the impossibility of a perpetual 
motion machine (perpetuum mobile), as those delusive devices are 
called which claim to do useful work, continually, without drawing 
on any energy source. In its application to purely mechanical engines, 
this principle had already been recognized by Stevinus, Newton, 
and Leibnitz. The discoveries of the middle of the nineteenth century 
(section 11) extended it to engines of any kind. 

Exercise 14. 300 g of mercury are dropped from a height of 600 cm into a 
calorimeter consisting of 500 g of water at 10 C. In this process the potential 
energy of the mercury is, first, converted into kinetic energy and then into heat. 
What will be the final temperature of the water and mercury? (The specific heat 
of mercury is 0.0334 cal g" 1 deg" 1 , the acceleration of gravity g - 980.7 cm sec" 2 ). 

Exercise 15. The latent heat of vaporization of water is / - 9730 cal mol~ l 
(per 18 g). What part of it is spent on the work of expanding the steam, as calcu- 
lated in exercise 8? 

14. Molal heat at constant volume and at constant pressure. 

We are going to apply the first law, as formulated in eq. (3.05), to the 
special case of a simple, chemically pure system. We shall refer 
Q, U, W to one mol of the material of the system, denoting the molaj 
quantities, by the small letters q, , w. Moreover, according to (2.04), 
the element of work is, for a simple system, Dw = pdv, whence 

Dq = du + pdv. (3.07) 

The molal heat is, according to the definition (2.19) of specific 
heats, given by 

c - Dq/dT (3.08) 

and depends upon the conditions under which the element of heat Dq 
is imparted to 1 mol of the system. Two cases are of special prac- 
tical interest: (1) the volume of the system is kept constant (dv 0) 
while heat is added to it; (2) its pressure is kept constant (dp ** 0) 
The internal energy u depends on the variables which define the 
system: in the case of a simple system, we can regard it as a function 
of any two of the three parameters p, v 9 T. To treat the case of 


constant volume it will be useful to describe it in terms of temperature 
and molal volume: 

u = u(T, v), (3.09) 

a relation which is often called the caloric equation of state. Its total 
differential has, then, the expression 

which, substituted into (3.07), gives 

If we put dv = 0, we obtain, comparing this with (3.08), the molal 
heat at constant volume 

c v = 

On the other hand, to treat the case of constant pressure, we must 
find an expression of Dq in terms of the differentials dT and dp. The 
simplest way of doing this is to recall that, because of the equation of 
state, v itself can be regarded as a function of T and p so that 

Substituting this into (3.11) 

Putting dp = 0, we find by comparison with (3.08) the molal heat 
at constant pressure 

The subtraction of (3.12) from (3.14) leads to the important relation 

The reason why c p is larger than c 9 is obvious: If the system is per- 
mitted to expand while the element of heat Dq (per mol) is imparted 
to it, this heat is used, not only to raise the temperature of the systen| f 


but also to do work, namely, work against outer forces to the extent 
p(dv/dT) p dT and against inner forces of the system itself in the 
amount (3u/3) r - (dv/dT) p dT. 

In the case of condensed systems (liquids and solids) the coefficient 
of expansion (3^/3 T) p is so small that it can be neglected for most 
purposes. Within this approximation it is unnecessary to distinguish 
between molal heats at constant volume and at constant pressure 
(c p c v = c). We postpone a detailed discussion of the molal heats, 
as to their experimental values as functions of temperature, until 
Chapter XVIII, but it will be necessary to say something about the 
approximate laws for permanent gases and for solids in the next two 

Exercise 16. The approximate value of Ow/dtOr for N 2 , at C and 1 atm, is 
0.0047. Estimate (dv/dT) p from the data of Table 1, and say by what percentage 
c p c v deviates from R. 

IS. The internal energy of perfect gases. The question in what 
measure the internal energy of gases depends on their volume can be 
decided in an experimental way. A very suitable experiment for this 
purpose is the expansion of a gas into the void mentioned in section 12. 
Provided that the system is adiabatically isolated, neither heat is 
imparted to it in this process nor does it do any external work, so that 
the terms Q and W in eq. (3.05) of conservation of energy vanish, and 
it is reduced to /2 U\ = or, referring it to 1 mol of the gas, 
Au = U2 ui = 0. If the expansion is small, we can write with the 
help of (3.10) and (3.12) 


If the internal energy is independent of the volume, (du/dv)r = 0, 
there follows Ar = 0: the expansion into the void is not attended 
by any change of temperature. It is possible to test this point by 
conducting the process while the system is in contact with a calorim- 
eter> instead of being adiabatically isolated. Joule carried out the 
experiment (1845) in the following form: two large, closed copper 
vessels were connected by a short copper pipe containing a stop cock. 
While the stop cock was shut, one of them was filled with gas at high 
pressure, the other evacuated and both were immersed into the same 
water calorimeter. When the stop cock was opened, the gas rushed 
into the empty vessel without doing work, but no change of temper- 


ature of the calorimeter could be discovered for any of the gases used. 
(A somewhat similar experiment, tried by Gay-Lussac as early as 
1807, had a quite different purpose). It may be concluded from this 
that the dependence of the internal energy of permanent gases upon 
their volume is small, but the large heat capacity of the calorimeter 
reduced the accuracy of the experiment too much to say more. To 
increase the accuracy, it is desirable to remove the calorimeter alto- 
gether and to measure directly the temperature of the expanded gas. 
However, this is impossible in the original form of the experiment: 
after the stop cock is opened, the gas is in a turbulent state with no 
uniformity of pressure or temperature. It comes to rest only after a 
period of time long enough to vitiate the results by heat losses through 
convection. A modification which avoids these difficulties was sug- 
gested by William Thomson (later Lord Kelvin) and carried through, 

jointly, by Joule and Thomson in a 
MI NI M 2 Nz series of investigations extending from 

- l p i' T i! B ! P 2' T 2 ["> 1852 to 1862 - The y maintained a sta- 

hi C A 2 tionary stream of gas through a tube 

FIG. 7. Joule-Thomson A \A^ (Fig. 7), obstructed in the middle 

process. by a cotton plug C. Owing to the friction 

in the plug, there was no turbulence in 

the flow and the gas was in a homogeneous state on either side of the 
obstruction. Before passing it, its pressure, molal volume, temper- 
ature, and internal energy were pi, vi, Ti, u\, afterwards p2, 22, T%, #2. 
The difference of temperatures T% T\ was measured by a thermo- 
electric couple. Heat exchange with the environment was prevented 
by protecting the tube with heat-insulating material. As to the 
kinetic energy of the gas, and the friction losses in the plug, they are 
proportional to the square of the velocity of the flow, which was chosen 
so low that they were entirely negligible, compared with the other 
energy items. Let us set up the energy balance for a portion of the gas 
which is contained (at a certain moment) between the cross-sections 
MI and Af2, for the time during which 1 mol of the gas flows through 
the tube. At the end of this time it will occupy the new position 
between NI and N2 and we can apply to this process the eq. (3.05) 
of conservation of energy which reduces here to C/2 U\ + W = 0, 
since no heat is imparted to the gas (Q = 0). As a result of the dis- 
placement, 1 mol of the gas (with the internal energy #2) appears on 
the right side of the plug and 1 mol (of the energy ui) disappears on 
the left: Uz U\ = u% u\. On the right side, work is done by 
this part of the gas in pushing forward the adjacent portions through 
the volume v% at the constant pressure pz\ on the left, work is done 


against it while it is displaced through the volume v\ (pressure pi): 
W = p2V2 p\v\. Our equation assumes, therefore, the form 

2 + p2V2 = U\ + piVl. (3.16) 

We shall postpone the analysis of this equation and of the results 
of Joule and Thomson until section 28. Suffice it to say here that, 
generally, the partial (du/Qv)T turned out to be small, but not equal 
to zero. Comparing the different permanent gases at room temper- 
ature, it decreases in the order in which the gases become more perfect 
(section 6) and is almost negligible for helium. It is, therefore, logical 
to define the perfect gas as characterized by an internal energy entirely 
independent of its volume: (3w/3z>)r = 0, so that, from (3.12) and 

du - c v dT. (3.17) 

This implies, of course, that c v is independent of v, because of the 
condition (2.10). As to its dependence upon the temperature, the 
experiments with real gases show, again, that it is the smaller, the 
nearer the gas approaches in its equation of state the behavior of 
perfect gases. In the definition of the perfect gas c v is, therefore, 
taken to be constant. Integrating eq. (3.17), there follows the expres- 
sion of the internal energy of perfect gases 

u - c v T + u , (3.18) 

where u (the so-called " zero point energy ") is an integration con- 
stant. This formula gives a good approximation to the internal 
energy of real monatomic gases, although it is not safe to extrapolate 
it to extremely low temperatures (section 105). In the case of diatomic 
and polyatomic gases, the approximation is fair only for a limited 
range of temperatures (on both sides of the room temperature): At 
very low temperatures the molal heat decreases because the rotational 
motions of the molecules require less energy; at very high temperatures 
it increases because the molecular vibrations begin to come into play. 
In view of this situation it is well to give the expression also for the 
case when the gas is thermally perfect (i.e. obeys the equation pv = RT) 
but calorically imperfect, to the extent that its specific heat, being in- 
dependent of v, is a function of T 

c, dT + ,. (3.19) 

In the special case c, = const, this becomes identical with (3.18). 



III 15 

In either case, eqs. (dv/dT) p = R/p and (9w/9^)r = hold, whence 
from (3.15): 

c p - c v = R. (3.20) 

This is the equation which J. R. Mayer used for the determination 
of the mechanical equivalent of heat, taking, for c p and c v , data 
expressed in calories (per degree and mol) and, for R, in mechanical 
units (compare section 12). Theoretically, his method was not free 
from objection, since, at the time, there was no foundation for the 
hypothesis that in the expansion of gases no energy is needed to 
overcome internal forces. In practice, however, the later work of 
Joule and Thomson showed that the formula is satisfied fairly well: 
Mayer would have obtained a pretty accurate value of the equivalent 
if good data of c v and c p had been available. The modern value of 
the gas constant in caloric units is R = 1.9864 cal deg mol" 1 . 

In the case of monatomic gases the molal heats are very close to 
c v = 3R/2, c p = 5-R/2, so that the ratio is 7 = c p /c v = 5/3. The 
data for the more important gases at 15 C are given in Table 4. 



7 = C P /C V 

c P /R 

(c p - c v )/R 


7 = C P /CV 


(c p - c v )/R 





C0 2 






N 2 O 






NH 8 






CH 2 




H 2 




H 2 S 




N 2 




SO 2 
















C 2 H 2 








C 2 H 4 








C 2 H 








A more detailed account of specific heats of gases will be found 
in Chapter XVIII. 

Exercise 17. For water vapor in the range between t = C and * = 650 C 
Nernst gives the empirical formula 

c p - 8.62 + 0,002 / + 7.2 X 10- / 2 cal mol" 1 deg- 1 . 

Assuming that eq. (3.20) is satisfied, find c v and determine the increase of internal 
energy c*o ~ o. 

Ill 16 



Exercise 18. For CO 8 between -75 C and +20 C the empirical formula is 

c p - 8.71 + 66 X 10-* * - 22 X 10-' 1* cal mol'i deg~ l . 
Find uto M-75. 

16. Molal heats of solids. We shall postpone the detailed theory 
and discussion of specific heats till Chapter XVIII and shall give here 
only two laws pertaining to solids. Though not rigorous, they are 
satisfied with a fair approximation at ordinary temperatures and 
are very useful because of their simplicity. 

(A) The law of Dulong and Petit (1819). The molal heats of all 
elementary solids have approximately the same value, close to 
6 cal deg" 1 mol^ 1 . (Compare Table 5). 






t C 















C (graph) 







C (diam) 




















(B) Neumann's law (1831). The components of solid chemical 
compounds have, approximately, the same heat capacity in the 
compound as in the free solid state. In other words, the molal heat, 
divided by the number of atoms n in the molecule, is roughly equal to 
six (Table 6). 





















































CaF 2 









Cu 2 S 










PbF 2 










PbCl 2 










Pb0 2 






17. The heat function and the isobaric process. On both sides of 
eq. (3.14) of Joule-Thomson appears the expression 

X - u + pv. (3.21) 

The quantity x is known as the (molal) heat function, also called heat 
content and enthalpy. Like the internal energy u, the heat function is 
completely determined by the variables of the system: It is another 
example of a * 'characteristic function" (section 13). Differentiating 
eq. (3.21), we find du = dx vdp pdv and, substituting this into 
the expression (3.07) of the element of heat imparted to a simple 

Dq = d x - vdp. (3.22) 

In a similar way, referring all quantities not to 1 mol of a substance 
but to the whole simple system, we define as the total heat function 
X SB JJ + pV and find DQ = dX Vdp. The heat function acquires 
a particular importance in the so-called isobaric process, i.e. a process 
which takes place without change of pressure (p = const, dp = 0). 
For an infinitesimal isobaric change the last equation gives DQ = dX. 
In the case of a finite reversible change of the system, at constant />, 
from the state (1) to the state (2), we obtain by integration 

Q = X 2 - Xi. (3.23) 

The heat imparted to the system in a reversible isobaric process is 
equal to the difference of its heat functions in the final and initial states. 
Under usual laboratory conditions, processes of change of the physical 
state (melting, vaporization) are conducted at constant pressure, as 
well as most chemical reactions. The formula (3.23) has, therefore, 
a wide range of applications. For instance, the latent heat per mol is, 
obviously, equal to the difference of the molal heat functions of the 
material in the two states of aggregation 

/ . X 2 ~ Xi. (3.24) 

If we choose T and p as the variables describing simple systems, 
we can write eq. (3.22) in the form 

and comparing with (3.13), (3.14), 



In the case of a perfect gas, the heat function can be given explicitly: 
According to (1.13), x = " + RT, whence from (3.19) and (3.20) 

x - / c p dT + u , (3.27) 

and if c p = const, 

X = c p T + u . (3.28) 

Exercise 10. One mol of water is vaporized at 100 C and the vapor heated to 
650 C. Calculate the increase of x in this process, using the data given in exercises 
15 and 17. 

18. The isothermal and adiabatic processes. It will be useful to 
discuss here the simplest thermodynamical processes, especially, with 
respect to the amounts of heat received and work done. Since the 
perfect gas is, so far, the only system whose behavior we know com- 
pletely, both as to its equation of state and its internal energy, we can 
give explicit analytical formulas only for perfect gases. The internal 
energy being a ' 'characteristic function" (compare section 13), its 
change is determined by the initial (1) and final (2) state of the system 
and is independent of the particular process. In the case of perfect 
gases it is, according to eq. (3.18), 

/ rj-\ rr< \ /^ OQ\ 

or if the gas is only thermally perfect, and calorically imperfect, 
according to (3.19), 



On the other hand, the work w depends on the special process and, 
consequently, also the heat q which can be determined from the first 
law of thermodynamics (referred to 1 mol of a chemically pure system), 

q = U2 #1 + w. (3.31) 

We have already discussed the isobaric process (p const) in the 
preceding section. The process consisting in heating a simple system 
without change of volume (V = const) is so trivial that little need be 
said about it. It is called "isochoric" and, obviously, takes place 
without external work: w = and q = #2 i- 

Of great importance is the isothermal process (T const). 
We have already used the concept of isothermal curves (section 5), 
regarding them as the loci geometrici of all possible states of a system 
corresponding to the same temperature. However, we can interpret 


them as the graphical representation of the following reversible process: 
The system is contained in a vessel with heat-conducting walls 
immersed in a large heat bath of the temperature T. The geometrical 
variables of the system (e.g. the volume V, if it is simple) are then 
infinitely slowly changed by external forces through their whole 
range of variability. If the compression or expansion takes place 
with finite velocity the temperature of the system will not remain 
strictly uniform, so that a process cannot be isothermal unless it 
is conducted in a reversible way. In the case of a thermally perfect 
gas, eqs. (3.29) and (3.30) give both u 2 u\ = 0, so that q = w. 
Moreover, p = RT/v, Dw = pdv = RTdv/v, and since T is constant, 

= RT log - - RT log - (3.32) 
. V V\ p2 

Equally interesting is the adiabatic process: This is the name for 
anything that can happen in a system surrounded by a heat-insulating 
or "adiabatic" (compare section 2) envelope which cuts off any inter- 
change of heat with the environment. This cover does not, in general, 
prevent the system from doing work against outer forces, as is illus- 
trated by the example of a gas in a cylindrical vessel closed by a slid- 
ing piston, both of adiabatic material. The gas could do work expand- 
ing and lifting by its pressure a load on the piston. Unlike the isother- 
mal, the adiabatic process can be conducted either reversibly or 
irreversibly. Since no heat is imparted in it to the system (0 = 0), 
the first law of thermodynamics (3.05) takes the form 

W = Ui - U*. (3.33) 

The system can do work only at the expense of its internal energy. 
In particular, the work of adiabatic expansion of a perfect gas from 
the state (1) to the state (2) is (per 1 mol), according to (3.29), 

w c v (Ti - T 2 ). (3.34) 

This is the formula which Sguin was trying to verify in his attempt 
to establish the first law experimentally (section 11). It implies the 
same tacit assumption which was made by Mayer: he took it for 
granted that the expansion of real gases proceeds without internal 

When the adiabatic process is conducted reversibly, to every infini- 
tesimal step of it applies the differential equation 

Q, (3.35) 


and in the particular case of the compression or expansion of 
a perfect gas, c v dT + Dw - 0. We substitute for Dw as above, and 
obtain c v dT + (RT/v)dv = 0. Dividing by c v T and using the nota- 
tion of section IS, 7 1 = R/c V9 

y + (7~ 1)*-0. (3,36) 

In the derivation of this formula only the thermal equation of state 
was used. It is valid also for the case where c v and 7 depend on 
temperature. However, when the specific heats are constant (calori- 
cally perfect gas) its integral is particularlv simple. The equation is, 
then, easily integrable and gives 

TV 1 = const. (3.37) 

We can substitute T = pv/R from the equation of state: 

pv y = const. (3.38) 

These two formulas are equivalent forms of the equation of the 
adiabatic for perfect gases, first derived by Poisson (1823) on the 
basis of the old ideas of a caloric fluid. Like the isothermals, the 
adiabatics can be represented as a family of curves in the 
(,p)-diagram. These curves have the same asymptotes as the iso- 
thermals pv = const, namely the axes p = 0, v = 0; but the slope of 
the adiabatic (dp/dv = yp/v) is always steeper than the slope of the 
isothermal (dp/dv = p/v) passing through the same point. 

One of the most accurate ways of determining the ratio 7 ** c p /c v 
experimentally is the measurement of the velocity of sound in the gas 
in question. The theory of elasticity gives for the velocity of sound a 
the formula (already derived by Newton) 

a2=s "T- 
M dv 

where /* is the molecular weight of the gas. The rate of compression 
and expansion in a sound wave is so fast that there is no time for any 
appreciable heat interchange by conduction and the process must be 
considered as adiabatic. 1 If the medium is a perfect gas, we find, 

a 2 ypo/n. (3.40) 

1 Compare section 122 with respect to the limitations of the method of sound 
velocities. *4 


Exercise 20. Calculate numerically the work done by a perfect gas in its 
isothermal expansion from i * V Q to v t - $t>o at C. Express it in ergs and in 

Exercise 21. Give the expression for the work of adiabatic expansion in the 
case of a Van der Waals gas. Taking the coefficients a and b from Table 2 on p. 14, 
calculate it numerically for H* and N 8 and v\ - v 0t v* = Jt> and obtain the difference 
between the work done by these two real gases and the perfect gas. 

Exercise 22. Give the general form of the integral of the eq. (3.36) of adiabatic 
expansion, when c v (and c p ) is a function of T. Apply it to the empirical formula for 
water vapor given in exercise 17. 

Exercise 23. Calculating the work of adiabatic expansion from the formula 
(3.38), check the result (3.34). 

19. The cyclic process of Carnot. The French engineer S. Carnot 
devised (in 1824) a process which represents a schematic approximation 
of the essential features in the operation of a heat engine. Suppose 
that we have two heat reservoirs of so large a capacity that their 
respective temperatures Ti and T 2 (<Ti) are not appreciably changed 
if we withdraw or add to them a finite amount of heat. They corre- 
spond in Carnot's scheme to the boiler and the cooler of a steam engine. 
We shall explain Carnot's process, giving to the operating part of his 
ideal engine, at first, a special construction: Let it consist of a cylin- 
drical vessel, with a sliding piston as its top, filled with N mols of a 
perfect gas. The cylindrical side walls and the piston are made of 
adiabatically insulating material while the flat bottom is heat con- 
ducting but can be overspread with a suitable adiabatic cover. In the 
beginning the gas is in a state represented by the point A (pi, vi, T\) 
of the (/>,iO-diagram (Fig. 8), and the process consists of four divisions: 
(1) The bottom is heat conducting and in contact with the first heat 
reservoir; by an infinitely slow reduction of the outer pressure on the 
piston the gas is expanded in a reversible isothermal way at the tem- 
perature Ti to the point A'(p'\> v'i, Ti). (2) The bottom is made 
adiabatic by applying the cover and the expansion continued, reversi- 
bly and adiabatically , to the point B'(p f 2, 1/2, T 2 ). (3) The bottom is 
again made conducting and brought in contact with the second 
reservoir; the gas is reversibly compressed along the isothermal 
T T* to the state B(fa t i> 2 , T*). (4) Finally, the bottom is adia- 
batically covered a second time and the compression conducted, 
reversibly, along the adiabatic BA. The point B is chosen in such a 
way that this adiabatic brings the gas back to its initial state A, thus 
completing the cycle. 

The general Carnot process has in common with that just described 
that it operates between the same two heat reservoirs and consists of 
four analogous divisions: Two of them are an expansion and com- 


pression of the working part of the engine, respectively, in contact 
with the two heat reservoirs (boiler and cooler) ; the other two are an 
adiabatic expansion and compression. However, it is less specialized 
in two respects. In the first place, the working substance need not be 
a perfect gas but can be any real or ideal thermodynamical system. 
In the second place, the four divisions of the cycle may be conducted 
not reversibly but with a finite veloc- 
ity (irreversibly). 

Since no heat is imparted to the 
working system in the adiabatic ex- 
pansion and compression, its total 
heat intake Q during the cycle con- 

<P 2 'V 

<P 2 ',v 2 '> 

sists, in all cases, of the heat Q\ 

(positive) received from the reservoir 

of higher temperature (T\) and the v 

heat Q 2 (negative) received from that FIG. 8. Carnot's cyclic process. 

of lower (r 2 ), so that Q = Qi + Q 2 . 

As the final state of the working system is identical with the initial, 

we can apply the equation of the first law in the form (3.06) : 

W = Q = Qi + Q 2 . 

From the standpoint of the engineer, Qi is the heat conveyed to 
the engine to make it operate and supplied by the consumption of fuel. 
Not all of it is transformed into work, because Q 2 is negative. The 
heat given to the cooler ( Qz) is, to his mind, a wasteful loss of energy 
which is, however, necessary in order to bring the working part of the 
engine back to its initial state and to ensure a continuous operation by, 
periodically, repeating the cycle. The ratio rj of the useful work W to 
the consumed heat Qi is called the efficiency or the conversion factor 
of the engine 

W Qi + 

We shall compute the efficiency for the special case when the cyclic 
process is conducted reversibly with N mols of a perfect gas as operat- 
ing substance (Fig. 8). From the expression (3.32) of the heat imparted 
to the gas in an isothermal process 

Q l - NRTi log (v'M, Q 2 - - NRT 2 log (v' 2 /v 2 ). 

On the other hand eq. (3.37) applied to one of the adiabatic divi- 
sions gives 7W" 1 * T 2 v 2 y " 1 or (vi/vtf* 1 - T 2 /Ti, and to the 


other: (i/i/fl^) 7 " 1 = T*/T\. There follows vi/v* = v'i/v' 2 or 
v'i/vi = v'z/V2. Consequently 

Qi + Q2 = NR(Ti - T 2 

QI + & Tt- T 2 

V = ^ - * ^ , (3.42) 

Qi Ti 

a relation which can, also, be written in the form 

+ -0 (3-43) 

1 1 12 

and will prove very important in the next chapter. 

Exercise 24. Use the formula of adiabatic expansion derived in exercise 22 to 
show that the relation v'\/v\ v'z/v^ holds also in that case. The efficiency of 
Carnot's process is the same when c v is a function of T. 

Exercise 25. What is the thermodynamical efficiency of engines having coolers 
at C and boilers at 100 C, 150 C, 400 C? 


20. Several formulations of the second law. The second law of 
thermodynamics is older than the first. It was discovered by S. Car not, 
who enunciated it in the same pamphlet (1824) in which he gave the 
theory of his cyclic process. The reinterpretation and extension of 
the second law on the basis of the energy concept is due to Lord Kelvin 
(1851) and Clausius (1850-1863). There exist several equivalent for- 
mulations of this principle of which we give here the following: 

1 st enunciation (Kelvin) : It is impossible to transfer heat from a 
colder system to a warmer without other simultaneous changes occurring 
in the two systems or in their environment. 

The reader will recognize that this is a more precise and amplified 
statement of one of the facts which we mentioned in the very beginning 
of this course (section 1), where we introduced the notions of equi- 
librium and temperature, and again in section 10, where we defined 
the concept of heat. We said there that heat flows spontaneously 
always from the higher temperature to the lower and never in the 
opposite direction. The second law goes beyond this and asserts that 
one cannot completely undo the results of the process of heat conduc- 
tion even in any indirect way. We shall show in the following sections 
that the above formulation is completely equivalent to the 

2 nd enunciation : // is impossible to take heat from a system and 
to convert it into work without other simultaneous changes occurring in 
the system or in its environment. 

This law does not prohibit a process like the adiabatic expansion 
of a system (in which work is done at the expense of heat in the form of 
its internal energy) because this process is attended by " another simul- 
taneous change", namely, the change of volume of the system. Neither 
does it forbid the operation of Carnot's engine (preceding section): 
In his cycle heat is taken from one reservoir, but its conversion into 
work is not complete. The simultaneous change, which makes this 
process permissible, is the transfer of part of the heat to another 
reservoir of lower temperature. The transferred heat is less available 
for conversion into work in a similar way because this would require a 



third heat reservoir of still lower temperature. It is, therefore, often 
said that this heat is "dissipated" or "degraded". In general, a 
difference of pressures or of temperatures is necessary to obtain work; 
therefore, the phenomena of internal friction and of heat conduction 
which tend to reduce or destroy these differences must be regarded as 
attended by dissipation of energy. The establishment of the first law 
of thermodynamics made an end to all hopes of building a perpetual- 
motion machine (of the first kind) which could continually do work 
without some other form of energy being supplied to it. However, for 
all practical purposes a machine which could convert into worjc the 
enormous stores of internal energy contained in the oceans, the atmos- 
phere, and the body of our earth would serve just as well. Ostwald 
called such a machine, which could work without temperature dif- 
ferences, a perpetual-motion machine of the second kind. It is obvious 
that its operation would be in violation of the second law, so that we 
arrive at the 

3 d enunciation (Ostwald) : // is impossible to construct a perpetual- 
motion machine of the second kind. 

This formulation makes the second law analogous to the first, which 
asserts the impossibility of a perpetual-motion machine of the first kind. 

41. Efficiency of heat engines. Equivalence of the formulations. 
We have calculated, in section 19, the efficiency y = (Q\ + (?2)/(?i of 
the Carnot engine when operated reversibly with a perfect gas as the 
working substance. We shall call this particular type of construction 
and operation the perfect Carnot engine. The second law of thermo- 
dynamics permits us to say something about the conversion factor of 
the general Carnot cycle, which is the schematic prototype of any heat 
engine. In the real case, when the working part is any thermodynami- 
cal system and the cycle conducted either reversibly or irreversibly, we 
shall denote the efficiency by vj' = (Q'\ + Q'd/Q'i- Let us suppose, 
for a moment, that 17' > t?. We shall prove that this supposition is 
impossible, by carrying through the cyclic process alternately with a 
real and a perfect Carnot engine which share the same two heat reser- 
voirs. The length of the isothermal divisions A A' and B'B (Fig. 8) 
of the perfect engine is at our disposal and can be adjusted in such a 
way as to make the work done by the two cycles equal : 

W W or Q'i + Q't - Qi + Q 2 . (4.01) 

Moreover, the perfect engine being reversible, it can be run back- 
ward, in which case the quantities Q\, (>2 W simply reverse their signs, 
so that work is consumed and heat produced in the process. What 


will now be the result of a double cyclic process in which the real 
engine goes through one cycle forward and the perfect backward? No 
work will appear in the net balance, because the first engine does as 
much work as the second consumes. The only effect will be that the 
heat Q'\ Q\ will be taken from the first reservoir and Q f 2 Qz 
from the second. These quantities are oppositely equal, according to 
eq. (4.01): Q f \ Qi = (Q f 2 (?2). Let us now recall our supposi- 
tion if > TJ or (Q'i + Q f 2)/Q'i > (Qi + (?L>)/(?I. Since the numerators 
are equal, this leads to 1/Q'i > l/(?i or Q'\ < Q\ and, consequently, 
Q f 2 > (?2- A positive amount of heat Q'% Q% is taken from the 
reservoir of lower temperature T% and transferred to that of higher Ti 
without any other changes occurring, either in the reservoirs or in 
the working parts of the engines. The supposition TJ' > 17 leads, 
therefore, to a contradiction with the second law of thermodynamics, 
and we must conclude that it is impossible and that only the possibilities 

*' ^ * (4.02) 

are permissible. 

In the particular case, when the process in the real engine is also 
conducted in a reversible way, we can run the perfect engine forward 
and the real backward. Repeating the same argument with transposed 
(primed and unprimed) symbols, we can, then, prove that 17' < 17 
is impossible and that 77' S must hold. This result is compatible 
with (4.02) only when we have the sign of equality: rf = 77. All 
reversible Carnot engines have the same efficiency, no matter what the 
construction of their working part is. The sign < in eq. (4.02) refers, 
on the contrary, to irreversible cycles. Summarizing these conclusions 
and making use of the expression (3.42), we can give the general law 
for the efficiency 17 of any engine carrying out the cyclic process of 


where the sign of inequality (<) refers to irreversible, that of equality 
{ = ) to reversible, operation. In the reversible case, there holds also 
the relation 

+ % = 0, (4.04) 

ll -12 

whicb it is only another way of writing eq. (3.42). 

An irreversible Carnot process is always less efficient^rfhQMaocr' 
wasteful than a reversible inasmuch as a smaller fraction/^ tHB Be*f (? 


taken from the first reservoir (boiler) is converted into useful work. 
However, eq. (4.03) with the sign < will also apply when only one 
of the four divisions of the cycle is conducted irreversibly and the 
three others reversibly. We can conclude from this that in the expan- 
sion of a system in contact with a heat reservoir less heat is received 
by the system and less work is done when the expansion is conducted 
in an irreversible than in a reversible way between the same initial 
and final states. In fact, let these two kinds of expansion be the first 
divisions of two Carnot cycles, while the remaining three divisions are 
identical and reversible in both. We denote the heat items in the 
first (partially irreversible) cycle Q'\, Q% and in the second Q\, Q%. 
The condition 17' < 77 or (Q'i + Qd/Q'i < (Qi + Qz)/Qi leads to 
Q'i < Qi, since 2 is negative. The change of internal energy is in 
both cases the same, because it depends only on the initial and final 
state, so that the smaller heat intake is attended also by the smaller 
production of work. 

We have used above the first enunciation of the second law given 
in section 20. However, the second leads to, precisely, the same 
results. Their equivalence is shown, in a general way; by the following 
simple reasoning. Consider a process which contradicts the second 
enunciation, that is, one which produces the work W taking heat from 
a reservoir at the temperature T^ without any other simultaneous 
changes. We can feed this work into a reversible Carnot engine 
which runs in the backward direction, adjusting it so that in a cycle 
just the work W is consumed, and using the reservoir T<z as the cooler. 
The net result will be that a certain amount of heat will be transferred 
from this reservoir to one of higher temperature TI (the heater of the 
Carnot engine), ^'without any other simultaneous changes". In 
other words, a process which violates the second law in the sense of 
enunciation 2, violates it also in the sense of enunciation 1. As to the 
third enunciation, it is in substance identical with the second being, 
merely, a restatement of it in different words. 

Exercise 26. Give an alternative proof of the relation (4.02), for the second 
enunciation, analogous to that given in the text, for the first. Instead of adjusting 
the two engines so as to satisfy (4.01), adjust them so that the total heat received 
by the cooler in the double cycle vanish. 

22. The concept of entropy. All that follows from the second law 
of thermodynamics for a reversible Carnot cycle is embodied in the 
formula (4.04). We now extend our considerations by asking: What 
follows from it in the case of any other reversible cyclic process (different 
from that of Carnot's) carried out with a thermodynamical system. 
Our arguments and formulas will refer to systems of the most general 

IV 22 



kind, but we shall illustrate them by a drawing (Fig. 9) relating to a 
simple system, the only one which can be represented in a two- 
dimensional diagram. We repeat what was said in section 13 about 
carrying out such a process. We must have an infinity of heat reser- 
voirs differing in temperature by infinitesimal amounts. The system 
is brought in contact with them, successively, and at the same time 
subjected to infinitely slow compressions or expansions. The cycle is, 
then, visualized by the solid closed curve of the figure. In its infinites- 
imal part corresponding to the segment A A' the system receives an 
element of heat DQ. Let us now draw through the point A the isother- 
mal AC and through the point A' the adiabatic A'C, and denote by 
DQi the heat which the system would receive if it were subjected to 
the infinitesimal isothermal process AC. 
The relation between DQ and DQ\ 
is obtained by considering the little 
cyclic process AA'CA and setting up 
the equation (3.06) of the first law 
for it: the isothermal part AC is 
carried out in the reverse direction and 
we have DQ - DQi = DW. The ex- 
pressions DQ and DQ\ are infinitesimal 
of the first order, but the work DW 
done in the little cycle is represented 
by the area of the triangle AA'C and is, 
therefore, infinitesimal of the second 
order, infinitely small compared with 
the other two terms. It must be omitted from the equation, leaving 
DQ = DQ\: we can, therefore, change our notations, denoting the 
work done in the segment A A' also by DQ\. This conclusion holds 
for any system and not only for a simple one: Although the work 
done in an infinitesimal cyclic process cannot, in general, be re- 
presented in a two-dimensional diagram, it is, nevertheless, always 
infinitely small of the second order. 

Let us now continue the adiabatic through the point A 1 and draw 
one through A until they intersect the closed curve a second time in 
B', B. According to the result just obtained, the heat DQ^ received 
by the system in the part of the process B'B is the same which corre- 
sponds to the isothermal segment B'D passing through B f . We see, 
therefore, that the two heat items DQi and DQ* are precisely the same 
which the system would receive from the boiler and cooler if it formed 
the operating part of a Carnot engine carrying out the reversible Carnot 
cycle ACB'DA. Denoting by T\ and T* the temperatures which the 

\ B 

FIG. 9. General cyclic process. 


system has in the points A and B', we can apply to DQ\ and DQ2 
the relation (4.04) for a reversible cycle of Carnot 


JL I J. 2 

If we cut up into strips the whole area of the closed curve repre- 
senting the cycle, by a system of adiabatics drawn at infinitesimal 
distances from one another (the dotted lines of Fig. 9), eq. (4.05) will 
hold for the pair of segments of the cycle in every strip. Integrating 
over all strips 

C no* C nn* 

= o. 

The first integral represents the summation over all elements of 
the cycle between the points M and N, the second term the summa- 
tion over the other half of the curve from N to M. We can, therefore, 
replace the two terms by the single integral 

0, (4.06) 

extended over the whole cycle. This result applies to any cyclic 
process, so that the integral of the linear differential expression DQ/T 
taken over any closed path vanishes. We know from section 8 that, 
in this case, the expression DQ/T is the exact differential of some 
function 5 of the variables of the system : 

DQ dU + DW J0 

-j - - J> - - dS. (4.07) 

In the language of that section we say that DQ, while not an exact 
differential, is integrable. The integrating multiplier is 1/T, the 
reciprocal of the absolute temperature as measured by a perfect gas 

Clausius who first introduced the function 5 called it the entropy 
of the system (derived from the Greek word Ivrpcn-ofuu = turn inside, 
an allusion to its one-sided character). If we integrate eq. (4.07) over 
a reversible process leading from state (1) to state (2), we find 

- Si - . (4.08) 

J\ T 

The integral depends only on the initial and final value and is 
independent of the path, provided that this path is reversible (compare 


section 8), and the entropy is a "characteristic function", i.e. com- 
pletely defined when the state of the system is defined. We can take 
as state (2) in formula (4.08) any state of the system and as (1) some 
state (n) which we arbitrarily choose as normal 



Owing to the fact that the entropy is defined by the differential 

equation (4.07), it contains a constant of integration. As such ap- 

pears in the expression (4.09) the entropy S n of the normal state. As 

.far as the first and second laws of thermodynamics are concerned, it 

cannot be determined by any theoretical reasoning. 

In the case of a homogeneous system, the heat DQ received in 
every element of a reversible process is distributed uniformly, so that 
the entropy S also acquires a uniform distribution over the system. 
We can speak, therefore, of the specific entropy and, when the system 
is chemically pure, of the entropy per mol or molal entropy which we 
shall denote by 5 (small). We may further ask: What is the joint 
entropy S of two systems which, individually, possess the entropies 
Si and 52? According to the defining equation (4.09), S is obtained 
by carrying out a reversible process with the combined system. The 
reversibility requires, however, the equilibrium of all parts at every 
stage of the process, so that our two constituent systems must have 
at every moment the same temperature T. If we denote the elements 
of heat imparted in an infinitesimal step of the process, respectively, 
by DQ, DQi, DQ 2 , there follows DQ - DQi + DQ 2 , DQ/T - DQi/T 
+DQz/T and, from eq. (4.07), 

dS = dSi + dS 2 , or S = Si + S 2 . (4.10) 

The entropy is additive. We repeat, however, that the additive 
property rests on the fact that two systems in thermodynamical 
equilibrium have the same temperature. If two systems are, initially, 
not at the same temperature but can be brought into thermal contact, 
it has a good sense to ask about their joint entropy, and we shall define 
it as additive. In other cases this question is without meaning. It is, 
of course, possible to imagine a case in which the two parts are adia- 
batically insulated from one another but can interact by means of 
movable pistons and similar devices. They could be in partial equi- 
librium without equality of temperature, but this is a dynamical 
problem rather than a thermodynamical. The notion of an entropy 


IV 23 

of such a combination is artificial and without interest in thermo- 
dynamics. 1 

The entropy difference (4.08) takes a very simple form when the 
initial and the final states are of the same temperature. We can, then, 
take an isothermal process as the path of integration (T = const). 

S2 Si = 7T 


(S r T 3 ) 

23. The entropy principle. Let us discuss how the entropy of a 
system changes in an adiabatic process. If the process is reversible, 
eq. (3.35) or DQ = applies to every infinitesimal step of it. It 
follows, then, from (4.07) 

dS = 0, and S = const. (4.12) 

The reversible adiabatic process leaves the entropy unchanged: it is, 
therefore, also called " isentropic". 

When we turn to the irreversible adiabatic process, we are on new 
ground. We cannot expect to get an answer to our question from the 

considerations of the preceding sec- 
tion because they were restricted to 
reversible processes. Therefore, we 
have to fall back, once more, upon 
the second law as enunciated in 
section 20. In the same way as in 
introducing the entropy concept we 
shall use a (p, V) -diagram (Fig. 10) 
as an illustration, although our ar- 
gument will be valid for the most 
general thermodynamical systems. 
Suppose that the system is sub- 
jected to an< irreversible adiabatic 
process which brings it from the 

state A (with the temperature T\ and entropy Si) to the state 
B(T%, ^2). The transition itself, being irreversible, is not capable of a 
graphical representation (compare section 12). What can be said 
about the entropy change AS = 2 ~ Si? We can bring the system 

1 The only known significant case in which the entropy is not additive occurs in 
the theory of radiation. When a pencil of rays is split by partial reflection into two 
(the reflected and retracted), they have, in general, temperatures different from each 
other's and from that of the parent pencil. At the same time the sum of their 
entropies is not equal to the entropy of the parent pencil. Compare: M. Von Laue, 
Ann. der Phys. 20, p. 365 f 1906; P. S. Epstein, Phys. Zs. 15, p. 673, 1914. 

FIG. 10. Entropy change in irre- 
versible adiabatic process. 


back into its initial state A in the following reversible way: First, 
we let it undergo a reversible adiabatic process EC roughly opposite 
to the first, until it comes to the temperature Ta, while its entropy 
remains constant and equal to 52. From the state C on, we conduct 
it in a reversible isothermal (CD) until the initial entropy Si is again 
reached; then we complete the cycle, bringing the system adiabatically 
back to the state A. The temperature T% can be chosen arbitrarily 
higher or lower than either T\ or 7^2, or between these values. It is 
important to have this freedom of choice because certain processes 
(as transformations) can be conducted only at special temperatures. 
The system receives no heat in the three adiabatic branches of the 
cycle, and the heat Q in the isothermal part. Applying eq. (3.06) of 
the first law to the cyclic process (from A to A) we write 

where W is the total work done by the system. If Q and W were 
positive, the result of the process would be that the heat Q is taken 
from the heat bath, maintaining the constant temperature T% in the 
isothermal branch CD, and converted into work " without any other 
simultaneous changes." This is a violation of the second law (enunci- 
ation 2), and we conclude Q > or Q ^ 0. The application of 
eq. (4.11) to the isothermal CD gives Q = Tz(Si 2), since Si is 
here the final and 2 the initial state. The inequality implies, therefore, 

Si - S 2 ^ or 

AS = S 2 - Si ^ 0. 

The same argument can be easily adapted to the more general case 
where the system consists, initially, of many parts which are not in 
equilibrium. At the end of the irreversible adiabatic process it may 
have the same parts in new states or, even, other parts since some of 
the parts may have become merged, others separated into several units. 
At any event, the system can be brought back to the initial condition 
by the following reversible procedure. The final parts are brought, 
separately, in a reversible adiabatic way to the same suitably chosen 
temperature TV At this temperature they are iso thermally rearranged 
into the original parts, unmixing some of them if necessary (compare 
section SO). Still at the same temperature T$ each of the part systems 
is separately brought to its initial entropy. The cycle is completed by 
bringing the parts back to their initial states in independent, reversible, 
adiabatic processes. The conclusions drawn above for the simpler 
case obviously apply without change in the more complicated one. 
They apply, of course, also to every part of the irreversible process for 
which a definite entropy change can be defined. Under no circum- 


stances can this change be negative. It follows from this that, in the 
case 52 Si = 0, the change must be zero for every smallest part: 
dS = 0. That is to say, A5 = refers to the reversible process and 
A5 > to the irreversible : The irreversible adiabatic process is always 
attended by an increase of entropy. 

The summary of the results of this and the preceding section is the 
"entropy principle", due to Clausius, which represents an analytical 
formulation of the second law of thermodynamics. It consists of two 

4 th enunciation: (a) There exists a characteristic thermodynamical 
function called the entropy. The difference of entropies of a system in 
the states (1) and (2) is given by the expression 


where the integral is to be taken over any reversible path connecting the 
two states. 

(b) In an adiabatic process the entropy either increases or remains 

AS ^ (4.14) 

where the upper sign (>) refers to the irreversible, the lower ( = ) to 
the reversible, case. The adiabatic processes include also the case when 
the system is "left to itself", meaning by this that it is completely 
cut off from any interaction with its surroundings and does not receive 
any energy from outside, either in the form of heat or of work. 

The parts (a) and (b) of the entropy principle contain all that can 
be deduced from the enunciations of section 20. Sometimes, a third 
general law is included in the entropy principle: the fact that all 
systems tend towards equilibrium so that the entropy always does 
increase when the conditions of the system permit it. We prefer, how- 
ever, to discuss this law separately in section 30. As enunciated here, 
the entropy principle is completely equivalent to the formulations 
given in section 20. In fact, their validity is a sufficient condition 
because we derived the principle from it. But it is also a necessary 
condition because the adiabatic processes prohibited in those formu- 
lations are attended by a decrease of entropy. (Compare exercise 27). 

Since a spontaneous irreversible process is one of increasing entropy, 
we can use the condition AS > as the criterion of irreversibility. 

Exercise 27. Show that, in the fictitious case when the efficiency of the Carnot 
engine is y f > ij, the expression on the left side of (3.43) is negative. This means 
that the sum of the entropies of the heater and cooler decreases. 


24. Entropy of the perfect gas. In the case of a simple homo- 
geneous and chemically pure system, the entropy can be referred to 1 
mol of its material (section 23), and eq. (4.07) takes the form 

(4 , s) 

The only system for which the expressions of u and p were given 
in the preceding chapters is the perfect gas: According to (1.13) and 
(3.17) we have du = c v dT and p = RT/v t 

<fe-*y + ** (4.16) 

This, really, is an exact differential of the function 

s = c v log T + R log v + s'o, (4.17) 

where s'o is an arbitrary constant, the so-called entropy constant of the 
perfect gas. Using the equation of state, we can express v by p and T, 
or T by p and v, and so obtain two other forms of the molai entropy 

s = c p log r - R log p + so, (4.18) 

s = c v log p + c p log v + s"o, (4.19) 

since c p = c v + R, according to (3.20). The constants are related 
in the following way: s'o = so R log R, s"o = so c p log R. It is 
easy to see that eqs. (3.37) and (3.38) of the adiabatic are equivalent 
to the condition 5 = constant. 

It is well to mention also the case when the gas is perfect, in thermal 
respects, but the molal heat c v is a function of temperature (end of 
section 15). Equation (4.16) leads then to the expressions 



J ~T~~ 

Exercise 28. Check the statement that the process of expansion into the void 
(section 15) from the molal volume Vi to v 2 is irreversible (i.e. As > 0) for the perfect 
gas, by computing the entropy difference $ 2 SL between the final and the initial 

Exercise 29. Two vessels contain each one mol of the same perfect gas at the 
respective temperatures Ti and TV They are brought in contact and the two 
specimens allowed to come into equilibrium through the heat-conducting walls 
without change of volume. Compute total entropy in initial and final states and 


show that it increases. (Remark: The arithmetical mean of two numbers J(a + b) 
is always larger than the geometrical * 

25. The reciprocity relation of thermodynamics. The require- 
ment that the expression (4.15) must be an exact differential imposes a 
restriction on the analytical form of the internal energy u: the caloric 
equation of state (3.09) is not quite independent of the thermal. In 
fact, the exact differential must satisfy the reciprocity relations (2.10) 
of section 8. In the case of a simple system, there are only two inde- 
pendent variables and only one reciprocity relation. Let us substitute 
for Dq the form (3.11) 


3 T.- dT + -- n ^ 

Then the reciprocity relation has the form 

~ir\() + T\' (4 ' 22) 

: oi LJ. \OV'T 1 Jt> 

In carrying out the differentiation, we shall remember, that u is 
here regarded as a function of v and T only, and drop the subscripts 
in the second derivative 

1^2 1 

O ** 1 ^, ^. 


l(*P\-\(*\ + J 
T\dTj v T 2 l\dv/T *T 

The order of differentiation in the second partial can be transposed 
and the two terms depending on it cancel out. There remains 


which gives, substituted into (4.21), 

(IX- (); 

In the case of perfect gases, (dP/dT) v = p/T and (du/3v) T = 0. 
We justified the assumption c v = f(T), for them, by the measurements 
of Joule and Thomson on real gases (section IS). We see that this 
assumption is consistent with the second law, as we have already 
checked in the preceding section. Vice versa, accepting the second law, 
we can conclude theoretically that u is independent of v and a function 
of T only, for substances satisfying the equation pv = RT. This fact 
is well supported by observations on real permanent gases. When 

IV 25 



the measurements are very accurate, the deviations from the law of 
perfect gases begin to tell. The incentive which caused Thomson and 
Joule to undertake their investigation was the fact that Regnault's 
data for (dp/dT) v , when substituted into (4.23), gave a finite value of 
(du/dv)r for real gases. 

In the case of gases strictly obeying the eq. (1.21) of Van der 
Waals, we find from (4.23) 

= J' < 4 ' 25 ) 

This partial is independent of T. Since du is also an exact differen- 
tial, we can apply the reciprocity relation to it and conclude that the 
other partial c v = (du/QT) v is independent of v. The internal energy 
has then the expression 


c v dT - - 


and the term a/v represents the inner potential energy of the forces 
of interaction between the molecules. 

Equation (4.23) can be used to simplify the expression (3.15) of 
the difference of molal heats c p c v , making it dependent upon easily 
measurable quantities 

This formula was extensively tested by measurements o/ the 
velocity of sound in gases (section 18) up to pressures of 200 atm and 
found to be in good agreement with observations. 1 

Exercise 30. Show that the entropy of the Van der Waals gas is 


Exercise 31. Derive by integrating (4.23) 

Exercise 32. Starting from (4.23) prove the relation 

P. P. Koch, Ann. Phys. 26, p. 551; 27, p. 311, 1908. 


Exercise 3d. A further relation can be derived by partial differentiation. Sub- 
stitute in u( T, v) the expression of T in terms of p and v so that u = u[T(p t v) t v]. 
By differentiating with respect to v derive 

(Q\ _ (^u\ ( ^ u \ (^T\ 
9v/ p \ Qv/7 1 \97/ v \9i/ /p 

Transform this with the help of (3.12), (4.23), and (4.27) into 

- P- (4.31) 

26. Measurement of entropy of differences. From the definitions 
of molal heat (3.08) and of entropy (4.15) there follows the relation 
c = Tds/dT, which is valid for all ways of imparting heat to the 
system. In the two simplest cases of constant volume and of constant 
pressure this leads to 

<4 - 32 > 

Dividing by T and integrating, we obtain the following two expres- 
sions for the entropy 


P), (4.34) 

whose equivalence with (4.32) can be checked by differentiating them 
partially with respect to T, at constant v and p, respectively. 

If we wish to find the entropy increase (per mol) experienced by a 
substance when its temperature is raised, at constant volume, from T\ 
to Tg, the first equation gives 


s(r a , r) - s(Ti, v) = / ^dT. (4.35) 

*/ Tl 1 

On the other hand, the increase at constant pressure is from the 
second equation 



, p) - s(ri, p) = / ~ dT. (4.36) 
*/ TI 1 

The above expressions give the entropy changes, as long as the 

system does not undergo any transformation of state. When a trans- 
formation occurs, it involves a latent heat / (per mol) and an increase 
of entropy which is, according to (4.11), 

*-*i-, (4.35') 


since the transformation takes always place at constant temperature. 
Here, too, the cases v = const and p = const must be distinguished. 
At constant volume no work is done and the latent heat is, simply, 
equal to the increase of the internal energy. At constant pressure it 
is equal to the increase of the heat function (compare section 17): 

/ = Au, l p = A X . (4.360 

We see from this discussion that the determination of entropy dif- 
ferences involves only measurements of molal and of latent heats. 
The same data which are necessary to find internal energies and heat 
contents give also the knowledge of entropies. 

We have tacitly assumed that the system we are dealing with is a 
chemically pure substance. If this is not the case, we can readily 
adapt eqs. (4.35) and (4.36) to any system in which no chemical 
reaction takes place, simply writing the total entropies and heat 
capacities (capital letters) instead of the molal. With respect to 
chemical reactions, two cases may occur: (1) The reaction is conducted 
at constant temperature. This case is completely analogous to that 
of a transformation, the entropy increase being 

S 2 -Si = |, (4.35") 


Q v = A 7, Q p = AX. (4.36") 

(2) There is a continuous displacement of chemical equilibrium as 
the temperature is raised. Then the heat of reaction must be included 
in the heat capacity; we shall consider a case of this sort in section 121. 

Exercise 34. Calculate the entropy increase in the following process: 1 mol 
of liquid water (18 g) at 100 C is vaporized at the pressure of 1 atm. and the vapor 
is heated (at constant pressure) to 650 C. Take data from exercises 8 and 19. 

Exercise 35. (a) The molal heats c p for Ag, Al, and graphite are: 

















Al . . 








C (graphite) . . 

. 0.13 






2. fed 

Calculate in a rough graphical way the increase of the heat function as 
temperature is raised from 50 to 298.l K (p = const). Ot the 

(b) Calculate from the same data (in a rough graphical way) the in' 
entropy between 50 and 298 .l K. 

Exercise 36. Calculate the heat and the entropy imparted to a v~** (4.43) 
the following two reversible processes: (a) starting from the state pj 
is conducted first at p PI to pi, , then at v vt to pi, PI. (V 'The introduc- 


at v v\ to pz, Vi, then at p pi to pz t vz. Suppose c v and c p to be constant. 
Show that the heat imparted to the system is different in the two processes but the 
entropy change is the same. 

27. The extended law of corresponding states. The relation 
(4.23) between the caloric and the thermal equations of state can be 
written in the integrated form 

= f [ T (|^ - p]dv + f(T). 


We have seen in section 6 that, within certain groups of substances, 
the thermal equation satisfies the law of corresponding states. It may 
be asked, therefore, if a similar law does not apply to the caloric 
equation of state (4.37). As the most precise expression of the law of 
correspondence we may take that following from the mean reduced 
equation (1.29) of Kamerlingh Onnes. According to that equation, the 
quantity K RT c /p c v c is a constant for a given substance and, at the 
same time, a universal function of the reduced variables ?r, <p, r. Conse- 
quently, it must be a universal constant, as also 

^ = ., (4.38) 

L c 

since K = R/K is another universal constant. If we substitute into 
the combination pv/T the expressions of the parameters in terms of 
the reduced variables (p = p c TT, v = v c <f>, T = T c r), we find im- 

^ = K^ = /i(ir,*>, r). (4.39) 

In words, the quantity pv/T has the same value for different substances 

in corresponding states. In a similar way, substituting TT, #>, r into 

the integral of eq. (4.37), we find that this integral, divided by T, 

is a universal function of the reduced variables. This fact suggests 

he hypothesis that the same applies to the second term and to the 

>ole expression (apart from the zero energy) 

jT- =/2(7r,^r). (4.40) 

iantity might be also the same for different substances in 

states. This hypothesis was, in fact, made by 

nnes and others. We shall call the simultaneous validity 


of eqs. (4.39) and (4.40) the extended law of corresponding states. Though 
far from rigorous, this law is satisfied with a fair approximation within 
certain ranges of temperatures by large groups of substances (compare 
section 47). It may be pointed out that, in the case of perfect gases, 
the two functions /i and /a have the simplest possible form : they 
are reduced to the constants R and c v . 

An immediate consequence of these equations is that the expres- 
sion (4.15) of the entropy differential is also a universal function Fof 
TT, <p, T, whence 

As = Ffa, <p2j r 2 ) - F(TTI, ?i, n). (4.41) 

The molal entropy difference between corresponding states is the same 
for all substances (obeying the extended law). 

There were physicists who went even beyond this in their extension 
of the law and applied it to more general systems, having other 
variables than />, V, T. Suppose, for instance, that the element of 
work is, in the sense of section 7, dW = pdV + ydX, whence the 
entropy differential (4.07) 

dU + pdV + ydX 
dS = - - -- (4.42) 

The generalized force y can be treated exactly on the same footing 
as p (exercise 38), and it is possible to argue that the expression yX/T 
should have similar properties (with respect to corresponding states) 
as pV IT and U/T. We shall see in sectioit 86 that this idea was 
fruitfully applied to the theory of surface tensions. 

Exercise 37. Substitute for p, v, T the reduced variables into the integral of 
eq. (4.37), divided by T, and show that the result is a universal function of IT, <p, r. 
Exercise 38. Substitute in (4.42) 

v,x /x,T \dX/ T .v 

and set up the reciprocity relations in analogy with (4.22), (4.23). 

28. Theory of the Joule-Thomson process. We are now in a 

position to subject the experiments of Joule and Thomson, described 
in section IS, to a closer analysis. We found there that the relation 
between the states of the streaming gas (Fig. 7) on both sides of the 
plug is given by eq. (3.16), which we can write in the form 

AX = X2 xi = 0, (4.43) 

making use of the definition (3.21) of the heat function. The introduc- 


tion of this function permits us to express the entropy differential in 
a new way: From eqs. (3.25) and (4.15) 

We apply the reciprocity condition (2.10) to this expression and 
obtain, in complete analogy with the way in which we found the for- 
mulas (4.23) and (4.15), 

, . 


Of course, this is not a new relation but only another form of the 
reciprocity condition (4.23) into which it can be transformed by 
applying the formulas (1.05) and (3.21). Owing to this equation and 
to (3.26), the differential of the heat function becomes 

[v - T (j) ]dp. 

d x = cJT + v - T dp. (4.47) 

Suppose that in the Joule-Thomson experiment the difference of 
pressures A/> = p2 pi and of temperatures AT 1 = T% T\ is so 
small that it is permissible to neglect squares and higher powers and 
use eq. (4.47) for expressing AX in terms of &p and A7\ The condition 
(4.43) gives then 

Ar v - T(Qv/dT) p 

" = -^ = - 7 P - ' (4 ' 48) 

This change of temperature per unit pressure drop, n = AjT/A/>, 
is, usually, called the differential Joule-Thomson effect. With the help 
of this formula its sign and magnitude can be calculated theoretically 
if the equation of state and c p are known. We see that the sign of the 
effect depends on that of T(dv/dT) p v, or, transforming this by 
means of (1.05), upon the sign of the inequality 

The differential Joule-Thomson effect has the sign opposite to that 


of the bracket expression (because dp/dv is negative). In the case of 
Van der Waals gases, this condition takes the form 

2a RTb > ^ 

2 2 < - 

v 2 (v b) 2 < 

When the density is low, the constant b in the denominator of the 
second term gives rise to corrections of the second order only, and can 
be neglected in the first approximation, leaving the condition 

T T^ Ti = 2a/Rb = 6.757;, (4.51) 

taking into account (1.24). Ti is called the inversion temperature: 
below it, the effect is positive and consists in a cooling of the gas; 
above, it is negative and the gas gets warmer in passing through the 
cotton plug. We see from (4.50) that the two Van der Waals constants 
have an opposite influence, so that the sign depends on which of them 
dominates. From the physical point of view, we have here an interplay 
of two phenomena: On the one hand, when a is different from zero, 
the expanding gas does work against inner forces (compare eq. (4.26) 
of preceding section) and this decreases its temperature accounting for 
a part (a/v 2 ) of the expression (4.50). On the other hand, the outer 
work p22 piv\ of the process does not vanish for Van der Waals 
gases, even when the temperatures remain unchanged. It can be 
positive or negative and contributes the remainder of (4.50). On the 
whole; the larger part of a positive effect is due to inner work. 

The interplay of the two effects is more complicated when the 
density is not low. The expression (4.50) with the sign of equality, 
determining the inversion point, is best written in terms of the reduced 
variables (1.25) 

-- < 4 - 52 ' 

Eliminating <f> from this expression and the reduced Van der 
Waals equation (1.26), we find 


This formula shows us that there are two inversion points. The 
coefficient /* is negative above the upper inversion point and below the 
lower, being positive in between. The position of these points depends 
on the pressure : the upper of them sinks as the pressure is increased, 
the lower rises. 



IV 28 

In a qualitative way these predictions of the Van der Waals theory 
are fulfilled by real gases. The lower inversion lies in the liquid state 
and has been observed in several substances. It is, further, borne out 
by the experimental data that, the more perfect a gas is, the lower its 
upper inversion point lies and that it is depressed by pressure, while 
the lower point is raised by it. However, the Van der Waals equation 
cannot be used for a quantitative calculation of the Joule-Thomson 
effect or of the inversion point because its approximation is insufficient 
in the range of pressures and temperatures here involved. This is 
apparent from the figures given in Table 7. 



p aim 







Roebuck and Osterberg 1934 

H 2 



Olszewski 1901 




Hoxton 1919 

< t 







Roebuck and Osterberg 1933 

4 4 






Hausen 1926 

4 I 




18 to 100 


Burnett 1923 

Hoxton 1 gives the following empirical formula for air, in the range 
from to 280 C and from 1 to 220 atm, 


1 J. ~ 

where p is measured in atmospheres. 

The cooling by means of the Joule-Thomson process has acquired a 
great importance in the technique of low temperatures and of liquefying 
gases. Mostly, it is not used as the sole agency of cooling but in com- 
bination with adiabatic expansion. Air, oxygen, and nitrogen can be 
subjected to the process at room temperature since their inversion 
points lie higher. For hydrogen, T> lies above the boiling points of 
liquid oxygen and nitrogen so that it can be made amenable to the 

i Hoxton, Phys, Rev. 13, p. 438, 1919. 


process by being precooled by liquid air. Again 7\ of helium is above 
the boiling point of hydrogen (20?5 K). Whether these gases are pre- 
cooled in this way (cascade system) or by adiabatic expansion, the 
process they are subjected to in technical applications is the integral 
Joule-Thomson effect, i.e. an expansion from a high pressure (p\) to a 
considerably lower one (pz) with an attendant temperature drop from 
Ti to T2. These changes are still governed by eq. (4.43) or 

x(02, r 2 ) - x(i, Ti) = 0. (4.55) 

We may ask under what conditions the maximum cooling is 
attained, (T% = min.), if we start from a given temperature TI. This 
question is, in fact, of practical importance, because the temperature 
Ti to which the system can be precooled is usually dictated by technical 
considerations. Let us suppose, for a moment, that the end pressure 
p2 is also given (Ti = const, p2 = const). Then eq. (4.55) is a 
relation between only two variables T^ and pi, and the condition for 
T2 = min. is obviously dT^/dpi = 0. The total differential of the 
equation is 

(IlMl;)*^ < 

Hence we conclude that, in order to obtain the lowest end tem- 
perature, the pressure pi must be chosen so as to satisfy the condition 

r = 0. (4.57) 

The fact that this condition is independent of p2 shows that our 
assumption of a fixed end pressure was unnecessary. The choice of pi 
as a root of eq. (4.57) assures the maximum of cooling for any end 
pressure, provided the difference Ap is not very small. However, 
owing to eq. (4.45), this condition is precisely of the same form as that 
for the differential inversion point, which we transformed into (4.49) 
(with the sign of equality). For a given (initial) temperature, the 
optimum in the integral Joule-Thomson effect occurs at the same pressure 
as the inversion in the differential effect. If the substance follows the 
law of Van der Waals this pressure is determined by eq. (4.53), or 
resolving it with respect to TT: 

Exercise 39. Starting from eq. (4.47), prove the relation 



Exercise 40. Using partial differentiation (as in exercise 33) derive the relation 

. , 

dp/ v \dp/T \dT/ p \dp 

With the help of (3.26), (4.45), and (4.27) transform it into 


Exercise 41. Calculate the differential inversion points, for the gases and 
pressures of Table 7 (excepting air), making use of formula (4.53) and of the data of 
Table 2 (on p. 14). Compare them with the measured values. 

29. Realization of the absolute scale of temperatures. The abso- 
lute temperature T was introduced in section 4 as that which would 
be measured by means of an ideal thermometer filled with a perfect 
gas as thermometric fluid. This temperature acquired a more general 
significance when it was found (section 21) that the efficiency of the 
reversible Carnot cycle is independent of the nature of the working 
system and, being always the same as in the case of a perfect gas, 
can be very simply expressed in terms of T. This fact led, in turn, to 
the possibility of defining the concept of entropy by its differential 
dS = dQ/T, in which the reciprocal of the absolute temperature 
appears as the integrating multiplier. 

The role which T plays in the analytical formulation of the second 
law makes it a quantity of general thermodynamical importance and 
gives a practical way of indirectly determining the absolute tempera- 
ture, although the perfect thermometer is only an abstraction. In 
fact, any thermodynamical equation derived from the second law 
represents a relation between T and the other variables of the system 
to which it applies. Theoretically it is, therefore, possible to express 
the absolute temperature T in terms of measurable characteristics of 
any system of nature. On the other hand, one can read directly, 
under the same conditions, the arbitrary temperature / on a real 
thermometer and, comparing it with T, graduate this thermometer in 
the absolute scale. This procedure was conceived and carried out by 
Lord Kelvin who used, for this purpose, the equation of the Joule- 
Thomson process. Let us take the form (4.48) which refers to the 
case of small pressure differences Ap. Suppose that we have estab- 
lished a correspondence between the scales of T and /, so that we can 
regard them as functions of each other. We wish to introduce into the 
equation the quantities A<, (9v/3/)pi C'P measured by means of the 
real thermometer. We obtain immediately, AT" = At*dT/dt, 
(dv/dT) p (9v/3/) p dt/dT; moreover, from the definition of molal 


heats we have, in general, c Dq/dT = (Dq/dt)dt/dT = c'dt/dT and, 
in particular, c p = c' p dt/dT. The substitution into (4.48) gives 

Ap dt 

dT __ Qp/30 

The right side of this expression contains only readily measurable 
quantities, so that the absolute temperature can be found by integrat- 
ing it. In practice it is convenient to measure the temperature t with 
a gas thermometer filled with the same gas which is used in the Joule- 
Thomson process. We have then, according to (1.08), v = i>o(l + a/), 
and this is the same v as in the denominator of (4.61), since p is there 
considered as constant. Further, (dv/Qf) p = VQOI, so that the integral 
of the right side becomes 



Suppose that to the point / = C corresponds T = TO, then we 
can adjust the scales so (compare section 4) that to t = 100 C corre- 
sponds T = To + 100. The integral of (4.61). is, therefore, 

log (r/7o) = K(t), (4.63) 

and for t = 100 one obtains an expression for To in terms of measur- 
able quantities 

To = 100[exp X(100) - 1]. (4.64) 

The number To indicates how many degrees of the absolute scale 
the zero point of absolute temperature (T = 0) lies below the point 
of freezing water (/ = 0). It is remarkable that the measurements of 
A//A/> need not be carried out at very low temperatures, in order to 
obtain it, but only between / = and t = 100. In this range the 
numerical value of the term c' p &t/vo&p is pretty small. In the case of 
helium, it is smaller than 0.001 in any part of the interval, so that 
neglecting it altogether would already give an approximation of better 
than 0.1%. If this is done, eq. (4.62) reduces to K(f) = log (1 + a/), 
whence To = I/a, as if helium were a perfect gas. With the experi- 
mental a = 0.003659 this gives To = 273.32, which is, in fact, pretty 
close to the true value. The most accurate determinations of TO were 
carried out by Roebuck (by this method *) and by Henning 

1 J. R. Roebuck, Proc. Am. Soc. Arts, Sci. 60, p. 537, 1925. 


and Heuse (by a different one *). The weighted mean of their 
results 2 is 

To = (273.18 0.03) K, 

where the symbol K indicates degrees in the absolute or Kelvin 
scale. Nevertheless, we shall have to adopt here the old value of 
To = 273M on which all current tables are based. A change in the 
definition of the zero point has far-reaching consequences and can be 
made only by international agreement. 

As stated in the beginning of this section, any phenomenon which 
satisfies the following two conditions can be used for establishing the 
absolute temperature scale: it must be (1) expressible by a theoretical 
formula as a function of T, and (2) measurable in terms of a relative 
scale /. Temperatures in the vicinity of K are being established by 
means of the magneto-caloric effect (compare section 136), the same 
effect by means of which these extremely low temperatures are reached. 

Exercise 42. In the range from to 100 C (at 1 atm) the Van der Waals 
equation gives a sufficient approximation for a rough estimate of the third term in 
the denominator of the integral (4.62). Calculate its maximum value in this interval 
from eq. (4.49) and the data of Table 2, for H 2 and N 2 . Estimate the percentual 
error in To, if this term is neglected. 

Exercise 43. How can the formula O/a)r = T(dp/dT) v - p be used to 
establish the absolute scale of temperatures? Suppose that (d/dt>)*, (dp/dt) v 
and p can be measured as functions of the relative temperature t. 

30. Statistical interpretation of the entropy concept. Matter con- 
sists of discrete particles (atoms, molecules, etc.), and the actions 
exercised by material systems upon one another can be traced to the 
effects of these ultimate parts. Thus the pressure of a gas upon a wall 
is due to its bombardment by the moving molecules of the gas which 
hit it at very short, but irregular, intervals and transfer to it varying 
amounts of momentum. From this point of view, the pressure is a 
statistical average which is automatically taken and presented to us by 
the integrating action of our measuring instrument, the manometer. 
The same is true with respect to other thermodynamical variables and 
functions. The temperature, in particular, can be brought in con- 
nection with the average of the kinetic energy which is distributed 
over the individual particles according to the laws of chance or the 
laws of probability. 

The mathematical theory of probabilities had its origin in the 
analysis of games of chance. Suppose that a deck containing an 

* F. Henning and W. Heuse, Zs. Physik 5, p. 285, 1921 ; W. Heuse, 37, p. 157, 1926. 

* R. T. Birge, Phys. Rev. Suppl. 1, p. 18, 1929. 


equal number of black and red cards is manufactured and shuffled so 
perfectly that the chance of drawing from it a black or a red card is 
exactly equal. In other words, if the drawing is repeated a large 
number of times Z, the card being returned and the deck reshuffled 
every time, the number of red and black cards drawn will be, relatively, 
the closer to Z/2 the larger Z is. We say then that the probability of 
drawing a red card is one half: PR = ^, and, in the same way, the 
probability of drawing a black card PB = ^- What is now the prob- 
ability PRB of obtaining, in two successive drawings, first a red and 
then a black card? There are four possible cases for the result of this 
double drawing: (1) red, red; (2) red, black; (3) black, red; (4) black, 
black. Since the chance of each of the four cases is the same, we 
conclude that the probability P RB = J or P RB = P/ePz*. This is 
an example of the multiplicative law of probabilities. In general, the 
probability Pi2 of a complex event, which consists of two simple 
events, is equal to the product of their respective probabilities PI, P% 

Pi2 = PiP 2 . (4.65) 

The multiplicative law holds only when the simple events are 
statistically independent, meaning by this that the outcome of the first 
in no way influences the chances of the second. 

The application of the laws of probability to the molecular structure 
of matter is treated in the branches of physics called statistical 
mechanics and kinetic theory. Here we shall give only a brief refer- 
ence to some notions about the entropy which are, in part, postulated, 
in part, deduced in these disciplines. A striking property of the entropy 
is its onesidedness. In any spontaneous process of nature it can only 
increase, and the state of equilibrium is the state of maximum entropy 
(compare also next section). On the other hand, when we regard a 
physical system as an assembly of interacting atoms and molecules, 
we say that it will change in the direction of a more probable state, as 
to positions and energy distribution of its ultimate units: The final 
state will be that of the highest probability. The question arises, 
therefore, whether it is possible to give a definition of the probability 
of a state which would bring out its parallelism with the entropy and 
permit to follow it through in a mathematical way. This is, pre- 
cisely, what is accomplished in statistical mechanics: The probability 
of a state is defined there as the number of ways in which this state can be 
realized. Let us illustrate this by the example of pure and mixed 
crystals. Consider a crystal consisting of Zi identical atoms in a reg- 
ular arrangement. We shall discuss here the probabilities, only in so 


far as they depend on the normal position of the atoms, and leave out 
of consideration the deviations and velocities due to their thermo- 
kinetic motions. If we could label and individually distinguish the Z\ 
atoms, every permutation of them would lead to a new way of realizing 
the crystal, and the total number of ways would be equal to the 
number of possible permutations P s i = Z\ !. This is called by Gibbs 
the specific definition of the probability. In modern statistical mechanics 
it is recognized, however, that it is not possible to tell the atoms apart 
so that all the permutations must be considered as one single realization 
Ppi = 1: the generic probability definition which was preferred by 
Gibbs himself. On the other hand, let us take the case of a mixed 
crystal, that is Z\ atoms of one kind and Zi of another arranged in a 
joint lattice. If we assume that every distribution of atoms of the 
second kind between those of the first represents still the same macro- 
scopic state of the mixed crystal, we find that the number of possible 
realizations is, in the specific definition, again equal to the permutation 
number P s i 2 = (Z\ + Z%)\. In the generic definition it is smaller 
because the permutations of atoms of each kind among themselves 
(Z\\ and Z2 ! .) do not count. The previous number, therefore, must 
be divided by the product Z\ \ Z* \ 

Let us consider the process of combining two pure crystals (Z\ 
and Z2) into a mixed crystal. The initial probability of the system, 
according to the multiplicative law (4.65), is P\Pi\ We see that it is 
increased in the process in the ratio given by the expression (4.66), no 
matter which of the two definitions we use. We repeat that the pre- 
ceding discussion implies that all the realizations which we counted 
belong to the same macroscopic state of the crystal. To what extent 
this is the case in reality, will be considered in sections 98 and 99. 

What is now the mathematical relation between the probability 
and the entropy function? The probability of complex systems follows 
the multiplicative law, the entropy the additive. We cannot, there- 
fore, make them simply proportional. But if we take the logarithm 
of the probability, the difficulty is removed, since eq. (4.65) then 
becomes log Pi2 = log PI + log P%. In view of this, Boltzmann 

S = * log P, (4.67) 

as a general principle which now bears his name. Here k = R/n A 
is the Boltzmann constant as defined by eq. (1.14). Applying this to 


the above example, we find that the entropy increase in the process 
of producing a mixed crystal out of two pure ones is 

AS = k [log (Zi + Z 2 )! - log Zi! - log Z 2 !]. (4.68) 

As stated above, the multiplicative law of complex probabilities 
applies only when the simple events are statistically independent. 
On the other hand, the entropy of a complex system is additive, only, 
when the equilibrium of its constituent parts is characterized by a 
uniform temperature (section 22). It follows from this that the 
thermodynamical criterion of two statistical systems not being inde- 
pendent is that they come in equilibrium with different temperatures 
(compare footnote on p. 60). 

This is not the place to enter into the applications of Boltzmann's 
principle. We shall only mention one or two of its immediate conse- 
quences which will be useful in interpreting and rounding out the 
thermodynamical treatment. We restrict ourselves to the case when 
the events are considered as statistically independent. 

(A) There is an alternative formulation of Boltzmann's principle 
which we give only for two special cases as we shall not need it in its 
generality. (1) If the particles of a system move in a field of conserva- 
tive forces, the probability of finding a particle in the space element 
Ar is 

Pdr = Cexp (-e po t/&7>r, (4.69) 

where e po t is the potential energy of the particle, when in the space 
element dr, and C is independent of the coordinates of the particles. 
(2) A similar relation exists for the total energy, its formulation is 
particularly simple in the quantum theory. If a system consists of 
identical elements (atoms, molecules, linear oscillators, etc.), each 
capable of assuming the quantum states 0, 1, 2, ... with the respective 
energies eo, ei, 62, . . ., the probability of finding an element in the 
state (j) is 

P,= C'exp(-e,/tr). (4.70) 

(B) In classical statistics (when quantum restrictions do not exist 
and all energy levels are permissible) Boltzmann's principle leads to 
the equipartition of energy. The mean kinetic energy of an element 
is equal to kT/2 per degree of freedom: 

Sun = nkT/2, (4.71) 

n being the number of (translational and rotational) degrees of 
freedom. As to the potential energy, this law applies to it only in 


the case when the forces producing it are elastic forces (i.e. are pro- 
portional to the distance). 

Exercise 44. A coin is tossed three times; what are the probabilities of (1) head 
being up all three times, (2) head being up twice and tail once? 

Exercise 45. Calculate from the formula (4.69) the relative number of mole- 
cules of air in a space element dr of the earth atmosphere at the height h above the 
surface of the sea. Use a mean molecular mass (m), and, substituting k R/n At 
show that one obtains in this way the barometric formula of Laplace. 



31. The direction of spontaneous processes. Let us consider a 
complex thermodynamical system whose parts may or may not be in 
equilibrium. We assume it prevented from heat exchange with the 
outer world by an adiabatic envelope but capable of doing work. If 
all its microscopic dynamical, electrical, etc., parameters were given, 
as well as the outer forces acting on it, its future states would be deter- 
mined and predictable. Can we foresee anything about the spon- 
taneous processes in it and their direction, knowing only its thermo- 
dynamical characteristics? The two laws of thermodynamics as 
stated in the preceding chapters will not lead us very far in this 
endeavor : the first law only states that one type of energy is converti- 
ble into another, but, as far as this principle is concerned, the con- 
version can go either way, provided the geometrical constraints of the 
system permit it. The second law tells us a little more about the 
direction of spontaneous processes: only such changes are permissible 
in which the entropy does not decrease. This excludes, at once, a 
large number of conceivable occurrences as incompatible with this 
law. Still, the second law is incomplete in that it leaves the question 
open whether the entropy actually will increase when conditions per- 
mit it. In fact, in the enunciations of the preceding chapter the fun- 
damental fact is not contained that there is in every thermodynamical 
system a tendency towards a definite state of equilibrium. 

Fortunately, the empirical evidence relating to these questions can 
be summarized in the form of two simple rules supplementing the first 
and the second law of thermodynamics. In the first place, the internal 
energy U has the nature of a potential energy. A dynamical system 
which is at rest, at a certain moment, will set itself in motion if its 
constraints permit a decrease of the potential energy. The potential 
energy has a tendency to diminish and to be converted into kinetic 
energy, so that equilibrium can ensue only when it has reached its 
minimum. In a similar way, the internal energy of an isolated thermo- 
dynamical system tends to decrease when the existing conditions give 



it a chance of being converted into forms of energy of non-potential 
nature. U tends towards its minimum, and the system will not be 
in equilibrium until this minimum is reached. 

U->U min . (5.01) 

In the second place, the entropy of isolated systems tends to increase. 
Processes attended by an augmentation of the entropy are not only 
permissible, as the second law states, but one of them will necessarily 
take place spontaneously. Equilibrium will be reached only when S 
has attained its maximum 

S-S ma *. (5.02) 

This condition can be regarded as a portion of the second law, a 
third part of the entropy principle, of equal importance with the first 
two enunciated in section 23. We speak of it separately in this place, 
in order to bring out its analogy with the rule (5.01) supplementing 
the first law. 

From this discussion follow the necessary and sufficient conditions 
of equilibrium of a thermodynamical system : The internal energy must 
have its minimum and the entropy its maximum value. It is known, 
from mathematics, that this will be the case if, for all virtual infinitesi- 
mal changes of state, consistent with the nature of the system, the 
variations 6 U and SS vanish : 

5 [7 = 0, 55 = 0. (5.03) 

There is a difference between the differential -and the variation. 
The differential dU refers to an actual infinitesimal change of state 
which the system undergoes in a specified real process. In thermo- 
dynamics, the initial and the infinitely close final state must be states 
of equilibrium if they are to define a differential. On the other hand, 
the variation has reference to all conceivable (" virtual' ') changes 
whether they conserve the equilibrium or not. A variation may start 
from a state of equilibrium and lead away from it. It may even con- 
nect two (infinitely close) thermodynamically unstable states. The 
reason for such a definition of the variation is that we do not know in 
advance which virtual changes are permissible, and we wish to select, 
a posteriori, those which are compatible with the conditions (5.03) 
and with other conditions, inherent in the nature of the system and in 
the concept of equilibrium. It is clear from this explanation that the 
virtual changes are the more comprehensive class: every actual change 
is contained among the virtual changes, but the reverse is not true. 


32. Variation of the internal energy. Partial molal quantities. 

The systems and states we are going to deal with must be capable of a 
thermodynamical description. We have to restrict our considerations 
to systems which can be divided into phases with the following prop- 
erties: though, possibly, not in physical or chemical equilibrium, each 
phase must be uniform with respect to temperature, pressure, and 
composition so that it has a definite internal energy and entropy. For 
all virtual changes in which no matter is added to the system, holds 
the fundamental formula (4.07), or 

dU = T5S - pdV. (5.04) 

We can choose the volume V and the entropy S as the thermo- 
dynamical variables. In addition, the system may be characterized 
by other parameters specifying its mass, chemical composition, etc. 
To fix our ideas let us assume that the system is not chemically pure 
but a mixture of Ni mols of a first chemically pure component, N% mols 
of a second, and so on. The internal energy is then a function of the 

The variation of U is the most general change which takes place 
while all the variables undergo virtual changes, 

We shall only consider virtual changes in which no matter is added 
to the system so that the variations dN k represent increases of some 
components at the expense of other components. In this case the 
formulas (5.04) and (5.06) are valid simultaneously and apply to all 
possible virtual changes satisfying this condition. In particular, we 
may consider the case when the composition remains altogether 
unchanged, SN k = (compare, in this connection, section 39). Com- 
paring the two equations, we find 

(H) - T - (If) --> < 5 -"> 

\OO/V.N \OV/S.N 

The subscript N is usually omitted because the symbol of partial 
differentiation implies that all the other variables are to be kept con- 
stant. Substituting this into the general eq. (5.06) and subtracting 
(5.04), we obtain the remarkable relation 


These results can be generalized for systems with several phases 
(provided T and p are uniform throughout), but we need not deal with 
this aspect of the question here since it will occupy us in section 42, 
in connection with the problem of equilibrium. The partials of 
eq. (5.08) should not be confused with the partial molal internal ener- 
gies which are defined as follows : 


In the case of a chemically pure substance, U = Nu, where the 
ordinary molal internal energy u is a function of temperature and pres- 
sure only and remains constant when p and T are kept constant. 
Therefore, Uk is a generalization of u for mixtures and solutions, 
becoming identical with it in the extreme case of pure substances. 
Another partial molal quantity worth mentioning is the partial molal 
volume. When several substances are mixed or dissolved in one 
another, the total volume is, generally, not additive but a complicated 
function of the mol numbers (compare also section 57) 

V = V(p, T,NL.. N 9 ). 
The partial molal volume of each component is defined as 



The relation of the functions (3 U/dN k ) s . v to partial molal quan- 
tities will become clear in the next sections. 

Exercise 46. The measurements by Wade (J. Chem. Soc. 75, p. 254, 1899) on 
the volume of aqueous solutions of NaCl can be represented by the following empiri- 
cal formula (due to Lewis and Randall) 

where N\ refers to water and Nt to NaCl. Calculate the volumes of solutions with 
0.1, 0.3, 0.5 mol of NaCl per 1000 g of water and the partial molal volumes of NaCl 
in all these cases. 

Exercise 47. Prove for the perfect gas the formula 

Directions. Calculate &U = d(Nu) from (3.18) and eliminate ST and Sv with 
the help of the conditions 55 5(Ns) = and dV - 5(Nv) - 0. For s, use the 
formula (4.17). 


33. Construction of new characteristic functions. Further prop- 
erties of the heat function. In addition to the characteristic functions 
U and 5, we had occasion to introduce in section 17 the heat function 
defined as 

X - U + pV, (5.11) 

while its differential, in the case of a simple system, is obtained from 
(3.22) and (4.07), 

dX = TdS + Vdp, (5.12) 

whence in an isobaric process (dp = 0) 

Q P = X 2 - Xi. (5.13) 

Instead of using the variables 5, V of the preceding section, we 
can describe the system also by the variables S, p, and use X as the 
appropriate characteristic function. The term pV in (5.11) has the 
effect that its differential d(pV) = pdV + Vdp cancels the term 
pdV of (5.04) and leaves in its place Vdp. In this way the differ- 
ential dp is introduced instead of dV. In mathematics the method of 
replacing a term ydx by the term xdy by means of subtracting the 
differential d(xy) is called a Legendre transformation. It is obvious 
that we can construct by this expedient still two more characteristic 
functions from the expression (5.04), namely, subjecting to Legendre 
transformations the first term, or both terms simultaneously. We 
shall discuss these functions in the next sections and reserve the 
remainder of this one for a few remarks about the variation of the heat 

As explained in the preceding section, we assume that the state of 
the system depends on the mol numbers Ni. NZ, . . . N ff in addition to 
the thermodynamical variables: 

X = X(5, p, Ni,... N ff ). (5.14) 

We can obtain, for the variation of X, two expressions closely 
analogous to those for dU in the preceding section. On the one hand, 
the relation (5.12) is valid also for the variations 

dX = TdS + Vdp. 
On the other hand, we find from (5.14) 


As in the preceding section, we conclude by comparing the two 

(i) - r - (HD - v - 

\dS/p,N \3P/S.N 

(The subscript N is usually omitted). Another form of the variation 
of (5.11) is dX - Vdp = dU + pdV. Hence and from (5.06) and 

We cannot conclude from this equation that the terms are equal 

9 X\ 

because the virtual changes 6N k are not independent: we have excluded 
changes in which matter is added to the system. However, the proof 
of the identity (5.17) can be easily supplied by partial differentiation. 
In fact, let us describe U by the coordinates S, p instead of 5, V. We 
can effect this change of variables by substituting for V its expression 
in terms of 5, p, N. We need not bring S in evidence since it is con- 
sidered as constant, and we write instead of (5.05) 

U - U[V(p, N), N]. 
By the rules of partial differentiation 

According to (5.07), the second term on the right is [d(pV)/dN] St p . 
Taking it to the left side, the equation becomes identical with (5.17). 

Exercise 48. Consider the general form of d U based on (2.02) 

Show that the number of characteristic functions which can be constructed from it 
by Legendre transformations is 2 n + l 1. 

Exercise 49. Prove that, for a perfect gas, 

Directions. Calculate dX = 8(N\) from (3.28) and eliminate ST with the help 
ol the conditions 6S d(Ns) and dp = 0. For s, use the formula (4.18). 


34, The work function. We subject to a Legendre transformation 
the first term of the differential of the internal energy (4.07) 

dU = TdS - DW. (5.18) 

Subtracting from it d(TS), we find 

D* - - SdT - DW, (5.19) 


* - U - TS. (5.20) 

The function ^ is useful in bringing out certain properties of iso- 
thermal processes. We have, in this case, dT = and 

d& = - DW. (5.21) 

Integrating this over a reversible process at constant temperature 
from the initial state (1) to the final (2), 

*! - * 2 = - A* = W. (5.22) 

The work done by a system in a reversible isothermal process is equal 
to the decrease of the function V. The case when the system remains 
in contact with a heat reservoir of constant temperature but the 
process is not conducted reversibly is sometimes, loosely, called an 
irreversible isothermal process. In this kind of a process the tempera- 
ture of the system is not strictly equal to that of the reservoir and is, 
in fact, not strictly uniform. We have seen in section 12 that the 
process is less efficient and the work done in it by the system is less 
than in the reversible case. We can, therefore, say that ^i 2 
represents the maximum work that can be obtained from the system in 
any isothermal process leading from the state (1) to the state (2). 

Applying eq. (5.20) to the initial and final states and taking the 

Ui - E/2 - *i - * 2 + T(Si - S a ). (5.23) 

The second term on the right side is, according to eq. (4.11), equal 
to Q, representing the heat transferred to the heat reservoir during 
the process. Helmholtz, who used the function ^ a great deal, called 
it the free energy because the difference ^f\ ^2 represents that part 
of the decrease of the internal energy U\ Uz in the process which 
is available for work while the bound energy T(S\ 2) is lost to the 
reservoir maintaining the constancy of temperature. However, the 
name cannot be regarded as very fortunate because it loses sight of 
the fact that there exist in nature endothermic processes in which 
T(S\ ^2) is negative. In this case, work is done by the system, in 


part at the expense of the heat reservoir, which supplies heat instead 
of receiving it. The decrease &i ^2 can, therefore, be larger than 
Ui - C/ 2 , as was first pointed out by Gibbs. To make matters worse, 
in a part of the chemical literature of this country the name "free 
energy" is applied to a different function (namely, to $ defined in the 
next section). We think it, therefore, advisable to avoid the term 
"free energy" altogether and shall refer in this book to the function ^ 
as the work function. This name was introduced by R. H. Fowler 1 
and is highly appropriate in view of the physical meaning of the 
quantity ^ as expressed in eq. (5.21). This relation means that, as 
long as the temperature is kept constant, the elements of work used in 
thermodynamics are exact differentials which possess the potential &. 
We shall discuss the very important question, under what circum- 
stances ^ tends to decrease, in section 36 and shall say here only a few 
words about its partial derivatives. In the case of a simple system, 
DW = pdV, and the variation becomes, in accordance with (5.19), 

6* =- SdT - pdV. (5.24) 

The appropriate coordinates for the description of the work func- 
tion are, therefore, T and V in addition to the mol numbers N k , deter- 
mining the composition of the system: 

V, N l9 ... NJ. 
In analogy with the preceding sections, we find 



\dTJv, N \OV/T,N 

& -P. 

(The subscript N is usually omitted). As to the partials with 
respect to Nk, it is easy to prove the identity (exercise 51) 


by means of which eq. (5.08) can be expressed in terms of derivatives 
of the function ^. 

It will be useful to give here the explicit expressions of the molal 
work function $ for perfect and Van der Waals gases. We take the 
more general case of the formulas (3.19) and (4.20) when the molal 
heat is not considered as constant. In the case of the perfect gas, 

* '(D - RTlogv, (5.27) 

1 R, H. Fowler, Statistical Mechanics, p. 96. 1929. 


of the Van der Waals gas, according to (4.26) and (4.28), 

* = co'(r) - - - RTlog(v -ft), (5.28) 

where o/(r) is an abbreviation for 

'(r) -y cjT - rj c -^ + */<>- TS' Q . (5.29) 

Exercise 50. Prove, for the perfect and for the Van der Waals gas, the relation 

Directions. Calculate 5* = d(Ntf with ST = from (5.27) and (5.28). Elimi- 
nate 5v by means of the condition 5 V = d(Nv) = 0. 

Exercise 51. Prove the identity (5.26) in the following way: In (5.05) replace 
5 by a function of V, T, N, i.e. U - U[S(T, N), N], and take the partial (dU/dN) T . v 
in analogy with the procedure at the end of section 33. 

Exercise 52. Consider the case of a system with many degrees of freedom 
when the element of work has the general form (2.02): DW = 2y k dX k . Write out 
the differential of the work function * = U - TS and, by applying the reciprocity 
relations (2.10), prove the equation 

i) . (5.30) 


35. The thermodynamic potential. Let us consider a system 
which depends on some non-mechanical (e.g. electrical, magnetical, 
etc.) variables, in addition to p and T. The element of work done by 
it can, then, be represented as DW = pdV + DW ', where DW is the 
work done by the non-mechanical forces. Accordingly, the differen- 
tial (5.04) of the internal energy takes the form 

dU = TdS - pdV - DW. (5.31) 

We subject the first and second term on the right side to Legendre 
transformations adding - d(TS) + d(pV). The result is 

d$ = - SdT + Vdp - DW, (5.32) 


$ = U - TS + pV (5.33) 

is known as the thermodynamic potential, 

In the laboratory, processes (like chemical reactions, etc.) are very 
often conducted at constant temperature and constant pressure. 
The temperature is maintained by a suitable heat bath while the pres- 


sure is that of the terrestrial atmosphere. Under these conditions 
(dT = 0, dp = 0), eq. (5.32) reduces to 

' d* - - DW, (5.34) 

or integrating this over a reversible path (at T =* const, p = const) 
from the initial state (1) to the final (2) 

$1 - $2 = - A$ - W. (5.35) 


The non-mechanical work done by a system in a reversible isothermal, 
isobaric process is equal to the decrease of the function <. When the 
process is led between the same two states in an irreversible way 
(i.e. when the temperature and pressure are imperfectly maintained), 
it is more wasteful. The difference $1 $2 gives, therefore, the maxi- 
mum of the non-mechanical work that can be obtained from the system in 
any process leading from state (1) to state (2) of the same temperature 
and pressure. 

The great importance of the thermodynamic potential arises from 
the fact (which we shall prove in the next chapter) that, in the equilib- 
rium state of a heterogeneous system, p and, T are the same in all its 
phases. They are, therefore, the most convenient thermodynamical 
variables, and eq. (5.32) shows that & is the appropriate characteristic 
function for this choice of variables. In the special case of a simple 
system DW f = 0, and the variation becomes 

*$ = - S5T + V6p, (5.36) 

while $ has the expression 

In analogy with the preceding sections 



(The subscript N is usually omitted). The partial derivative with 
respect to Nk must be taken at constant p and T: it is, therefore, the 
partial molal thermodynamic potential, according to the definition of 
the partial molal quantities given in section 32. 



It is easy to prove the identity (exercise 54) 

or from (5.17) and (5.26) 

We shall have occasion to use the expression for the molal thermo- 
dynamic potential of a perfect gas. From (5.33), (4.20), and (3.19) 



In the particular case c p = const, 

co(r) = c p T(l - log r) + uo - TSQ. (5.43) 

Exercise 53. Derive the explicit expression of <f> for the Van der Waals gas: 

V"(r)+*r log (/>+") --+-^~ (5.44) 

\ V l / V V 

Exercise 54. Prove the identity (5.39), making in exactly the same substitu- 
tion for V which was made at the end of section 33 in U (to prove 5.17). 

Exercise 55. Consider the case of a system with many degrees of freedom when 
the element of work has the general form (2.02): DW HyidXk. Write out the 
differential of the generalized thermodynamic potential & = U TS + Sy^Yjfc, and 
prove, by applying the reciprocity relations (2.10), the equation 

36. The decreasing tendency of the thermodynamic potential and 
of the work function. We have found in eqs. (5.22) and (5.35) that 
work is done by a system (at T = const) at the expense of the work 
function, and non-mechanical work (at T = const, p = const) at the 
expense of the thermodynamic potential. The question arises, how- 
ever, as to the tendency of these functions to decrease: if the differ- 
ences 1 ^2 or $1 $2 are available in a system, will they be spon- 
taneously converted into work, and under what conditions? 

Let us start our considerations from the work function ^ U TS. 
It is obvious that ^ would decrease (if possible) in a system " left to 
itself " or " isolated", but remaining at constant temperature. In fact, 


we know from section 31 that the positive term U has a tendency to 
decrease and the negative TS to increase (in absolute value). How- 
ever, in the processes which we wish to consider the systems are not 
isolated: in the first place, there is the heat bath by means of which 
the temperature is maintained; in the second, the environment may 
do work against it through the forces of pressure. Let us try to make 
the system isolated by including in it both these influences. With 
respect to the heat bath, this is easy since it does not do any work of 
either a mechanical or a non-mechanical nature. According to eqs. 
(5.21) and (5.34), the values of the functions < and ^ characterizing 
the heat reservoir remain constant in view of this. We may regard it, 
therefore, as a part of the system without changing anything in the 
available differences ^i ^2 and 3>i <i>2, or, in other words, we 
may base our discussion on ^ = U TS of the enlarged system (with 
inclusion of the heat bath). The conditions are simplest when other 
interactions with the outer world are precluded. This is the case when 
the volume of the system is constant, for instance, when it is surrounded 
by a rigid heat-conducting envelope and immersed in the heat bath: 
when the temperature and the volume of a system are kept constant, its 
work function has a tendency to decrease. Any spontaneous isothermal 
and isochoric process goes on until ^ reaches its possible minimum. 
The necessary and sufficient condition for the starting and proceeding 
of a process is that the work function must be larger in the initial 
state than in the final (^i ^2 > 0). This can be used as a criterion 
whether a chemical reaction will go. At constant temperature 
and volume the maximum possible change of V is a measure of the 
chemical affinity of the substances entering into the reaction. It is to 
be noted that the work done by this reaction, if any, is non-mechanical 
work: from dV = 0, there follows DW = DW. 

The conditions are different when the pressure is maintained con- 
stant, instead of the volume. The practical method of accomplishing 
this is to subject the system to the hydrostatic pressure either of the 
atmospheric air, or of another medium (liquid or gas) compressed by 
a suitable load. If the system expands under this pressure from the 
initial volume V\ to the final V^ the work of expansion p(V^ Vi) is 
converted into the potential energy of lifting the load or the air column 
resting on the system, and pV is a measure of this potential energy at 
any moment. In addition to 7, having the nature of a potential 
energy, we have here still the item pV which tends to be converted 
into energy of a non-potential character. We have, therefore, to use 
the thermodynamic potential $ = U TS + pV: including the heat 
bath into the system as in the former case, we see that this function has 


a tendency to decrease (at T = const and p = const), since the two 
positive terms tend to decrease, the negative to increase (in absolute 
value). The criterion for a process taking place under these conditions 
is that the thermodynamic potential must be larger in the initial state 
than in the final ($1 $2 > 0). This applies in particular to chemical 
reactions: they will continue as long as they can produce non-mechani- 
cal work. The measure of chemical affinity in a reaction conducted 
in an isothermal and isobaric way is the maximum non-mechanical 
work W = $1 $2 which can be obtained from it. 

All three terms of $ tend to decrease, but it would be wrong to 
conclude that they always diminish separately. The functions U, S, V 
are all interdependent, and the reduction of one may force the growth 
of another. Only the sum <i> always decreases, each individual term is 
only then sure not to increase when the other two remain constant. 
From (5.35) and (5.33) 

W' = - A* = (Ui - U 2 ) - r(Si - 5 2 ) + p(Vi - F 2 ). (5.46) 

The term T(S\ 62), representing the heat imparted to the heat 
bath, is often negative (endotermic reaction), and the third term 
p(Vi 2) is also sometimes negative (reaction with expansion). 

In later chapters we shall have to consider the case when the sys- 
tem is in partial equilibrium, its phases having the same temperatures 
but standing under different pressures pi, p2, p a * Since U and 5 
are additive, the thermodynamic potential has, then, the form 

* = U - TS + piVi + . . . + p a V a . (5.47) 

It is obvious that it tends to decrease also in this case, because the 
terms pjVj are the potential energies due to the expansion of the indi- 
vidual phases. 

37. The Gibbs-Helmholtz equation. Galvanic cells. Another 
form of the relation (5.46) is 

A$ = X 2 - Xi - T(S 2 - Si), 
or substituting for S from eq. (5.37) 

This relation is known as the Gibbs-Helmholtz equation. It applies 
to any isothermal difference A< as it is based only on the definitions of 
the functions $ and X. In the special case of an isothermal-isobaric 


process, we have A$ = W, and from (4.36"), AX = Q p (isobaric heat 
of reaction), 

(s - 49) 

The left side of the equation is identical with T(S2 - Si) and 
represents the energy gained by the system from the heat reservoir, 
in the isothermal-isobaric process we are considering. It is interesting 
to note that this heat is proportional to the temperature coefficient 
of W. The Gibbs-Helmholtz equation has many uses; as an example 
we mention here its application to the electromotive forces of galvanic 
cells. The active substance in a galvanic cell is an electrolyte whose 
negative ions react with the material of the positive plate, transferring 
their electric charges to it. At the same time, an equal number of 
positive ions is deposited upon the negative plate and neutralized by 
the (negative) current flowing to it through the outer circuit of the 
cell from the positive plate. According to Faraday's law, the electric 
charge supplied to the circuit by 1 mol of the electrolyte is <rF, where 
F is Faraday's equivalent, (9648.9 0.7) abs e-m units, and o- the 
valency of the ion. The electric energy w f of the current (per 1 mol 
of the electrolyte) is obtained multiplying this by the electromotive 
force E, 

w r = 0FE, (5.50) 

or substituting into (5.49) 

r (af) - E + >/** < 5 - 51 > 

where q p is the heat of reaction, referred to 1 mol. When q p is 
expressed in calories and E in international volts, the numerical value 
of the Faraday becomes F = 23 046. 

Let us apply this to the Weston cell in which the electrodes consist 
of Cd-amalgam and of pure mercury. It contains the following sub- 

Hg-Cd | CdS0 4 (solid) | CdS0 4 (sat. sol.) | Hg 2 S0 4 (paste) | Hg 

arranged as shown in Fig. 11. The following empirical formula has 
been found (and internationally adopted) for the electromotive force 
of the Weston cell 

i [1.01827 - 4.06(/ - 20) X 10" 5 - 9.5(* - 20) 2 X 10~ 7 

+ (20 - O 3 X 10~ 8 ] int volt. (5.52) 




Saturated Solution 

of Cadmium Sulphate 




Platinum Wire 

Equation (5.51) permits to calculate the heat of the reaction 
(Cd + Hg 2 SO 4 = 2Hg + CdSO 4 ) in the cell. It gives, with <r - 2, 
q p = 47 482 cal mol"" 1 , while direct thermochemical measurements 
yielded 47 437 cal mol" 1 . In the Weston cell the temperature coeffi- 
cient of E is negative and represents an energy loss. There are, how- 
ever, other cells (see exercise 56) where it is positive, so that a part of 
their energy and electromotive force is supplied by the heat bath 
(surrounding air). 

It is clear from this discussion that in a galvanic cell the heat Q p 
of an exothermic reaction is converted into (electrical) work in a 
practically complete way. Its 
operation is not cyclic but can 
be made continuous, [since the 
working substances can be fed 
in, at the positive plate, and 
the waste products taken out, 
at the negative. If the same 
reaction were used for heating 
the boiler of a cyclic heat engine 
(section 21), not more than the 
fraction (7\ - T 2 )/Ti of the 
heat Q p could be utilized, ac- 
cording to eq. (4.03). In view 
of this superiority, there was a 
great deal of speculation whether 
it is feasible to construct a galvanic cell working with the most com- 
mon industrial fuel, coal (i.e. carbon). But it proved impossible 
to bring carbon into electrolytic solution. 

For some of the applications it is useful to integrate eq. (5.49) and 
to express W = A< in terms of Q p . An integrating multiplier 
of the equation is 1/r 2 , and the multiplication by it gives 

[d(W'/T)/dT] P = &/r 2 , 

whence by partial integration 

y (5.53) 

Exercise 56. The Pb-Hg cell is constructed as follows: 

Pb-amalgam | PbSO 4 (solid) | NajSO 4 (solution) | HgjSO* (solid) | Hg 

Its electromotive force is 

E - [0.96466 + 1.740 - 25) X lO" 4 + 3.8(* - 25)* X 10~ 7 ] int volt 

Platinum Wire 

FIG. 11. The Weston galvanic cell. 


What part of E is supplied by the heat reservoir, and what is the heat of reaction at 
25 C? 

Exercise 57. In the Cu-Hg cell 

Hg-Cu | CuSO 4 (solid) | CuSO 4 (sat. sol.) | Hg 2 SO 4 (paste) | Hg 

the molal heat of reaction is, at 20 C, q p = - 39 596 cal mol" 1 , the electromotive 
force E = 0.3500 int volt. Find the temperature coefficient of E. 

Exercise 58. In the same way as (5.48) is derived for A4>, derive for * the 


Hence prove the relation 


* -- T f^ dT + Tf(P).l (5.55) 

Remark. In the differentiation and integration with respect to T the pressure 
is regarded as constant. Therefore, the integration constant may depend on p. 
In (5.53) the function of p cancels out when the isobaric difference $2 $1 is formed. 

Exercise 59. Prove from (5.20) and (5.25) the analogous equations 


~\ \J *. r y Jt I- (J J. -M f 


-dT + TMv). (5.57) 

Forming the difference A = 2 1 leads to the relation 

T (^f) v =**-* u ' (5.58) 

which sometimes is also given the name of a Gibbs-Helmholtz equation. 



38. The method of virtual displacements. The theory of thermo- 
dynamical equilibrium was developed by Gibbs and patterned by him 
after the mechanical theory of statics of Lagrange. We have explained 
in section 31 what virtual changes are, but it will be well to say here 
a few words about how they are used. Let the parameters of the 
system be denoted by 1, fe, n , which may include as well the 
thermodynamical variables as those defining the composition of the 
system; and let the equilibrium be determined by the condition that 
the variation of a certain characteristic function -F(i, . . . { n ), 

vanish for all virtual changes dl-k consistent with the conditions of the 

If the parameters are not subject to any further restrictions, the 
virtual changes are entirely independent of one another and can be 
chosen in any arbitrary way. In particular, we could choose all but 
one of them equal to zero ($* ^ 0, and when i ^ k, $ = 0). Then 
the condition (6.01) is reduced to (3F/3{*)% - Oi or 

- = 0, (* - 1, 2, . . n). (6.02) 


Since 5* was selected at random, this must hold for each, of the 
parameters. We have, therefore, n equations from which the equilib- 
rium values of the n parameters & can be determined, and this consti- 
tutes a complete solution of the problem. 

However, usually, the parameters are not quite independent but 
subject to constraints or subsidiary conditions which take the analyti- 

1 Suggestion to teachers. Chapters V and VI are both somewhat abstract. It is, 
therefore, recommended to interpose between them sections 134, 135, and 136 of 
Chapter XX, which contain applications of the concepts introduced in Chapter V. 



VI 38 

cal form of equations imposed upon them. Suppose that there exist 
m (< ri) subsidiary conditions 

/ifti, ...) 
/2i, . . 6.) 


/mftl, ...*)- 0. 


In this case, only n w of the f A; are independent and can be chosen 
arbitrarily. The remaining m parameters are determined by these 
through eqs. (6.03). The same is true of the virtual changes 5 fc : only 
n m of them are within our free choice. In order to find the final 
conditions of equilibrium, it is necessary to eliminate the m dependent 
virtual changes from the variation dF and to represent it as a linear 
form of the independent $ *, only. This could be done by eliminating 
m of the parameters from the function F, with the help of eqs. (6.03), 
before taking its variation. However, this is seldom convenient, and 
it is preferable to take 5F in the form (6.01) and to eliminate the 
dependent dfa from it, using a method worked out by Lagrange. The 
subsidiary conditions can be, also, thrown into a variational form; in 
fact, they apply as well to the values of the parameters & as to & 


= 0. 


The difference between (6.04) and (6.03) is, for infinitesimal 


f& = 0, (* = 1,2, . . . m). (6.05) 

Often the subsidiary conditions are given directly in the variational 


which cannot always be integrated. However, it is irrelevant for the 
further treatment whether the form is (6.05) or (6.06). To fix our 
ideas we use eqs. (6.05) and multiply them, respectively, by Lagrangean 
multipliers A,(i, . . . () and add them to eq. (6.01) 

w i At 


The m functions \i can be chosen in such a way as to make the m 
first parentheses of this equation vanish 

f or k = 1,2, ... m. The remaining expression 

Z" ( dF + i 9 ^+ -LX 9 - 
\*t + Xl 5T + ' ' ' + Xm ~ 

*-m+l ^3** 3* 

contains now only n m independent $*, so that the problem is 
reduced to the case of independent virtual changes treated above: this 
equation is satisfied only then when all the coefficients in parentheses 
vanish. The conditions (6.01) obtain, therefore, for all values of 
k( 1, 2, . . . n). There are n such equations, and together with 
the m eqs. (6.03), we have m + n relations between the variables 
1, . . . n; Xi, . . . X m , i.e. the number necessary to determine all of 
them and to obtain a complete solution of the problem of equilibrium. 

Exercise 60. A heavy material point moves in the field of gravity of the earth. 
It is constrained to remain in the curve (ellipse) 

(ax + Py + yz) 2 + ey* = 1, ax - by = 0, 

where x t y, z are cartesian coordinates, z being the vertical. The point comes to rest 
when its potential energy has its minimum (z min.) What is its position of rest? 
(Use Lagrangean multipliers). 

Exercise 61. Make the same calculation for a point which is constrained to 
move in the surface 

x* H- (y cos a + z sin a) 2 -f 2p(y sin a z cos a) = 0. 

39. Auxiliary constraints. Homogeneity of the thermodynamical 
functions. It follows from the considerations of the preceding sections 
that the equations of equilibrium (6.07) hold for any system of virtual 
changes $& satisfying the main condition (6.01) and the subsidiary 
(6.05). In some cases, it is useful to treat the problem of equilibrium, 
not in its entirety, but only in part, and to select a special system of 
variations 5& compatible with those conditions. It is trivial that the 
requirements (6.01) and (6.05) are satisfied if all the virtual changes 
are taken equal to zero (3& = 0; k = 1, 2, ...). If only some 
arbitrarily chosen of them are equal to naught (provided the subsidiary 
conditions permit it), the result is quite consistent with the general 
solution but represents only a part of it. In fact, in this case, the 


corresponding terms of the expressions (6.01) and (6.05) vanish: the 
method of the Lagrangean multipliers leads, then, simply to an incom- 
plete set of eqs. (6.07). 

More generally, we can say that it is permissible to impose upon the 
parameters any additional constraints which do not disturb the equilibrium 
of the system. At first sight it may appear that this principle is of 
little practical use because, apart from such simple cases as that just 
mentioned, we must first know what the state of equilibrium is before 
we can judge whether a constraint is compatible with it or not. In 
practice, however, the simple cases are the most important ones, and 
frequently the knowledge of the nature of the system and of the 
physical circumstances of its equilibrium gives enough criteria to 
decide this question. In fact, the introduction of auxiliary constraints 
is a very useful and widely applied expedient. 

In this chapter we are going to consider a system consisting of 
several phases, each described as well by thermodynamic variables as 
by the mol numbers NI, N 2 , . . . N, defining its composition (compare 
section 32). We shall make use of auxiliary constraints in two con- 
nections. The first application consists in breaking up the problem 
of equilibrium into two partial problems by first considering the case 
when the composition remains unchanged and dealing with the ther- 
modynamic variables alone. Consequently, the auxiliary constraints 
take here the simple analytical form dN\ = $N2 . . . = dN ff = 0. It 
is obvious that this assumption is compatible with the notion of 
equilibrium because the most perfect equilibrium obtains then when 
no process at all is going on in the system. The conditions, the 
parameters Nk are subject to, are found separately as the second par- 
tial problem into which the treatment is broken up. 

In the other application the auxiliary constraint takes a geometrical 
form. Let us imagine that, anywhere, in the system a thin, closed 
layer of its substance is removed and replaced by a rigid and adiabatic 
envelope. Will the presence of this envelope disturb the equilibrium? 
Strictly speaking, it will. The state of a substance, at any point, is 
influenced by all the molecules in a small radius around it (radius of 
molecular action). Therefore, a very thin layer of matter, immediately 
adjacent to the imaginary envelope, will find itself in a new condition, 
and this will influence the system as a whole. However, when the 
volume within the envelope is sufficiently large the influence of the 
surface layer can be neglected. In fact, the volume increases with 
the third power of the linear dimensions, the surface with the second, 
so that the influence of the surface becomes relatively less and less 
important as the size increases. We shall treat the properties of the 


surface layer separately in Chapter XII, and we are going to assume 
here that all the phases of the system are so large that the surface 
effects can be neglected (as well at the boundary as at the surfaces of 
discontinuity between the phases). When this assumption is made, 
the equilibrium is not disturbed by the rigid envelope just mentioned, 
whether we think it surrounding the whole of the system or a part 
of it. It is inherent in the notion of a system in equilibrium that 
there is in it no relative motion of the parts and no heat transfer 
between them. The imaginary envelope will not interfere, therefore, 
with any actual process going on in the system. As to virtual changes, 
some of them are precluded or restricted, but no new virtual changes 
are made possible, by the presence of the envelope, which did not exist 
without it. According to the above argument, it represents a con- 
straint which does not change the form of the conditions of equilib- 
rium. This fact, immediately, leads to some physically interesting 
conclusions: (1) The envelope can be laid in such a way as to cut off 
a part of any of the phases. Therefore, the conditions of equilibrium 
cannot depend on the total masses and volumes of the phases but only 
on their specific properties like temperature, pressure, composition, 
etc. (see below). (2) It can be laid so as to include only two of the 
phases and exclude all the rest. Therefore, the equilibrium of two 
phases is determined only by their own properties and is not affected 
by the presence of other phases. (3) Since this is true for any pair of 
phases, the problem of equilibrium of several of them can be reduced 
to the simpler one of equilibrium between two. It also follows from 
this that two phases which are in equilibrium with a third are in 
equilibrium between them. (4) In general, a thermodynamical sys- 
tem is not completely defined unless it is known how it is delimited at 
its boundary. However, it is clear from the preceding discussion that 
any boundary conditions can be replaced, without loss of generality, 
by rigid, adiabatic walls enclosing the system. In this case, the con- 
straints introduced by the walls are not only auxiliary but necessary 
as forming part of the definition of the system. In the following sec- 
tions we shall use the expedient of the rigid, adiabatic envelope for all 
these purposes. 

A further example of an auxiliary constraint in geometrical guise 
is a semipermeable partition or semipermeable membrane, interposed 
between two phases, which lets through one kind of molecules but is 
impenetrable to all other kinds. This device can be also regarded as 
a part of the definition of the system, inasmuch as it is the thermo- 
dynamical criterion for the difference or identity of molecules: two 
particles are different if it is possible to find a semipermeable membrane 


which lets through the one but holds back the other. 1 It is, there- 
fore, perfectly legitimate to introduce semi permeable partitions which 
are selective not only with respect to chemical differences of mole- 
cules but also with respect to differences in the physical (quantum) 
state of otherwise identical particles. 2 

The notion of specific (also called intensive) properties used above 
requires some amplification. It is easy to define it accurately for 
phases which are described by the variables />, T, N\, N 2 , . . . N ff . 
Any property which does not change when all the N k are increased in 
the same proportion e (p and T remaining constant) is a specific prop- 
erty. For instance, the composition is best described in a specific way 
by the mol fractions 

oc k - N k /N, | 

N = Nl + N 2 + . . . + N,. J (6 ' 8) 

It follows from the definition of the mol fraction that 

*i + *2 + . . . + x a = 1. (6.08') 

In general, a function F n (N\, . . . NJ which changes in the pro- 
portion e n , when all the variables N k are increased in the proportion e, 

F n (eNi, . . . eN.) = e"F n (Ni, . . . NJ, (6.09) 

is called, in mathematics, a homogeneous function of the n-th degree. It 
satisfies Euler's equation 

which is easily derived (compare exercise 62) from the definition (6.09). 
Mathematically speaking, the specific quantities are, therefore, homo- 
geneous functions of the degree zero in the mol numbers Nk, since they 
satisfy the condition 

. . . NJ. (6.11) 

Of course, F depends also on p and T, but we shall compare only 
states and systems at the same temperature and pressure and need 

*An interesting discussion of these matters will be found in a paper by 
L. Szillard, Zs. Physik 32, p. 840, 1925. 

1 Einstein pointed out the paradox that a semipermeable membrane must dif- 
ferentiate between two particles, be the difference in their states ever so little. 
This difficulty disappears in the quantum theory where the states are discrete 
(compare: J. Von Neumann, Math. Grundlagen d. Quantentheorie, p. 197. 
Berlin 1932). 


not bring these variables in evidence. The factor e may have any 
value, and, in particular, we may put e = l/N, whence 

Fo(Ni, ...#.)- FQ(XI, . . . *.). (6.12) 

In a specific quantity we may replace the mol numbers Nk by the mol 
fractions Xk without changing it. 

On the other hand, the volume F, the internal energy Z7, and the 
entropy 5 are additive quantities. When all the mol numbers Nk are 
increased in the proportion e, the mass of the whole system increases 
in the same proportion, without change in its composition. The quan- 
tities F, U, S are proportional to the mass and also increase in the 
proportion e (p and T remaining constant). The thermodynamic 
potential is defined as $ = U TS + pV and has, obviously, the 
same property 

. . . N 9 ). (6.13) 

The thermodynamic potential, like all the other additive functions 
of thermodynamics, is a homogeneous function of the first degree. Hence 
its derivative with respect to Nk, or the partial molal thermodynamic 
potential, <pk = d$/dN k (compare section 35), is a homogeneous function 
of the degree zero, as is easily shown by differentiating (6.13) with 
respect t& Nk. Therefore, $ and ^* satisfy Euler's equation (6.10) 
with n *= 1 and n = 0. 

+ ...+ 


As 3^*/3^y = d$j/dNk, this can be also written 

Every term of this expression is a homogeneous function of the 
degree zero. We can, therefore, make use of the property (6.12) and 
replace the mol numbers by the mol fractions 

Although this relation was already given by Gibbs, it is usually 
called Duhem's equation. 


Exercise 62. Prove Euler's eq. (6.10) from (6.09). (Directions. Differentiate 
(6.09) with respect to e, denoting, for short, eN k - *. Then put e = 1; & = Nk)> 

Exercise 63. Check by direct calculation the statement that Euler's equation 
is correct for the homogeneous (inx t y,z) function F n - x a y*z n ~ a ~ b . Therefore, it 
must, be true for any sum of such terms of the degree n. 

40. The temperatures and pressures of phases in equilibrium. 

Let us consider a heterogeneous system consisting of simple phases. 
Its total internal energy U is the sum of the internal energies C7 (i) of the 
individual phases 

u = u m (6 * 17) 


In the same way, the total entropy S is additive with respect to the 
individual entropies S (i) 

>, (6.18) 

and we define as the total volume V the sum of the volumes F (0 of the 


**. (6.19) 

We can choose 5 (i) and V (i) as the two thermodynamical variables 
characterizing the state of the phase (i). It will, further, depend on a 
number of parameters describing its chemical composition. In gen- 
eral, the phases will not be chemically pure substances but mixtures 
of several or many chemically pure constituents or components. Sup- 
pose that they contain Ni mols of the first component, N<z of the 
second, and so on, the number of components being <r. The internal 
energy of the phase (i) is, then, a function of the following variables 

. . #<*>), (6.20) 

although a part of the mol numbers N may be equal to zero in some 
of the phases. This is the form we have considered in section 31 : the 
variation of the total internal energy U is, therefore, in accordance 
with eqs. (5.06) and (5.07), 

Applying auxiliary constraints, as explained in the preceding sec- 
tion, we shall at first suppose that the mol numbers do not change 


(dN k (i} = 0) and shall find only the conditions controlling the thermo- 
dynamical variables. 

The conditions of equilibrium are, according to eqs. (5.03) : dU = 0, 
8S = 0. It is mathematically convenient to regard dU = as the 
primary condition. It now takes the form 

dU = > (T (i} 5S (i) ( '>$F (i) ) = 0. (6.22) 

The condition 


55 = ^ SS (i) = (6.23) 


we regard as subsidiary. Another subsidiary restriction is obtained 
if we replace the border conditions of the system by the assumption 
that it is surrounded by a rigid adiabatic envelope (compare the pre- 
ceding section) : We have then F = const, or 

In the case when the several phases are separated by natural sur- 
faces of discontinuity and each of them can expand, virtually, at the 
expense of the others, the variations <5S (i) , 5V (i) are not subject to any 
other restrictions so that eqs. (6.22), (6.23), and (6.24) exhaust the 
problem. Multiplying eqs. (6.23) and (6.24), respectively, by \i and 
\2 and adding them to (6.22) 



= 0, (6.25) 


where the dS (i) and 6F (i> must be considered as independent (compare 
section 38): 

The temperatures of all the phases are equal to the same function 
which has, therefore, the physical meaning of the uniform temperature 
T of the whole system (Xi = T). In the same way, the pressures 
of all the phases are equal to p 

p M =p. (6.26) 

In short, the temperature and pressure in a system in equilibrium are 
uniform. However, this applies only to the case when each of the 
phases can change its volume. In the opposite case, when the con- 


straints are such that the volume of each phase remains unchanged 
(y> = const.), the condition (6.24) must be replaced by a separate 

57>=0. (6.27) 

This will occur, for instance, when each of the phases is enclosed in 
rigid heat-conducting walls, but also in some cases which are less 
trivial, as we shall see in sections 107 and 127. In applying the 
method of Langrangean multipliers, we take eq. (6.23) with the factor 
Ao and (6.27) with the factors X (i) and add them to eq. (6.12) 


= o, (6.28) 


r = - Ao = T, />> = A (i) . (6.29) 

The temperature is uniform throughout, as in the previous case, but 
the pressures of the phases are all different. It is quite obvious how 
these considerations are to be extended to the mixed case when the 
volume of some of the phases is constant while others can expand. 
The case when some of the phases are adiabatically enclosed and their 
entropies are constant (dS (i) = 0) is also possible but hardly worth our 
consideration, since it is one of dynamical, rather than thermodynami- 
cal, equilibrium. 

We have obtained a partial solution of the problem of equilibrium, 
with respect to the thermodynamical variables T and />, and can now 
turn to the other side of the problem relating to the virtual changes 
of the mol numbers. In order to deal with it, we shall again impose 
upon the system suitable constraints. We already know that, in 
equilibrium, the temperature T is uniform while the pressures p (i) of 
the phases may or may not be equal. We shall now consider only 
virtual changes in which dT = and dp (i) = 0. It was shown in sec- 
tion 36 that, under these circumstances (when the temperature and 
the local pressures are kept constant), the function <f> tends to decrease 
and equilibrium is possible only when it has reached its minimum 

(&typ. T = 0. (6.30) 

The total thermodynamic potential is represented by the sum over 
the phases 


In forming the variation T and p are to be kept constant, whence 

In general, the virtual changes 5N t (i) are not independent but sub* 
ject to subsidiary conditions determined by the nature of the system. 

Exercise 64. Consider eqs. (6.22), (6.23), and (6.24) for the case of only two 
phases and solve them without use of Lagranean multipliers. Show that this 
method can be used to obtain the general result by the reasoning of section 39. 

Exercise 65. Take the case of p and T being uniform throughout the system 
and generalize the argument of section 32 which led to the formula (5.08) for systems 
with several phases. Show that this is an alternative way of obtaining the condition 

41. Number of phases in equilibrium. The phase rule. In order 
to derive the explicit equations of equilibrium from the condition 
(6.31), it is necessary to eliminate from it the dependent virtual 
changes dN k (l) by means of the subsidiary conditions. It was pointed 
out in section 38 that this can be done either before taking the varia- 
tion (6.30) of the function <J>, or after. The former procedure consists 
in expressing a part of the arguments N k (i) of the function $ by the 
remainder and so reducing their number as far as possible. We say, 
then, that the thermodynamical potential is expressed in terms of the 
mol numbers of the independent components of the system. To illus- 
trate this by an example, let us take the case that N\ mols of hydrogen 
(H2) and N% mols of iodine (12) are brought into the same vessel. 
The two gases will partially react and form a certain amount (N% mols) 
of hydrogen iodide which will be the third constituent of the mixture 
but not an independent one: in the state of equilibrium N& is com- 
pletely determined by N\ and N%, so that hydrogen and iodine can be 
taken here as the independent components. We see, moreover, from 
this example that the method is of no help in the treatment of con- 
crete cases, because we must have solved the problem of equilibrium 
and found the expression of Na, in terms of NI and N2, before we can 
eliminate it from the thermodynamical potential. This is, in fact, 
characteristic of all thermodynamical systems: the main part of the 
subsidiary conditions between the virtual changes 6N k (i) are given in 
the variational form (6.06), and not in the integral (6.03), and often 
there is no practicable way of eliminating the dependent mol numbers 
N k (i) except by means of the final solution of the problem. However, 
some interesting conclusions can be drawn from the theoretical possi- 
bility of this elimination: we simply suppose that the system is reduced 


to its independent components, without attempting to carry out this 
reduction in any special case. 

Generally, those constituents of a system should be chosen as its inde- 
pendent components which cannot be converted into one another by any 
reaction going on in it. In the above example hydrogen and iodine 
satisfy this requirement, but, in general, the number of independent 
components of a system is, by no means, identical with the number 
of chemical elements of which it consists. For instance, if all the 
phases of a system consist of water in different states of aggregation 
(ice, steam, liquid water), hydrogen and oxygen are not independent 
components, because the mol number of the one is defined by that 
of the other, viz. stands to it in the constant ratio 2:1. We have 
here a system of only one independent component. Strictly speaking, 
the number of components depends on the accuracy with which we 
wish to describe the system : there is always a certain amount of disso- 
ciation in steam, and, because of the different solubility of hydrogen 
and oxygen, the phases will not contain exactly equal mol numbers 
of the two elements. However, these effects are entirely negligible 
at ordinary temperatures. An example of a system in which the num- 
ber of independent components is larger than the number of elements 
is presented by a mixture of hydrocarbons. There are many hydro- 
carbons which do not react under ordinary conditions, so that the mol 
numbers of hydrogen and carbon are quite insufficient for a description 
of such a system. The ultimate criterion for the choice is, of course, 
that a virtual change of the mol number in one of the independent com- 
ponents should not necessitate any virtual changes in the others. 

The notion of independent components is sufficiently clear from 
this discussion. Suppose that their number is ft and that the mol 
numbers of all the constituents, the independent as well as the depen- 
dent, are eliminated from the functions <> (i) and replaced by the masses 
Mi (i \ M2 (i \ . . . Af e (i) of the independent components in the several 
phases. Equation (6.31) becomes, then (if we drop the subscripts 

(6 ' 32) 

The only subsidiary conditions which remain are now those which 
apply to each independent component separately. We shall restrict 
ourselves here to the case that the phases are in direct contact, so that 
a component can pass freely from one phase into another. As was 
pointed out in the preceding section, the border conditions of the 
system as a whole can be replaced by an imaginary rigid envelope 


surrounding it. This implies that the total mass of each component 

a _ 

must remain constant. /, Af* (0 = const, or taking the variation, 

0, (* = 1, 2, . . . ft). 

Multiplying these equations, respectively, by the Lagrangean mul- 
tipliers X* and adding them to (6.32), 


). -ft 

- +X* 

where the virtual changes BM k are completely independent. There 
follows the system of equations 

= - X*. (6.33) 

The index k goes here from 1 to ft, so that this formula represents 
aft relations. If we eliminate from them the functions X*, there 
remain Zeq = aft ft equations between the partials of the functions 
3> (i) . What is, on the other hand, the number of variables on which 
these partials depend? It was shown in section 39 that the partials 
of <i> (i) , with respect to N k (i \ represent specific quantities. The reason- 
ing used in that demonstration is in no way affected, if the masses 
M k (i) are regarded as the arguments of these functions, instead of the 
mol numbers. The partials 9* (i) /3^ (0 are, therefore, homogeneous 
functions of the degree zero in the masses. In analogy to the mol 
fractions we introduce the mass fractions 


and conclude from eq. (6.12) that 3* ( * ) /9Af* (0 can be represented in 
terms of the variables y k (i) . Only ft 1 of these fractions are inde- 
pendent because there exists the identical relation 

The number of variables defining the composition of the phase (i) is, 
just ft 1, and the number of them in all the a phases is aft a. To 
this must be added the two thermodynamical variables, the common 
temperature T and pressure p of all the phases. The total number of 


variables by which the partial derivatives of the system (6.33) are 
determined is, therefore, Z var = a/3 a + 2. 

A system of equations is only then compatible and capable of 
being satisfied when their number does not exceed the number of 
variables on which they depend (Z eq g Z var ). This condition restricts 
the number of phases which can be simultaneously in equilibrium: 
from a/3 (I afl a + 2 there follows 

a ft + 2. (6.36) 

This inequality expresses the famous phase rule of Gibbs: The 
number of phases coexisting in equilibrium cannot exceed the number of 
independent components by more than two. 

Let us consider the main cases that can arise. Examples of them 
will be treated in the following chapters. The maximum which a can 
reach is 

a = + 2, (6.37) 

it will be attained when the number of variables is just equal to the 
number of equations. All the variables are then completely deter- 
mined, the equilibrium can exist only at one definite temperature and 
one definite pressure, and the composition of all the phases is also 
quite definite. This state is often called the fundamental state of the 

In the case 

a = /3 + 1 (6.38) 

the number of the variables exceeds that of the equations just by one: 
the value of one of the variables can be chosen arbitrarily, and this 
fixes the values of all the others. For instance, if we specify the tem- 
perature, the pressure and the compositions are completely deter- 
mined by it, etc. 
Similarly, when 

a = 0, (6.39) 

two of the variables can be assigned arbitrary values: if p and T are 
specified, the compositions of the phases are definite. 

Finally, in the cases a < /3, the compositions of the phases are not 
yet defined by the temperature and pressure. There are, then, so 
many cases of equilibrium possible that hardly any rules of general 
validity can be given. 

In conclusion, it will be well to emphasize that the phase rule deals 
only with the number of phases in equilibrium. We have mentioned 


water in its different states of aggregation as an example of a system 
with one independent component (ft = 1). According to the phase 
rule not more than 1+2=3 phases of it can coexist (viz. vapor, 
liquid water, ice), and the simultaneous equilibrium of all three is 
possible only in the fundamental state at a definite temperature and 
pressure (compare also section 48). However, everybody knows that 
he can throw a lump of ice into a bucket of water and that there will 
be, at all temperatures, water vapor on top of the system. In this 
case, the three phases are not in equilibrium, and, in the long run, one 
of them will disappear, but the coexistence without equilibrium may 
last hours. In applying the phase rule one must be quite sure that 
equilibrium is, actually, established. 

42. Explicit equilibrium conditions. We return now to the 
description of the phases by the mol numbers of all their chemically 
distinct constituents and not only of the independent components. 
We shall make use of the notation (5.38) defining the partial molal 
thermodynamic potential 


(6 - 40) 

It will be well to recall here that it is a generalization of the ordinary 
molal potential. In the special case, when the phase is chemically 
pure, there is only one constituent, 3> (i) = N (i) <p (i \ where <p (i) is inde- 
pendent of N (i \ Then ^ (i) becomes identical with <p (i) . 

The condition (6.31) now takes the form 

' = 0, (6.41) 

and is valid for all possible virtual changes dN k (i) consistent with the 
nature of the system. 

The virtual processes in the system can be reduced to two simple 
types. In the first place, a small amount of the constituent k can pass 
from a phase (1) into an adjacent phase (2). The molecular weights 
MA: (I) and jufc <2) of this component in the two phases need not necessarily 
be the same because there is the possibility of association. The mass 
added to the first phase is, then, M* (1) &W 1} and that added to the 
second n k (2) 5N k (2} . These masses must be oppositely equal 


W> : 6N k - - -1- : -^ - *> : ,*>. (6.42) 

(1) (2) 


The symbols v k are integral numbers with the following meaning. 
Suppose that j>* (1) mols of the component k are removed from phase 
(1): they are converted into v k (2} mols belonging to phase (2). If no 
association takes place in either phase, we can put 1>* (1) = v k (2) = 1. 

There are no conditions in the system which should prevent us 
from putting all the other virtual changes bN k equal to naught, in the 
way of an auxiliary constraint. Therefore the condition (6.41) reduces 

A$ SE v k Tp k + v h *W> = 0. (6.43) 

A<i> can be interpreted as the change of the total thermodynamic 
potential which takes place in the finite virtual process of reversibly 
transferring v k m mols of the component k from the first phase into 
the second (where they are converted into v k (2) mols). The temperature 
and pressure are supposed to remain constant, maintained so by a 
suitable arrangement of heat baths and loads. Of course, the process 
will be attended by the system receiving (positive or negative) heat 
from the heat baths and doing work against the loads. 

Sometimes it is convenient to express the composition not in terms 
of the mol numbers N k (i) but in terms of the masses M k (i) = n k (i} N k (i) of 
the individual components. We shall denote the partial specific 
thermodynamic potentials (referred to unit mass) 

Because of (6.42), /** (1 W n = Mt (2 V 2) , and (6.43) reduces to 

?Af* (1> - ?W 2) = 0. (6.45) 

The second simple type of process we have to consider is a chemical 
reaction between several constituents of the same phase. Let us take, 
as an example, the equation of the reaction between hydrogen and 

2HI = H 2 + I 2 , 

2HI - H 2 - I 2 = 0. 

Like the equations of all chemical reactions it is of the general type 
viGi + v 2 G 2 + . + v 9 G 9 = 0, (6.46) 

where GI, . . . G are the chemical symbols of the different substances 
(components) an4 ?i, . . . v 9 the numbers of mols with which they 
take part in the reaction. We may observe here that the process of 


transferring matter from one phase into another, as considered above, 
can be represented by the same sort of a symbolical equation 

>=0. (6.47) 

Both equations are included in the general form 

> = 0. (6.48) 

We shall usually consider a definite direction of the reaction, namely, 
that in which it is endothermic and the heat of reaction positive. Con- 
sequently, we shall attribute positive signs to the numbers vk of those 
components which are produced, and negative to those which are 
consumed, in the endothermic reaction. 

When eq. (6.41) refers to a process of the type (6.46), the corre- 
sponding virtual changes dNk must be taken proportional to the num- 
vers vk 

v%: . . . : v ff . 

(The numbers vk and variations dNk for the components not taking 
part in the reaction can be put equal to zero because of the principle 
of auxiliary constraints). The condition becomes 

A$ E= vjQt + . . . + Vj p 9 = 0. (6.49) 

As in the preceding case A<i> must be interpreted as the total change 
of the thermodynamic potential in the finite virtual process corre- 
sponding to the transformation of *>i, v v mols of the several com- 

Equations (6.43) and (6.49) represent partial solutions of the prob- 
lem of equilibrium. The complete solution is given by the simulta- 
neous system of equations corresponding to all processes of these two 
types possible in the thermodynamical system. Both these equations 
are included in the general form 

in which a part of the coefficients v k (i) are equal to those of (6.43) or 
(6.49) and the rest vanish. It will be useful to find the partial deriva- 
tives of A< with respect to the pressure and temperature: from 
eqs. (5.37) 


and (3A$/9r) p = AS, where AFand AS represent the changes of the 
total volume and the total entropy in the process. This process being 
isothermal (as well as isobaric), we can apply to AS the formula (4.11), 
AS = Qp/T, where Q p is the isobaric heat of the reaction 


Moreover, taking into account the condition A3> = 0, we can write 

Qp (6.53) 

Exercise 66. What is the structure of A 7 and Q p in terms of partial molal 
quantities? Show that 

, Q, 



43. Equilibrium of two phases. If all the phases of a system con- 
sist of the same chemically pure substance, in different physical states, 
it contains, according to section 41, only one independent component 
(ft = 1). The phase rule (6.36) tells us, then, that no more than three 
phases can be simultaneously in equilibrium. There exist substances 
which are known in more than three states. For instance, sulfur 
occurs in the vaporized, liquid, monoclinic crystalline, and rhombic 
crystalline states, but only three of them can coexist in equilibrium. 
The problem of thermodynamics in relation to pure substances is, 
therefore, reduced to the consideration of the equilibrium of one phase, 
two phases, or three phases. The properties of a single phase were 
sufficiently discussed in Chapters I to IV, and we shall restrict ourselves 
here to the cases of two and three phases, starting with the two-phase 
system as the simpler problem. 

The only possible virtual process in such a system is that repre- 
sented symbolically by eq. (6.43): v (1) G (1) + v (2) G (2) =0, viz. the 
transformation of v (2} mols of the second phase into v (l) of the first. 
It will be more convenient to define the numbers v (1) and v* both as 
positive and to indicate the fact that the second phase is consumed in 
the process by writing its equation as v (l) G (l) j/ (2) G (2) with 

instead of (6.42). It was pointed out in the preceding section that, for 
a pure substance, the partial thermodynamic potential and the ordi- 
nary molal are identical ^ = <p. The eq. (6.31) of equilibrium takes, 
therefore, the form 

A* = (1 V (1) - v (2 V 2 >, (7.02) 

or, in the special case when there is no association and the molecule 
is the same in both phases (> (1) = v (2) = 1), 

<<> - > (2) . (7.03) 

In words, the molal thermodynamical potentials in two adjacent 



phases are equal. The molecular constitution of the condensed phases 
is not always known and, in some cases, even not well defined. For 
this reason, many authors refer the quantities <p (i) of all the phases to 
1 mol of the vapor. In particular, this is unavoidable in the chemical 
literature which has to present results of measurement on all sorts of 
substances, including those whose molecular structure is not yet suf- 
ficiently investigated. The purpose of this book is, however, a different 
one: we are not so much interested in the presentation of a large 
amount of experimental material as in its theoretical penetration. 
The questions of association and dissociation play an important role 
in the interpretation of the experimental behavior of substances, and 
they can be conveniently studied if the fundamental equation of 
equilibrium is written in the general form (7.02). We shall, therefore, 
make this formula the basis of the following discussions (compare also 
section 60). 

In general, the coefficients *> (i) are integers, completely determined 
by the nature of the system, while the functions <p (i) depend only on 
the common temperature and pressure of the two phases, p and T 
are, therefore, the only variables which enter into eq. (7.02) or (7.03), 
and, if one of them is arbitrarily chosen, the other is determined by 
this relation. It was pointed out in section 41 that this is always the 
case when the number of phases is larger by one than the number of 
independent components (a = ft + 1). When the equilibrium of the 
liquid and the gaseous phase is considered, the temperature which 
corresponds to a given pressure, in virtue of the condition (7.02), is 
called the boiling temperature or the boiling point. Vice versa, if the 
temperature is regarded as given, the pressure corresponding to it is 
known as the boiling pressure. Similarly, one speaks, in the case of the 
coexistence of the liquid and the solid phase, of the freezing point and 
the freezing pressure. More general terms including all cases of a pair 
of pure phases are transformation temperature and transformation 

Let AC and DF in Fig. 12 represent, respectively, the isothermals 
of a pure substance in its liquid and gaseous states. Suppose we have 
a single phase of it corresponding to the point A of the diagram. 
Keeping the temperature constant we increase the volume reversibly, 
thus allowing the pressure slowly to drop: the representative point 
of the system will move along the isothermal towards C. Let the 
point B correspond to the boiling pressure for this isothermal at which 
the coexistence of the liquid and the vapor phase is possible. The 
question arises now: will the substance, after reaching this point, 
remain in a single liquid phase and continue to move along the branch 



B, or will it begin to vaporize at constant pressure and be converted 
into a two-phase system whose states are represented by the horizontal 
BE? What is the relative stability of the states M and N lying 
on a vertical and corresponding to the same volume? The results of 
the preceding chapter are insufficient to solve this problem because 
they regard the number of phases as given: the criteria given there 
are satisfied in both cases. To obtain an answer, we must fall back 
on the fundamental principles governing thermodynamical equi- 
librium as expounded in Chapter V. Since the processes BM and BN 
are both isothermal and lead to the same 
end volume, we have to use the criterion 
adapted to this case (section 36): The 
work function ^ tends to decrease, and, 
of two states of the same temper- 
ature and volume, that with the lower 
is more stable. Let us now evaluate 
the work functions &M and ^N in the 
states M and N by comparing them 
with ty B in the state B. We make use 
of eq. (5.22) which applies to any iso- 
thermal process: ^B ^M = WBM a 
the right sides of the equations repr 
system, respectively, in the processes 
find Vtr VM = WBM WBN. In 
represented by the areas under the 

. Isothermals in equi- 
lum of liquid and vapor. 

M lies lower than N, we have 
^fN < ^M- The states on the h< 
work function and of higher stabi 
that the states represented by 
impossible but sometimes occ 
(supercooling). However, they 
and the system is always read' { 
This reasoning can easily be 
transformation pressure or 
divides into two phases. J 
tional conditions. 


it is 

VB ^N = WBN, where 
. the work done by the 
and BN. Subtracting we 
iiagram WBM and WBN are 
and BN\ and, since 
We find, therefore, 
E are the states of lower 
<?-have mentioned in section 5 
anch BC are not absolutely 
' exceptional circumstances 
d to an unstable equilibrium, 
\ver into the two-phase state, 
o all analogous cases: at the 
i a pure substance, generally, 
5 a single phase only in excep- 

Exercise 67. When the 
and vapor), there are two ; 
represented by the curve an^ 

prove that the two shaded , ___ 

* areas 

44. The 


be directly used for the call u ' at * on 

^ o /fc . equation applies (continuity of liquid 

,O/ cesses between B and E (Fig. 1, p. 10) 
n ' , line. Applying eq. (5.21) to both of them, 
1 are equal. 

equation. Equation (7.02) cannot 
transformation temperatures 



VII 44 

because the explicit form of the functions <p (i) is known in only very 
few cases. However, its differential form gives a relation between 
easily measurable quantities. Let us consider the equilibrium of our 
two-phase system in two cases: at the temperatures and pressures 
r, p and T + dT, p + dp. The respective equations (7.02) are 
A$>(r, p) = and A<i>(r + dT, p + dp) = 0. According to the 
definition of the differential of a function, the difference between these 
two equations is 

v -(*)+&).-* <"*> 

We can s-X^';iite the expressions (6.51) and (6.52) for the partial 
derivatives. heat of the process (Q p ) is, in this case, the latent 

heat of trans *ion L (compare section 10). We find, therefore, 

. V, 


This relation was fc 
the discovery of the first 
from the modern point o* 
called the Clapeyron-Clau. 
transformed in the process 
we shall denote the latent he 
AFj (v (1) i; (1) j/< 2) z; (2) )/j/< 
and before the transformat 


^ ' 

"* ^French engineer Clapeyron before 

aTermodynamics (1832). Its deduction 

^ ^ due to Clausius. It is, therefore, 

& ^ion. If the mass of the material 

^ e ^s to 1 mol of the vapor phase, 

^ , hile the change of volume becomes 

"Terence of the molal volumes after 

' referred to 1 mol of the vapor) 

(7 - 06) 

This formula has been ext 
to be accurate within the limi* ve ^ ' 
we may take water at the no sf ex ?? n 
of volume is AF 2 - 30 186 c/ boiling 
mm/deg. The conversion lo1 . 

ested by experiment and found 
'mental error. As an example 

point (373. 1): the change 
T lg Q1S ^ dp/dT = 2? 12 

e . mm/deg = (1.01325/760 

X 4.1852 X 10) cal deg" 1 cm 

' 3 * 

The formula (7.06) gives, there 
direct measurement gives / = 972v, . t 

We give in Table 8 the normal ^ *** C 
melting points (7 of the more 
together with the latent heats of 
at these temperatures. 

X 10- 5 cal deg- 1 cm" 3 . 
'6 cal mol- 1 , while the 

L Ailing points (T B ) and 
e | c -mentary substances 

and of fusion (W 





T B 

IB (cal mor 1 ) 



IF (cal mol- 1 ) 













1 540 
















H 2 






N 2 

















C1 2 






Br 2 


7150 20.9 



I 2 













































. 1480 



















I, < 
























223000 36.2 




90700 28.6 





167000 36.6 




In the case of the transformations solid liquid and liquid 
* vapor, the latent heat L is positive. The volume of the vapor is, 
moreover, always larger than the volume of the same mass of liquid, 
AF> 0, so that the boiling temperature rises with the pressure 
(dTB/dp > 0). For instance, in the case of water we have the follow- 
ing conditions: 




6.025 X 10- 

50 /''17 




370.4 C 

On the other hand, the specific volume of the liquid phase may be 
either larger or smaller than that of the solid in equilibrium with it. 



VII 45 

For instance water (at C and 1 atm) has a 9% smaller volume than 
ordinary ice. Therefore, AF and dT/dp are in this case negative, as 
was pointed out by James Thomson (elder brother of Lord Kelvin). 
The melting point falls, therefore, with increasing pressure: when 
ice at a temperature slightly below C is subjected to pressure, it is 
brought into a state in which its T is above the melting point. It is, 
therefore, converted into water but freezes again as soon as the 
pressure is released. This phenomenon is called regelation, and it 
accounts for the plasticity of ice which permits, among other things, 
the flow of glaciers. The ease of skating on ice is also usually attrib- 
uted to regelation : enough water ir' '.elted under the pressure of the 
sharp edges of the skates to k*> / 

In the case of vaporization, c 
the formula (7.06) can be simplih 
so low that the equation of state ( 
sufficient approximation, to the gasv 
these conditions the molal volume . 
neglected compared with V2. In fact, 
ratio V2/vi for water (e.g.) is 1676, and 
law of perfect gases can be used, it is 
obtain an approximate form of eq. (7.C 


are sometimes such that 
0^ the vapor pressure is 
perfect gases applies, with 
ihase: V2 = RT/p. Under 
the liquid phase can be 
at 100 C and 1 atm, the 
* temperatures, when the 
arger. In this way, we 


which is useful for many purpose* 

Exercise 68. Calculate the latent / of water, at 100 C and 760 mm Hg, 
from eq. (7.07). Use the value dp/d /> 27.12 mm/deg. Compare the results 
with the exact value as given in text. 

Exercise 69. Calculate the latent heat of mercury at its normal boiling point 
(tB - 357.3, p - 760 mm). Use the value dp/dt = 13.81 mm/deg and eq. (7.07). 

Exercise 70. Calculate the coefficient dT/dp for the fusion of ice at C and 
1 atm, using the following data (per 1 g of the substance): l g = 80 cal g" 1 , 
v\ = 1.000 cm 8 g" 1 , j = 1.091 cm 8 g" 1 . Apply suitable conversion factor to p. 

Exercise 71. Naphthalene melts at 80. 1 C, its latent heat of fusion is 4563 
calories per mol, and the increase in volume on fusion is 18.7 cm 8 mol" 1 . Find the 
change of the melting point with pressure. 

Exercise 72. The transition point of mercuric iodide, red > yellow, is at 
127 .4 C. The latent heat absorbed in the change is .150 cal mol" 1 , and the approxi- 
mate change of volume 5.4 cm 8 mol" 1 . Find dT/dp for the transition point. 

45. Temperature dependence of the latent heat Another rela- 
tion between measurable quantities can be obtained by differentiating 


eq. (3.23) of section 17, dL = d(X (1) - X (2) ) = </(*> (1) x (1) - ^ (2) x (2> ) and 
substituting the expressions (4.47) for dx- We find 

dL = AC p <*r 

where AC P and AF denote the changes of capacity and volume in the 
transformation : 

AC P = v (1 V 1} - v (2 V 2) > AF - iW - / 2 V 2) . (7.08) 

Dividing the equation by dT and replacing dp/dT by (7.06) 

In the special case of vaporization at comparatively low pressure 
we can again make the simplification of neglecting v (2) (of the con- 
densed phase) and applying to v (1) the equation of perfect gases: 

log AF/3r) p = 1/r. The relation is then reduced to 

- *Cr (7.10) 

Exercise 73. Calculate dl/dT (per 1 mol = 18.0) for boiling water (at 100 C, 

1 atm) with the help of the following data: in liquid water c pz 18.0 cal, in vapor 
c Pl = 8.8 cal (both per 18 g, at 100 C), AF f = 30 186 cm 3 mol- 1 , (QAKj/aDp = 
84.66, I = 9720 cal mol" 1 . Compare the results given by the two formulas (7.09) 
and (7.10). (The measured value is -11.0). 

46. Approximate and empirical expressions for the transforma- 
tion pressure. The temperature dependence of the latent heat is, 
usually, slight. Especially, in the case of vaporization AC P in the for- 
mula (7.10) is small compared with L. As a first rough approximation 
one can, therefore, put L = LQ = const, or / = /o = const. Sub- 
stituted into eq. (7.06) this gives for the pressure of vaporization 


(B being a constant), a formula due to Van der Waals, who wrote it 
(for liquids) in the form 


which for a number of substances gives an approximate representation 
of the measurements even at high pressures. We shall return to this 
equation in the next section. 



VII 46 

It has been found since that the formula (7.11) is insufficient for 
an accurate representation of the vapor pressure of liquids but can be 
used for that of crystalline solids. Even in this case better approxima- 
tions are obtained by substituting into (7.07) the expansion 
/ = / + 1 V T + faT 2 + . . . . For instance the vapor pressure of ice 
between 90 and C is represented by l 
944.C c 

+ 8.2312 

- 0.0167006r 

+ 1.205 X 10~ 5 r 2 - 6.757169. (7.13) 

As to the vapor pressure of liquids, the very accurate modern 
measurements cannot be adequately represented even by formulas 
with many terms. 

For the dependence of the pressure of fusion upon temperature a 
theoretical formula has not yet been advanced, and one has to rely 
on empirical expressions. According to Simon and Glatzel 2 the follow- 
ing formula gives satisfactory results: 

logio(/> + a)=c logio T F + 6. (7.14) 

They give the following values for the constants (Table 10), pro- 
vided the pressure is measured in kg/cm 2 = 0.9678 atm. 








1 . 5544 











4 100 











CO 2 





3 100 




5 100 




3 900 




5 100 











Chloroform . . ... 












* The data for helium are taken from Simon, Ruhemann, and Edwards, Zs. phys. Chemie 
(B) 2, p. 430, 1929. 

1 E. W. Washburn, Monthly Weather Review 52, p. 488, 1924. 
* F. Simon and A. Glatzel, Zs. anorgan. Chemie 178, p. 309, 1929. 


Exercise 74. Calculate the latent heat / of sublimation of ice and the differ- 
ence Ac p of specific heats (between 1 mol of vapor and 1 mol of solid), at C. Use 
eqs. (7.13), (7.07), and (7.10). 

Exercise 75. At its normal point of fusion (-38.9 C) mercury has the molal 
volumes vi = 14.65 (liquid) and v, = 13.90 (solid). Its latent heat is / = 556 cai 
mol" 1 . Check whether the formula of Simon and Glatzel is in satisfactory agree- 
ment with (7.05). 

47. Vaporization as a corresponding state. It was stated in 
section 27 that certain groups of substances approximately satisfy 
what was called there the extended law of corresponding states. It is 
meant by this that within such a group the quantities (u o)/r, 5 
and pv/T are universal functions of the reduced variables. Since the 
molal thermodynamic potential is defined as <f> = u Ts + pv, the 
quantity (<f> uo)/T must also be expressible as 

<p UQ . ^ /T i r\ 

^ =/(T,T), (7.15) 

where / is universal for the whole group. The reduced volume does 
not appear as a variable, because it can be eliminated by means of the 
equation of state. 

It must be assumed that the equations of state apply equally to 
the gaseous and to the liquid phase, bringing out their continuity, in 
the sense of section 5. Therefore, the formula (7.15) is also valid for 
both phases. It follows then from (7.03) that, for substances which 
do not associate, the boiling condition is determined by the universal 


/d)( 7r , T ) =/O>(T,T). (7.16) 

Substances boiling at corresponding pressures have corresponding 
boiling points. In practice the boiling point changes but little with 
pressure. For instance, we see from the data for water on p. 119 that, 
while the pressure is increased from 1 atm to 2 atm, the temperature 
changes only by 20 or about 5%. For all the liquids satisfying 
eq. (7.16) there obtain, in this respect, similar conditions. In fact, 
it follows from it that the ratio 

dT . dP _ dir . 

~T ' p ~~ IT ' T 

must be universal. This explains the fact noticed already by Guldberg 1 
that many liquids have roughly the same reduced boiling temperatures 

1 C. M. Guldberg, Zs. phys. Chem. 5, p. 374, 1890. 



VII 47 

even when they all boil at the pressure of 1 atm, as appears from 
Table 11. 



T c 

T B 






Ethyl acetate 




Ethylene chloride 












Hydrochloric acid .... 
Oxygen . 




Carbon disulfide 












Sulfuretted hydrogen. 
Nitrous oxide 




Stannic chloride 








Acetic acid 




According to (4.11), the quantity l/T B represents the entropy 
increase on vaporization 

= s<i> - s< 2 >. 

It is an entropy change between corresponding states and must be, 
therefore, a universal function of TT, r according to (4.41). There 
follows that I/TB must be the same for substances which boil at 
corresponding pressures. However, both / and T B change but little 
with the pressure so that, in practice, l/T B measured at any pressure 
of the order of 1 atm, is roughly the same for many different sub- 
stances (Table 12). This fact is known as the rule of Deprez and 

The table contains of course a group of selected substances. If 
picked at random, the variations of Trouton's ratio are larger. Yet a 
very marked deviation of the ratio IB/TB from the average value is, 
usually, traceable to association. For instance, in acetic acid 
IB/TB = 14.8, and this is due to the abnormally high molecular weight 
of the vapor (97 instead of 60 corresponding to 2^02). Ethyl alcohol 
shows the opposite deviation, IB/TB = 26.8, due to a normal vapor 
but partly associated liquid. Equation (7.12) of Van der Waals is 

VII 48 




t B C 

IB (cal) 

h/T B 














155 9 











Sulfuretted hydrogen. 
Stannic chloride 


7 900 


Ethyl bromide 





71 2 

7 120 


only a special approximate form of the general equilibrium condition 
between a liquid and its vapor. To the extent to which the reduced 
equation (7.16) is valid, the special form can claim to have universal 
validity. In fact Van der Waals expected the coefficient a to 
be a universal constant (for all substances). However, two things 
must be borne in mind. In the first place, (7.12) is only a rough 
approximation and a is, in reality, not a constant but a function of 
temperature. In the second place, we should expect, at best, a uni- 
formity of a, not generally, but only within the groups of substances 
satisfying the law of corresponding states. The analysis of the experi- 
mental material shows that the substances of such a group can be 
arranged in a series so that a is nearly the same for any two successive 
members of the series, but it changes, systematically and considerably, 
from one end of the series to the other. 

The emphasis on Trouton's rule, as on the law of correspondence 
in general, has changed in the course of time. Soon after the discovery 
of these regularities their importance and accuracy were being over- 
rated. Then the pendulum swung to the other extreme: the insistence 
was altogether on the lack of rigor of these laws and on the numerous 
exceptions from them. However, as long as it is borne in mind that 
these rules are only rough approximations of limited range, they are 
interesting and useful. For instance, deviations from Trouton's rule 
are helpful in indicating association, as was pointed out above. 

48. The fundamental or triple point. There remains to say a few 
words about the third case mentioned in the beginning of this chapter. 
It follows from the general considerations of section 41 that the 
coexistence of three phases is possible only at a definite temperature 
and pressure which jointly define the fundamental or triple point. In 



VII 48 

fact, if we label the three phases by the numbers (1), (2), (3), we can 
apply the results of the preceding sections to each pair of them: For 
the equilibrium between (1) and (2), on one hand, and between (1) 
and (3), on the other, we have the two equations of equilibrium of 
the type (7.02) 

the third y (2 V 2) = p (3 V 3) being a consequence of these two. Corre- 
spondingly, of the three Clapeyron-Clausius equations following from 
these three relations 


only two are independent, while the third is a consequence of them. 

If we plot the pressure of the system against its temperature, 
the three equations (7.17) will represent three curves OA, OB, OC 
(Fig. 13). All points of the p, T-plane which 
do not lie on one of these curves correspond to a 

A / single-phase state of the system. The points on 

I I .*& ^n the curves OA, OB, OC represent the equilib- 

rium of two phases, and the point of intersection 
O is the triple point. As an illustration of the 
(p, r)-diagram, we have taken the most com- 
mon case when the three phases in question are 
vapor, liquid, and crystalline solid. The branch 
OA representing equilibrium of the liquid and 
gaseous phases ends abruptly at the critical 
point A, since there is no difference between 

these two states at higher temperatures. 1 Suppose that we have 
vapor and liquid in equilibrium and slowly change the boiling pressure 
and temperature so as to move the representative point of the system 
along the curve of boiling AO towards the triple point 0. When the 
point is reached, a contingency may arise similar to that discussed 
in section 43 with respect to the expansion of a liquid. Under special 
conditions of great purity of the material and of very careful manipu- 
lation, the solid phase may fail to appear and the vapor-liquid combina- 
tion may get into a state of supercooled and unstable equilibrium 
represented by the dotted continuation of the curve A 0. The slightest 
shock may cause then a sudden crystallization. 

1 With respect to attempts of extending the branch OA beyond the critical point, 
compare: Trautz and Ader, Phys. Zs. 35, p. 446, 711, 1934; Eucken, ibidem, 
p. 708. 

FIG. 13. Triple point 

and equilibrium of solid, 

liquid, and vapor. 



The triple point of most substances lies at pressures which are low 
compared with the normal atmospheric. When they are heated at 
constant atmospheric pressure, their state changes along a vertical 
straight line drawn to the right of the triple point. The crystal is first 
converted at a certain temperature (of fusion) into a liquid ; at a higher 
temperature the liquid is vaporized. It is different, when the pressure 
of the triple point is high or when the substance is heated at a low 
pressure: then, the vertical lies to the left of the point and the 
crystal is directly sublimated into vapor without passing through the 
liquid state. 

The best-known example of the coexistence of three phases is that 
of steam, water, and ice at p = 4.579 mm Hg and / = 0.0075 C. 
However, researches by Tammann l and Bridgman 2 have disclosed 
that, in addition to the ordinary ice (I), there exist at least four other 
crystalline states of water (ices II, III, V, VI) which are all denser 
than water and observable only at very high pressures. (The existence 
of a modification, formerly labeled ice IV, is considered as doubtful). 
The table of the several triple points, as far as they have been observed, 

is as follows 





FIG. 14 

Ice I, liquid, vapor. . . . 

0.0075 C 

4.579 mm Hg 

Ice I, liquid, Ice III. . . 


2115 kg/cm 


Ice III, liquid, Ice V. . . 


3530 ' ' 


Ice V, liquid, Ice VI ... 

-f OM6 

6380 ' ' 


Ice I, Ice II, Ice III... 

-37. 7 

2170 " 


Ice II, Ice III, IceV... 

-24. 3 

3510 " 


A graphical representation of the measurements carried out with 
water in its different phases is given in Fig. 14. 

An interesting situation was disclosed by Tammann 's investigation 
of the equilibrium curves of phosphonium chloride, PH4C1 (Fig. 15). 
The curve of fusion could be traced to temperatures considerably 
higher than the critical point of the liquid-vapor equilibrium. It seems 
paradoxical that the crystalline phase can be observed at temperatures 
at which the liquid does not exist or, rather, cannot be distinguished 
from the vapor. A similar behavior is shown by carbon dioxide (62) 
and by helium (He). The curve of fusion of CO2 was observed by 

1 G. Tammann, Ann. d. Phys. 2, p. 422. 1900. 
* Bridgman, Proc. Amer. Acad. 47, p. 441, 1912. 



VII 49 

Bridgman up to / = 93.5 C and p = 12 000 kg/cm 2 , while the critical 
data are t c = 31.4C, p c = 72.9 atm. The case of helium is even 
more striking: Simon, Ruhemann, and Edwards 1 observed its curve 






^0-40-30-20-10 10 


FIG. 14. Phase equilibrium in water. 


of fusion up to T = 42 abs and 5600 atm, i.e. solid helium can be 
prepared at a temperature fully eight times as high as the critical 

point of liquid helium (t c = 5.2 
abs, pc = 2.25 atm). The question 
is still open whether the curve of 
fusion extends indefinitely to still 
higher temperatures or is limited, 
either by a solid-liquid critical 
point or in some other way. Its 
experimental investigation is very 
difficult as it requires the applica- 
10 20 30 40 50 tion of ex tremely high pressures. 






FIG. 15. Phase equilibrium in phos- 
phonium chloride. 

49. Phase equilibrium of higher 
order. 2 The usual case of two 
phases in contact treated in section 
1 may be called equilibrium of the 

first order. Influenced by observations on liquid helium, Ehrenfest 3 
discussed another possible case which he called equilibrium of the second 

1 Simon, Ruhemann, and Edwards, Zs. phys. Chem. (B) 2, p. 340, 1929; 6, 
p. 62, 1930. 

2 This section may be skipped without loss of continuity. 

3 P. Ehrenfest, Communications Leiden, Suppl. 756, 1933. 



order. It may happen that the transformation takes place without 
development of latent heat and without change of volume. Supposing 
the molecule to be the same in the two phases, 

X2 Xi = 0, AF = V2 v\ = 0. 


This means that the two phases have the same molal heat function 
and the same molal volume. Because of the fundamental relation of 

A$ = 0, 2 - <PI = 0, (7.19) 


they have then also the same entropy (AS = $2 si == 0), 

A5 = Qp/T, according to (4.11). Yet the phases are different because 

of their other properties (for instance, the molal heats) changing dis- 




C 15 



2.0 2.2 2.4 2.6 2.8 3.0K 
FIG. 16. Specific heat of helium. 

continuously in the transformation. This seems to happen with 
liquid helium at the temperature of 2.2 K: the discontinuity in its 
molal heat is given in Fig. 16 as measured in Leiden l while changes 
of entropy and volume were not observed. Similar conditions are 
known to obtain in many other transformations: solid methane at 
20.4 K, solid oxygen at 44 K, HBr at 87 K, ammonium chloride 
and bromide, etc. 2 The accuracy of these observations was in no case 
sufficient for a conclusive proof that the conditions (7.18) were strictly 
satisfied, but it is interesting to follow up the theoretical implications 
of such a contingency. 

1 W. H. Keesom and A. P. Keesom, Comm. Leiden 222 d; W. H. Keesom and 
K. Clusius, 219 e; K. Clusius and A. Perlick, Zs. phys. Chemie 24, p. 313, 1934. 

2 Compare reviews by M. Von Laue and A. Eucken, Phys. Zs. 35, pp. 945, 954, 


Taking account of (6.51) and (6.52), the conditions (7.18) can be 
also written in the form 

0, A*,--0, (7.20) 

which converts the Clausius-Clapeyron equation (7.04) into an iden- 
tity. In order to obtain a relation between the differentials of tempera- 
ture and pressure, Ehrenfest proceeds to differentiate these two 

where the partial differentiation with respect to T or p is indicated by 
writing these variables as subscripts. He further postulates that these 
two relations be compatible. It should be remembered that, according 
to (5.37), (4.32) 

~~ C PI 


The elimination of dp/dt from eqs. (7.21) gives Ehrenfest's relation 
for equilibrium of the second order 


Serious objections against these conclusions were raised by Keesom 1 
and by Von Laue. 2 They can be brought out most clearly by a geo- 
metrical interpretation of eqs. (7.19) and (7.20). If we regard T, p, <p 
as cartesian coordinates of a three-dimensional space, the equations 
<p <pi(p, T) and <p = <f>z(p, T) represent two surfaces in this space. 
In the ordinary first order equilibrium the condition (7.19) is satisfied 
along the line of intersection of these two surfaces, as represented 
graphically in Fig 17a, which gives the trace upon the (<p, T)-plane. 
The relation between the transformation values of p and T, given 
by the Clausius-Clapeyron equation (and represented graphically in 
the figures of section 48), is then the projection of this line of inter- 
section onto the plane p, T. The additional conditions (7.20) for the 
second order transformation mean that the two surfaces are in contact 

1 W. H. and A. P. Keesom, Physica 1, p. 161, 1933. 

2 Footnote on p. 129. 


(Fig. 176). Von Laue points out that in this case no transformation 
at all is possible. In fact, the theory of equilibrium requires A4> g 
(section 36) and, therefore, the states of the system represented by the 
upper branches of the curves are unstable. It will be found in the 
more stable state (1) both above and below the temperature of the 
contact. Von Laue suggests, therefore, that the phases in question 
may have an equilibrium of the third order, i.e. the surfaces <p = <p\ 

(a) (b) (0 


Xv fi 


FIG. 17. Equilibrium of first, second, and third order. 

and <p w have contact with penetration (Fig. 17c). The mathe- 
matical conditions for this are 

A^JT = A$ pr = A$ pp = 0, (7.24) 

in addition to (7.18) and (7.19). They represent so much of a restric- 
tion upon the parameters of the two phases that Von Laue's suggestion 
was not favorably received by the other workers in this field. 

We think, however, that the topic should not be dismissed without 
a fuller investigation of the possibilities of the second order equilibrium. 
Ehrenfest and Von Laue make an assumption which, in our opinion, 
goes too far. They differentiate eqs. (7.20), implying that the contact 
exists along the whole line of intersection of the surfaces <pi, w or, at 
least, along a finite part of it. There is nothing in the experimental 
observations to justify this assumption. Theoretically it is also a 
rather remote case, because it cannot occur, unless the properties of 
the two phases happen to satisfy Ehrenfest's eq. (7.23) or Von Laue's 
even more restrictive conditions (7.24). It is a far more common 
occurrence that the two surfaces are in contact in a singular point, i.e. 
eqs. (7.20) are satisfied just in one point whose coordinates we shall 
denote by po, TQ. What is then the value of A< in a neighboring point, 
say, PQ + dp, TQ + dT? Expanding with respect to dp, dT to terms 
of the second order 

dT, po + dp) 

. . . (7.25) 



VII 49 

Because of the conditions (7.19) and (7.20), equilibrium in the 
vicinity of p, T is then determined by the condition 


The expression D of formula (7.23) is the discriminant of this 
equation. When D < 0, the equation has no real solutions so that a 
change of phase is impossible. The limiting case, D = 0, is that of 
Ehrenfest's and must be excluded because of the reasons advanced 
by Von Laue. But when D > 0, the equation has two real solutions. 
Because of (7.22), 



lA QT J p 



Consequently, transformations are possible, if not in the point 
po, To itself, in its immediate vicinity as is illustrated in the (p, T)- 
diagram of Fig. 18. There are two lines in which the phases (1) and (2) 







\] (1) 

FIG. 18. Vicinity of a point where 
equilibrium is of second order. 

FIG. 19. Special cases of first order 

can coexist, and they intersect in the singular point po, TQ. When 
the substance is cooled at a pressure exactly coinciding with that of 
the intersection (dotted line AB), no transformation takes place. If 
it is cooled at a slightly different pressure (line A'B') it is transformed 
first from the phase (2) into the phase (1) and, at a lower temperature, 
back again into (2). Some of the experimentally measured anomalies 
present features compatible with this picture: as the temperature is 
changed, the specific heat jumps discontinuously and, at a slightly 
lower temperature, jumps back to the initial value (Fig. 16). How- 
ever, it would require a more detailed experimental investigation to 
establish the existence of this type of transformation beyond doubt 
in view of the following considerations. 


More important than the direct application of the preceding 
results is the light they throw, indirectly, on transformations of the 
first order with small but finite latent heats and volume changes. 
Let po, To be a point in which A$r and A$ p are very small but finite. 
In the infinitesimal vicinity of the point po, To there will hold, of 
course, the Clausius-Clapeyron equation. But if we consider, instead 
of the differentials dp, dT t small finite increments A/>, AT, the second 
order terms in (7.25) may no longer be neglected and the equation 

= 0. 

This represents an ellipse or a hyperbola with its vertex near the 
point po, To, according to whether the discriminant (D) is negative or 
positive (Fig. 19). The dotted parts of the curves have no reality 
because the approximation breaks down when A/?, AT become appreci- 
able since then terms of third and higher order must be taken into 
account. The gist of the matter is that these />, jT-lines are strongly 
curved. In cooling the system along the lines A'B' we meet with 
conditions quite similar to those of Fig. 18. In fact, the transition 
from the first to the second order of equilibrium is a continuous one 
and the two cases cannot be distinguished without a complete experi- 
mental investigation of the whole region around the point po, TQ. 
Some of the cases enumerated in the beginning of this section are, 
probably, examples of this kind of equilibrium of the first order. 1 
Another interesting illustration will be presented in section 137. 

1 The same conclusions were drawn by A. Eucken (footnote on p. 129) from the 
study of the experimental material. 


50. The thermodynamic potential of a gas mixture. An example 
in which the general theory of equilibrium can be completely carried 
through in detail is the mixture of perfect gases. This is largely due 
to the fact emphasized in section 4 that the pressure of such a mixture 
is equal to the sum of the partial pressures of the individual gases 

p = pi+ P2 + . + p., (8.01) 

where pi, . . . p 9 are computed for each gas as if it were filling the 
available volume alone and the other gases were not present. Suppose 
that we have in a vessel of the volume V several gaseous constituents 
in the respective amounts of N\, . . . N 9 mols. Let their partial 
pressures be expressed according to eq. (1.12) 

Vh being the molal volume of the gas fe. In order to apply to this system 
the theory of Chapter VI, we must find the thermodynamic potential 
$ = JJ TS + pV, where T and p are the temperature and the total 
pressure of the mixture. The physical inference from the formula 
(8.01) is that there is no interaction between the gases, so that the 
thermodynamic potential of each of them must be computed, as if it 
were alone in the vessel, and all these potentials must be added to 


*=y%*. (8 - 3) 


A simple and convincing way of seeing this is opened by an ideal 
experiment devised by Gibbs. We give the vessel containing the 
system the following construction (Fig. 20). The cylinder C fits tightly 
into the cylinder C 1 and can slide within it without friction. The outer 
ends of both cylinders are closed by tops AB and A'B' impenetrable for 
the gases and adiabatic like the side surfaces. They have also bottoms 



MN and M'N' which are kept by a suitable arrangement at constant 
distances / from the tops while the cylinders are sliding. However, 
these bottoms are semipermeable membranes: The membrane M'N' 
lets through freely the component h but is impenetrable for all the 
others. On the other hand, M'N' is impenetrable only for h, offering 
no resistance to the rest of the components. The volume of each 
cylinder (ABMN and A'B'N'M') is the same and equal to V. We 
start our considerations from the state when the cylinder ABMN is 
pushed completely in, so that MN touches A'B' and the gas mixture 
fills the joint volume V. We begin now to pull out this cylinder in a 
reversible (infinitely slow) way. Three part volumes of the system 
must now be distinguished: (1) the section above M'N' is filled with 
the pure h component because this membrane is not permeable to the 
other gases. (2) The middle part between M'N' and MN contains all 
components. (3) The section below MN is free 
of the component h because this gas is pushed up 
by the membrane MN which is impermeable to it. 
The gas h is, therefore, present only in sections (1) 
and (2) (or in the upper cylinder). We have 
emphasized that its partial pressure p\ is inde- 
pendent of the presence of other gases: It will 
be, therefore, the same in section (1) where it is 
alone and in (2) where it is one of the constitu- 
ents of the mixture. Only the component h A' 

exercises pressure upon the top AB and the FIG. 20. Device for 
bottom MN of the upper cylinder. The rest of unmixing gases, 
the gases do not produce any pressure effects 
on the two surfaces: on AB, because no other component is 
present in the part volume (1) adjacent to it; on MN, because this 
membrane is completely permeable to the other gases. We see, 
therefore, that the forces of pressure on top and bottom are oppositely 
equal so that the resultant force on the whole cylinder vanishes. The 
process of reversibly pulling out the cylinder does not involve any 
work (W = 0), provided there are no outer forces upon the system 
(e.g. air-free outer space). Since the outer envelope of the system is 
adiabatic (Q = 0), the reversible process goes on without change of 
entropy (section 23). Moreover, it proceeds at constant internal 
energy and temperature. In fact, in the general eq. (3.05) of the first 
law, Q = t/2 Ui + W, the terms Qand W vanish leaving C/2 = E/i, 
and in the case of perfect gases, this is equivalent to T% Ti, accord- 
ing to eq. (3.18) or (3.19). 

Continuing the process until the membranes MN and M'N' touchy 


so that the middle section (2) disappears, one can entirely separate the 
component h from the mixture. In a similar way, the other compo- 
nents can be separated, one by one, so that, in the end, the system is 
completely unmixed and each component occupies, separately, a 
volume equal to V. The whole process is carried out without change 
of U and S. It follows from this that the internal energy and the 
entropy of a gas mixture are additive U = 2C/A, S = SS*, provided 
the entropies Sh of the individual perfect gases are computed as if 
each of them occupied the volume V alone. Therefore, the first two 
terms of the thermodynamic potential $ = U TS + pV are addi- 
tive, while the additivity of the third follows from the relation (8.01), 
thus completing the proof of eq. (8.03). 

If we use eq. (4.20) for the molal entropy of a perfect gas, the 
entropy Sh of the component h has the expression 

S h - N k j<ir - R log p h 
and the total entropy of the mixture becomes 

5 - 2] N " - R log pk + s * (8 - 04) 

Sometimes it is preferable to express 5 in terms of the total pressure 
p rather than the partial pressures ph. In order to carry out the trans- 
formation, it is convenient to use the mol fractions defined in section 39 

x h = Nk/N, (8.05) 

where N = Ni + . . . + N 9 represents the total number of mols (of 
the different constituents) in the mixture. In the case of perfect gases, 
equations p h V = N h RT and pV = (N\ + . . . + NJRT hold, so that 
ph/p = x h or 

ph = x h p. (8.06) 

Substituting this into (8.04) 

. (8.07) 

The first three terms of this expression are linear in the mol numbers 
while the last shows a more complicated dependence upon them. 


In a similar way, there follows from eq. (5.41) for the molal thermo- 
dynamic potential 

7k = RT log p h + a*(TO, (8.08) 


, N h [RT log p k + (DL (8.09) 


/T s* c 

c ph dT - T I ~ dT + u Qh - 7*0*. (8.10) 
J l 

or an alternative form after the substitution of (8.06). 

We denote the molal potential (8.08) by the symbol ^ instead of 
<ph for reasons which will become clear in the next chapter. This is 
entirely permissible because, in this case, the partial thermodynamic 
potentials of the mixture are identical with the molal of the pure com- 
ponents, as a consequence of the additivity expressed in eqs. (8.03) 
and (8.09). In fact, as determined by eq. (8.08), <pn satisfies the 
definition (5.38) of the partial molal thermodynamic potential. 

Exercise 76. Derive the expressions of the heat function X and the work 
function ^ for mixtures of gases. 

Exercise 77. Check the statement that the derivative (3*/3^)p f r is identical 
with (8.08). In the differentiation the total pressure is to be kept constant. 

51. The mass law. If the gases of the mixture can react chemi- 
cally with one another, according to the formula (6.46), S^G* = 0, 
the condition of equilibrium is given by eq. (6.49) or 2*>^ = 0. 
Substituting for <ph the expressions (8.08), we obtain 

log p h = - 

We recall that the coefficients j>& of the components consumed in 
the reaction have the negative sign, of those produced in it the positive. 
Therefore, the physical meaning of the expressions on the right side, 
when (8.10) is substituted, is as follows: 

= AC P (8.12) 

represents the difference of the heat capacities (at constant pressure) 
of the gases produced and consumed in the reaction, or the total 
change of the heat capacity due to it. According to eq. (3.27), 


c p hdT + UOH = *h is the heat function of the component ft. For 
this reason 

Q P (8.13) 

represents the total change of the heat function of the mixture AX 
which is equal to the heat of reaction Q p , as we know from eq. (3.23). 
The last term depending on the entropy constants of the constituents 
of the mixture, we shall denote by 

= R log L (8.14) 

We further introduce for the whole right side of eq. (8.11) the 
notation RT log K pt so that 

!<* Kp = / ^ dT - Jfr + ^g L (8.15) 

Within the range of temperatures, where the molal heats c p h can 
be considered as constant, this becomes 

log K P = ~2 log T - QL + i og /. (8 . 16) 

Strictly speaking, / is well defined only in this case, unless the 
lower limit of the integral in (8.15) is specified. 

The condition of equilibrium (8.10) takes, therefore, the form 

KW .../' = p. (8.17) 

An equivalent form is obtained by replacing the partial pressures by 
the molal fractions by means of eq. (8.06), ph = Xhp 9 

xi*xf . */* = K, (8.18) 

where K = K p /p" or 

log K = log / - j, + dT - v log p t (8.19) 

with the abbreviation 

*=Vl + 9* + ... + 9r (8.20) 

The relation (8.17) between the partial pressures of the constituent 
gases or the equivalent eq. (8.18) between the molal fractions is known 
as the mass law. Whatever the constitution of the mixture may be, 


at a definite temperature and pressure, the function pi*p* ... (or 
Xi n x 2 n . . . ) is constant. The quantity K p (respectively, K) is, 
therefore, called the equilibrium constant of the reaction. 

The mass law was first deduced by the Norwegian chemists 
Guldberg and Waage (1867) from statistical considerations which it 
will be instructive to reproduce here. Let us write the equation (6.46) 
of the reaction in the form 

vid + v 2 G 2 + . . . <=* v'iG'i + v' 2 G' 2 + . . . , 

where the symbols, on one side, refer to the substances entering the 
reaction and, on the other, to those resulting from it. In order that 
vi, i> 2 , . . . molecules of the substances GI, G 2 , . . . enter the reaction, 
they must all meet, i.e. be simultaneously present in a small volume 
TO. What is the composite probability of such an event? The prob- 
ability of one molecule of the kind GI being in the volume TO is pro- 
portional to the total number of such molecules available, which, in 
its turn, is proportional to the mol number N\. The probability of vi 
molecules being there simultaneously is proportional to N\ l (accord- 
ing to the rules expounded in section 30), provided that the molecules 
are statistically independent. This statistical independence implies 
that there are no forces of interaction of any kind between them, and 
this is true only for perfect gases. Since we have to restrict ourselves 
to this case, we may make use of the property of perfect gases that 
the mol number is proportional to the partial pressure and conclude 
that the above probability is proportional to pi* 1 . In a similar way, 
we find for the probability of v 2 molecules of the second component 
being in the volume T O the expression p<P> and so on. The probability 
of the composite event is, therefore, 

where the constant of proportionality a may depend upon the prop- 
erties of the system, especially its temperature and pressure. The 
reaction may, however, go also in the opposite direction producing 
Pi t v 2 , . . . molecules of the substances GI, G 2t ... It is necessary for 
this that i/i, 1/2, molecules of the substances G'i, G' 2 , . . . enter 
the reaction and meet in a limited volume. The probability for this 
is, obviously, 

P' = a'fi'yS* . . . 

In the state of equilibrium the number of molecules of every kind 


is stationary and the process must go as often one way as the other. 
The probabilities P and P' must be equal 

(K p = a'/a) an equation which is identical with (8.17). 

The thermodynamical derivation given above is due to Gibbs (1871) 
and leads a good way beyond the result of Guldberg and Waage inas- 
much as it contains in eqs. (8.15) and (8.19) the explicit dependence of 
the equilibrium constant upon p and T. It is sufficient to make one 
measurement at a definite pressure and temperature in order to 
determine the constant /. Knowing /, one can predict the equilibrium 
conditions for all possible states of the mixture. 

Exercise 78. Find the form of the mass law (8.17) for the following reactions: 
N 2 + 3H 2 - 2NH, = 0; 2HBr + I, - 2HI - Br a = 0; 2O - O 2 - 0; 

2O 8 - 3O 2 - 0. 

Exercise 79. Show that, in view of eqs. (4.36") and (3.26), the expression (8.15) 
can be reduced to 


J JKJ. " 

which is consistent with (6.53). 

52. Examples. Degree of dissociation. We have obtained two 
forms of the mass law in eqs. (8.17) and (8.18). The first form is 
simpler inasmuch as the quantity K p is a function of the temperature 
only. All the quantities on which eq. (8.17) depends are completely 
conserved if we add neutral gases to the mixture, because the partial 
pressures pi, . p 9 , as well as K p , remain unaffected by this. The 
second form has the advantage that it can be more readily generalized 
to include systems which are not perfect gas mixtures. 

The most interesting question is, usually, as to the fraction of the 
original substance or substances transformed in the reaction. Let us 
discuss, for instance, the oldest reaction to which this theory was 
applied by Gibbs, the dissociation of nitrogen tetroxide 

2NO 2 - N 2 O 4 = 0. (8.22) 

The coefficients v are in this case v Nl04 1> ^NO* 2, v = 1. 
Suppose that No moles of N 2 C>4 are brought into a vessel at a very 
low temperature when it is quite undissociated : what mol numbers 
shall we observe after heating the gas to the temperature T at the 
pressure p? If we denote by { the fraction of the N 2 C>4 molecules 
which are dissociated (degree of dissociation) the mol number of this 


component is ^ N2 o 4 = -^o(l ) To every dissociated molecule 
there appear in the mixture two molecules of NC>2. The mol number 
of nitrogen dioxide is, therefore, jV N o, = 2No, and the sum 
N = # Nl04 + WNO, = N (l + ) Recalling the definition (8.05) of 
the mol fractions and substituting into (8.18), we find 

K = K p /p. (8.23) 

I-? 2 

We see from this that the degree of dissociation is small when K is 
small, i.e. according to (8.19), at low temperatures or high pressures. 
Under the opposite conditions of high temperature or low pressure, 
approaches unity. 

The degree of dissociation being known, it is easy to compute the 
density, which is also accessible to direct measurement. The density 
of the mixture A, obviously, stands to the density 5 of the undissoci- 
ated component N2O4 in the inverse ratio of the numbers of molecules: 
A/6 = No/N, whence 

A = 

Accurate measurements on the dissociation of nitrogen tetroxide 
are due to Bodenstein. 1 This gas can be regarded as approximately 
perfect only when the pressures are not too high and the temperatures 
not too low. From observations under these conditions Bodenstein 
derives the formula 

logic K p =- - ~ + 1.75 logio T + 0.00483r 

- 7.144 X 10- 6 r 2 + 5.943, (8.25) 

when the pressure is expressed in mm Hg. The agreement with obser- 
vations can be judged from Table 14. 


T(abs) 282. 5 285. 7 289. 5 293. 305. 9 323. 9 

logio K p (obs.) 1.487 1.608 1.745 1.859 2.294 2.838 

logio Kp(calc.).... 1.475 1.600 1.741 1.889 2.286 2.838 

T(abs) 334. 8 342. 7 352. 5 361. 9 373. 9 387. 4 

logio K p (obs.) 3.135 3.326 3.577 3.784 4.027 4.293 

logio K p (calc.). ... 3.135 3.341 3.577 3.794 4.032 4.323 

1 M. Bodenstein, Zs. phys. Chemie 100, p. 74, 1922. 



VIII 52 

Bodenstein's formula represents also fairly well the older measure- 
ments by Natanson. 1 Although not very accurate, they offer us exam- 
ples of how the degree of dissociation varies with the pressure at 
constant temperature. One of the series was measured at / = 49. 7 C, 
for which temperature (8.25) gives K p = 643. The normal density 
of N 2 O4 is d = 3.180, whence the actual density (reduced to normal) 
is calculated by means of eq. (8.24). 





A (calc.) 

A (obs.) 




























A reaction of great practical interest is the dissociation of water 

2H 2 + O 2 - 2H 2 O = 0, 

with the coefficients V H , = 2, v^ = 1, v Ht0 = 2, v = 1. We suppose 
again that we put into the reaction vessel JVo undissociated water 
molecules (at a low temperature). At a higher temperature dissoci- 
ation ensues and the mol numbers of the three components become 

Equation (8.18) gives, then, 



2 (i - )2 (1 + f 

According to Siegel 2 the heat of this reaction is 
Q 9 - 113 820 + 2.6Sr - 4.41 X 10~ 4 r 2 + 1.252 X 10~ 6 r 3 

-9.12 X 10- 10 r 4 + 4.36 X lO-^r 6 , (8.27) 

1 E. and L. Natanson, Ann. Physik u. Chemie 24, p. 454, 1885. 
*W. Siegel, Zs. phys. Chemie 87, p. 641, 1914. The value of the constant 
1.08 is due to Nernst and Wartenberg. 

VIII 53 



and the equilibrium constant 

logic K P = ~ 

+ 1.335 logio r - 0.965 X 10~ 4 r 

+ 0.137 X 10~ 6 r 2 - 0.665 X 10- 10 r 3 + 0.191 X 10- 17 r 6 - 1.08. (8.28) 

Nernst 1 gives the following comparison of observed data of the 
degree of ionization with those calculated from this formula (at 
atmospheric pressure) : 



T (abs) 

100$ (calc.) 

100 (obs.) 

T (abs) 

lOOt (calc.) 

100$ (obs.) 


4.66X10- 2 ' 

4. 6 - 4. 8 X 10~ 2 





5.4 X10- 






































The agreement is remarkable in view of the width of the range and 
of the fact that the observations were obtained by many different 
observers and methods. 

Exercise 80. Find the equation for the degree of dissociation, analogous to 
(8.23) and (8.26), for the reactions: 3H 2 + N 2 = 2NH,, Hg + I 2 - HgI 2 . 

Exercise 81. Calculate for hydrogen iodide from eq. (8.30) at the pressure 
p = 100 mm and at the temperatures of Table 17. 

53. Influence of the excess of one component. Another inter- 
esting reaction is the dissociation of hydrogen iodide 

H 2 + I 2 - 2HI - 0, 
?H, = 1, "i, = 1, v 

- 2, v = 0. 


The equation v = means that the number of molecules is not 
changed in the reaction. The equilibrium constant (8.19), then, does 

1 W. Nernst, Theoretische Chemie, p. 775. llth edition. Stuttgart 1926. 


not contain p, so that the molal fractions and the degree of dissociation 
are entirely independent of the total pressure. Suppose we bring 
into a vessel (at a low temperature) N\ mols of HI and N% mols of Ha 
and heat the mixture to the temperature T. This assumption will 
permit us to study the influence of the excess of one component (Ha) 
upon the degree of dissociation . As in the preceding case, 
N m = Ni(l ). Each dissociated molecule of HI produces half 
a molecule of Ha and la, respectively: therefore, JV Ia = %Ni and 
-^Ha = 2^1 { + A/a, while the sum, N = NI + NZ, is independent 
of the degree of dissociation. The substitution into (8.17) gives 


-* -*' (8 ' 30) 

When the temperature is low, K is small and, with it, the degree of 
dissociation . We can take then for the denominator the approximate 
value 1 and see that the numerical value of greatly depends upon 
the term 2Nz/N{. When there is no excess hydrogen (N% = 0), we 
find = 2K*\ on the other hand, when 2Nz/Ni is not small, we can 
neglect , compared with it, obtaining = 2KN\/N%, a number 
which is much smaller: The excess of one of the products of dissociation 
seriously depresses its degree. When K is very large (high temper- 
atures), approaches unity. As an approximation, we can substitute 
= 1 in the numerator with the result 1 = (1 + 2N 2 /Ni)*/2K*. 
Qualitatively the presence of the term 2Nz/N\ has still a depressing 
influence. However, quantitatively sinks appreciably below 1 
only when 2N%/N\ begins to approach K in its order of mag- 

The physical reason of this depressive action is, of course, that the 
excess of Ha gives to the iodine, produced in the dissociation process, 
a larger chance of reacting with hydrogen and of being transformed 
back into hydrogen iodide. This simple explanation makes it clear 
that we have here a general phenomenon which occurs, under similar 
conditions, in all cases of dissociation. 

The equilibrium constant of the reaction (8.29) is well represented 
by the formula 1 

540 4 
logic K = logio K p = -- - + O.S03 logio T - 2.350. (8.31) 

1 Nernst, Zs. Electrochcmic 18, p. 687, 1909; K. Vogcl v. Falkenstein, Zs. phys. 
Chcmie72, p. 113, 1910. 


Hence and from (8.30) we obtain Table 17. 



Nt/Ni = 

Nt/Ni - 1 


logio K 

100$ (calc.) 

100$ (obs.) 

100$ (calc.) 

















- 1 . 202 









The depressive action is very marked, indeed. 

Exercise 82. Calculate from (8.31) with the help of (8.21) the heat of the 
reaction (8.29) at 1000 K. Do the same for the reaction (8.22) by means of the 
formula (8.25). 

Exercise 83. Derive the generalization of the formula (8.26) when there is an 
excess of H a , and estimate its depressive influence. 

54. Influence of neutral gases. In the absence of neutral gases 
the mol fraction of each component is given by the expression 

x k = N k /(Ni + ...+NJ. (8.32) 

On the other hand, if there are in the mixture other gases than 
those taking part in the reaction and the sum of the mol numbers of 
these neutral gases is NQ, the molal pressure of the same component 

Ni+...+N.). (8.33) 

Comparing these two relations we find 

*>k = *k(Ni + . + N,)/(N + Ni + ...+NJ. (8.34) 

The expression of the mass law in the presence of neutral gases is 
in accordance with eq. (8.17) 



or substituting (8.34) 


The addition of neutral gases at P = const has, therefore, the same 
effect as an increase of the equilibrium constant. This influence is 
opposite to that of the excess of an active component because to a 
larger constant K corresponds a higher degree of dissociation. The 
presence of neutral gases in general advances the degree of dissociation, 
provided that the total pressure p remains the same. Of course, in the 
particular case v = 0, neutral gases have no effect. 

We see from the definition (8.19) of the function K that the same 
result could be achieved by reducing the total pressure p of the mixture 
in the proportion (Ni + . . . + N ff )/(N Q + . . . + N ff ). In a certain 
sense, the influence of the neutral gases is, therefore, only apparent. 
It comes from the fact that we refer the effect to constant total pressure 
to which both the active and the neutral gases contribute, so that the 
partial pressures of the active components are reduced in precisely 
the same proportion. This corresponds to the usual experimental 
conditions in which the total pressure p is the fundamental datum. 
If the question were put as to the degree of ionization in a box of 
constant volume, the introduction of additional neutral gases into 
that box would have no influence whatever on it. 

Exercise 84. What is the increase of volume of the gas mixture in a reaction 
in which the sum of the mol numbers increases by v? Use the equation of perfect 
gases (1.18) and check that the result is consistent with (6.51). 

55. Mutual influence of two simultaneous reactions. Suppose 
that we mix at a low temperature N\ mols of hydrogen iodide (HI) 
and N2 mols of hydrogen bromide (HBr) and slowly heat the mixture. 
What will be its composition at a temperature T? In addition to the 
reaction (8.29), there can take place also the reaction of dissociation 
of hydrogen bromide 

H 2 + Br 2 - 2HBr = 0, 


1. ) 

"Hj = 1 "Bri = 1 ^HBr = 2, 

Accordingly there must be simultaneously satisfied two equations 
of the type (8.18) 

(8 37) 


HI x HBr 

We can use the expressions of section 53 for the mol numbers of 
HI and of 12, in terms of the degree of dissociation 1 of hydrogen 
iodide: N m = Ni(l - 1), N lt = jAih- Similarly we can write 
(denoting the degree of dissociation of HBr by {2) : NSB T =* JVa(l {2), 
, since the two reactions are precisely of the same type. 

VIII 55 



As to the mol number of hydrogen, it comes from two sources: each 
dissociated molecule of both HI and HBr produces half a molecule of 
H2, whence JV H , = i(^ii + N 2 2 ). Substituting into (&.QS) and 

4 (1 i) 2 ' 4 (1 &) 2 

It is apparent from these equations that the mutual influence of 
the two reactions results in the decrease of the degrees of ionization of 
both HI and HBr. The reason of this is easy to understand: either 
of the reactions liberates hydrogen and creates an excess of this com- 
ponent for the other process, exercising upon it a depressive action 
(compare section S3). In general, there exists still another mutual 
influence: If the total pressure is kept constant, the presence of the 
second gas (and of the products of its dissociation) decreases the partial 
pressure of the first and, in this way, produces the same effect of 
enhancing the dissociation as a neutral gas. We have already said in 
the preceding section that this effect is trivial: Therefore, we have 
selected here two reactions in which it does not exist because they are 
not pressure sensitive. They permit us to study the mutual depressive 
influence in its pure form. 

The equilibrium constant of the dissociation of HBr can be repre- 
sented as follows 

logio K 2 = - - + 0.553 logic T - 2.72. (8.38) 

At temperatures below 2000 K, 2 is, therefore, very much smaller 
than ft. If NI and N 2 are of the same order of magnitude, the disso- 
ciation of HI is not appreciably affected by the presence of HBr, and 
the values of 1 remain the same as in Table 17. On the other hand, 
the dissociation of HBr is greatly depressed, as is illustrated by 
Table 18. 



Ni = 

tfl - Nl 

T (abs) 

logio KI 






1.8 X 10- 




3.4 X 10-* 






Exercise 85. N mols of HI are brought in a vessel and heated to a high tem- 
perature where the reactions (8.29) and 21 - I* = must be taken into account 
simultaneously. Denote the degree of dissociation of HI by 1 and that of 1 2 by 2 . 
The equilibrium constant for the dissociation of I 2 is (Bodenstein, Zs. Electrochem. 
22, p. 338, 1916): 



K p - - ^~ + 1.75 log, T - 4.09 X 10-* T + 4.726 X 10~ 8 T - 0.440. 

Calculate gs under the above conditions and in pure It at the temperature 2000 K. 


56. The concept of a solution. We call a solution any homogene- 
ous system which consists of two or more chemically pure substances, 
no matter whether it is solid, liquid, or gaseous. Solutions must be 
distinguished from chemical compounds, on one hand, and (in general) 
from mixtures, on the other. A chemical compound consists of mole- 
cules of only one kind; therefore, the masses of the constituents of 
which it is formed stand in a definite ratio. From the point of view 
of the phase rule (section 41) it is a system with one single independent 
component or a pure substance: if the amount of one constituent is 
given, those of all the others are uniquely determined by it. It is 
quite different with the solution; the molecules or atoms of the con- 
stituents do not lose their individuality in it, and their amounts can 
assume any arbitrary ratio, within certain limits. The composition 
of the solutions being continuously variable, they must be regarded as 
homogeneous systems with several independent components. 

The difference from mixtures is most clearly brought out in the 
case of solid solutions. We know, at present, that a solid is always 
crystalline, and we say that two substances are mutually soluble, in 
the solid state, when they form a joint crystalline lattice, or a mixed 
crystal in which the lattice points are occupied, in part, by atoms 
(or molecules) of the one substance, in part, of the other. The solu- 
bility of solids is an exception, although several pairs of metals are 
known which are mutually soluble in all proportions, e.g. Ag Au, 
Ag Pd, Co Ni, Cu Ni, Cu Pd, Au Pt, Bi Sb, etc. The metals 
of each pair are of a closely analogous chemical nature belonging to the 
same group of the periodic system (or sometimes to two adjacent 
groups). Of non -elementary solids which are mutually soluble in 
all proportions we could mention AgCl NaCl, PbBr2 PbCb, 
SnCb PbCb, etc. More common is the case of limited solubility: 
the crystalline structure of a metal is preserved if a fraction of the 
lattice points (up to a certain limit) is replaced by atoms of another 
metal, usually from the same or the adjacent group of the natural 
system. The lattice cannot accommodate more than the limiting 
fraction without breaking down. Such cases are Cu Ag, Au Ni, 



Bi Pb, Sn Pb, etc. Of course, limited mutual solubility occurs 
also with pairs of solids of a non-metallic nature: KNOa NaNOa, 
KNOa TiNOa, etc. All other cases, viz. those in which the con- 
stituents of an alloy do not form a joint crystal, must be classed as 
mixtures. For instance, the microscopical examination of Ag Pb 
alloys shows that they are agglomerations of tiny crystals of pure silver 
and pure lead. Similar conditions prevail with most other alloys 
(e.g. Bi Cd, Pb Sb, also KC1 AgCl, ice AgCl, etc.). We have 
seen in section 39 that, as far as the thermodynamical applications and 
conclusions are concerned, the size of the phases is immaterial; there- 
fore, an alloy or other mixture of this sort must be regarded as a sys- 
tem of two or more coexisting phases, even if these phases are of 
microscopical dimensions. Unlike a mixture, a true solution is homo- 
geneous and represents a single phase. 

The conditions in a liquid solution are similar. A liquid can, in 
general, dissolve other liquids, solids, or gases, the true solution being 
homogeneous and characterized by a uniform distribution over its 
whole volume of the molecules or atoms of all the constituents. In 
the case of liquids mutual solubility is not an exception but a common 
phenomenon. At the same time there are numerous pairs of liquids 
which exhibit in certain ranges of temperature a complete (e.g. water 
carbon tetrachloride) or a limited (water ether, water phenol, etc.) 
mutual insolubility. The true or molecular solutions must be distin- 
guished from the mixtures which in the case of liquids are called sus- 
pensions, emulsions, and colloidal solutions. Like the solid mixtures 
they are inhomogeneous and represent two or more phases intermixed 
in a finer or coarser degree of dispersion. 

In the case of gases, the difference between a solution and a mix- 
ture disappears, since all gases are mutually soluble or miscible, with- 
out restriction, forming a homogeneous system in which the molecules 
of all constituents are uniformly distributed. 

The process of forming a solution is accompanied by mechanical 
and thermal effects which show that there is a certain degree of inter- 
action between the molecules of the different constituents. When we 
dissolve ethyl alcohol and water in equal parts, there is a considerable 
contraction, the volume of the solution being about 5% smaller than 
the sum of the volumes of the free components (compare section 32). 
At the same time, a certain amount of " heat of solution " is devel- 
oped, because the internal energy of the solution is not equal to the 
sum of the internal energies of the alcohol and water forming it. 
Both effects are characteristic of the general case although their 
numerical values vary widely, so that the heat of solution may be 


either positive or negative. There are certain pairs of liquids known 
(e.g. benzene-toluene) which form solutions with very little diminution 
of volume and development of heat. This usually occurs in the case 
of a close chemical resemblance of the constituents so that the inter- 
action between atoms of different kinds is nearly the same as that 
between atoms of the same kind. We can, therefore, imagine, as an 
ideal case, a solution in which the volumes, as well as the internal 
energies, are strictly additive and no change occurs in these quantities 
when the solution is formed. Following G. N. Lewis 1 we shall call it 
a perfect solution: although the perfect solution may not occur in 
nature in its pure form, it is a useful fiction as it will help us to bring 
out, theoretically, some typical properties of real solutions. 

57. Dilute solutions. One of the components of a solution, usually 
the most plentiful is called the solvent, the others the solutes. In this 
chapter we shall discuss the case when the mol number of the solvent 
JVo is much larger than those of the solutes (Ni, N 2 , . . . N ). The 
thermodynamical properties of every system are determined by the 
characteristic functions, especially by the thermodynamic potential 
which, in turn, depends on the internal energy and the volume. A 
theoretical treatment of the problem requires, therefore, some knowl- 
edge about the analytical structure of the functions U and V, par- 
ticularly with respect to their dependence upon the mol numbers Nk 
of the components. We have seen in section 39 that the functions 
U, V, 5, $ are homogeneous of the first degree with respect to the mol 
numbers. There obtains, therefore, the equation analogous to (6.13) 

, T, No,... N.) = U(p,T, 
Taking for e the special value e = I/No, we find 

This equation (and a similar one for V/No) is quite general and 
does not contain any restrictive assumptions. However, we shall 
introduce now a restriction limiting the generality of the function /, 
in that we assume that it can be expanded into a convergent multiple power 
series with respect to the ratios NI/NQ, . . . N ff /No* 

U _Ni , .N. 

- WO + TT 1 + + Tf U, 

1 G. N. Lewis, J. Am. Chem. Soc. 30, p. 668, 1908, 


We do not claim that this condition is always satisfied, and we 
shall treat in section 115 a notable exception to it. However, the large 
majority of solutions do conform to eq. (9.02), and the present chapter 
will be devoted to this case. The coefficients u h , u h i, ... of the expres- 
sion (9.02) are functions of p and T independent of the mol numbers. 
In the analogous expansion of V/No we denote the coefficients by 

Vh, Vhi, etc. 

We call the solution dilute if, within the desired accuracy, the terms 
of second and higher order of the expansion can be neglected so that 
U and V are expressed by the linear forms 

[7 = NOUO + NM + ... + N ff u ff , 
V = NOV O + Ni vi + . . . + N 9 v w . 


These equations contain, of course, the limiting case of the pure 
solvent (Ni = ... = N ff = 0), when they take the form U = NQUQ, 
V = NQVQ. The physical meaning of the coefficients UQ, VQ is, there- 
fore, the molal internal energy and the molal volume of the pure solvent. 
The mol number NO, representing the solvent, cannot be put equal to 
zero: therefore, the simplest form of the equations in which N h does not 
vanish is U = NQUQ + NhUh- It corresponds to a solution of only 
the component (ti) in the solvent. Consequently, we can say that the 
coefficients UH must depend on the properties both of the solute h and 
of the solvent. They may be called the molal energy and volume of 
the solute in the solution, but they will be, in general, different from 
the same quantities characterizing the solute in its free state. When 
another mol of the solvent is added to the dilute solution (9.03), its 
internal energy and volume become 

U' = (No + l)o + Niui + . . . + N u ff , 
V = (No + l)0o + Nivi + . . . + 

so that U' = U + wo, V = V + VQ. These are exactly the same values 
which the solution and the extra mol of the solvent have before they 
are brought together: there is no development of heat of solution and 
no change of volume when the solution is diluted still further. This 
additivity would not obtain if the second order terms of eq. (9.02) had 
to be taken into consideration: it can be taken as the experimental 
criterion of the diluteness of a solution. 

According to the definition of perfect solutions given at the end of 
the preceding section, the expressions (9.03) apply also to them. 
However, these solutions need not be dilute and the distinction 
between solvent and solutes disappears in them. Any of the compo- 


nents can be regarded as the solvent and, therefore, all the coefficients 
u h , Vh are identical with the molal internal energies and volumes of the 
pure components. This fact guarantees the additivity of the func- 
tions U and V which is the essential characteristic of a perfect solution. 
58. Entropy of dilute and perfect solutions. The differential of 
the entropy is defined by the second law of thermodynamics as 
dS = (dU + pdV}/T. The differentials dU and dV are taken for a 
definite composition of the system, so that the mol numbers N h must be 
regarded as constants with respect to the differentiation. Substitut- 
ing the expressions (9.03), we obtain 

- N h (du h + pdv h ) = N ds > ( 9 - 04 ) 


if we denote, for short, ds k = (du h + pdv h )/T. With respect to s h we 
must repeat what was said in the preceding section with respect to 
u h , v h - The quantities s h are functions of pressure and temperature 
and may be called the molal entropies of the components while in the 
solution. They coincide with the molal entropies in the pure state 
only for the solvent and for the components of perfect solutions, and 
are different from them in all other cases. 

The total entropy 5, obtained by integrating eq. (9.04), will con- 
tain a constant of integration C, which is constant only in so far as it is 
independent of the variables p, T. However, it may depend on the 
mol numbers Nh (which are not variables as far as the integration is 
concerned), and it can be interpreted as the entropy of mixing the 


N h s k (p, T) + C(N Q , Ni,... NJ. (9.05) 

This expression is entirely independent of the state of aggregation 
of the system and applies to solid and liquid solutions and to gas 
mixtures. If we had a complete knowledge of the equations of state 
and of the energies involved, including the continuity of the states, we 
could give the explicit analytical form of the functions s h (p, T) cover- 
ing all cases : to go from one state of aggregation to another, it would 
be sufficient to change the value of the arguments p and T, as we do it 
in an inaccurate way, limited to pure substances, in the Van der Waals 
theory. These considerations show us that the form of the function 
C(N , . . . NJ determined for one state of the system will be valid 
for all other states. We did find, however, the expression of the 
entropy of a mixture of perfect gases: If we admit that all substances 


of nature have a finite vapor pressure, be it in some cases ever so 
small, there is the theoretical possibility of vaporizing any solution 
into a mixture of gases. Moreover, by reducing the pressure, these 
gases can be made to obey, with any degree of approximation, the laws 
of perfect gases. It follows from this argument that the expression 
(8.07) of the entropy of a mixture of perfect gases is only a special case 
of the formula (9.05) and must have the same analytical structure. 
This is, in fact, borne out by comparing them : there are in (8.07) terms 
linear in the mol numbers Nh which correspond to the sum in eq. (9.05), 
and terms containing the factors log x h = log (N h /N). The totality 
of the latter terms must, obviously, be identified with the function C 


C(N , Ni, ... NJ = - RY] N h log x* (9.06) 

The possibility of this identification rests, entirely, on the hypoth- 
esis made in the preceding section that the functions U and V can be 
represented as the power series (9.02) and do not contain logarithmic 
terms. This assumption about the analytical form of U and V is, for 
the purposes of the present theory, equivalent to the conclusions which 
we shall deduce from it in the following sections, especially to Henry's 
law. Van't Hoff who originated the theory of dilute solutions based 
it on the validity of Henry's law. We prefer the above formulation 
(which is due to Planck) because, on the one hand, it is more general, 
being capable of extension by taking into consideration higher powers 
of the series (9.02). On the other hand, it paves the way to the under- 
standing of the reasons why in certain dilute solutions the laws deduced 
in this chapter are not satisfied (section 115). 

59. General conditions of equilibrium. Equations (9.03) for the 
internal energy and volume, together with the entropy formula (9.05), 
(9.06), are sufficient to obtain the expression of the thermodynamic 
potential $ = U - TS + pV, viz. 


* = 2 Nk(vk + RT log **> (9 - 07) 


the quantities <ph being defined as 

9h = Uh Ts h + PVK, (9.08) 

they are functions of p and T, independent of the mol numbers N h . 
Being built up from the functions A, s*, VA, they have the properties 
which we have pointed out as pertaining to these quantities. We may 
call them the molal thermodynamic potentials of the components in 


the solution. However, they are identical with the potentials of the 
pure substances only in the cases of the components in perfect solutions 
and of the solvent in dilute ones. In the case of solutes in dilute 
solutions they depend also on the properties of the solvent and are, 
therefore, different from the potentials of the same substances in their 
free state. 

The form (9.07) of the thermodynamic potential applies not only 
to dilute and perfect solutions but also to mixtures of perfect gases 
and to single phases of chemically pure substances. In fact, the com- 
parison with the expression (8.09) shows that, in the case of perfect 
gases, <ph must be defined as 

<p h = <p h - RT log x h = RT log p + u h (T). (9.09) 

It differs from ^ of (8.08) in that p is substituted instead of the 
partial pressure ph- On the other hand, a chemically pure substance, 
can be regarded as the limiting case of a solution without solutes 
(XQ = 1, log #o = 0). We can, therefore, apply our formulas to a 
heterogeneous system consisting of any number (a) of phases in equi- 
librium, each having one of these four characters: dilute solution, per- 
fect solution, mixture of gases, pure substance. If we designate the 
phase by an upper index j, the total thermodynamic potential of the 
system will be 

izr log **<*>]. (9.10) 

T^T ft^O 

The comparison with the general expression (6.14) shows that the 
partial thermodynamic potentials are generally $^ = <p^ } -{-RT log x^** 
We can apply to the last equation the general theory of equilibrium of 
section 42. The most general process possible in the system is given 
by eq. (6.48) 

The equation of equilibrium corresponding to it is, according to 

r- Ct 9 1 

A* - RT I ]T ]T v h (f) log xP - log K\ = 0, (9.12) 

L j-1 ft-0 J 


a ff 

/ j / ^ vt^ logXh^ = log K, (9.13) 

j-l /-0 


where the equilibrium constant K is an abbreviation defined by 

V, (9.14) 

which, in the particular case of a mixture of perfect gases, can be 
reduced to the form (8.19), as we shall see below. In application to 
dilute solutions the formula (9.13) is due to Van't Hoff. 

The left side of eq. (9.13) does not explicitly depend on temper- 
ature and pressure, while K is a function of p and T only, being inde- 
pendent of the mol numbers. An indication as to the form of the func- 
tional dependence of the equilibrium constant upon p and T is con- 
tained in eqs. (6.51) and (6.53) which give the partial derivatives of 
A$ with respect to T and p. Differentiating the expression (9.12) 
partially, we find 


RT 2> 

where AF and Q p are the change of volume and the heat of reaction 
in the process characterized by eq. (9.11). 

These equations are often called rules of displacement of equilibrium, 
inasmuch as a change in the equilibrium constant leads to a displace- 
ment in the mol fractions of the components, according to eq. (9.13). 
The first of them expresses the following fact: 

When a system is compressed at constant temperature, the equilibrium 
is displaced in the direction of a decrease of volume. I.e. the less volu- 
minous components are enhanced at the expense of the more volu- 

The other equation contains the rule: 

When a system is heated at constant pressure the equilibrium is 
displaced in the direction of heat absorption. Le. the components of 
higher heat function are enhanced at the expense of those of lower. 
This rule was first pointed out by Van t'Hoff and called by him the 
principle of mobile equilibrium. 

Exercise 86. From the definitions (9.08), (9.04), (9.03) of the quantities 
<?h, Sh, Uh, Vh check the statement that they satisfy the same differential relations as 
the corresponding functions in the free state of the substances, in particular that they 
obey eqs. (5.37). Having ascertained this, express A V and Q p in terms of t% and 5% 
from (9.15), (9.16), and (9.14). 


60. Nernst's distribution law. Let us consider the particular 
case when the process (9.11) consists in the transfer of j>& (2) mols of the 
component h from the phase (2) into the phase (1), where they may 
form a different number of mols v h (l \ owing to association or dissocia- 
tion. It will simplify matters and lessen the chances of misapprehen- 
sion if we define both v h (l) and ^ (2) as positive and recognize the fact 
that the phase (2) undergoes a loss by writing 


A (I) log* A < l) -^ (2) log**< 2 >=log^ = -[^W^-^W 2 ']/^. (9.17) 

Moreover, we shall often drop the superscript (2) to simplify writ- 
ing. With the abbreviation, 

^=^ (1) /^ (2) =M^ (2) /M. (1) , (9.18) 

denoting, according to (6.42), the ratio of molecular weights, this 

log [(xW/Xk] = (log^)A (2) = - [W n - * h (2) ]/RT. (9.19) 

In particular, when the process takes place without change of 
molecule, we have ^ (1) = Vh (2) = gh= 1 and 

/** (2) ) = log K h = - fo< - ^ (2) )/#r. (9.20) 

In many textbooks these equations are given only in the simpler 
form (9.20). Indeed, this can be justified on the following grounds. 
Even if the vapor of the component h has a different, say dissociated, 
molecule, the dissociation is theoretically never complete: according 
to Chapter VIII, there must exist a certain fraction of undissociated 
molecules, be it ever so small. Equation (9.20) gives then the equi- 
librium of the solution with the undissociated portion of the vapor 
and permits to determine its mol fraction, whereas the mol fraction 
of the main (dissociated) part of the component h can be obtained, in 
a secondary way, by using the theory of Chapter VIII. However, the 
more general eq. (9.19) is preferable from the practical point of view, 
as it gives directly the equilibrium with either of the two fractions. 
It permits one to describe by a single formula a much wider variety of 
observational results, a fact which more than compensates for its 
slightly more complicated form. 

Equations (9.19) and (9.20) are the fundamental relations in the 
theory of equilibrium of phases of dilute and perfect solutions. A num- 


ber of important laws can be derived from them. The second equa- 
tion can be written in the form 

*A a) /*A (2) = K h . (9.21) 

The ratio of the molal fractions of any component in two phases is a 
function of temperature and pressure only, provided the molecule of the 
component is the same in both phases. Suppose, for instance, that 
the phases (1) and (2) mainly consist of two solvents spread one on 
top of the other (as would be the case with water and carbon tetra- 
chloride). Let the same solute h be dissolved in both solvents: the 
quotient ^ (1) /^ <2) is then called the coefficient of distribution in the two 
phases. We see from eq. (9.21) that the coefficient of distribution is 
independent of the concentration of the solute in either phase, a law 
first pointed out by Nernst in 1891. 

The experimental data are, usually, expressed not in mol fractions 
but in mols of the solute per 1000 cm 3 of the solution, denoted by A^ 
Because of the dilution, Ah can be regarded as proportional to XH\ 
therefore, the ratio K' = Ah (1) /A h (2} must be also constant. As an 
illustration we may take iodine (12) dissolved in carbon tetrachloride 
(2) and in water (1) at 25 C. 1 


A ........ 0.02 0.04 0.06 0.08 0.09 0.10 

#'0.5 ____ 85.1 85.2 85.4 86.0 86.4 87.5 

In the general case, when the molecular weight of the solute is dif- 
ferent in the two solvents, eq. (9.19) gives 

(acny/scn. K I/V( *\ (9.22) 

For instance, benzoic acid has in benzene (1) a molecular weight 
twice as high as in water (j> (1) = 1, j/ (2) = 2, g = J). Another example 
is trichloroacetic acid (C2H ClaC^) dissolved in water (2) and in ethyl 
ether (1) at 25 C. The measured data are as in Table 20. 2 


A .............. 0.001 0.005 0.010 0.015 

A .............. 0.0021 0.0048 0.0068 0.0083 

15.0 14.7 14.7 14.7 

1 Washburn and Strachan, J. Am. Chem. Soc. 35, p. 681, 1913; Linhart, ibidem, 
40, p. 158, 1918. 

1 Smith, J. Phys. Chemistry 25, pp. 605, 616, 1921. 


It appears from this that g | or ji (1) = ZM^. The molecule is 
normal in the aqueous solution but has twice the normal weight in 
ether as a solvent. The results are only rarely as clean cut as this: in 
many solvents the association is partial, resulting in an effective 
molecular weight which is a non-integral multiple of the normal. 

Exercise 87. The following values were measured in the case of Brj dissolved 
in bromoform (1) and in water (2) at t = 25 C. 

4 (1) = 0.125 0.25 0.50 0.75 1.00 
100 X A = 0.193 0.382 0.750 1.11 1.47. 

Show that the distribution law is approximately satisfied. 

61. Influence of a neutral atmosphere on the vapor pressure. 

We label the condensed phase (2) and the atmosphere above it (1), 
and we assume that the gases of the atmosphere are but little dissolved 
in the condensed phase so that it can be considered as pure (x^ 1), 
with a sufficient approximation. The fundamental eq. (9.19), applied 
to the vaporization of the condensed substance, gives 

log *,> = fak>/ft - n]/RT. (9.23) 

The potential <p h (l> relating to the vapor has the form (9.09) and 
contains the term RT log p which can be brought to the left side and 
combined with x h (l \ Noticing that x h ( "p = ph (l) (the partial pressure 
of the vapor in the atmosphere), we have 

log p k - W/ft - k(T)]/RT t (9.24) 

where w A (r) is the remaining part of ^ (1) and a function of the 
temperature only. 

The terms on the right depend only on the properties of the com-, 
ponent h and not on the rest of the atmosphere. In the case when 
the condensed phase is in equilibrium with its pure vapor, the left side 
goes over into log p (l) (where p (l) is the total pressure) while the right 
side remains the same. We see, therefore, that the atmosphere does not 
influence the vapor pressure: the partial vapor pressure in the atmos- 
phere is identical with the total vapor pressure in its absence. 

62. Henry's law. In this section we shall apply the fundamental 
formulas (9.19) and (9.20) to the equilibrium of a liquid and a gaseous 
phase labeled, respectively, (2) and (1). The question we shall ask 
here is about the mol fraction x h of a dissolved component (gas) in 
the condensed phase if its partial pressure in the adjacent atmsophere 
is given. We may say that the condensed phase (2) is here preferred 
inasmuch as we are particularly interested in what occurs in it. It 


will be well to bring the preferred phase into prominence by dropping 
the superscript (2) relating to it: we shall write, therefore, XH instead 
of Xk (2 \ The process to be considered is the transfer of v h mols of the 
component h from the liquid solution into the gaseous phase where 
they may form v h (l) mols (because of dis- or association). 

By approximation we can neglect the change (in the process) of the 
volume of the condensed phase by comparison with that of the vapor. 
The quantity AF in eq. (9.15) denotes, then, simply the increase of 
volume of the gaseous phase due to VK (I) additional mols of the com- 
ponent h. According to eq. (1.18), this volume, before the beginning 
of the process, is V = (W (1) + . . . + N a ( ")RT/p\ after it takes place, 
the mol number Nh (l) is increased by VH (I} , the difference is, therefore, 

AF = VK^RT/p. (9.25) 

Of course, we suppose that N h (l) is very large compared with v h (l) 
so that there is no appreciable change in concentration during the 
process. (Or else the mol number changes are not the integers Vh 
themselves but infinitesimal numbers 5N h , proportional to v hl as in 
section 42). From the substitution into eq. (9.15) 

v- <> 


a relation which can be integrated. Since T is regarded as a constant 
in the partial derivative, the constant of integration with respect to p 
will be independent of p but, in general, not of the temperature T: 

log K = - Vh log p + v h log k k (T), (9.27) 

where the second term represents the constant (or, rather, function 
of T). 

Applying this to the special case of eq. (9.20), with v h (l) = 1, we find 

If we denote by p h (1) the partial pressure of the component h in the 
gaseous phase, we can write # A (1) = p h (l) /p, according to eq. (8.06), 

x k = ph (1) /k h . (9.29) 

When the mol fraction of a component in a liquid (dilute or perfect) 
solution is varied at constant temperature, the partial pressure of its 
vapor varies in the same proportion, and vice versa. In particular, we 
can obtain from this equation some information about the solubility 


of permanent gases in water and in other liquids: with neglect of quan- 
tities of the second order in N^/No, the amount of the gas absorbed by 
the solvent is proportional to #* (l) and, therefore, proportional to the 
partial pressure of the gas in the atmosphere over the solution. This 
law was established empirically by the English chemist William Henry 
(1803) and, accordingly, is known as Henry's law. 

The amount of the absorbed gas (denoted anph) is usually expressed 
in terms of the volume which it would occupy at C and 760 mm Hg. 
The quantity a h is then called Bunsen's coefficient of absorption (if p is 
given in mm Hg). The constancy of the coefficient a can be judged 
from the following data referring to the absorption of nitrogen (N2) 
in water at 25 C. 1 


/>Ar 2 (mm) 270 300 400 500 601.6 700 800 830 
14.38 14.37 14.31 14.26 14.20 14.28 14.36 14.38 

The trend of the dependence of kh upon T can be deduced from 
eq. (9.16). Substituting the expression (9.27) into it, we find 

d log k h Q 

The latent heat Q of driving out an absorbed gas from the solution 
is positive. Therefore, kh increases as the temperature is raised, so 
that solvents absorb gases and vapors better when they are cold than 
when they are heated. As an example we give the values of k for 
(argon-free) nitrogen dissolved in water. 2 


tC 5 10 15 20 25 30 35 40 45 50 

X10- 7 4.084.57 5.07 5.55 6.00 6.43 6.85 7.23 7.61 7.99 8.37 

In the general case, when the possibility of association (or dissocia- 
tion) is taken into account, eqs. (9.19) and (9.27) give 

*A = (kk/P)*; p h ( "/x*' OK = ** (9.31) 

A case in point is the solubility of hydrogen in molten copper 
(at / = 1123 C). Let M be the mass of hydrogen, in mg, dissolved 
in 100 g of copper. 3 

1 Drucker and Moles, Zs. phys. Chemie 75, p. 405, 1911. 

2 1. C. T. from measurements by Fox (Trans. Faraday Soc. 5, p. 68, 1909). 
A. Sieverts and W. Krumbhaar, Zs. phys. Chemie 74, p. 294, 1910. 



/>(mm).... 281 403 606 775 883 971 1046 

M ......... 0.380 0.443 0.549 0.610 0.680 0.705 0.745 

p*/M .... 43.5 44.2 43.,7 45.1 45.1 45.3 44.2 

Hence g = | or M = iM (1) the hydrogen dissolved in copper must 
be atomic. The constancy of p*/M has been ascertained for other 
gases dissolved in molten metals (62 in Ag, H2 in Ni and Fe, etc.). It is 
in keeping with this that the absorption increases with temperature. 
According to (9.30) this means that the process of driving the gas out 
of the metal is exothermic. 

In many organic solvents we find association of the gas molecules. 
However, it is usually incomplete, leading to irrational values of g*. 

Exercise 88. From the data about nitrogen in water and from eq. (9.30) calcu- 
late the approximate heat of solution Q. (Per 1 mol, v^ = 1). 

Exercise 89. Suppose that the same gaseous phase (1) is adjacent to two con- 
densed solvents. Equation (9.31) is then valid for either of the solvents. Show 
that the two relations so obtained are compatible with Nernst's law of distribution 
for the two condensed phases. 

63. Raoult's law. The symbol p in eq. (9.31) represents the joint 
pressure of the liquid solution and of its vapor. According to our dis- 
cussion in section 41, it is completely determined when the composition 
of the system and its temperature are given. It is called, therefore, the 
boiling pressure of the solution under these conditions. Equation 
(9.31) is true for every component: let us apply it to the solvent 
(h = 0) of a dilute solution. (As in the preceding section the con- 
densed phase is " preferred/ 1 we shall emphasize it by dropping its 
superscript 2 while reserving the superscript 1 for the vapor phase). 

rn- - . <*> - MO/*> (I) ). (9.32) 

This relation holds for every composition and, in particular, for the 
limiting case of the pure solvent characterized by N\ =* . . . = N 9 = 
and#o = #o (1) 1. The pressure corresponding to this limiting case is, 
then, the boiling pressure p$ of the pure solvent. We can write, there- 
fore, 1 = PQ/HQ or po = feo whence 

This relation takes a very simple form when the solutes are non- 
volatile (e.g. most salts) so that their partial pressures in the vapor 


are negligible. The gaseous phase can, then, be regarded as pure 

(*o (1) = 1) 

p/po = so 17 ' . (9.34) 

On the other hand, the mol fraction *o = No/N can be represented 
as 1 (Ni+ . . . + N ff )/N, or by approximation 

so _ 1 _(#! + ...+ NJ/No. (9.35) 

Whence, within the accuracy of the present theory which neglects 
terms of the second order in N h /No: 

PO gO No 

The presence of non-volatile solutes lowers the vapor pressure of a 
liquid. This fact was known to the chemists of the eighteenth cen- 
tury. In 1847 it was noticed by Babo that the relative lowering, 
(p Po)/Po> is independent of the temperature. A few years later 
(1858) Wiillner found it to be roughly proportional to the concentra- 
tion of the solution, and his observations were amplified by Ostwald. 
However, the most extensive investigation of this subject is due to 
Raoult (1886-1887), who established experimentally the law (9.36) 
according to which the relative lowering of pressure is determined 
solely by the mol fractions of the dissolved salts, being quite indepen- 
dent of the nature of the solvent and of the solutes. In earlier work 
(1878) Raoult also formulated the laws according to which the boiling 
and melting temperatures are changed by the presence of solutes. 
They form the subject of the next section. 

In the case of a single solute (1) eq. (9.36) can be written in either 
of the two alternative forms 


, m 

go mipo 10 

where m\ represents the number of mols of the solute in 1000 g of the 
solvent, called its molality. The second form follows from the first by 
substituting x\ = N\/No and noticing that in 1000 g of the solvent 
noNo = 1000 and N\ = mi . In the case of aqueous solutions MO (I)S=: 18: 
the quantity k m should, therefore, be numerically equal to 1.80. How- 
ever, in the tabulated data N\ is, of necessity, not the actual mol 
number of the solute (which is usually unknown) but "the gram- 
formula weight", i.e. the mol number as it would be if the solute were 
neither dissociated, associated, nor hydrated. The measured values 




of k m at low concentrations are for many substances close to 1.8, as 
appears from Table 24. 









HsPO 4 .. 


1 80 

Citric acid 


1 90 



1 775 

Pb(C2H 8 O 2 )2 


1 87 


1-0 3 

1 82 

H 3 AsO 4 


1 93 



1 78 

In general, electrolytic solutes have tabulated coefficients k m larger 
than 1.8 because they are often dissociated. Integral and half-integral 
multiples of 1.8 (K 2 S 2 O 3 3.65, Na 2 WO 4 3.64, Ba(NO 3 ) 2 3.55, etc,) are 
fairly common in them. On the other hand, organic solutes show 
mostly values of k m under 1.8, and this may be due to association. 

For non-aqueous solutions, the data are usually given in the first 
form (9.37). When go = 1 (the solvent does not associate), the 
formula requires k x = 1. This requirement is satisfied in a large 
number of cases, provided the concentration is sufficiently low, and 
even for values of #, from 0.05 to 0.10, it is often still approximately 






k x 





Ethyl alcohol 




Mercury * 




4 * 




4 1 




< 4 




*Hildebrand and Eastman, J. Am. Chem. Soc. 37, p. 2452, 1915. 

Of course, here too, examples of association (and dissociation) of 
the solute are not missing, as is indicated by the figure 0.55 relating to 
gold : this metal must have in mercury the molecule Au2. 

Exercise 90. In the last column of the following table are listed the measured 
values of the relative lowering of vapor pressure (po - p)/p* per 1 g of solute in 
1000 1 of solvent. 




M. W 




i < 



Br 2 






t W(pQ - p)/pQ 

20 C 0.978 


Ethyl alcohol Benzil 46 . 05 15 2.14 

Calculate the approximate molecular weights of the solutes from formula (9.37). 
The accuracy is not very high. 

64. Influence of solutes on the temperature and pressure oi 
transformations. Let the temperature and pressure of equilibrium 
of two phases of the pure solvent be TO, po. If small amounts of solutes 
are added to the two phases, the equilibrium temperature and pressure 
will be slightly changed, assuming the new values T, p but little dif- 
ferent from To, po- The function log K depends only on p and T, and 
we can expand it into the double Taylor series 

log K(p, T) = log K(po, To) + (p- po) 

neglecting terms of higher order. 

Equation (9.17) applied to the solvent (h = 0) gives 

i>o (1) log *o (1) - "o (2) log * (2) = log K(p, T). 

When the solvent is pure in both phases, the mol fr 'ty 
#o (1) = # (2) = 1 and the left side vanishes, so that log f *' 
For the partials of log K we can substitute from eqs. (9, ' 0^55 

AF 'If/ weights of 


1 7V/fc of eq. (9.43) 

We shall restrict ourselves to the foil from these data the 

vaporization of a liquid solution (2) conti - acid (MO 60.03). 

utes. Second, freezing of a liquid solutic 

not soluble in the solid solvent. In bot 1 b V the physiologist 

in our formulas, will be pure (*o (1 > = T on a semipermeable 

or freeze out while the solutes remain i , '' through the solvent 

phase is the preferred one, in the sense " and is denoted by P. 

we shall bring it into prominence b / dr ' er P a * of a cylindrical 

"*. For simplicity, we 


0(2) SBJ-Q we wr ite, as in the preceding section, l (N\+ . . . +N v )/No t 

log *o - - (Ni + . . . + NJ/No, log *o (1> - 0. (9.40) 

As we know from the theory of the phase rule (section 41), only 
one of the two variables T and p can be chosen arbitrarily when the 
composition of the liquid phase is given. We have, therefore, to dis- 
t nguish two cases in analyzing eq. (9.39). 

(1) We set arbitrarily T = TQ and find the difference of the vapor 
pressures of the solution and of the pure solvent, at the same tempera- 
ture To, 

RTo Nl + ' + N * 

The sign of the difference p po is opposite to that of the change 
of volume AF in the transformation. In the case of vaporization, we 
can use for A V the expression (9.25) and obtain with (9,18) the equation 

(9 . 42) 

(go = j>o (1> Ao) which is identical with (9.36). 

(2) We set p = po and obtain the difference of the boiling points 
f the solution and of the pure solvent, at the same pressure po. The 
(accc Miing of the heat of reaction Q p is here the latent heat of vaporizing 
Th, freezing VQ mols of the pure solvent; we shall denote it Q p = volo, 
aratinr % is the latent heat referred to 1 mol of the liquid phase. 
in the, 

rtdy* W1T.+ +* 

the solvef^ *o NO 

where A V 

1 mo i O f t j^ of vaporization the latent heat is positive, in that of 

p&V-dNo t>\* ^here exists, therefore, an elevation of the boiling 

the solution wii '*' iing &* ni ' both due to the solutes - As 

its contact with ' \ s much larger in absolute va lue, the 

since its boiling pr, *o.^:derably less than the melting point. 

its original position ^ e quantities N * in formula ( 9 - 43 ) re P r e- 

Po is - RTgndN 1 ' lumbers (taking into account association, 

that dN mols of the r nd nOt the ^ ram " formula weights. The 

then on, condenaatV computed for every solvent. Usually, 

remainder of the ' ^ ^^^ ^ n ^^ ^ ^ s l ven t (normal 

corresponds to 1 md^f numerical values for several solvents 

the change of volur 




T B 


T M 




0521 C 



Ethyl ether 

307 6 

2 18 

1 79 

Ethyl alcohol 

351 6 



353 35 


278 5 

- 5 10 


334 3 

3 82 

209 6 

4 80 




Acetic acid 

391 2 

3 11 

289 7 

3 90 

Aniline . . .... 

457 5 

3 61 

266 1 

5 87 



3 54 


7 4 



5 73 

278 8 


This effect offers a convenient and powerful method for the deter- 
mination of unknown molecular weights of solutes which was of great 
help in the development of our chemical knowledge. The theoretical 
treatment of these phenomena was due to Van t'Hoff. 

Exercise 91. In the last column of the following table are given the measured 
boiling-point elevations corresponding to a concentration c of the solute in g pf 
1000 g of solvent given in the third column 

Ethyl alcohol 
Acetic acid 







Picric acid 



Calculate from Table 26 the opproximate mr* 
the solutes. 

Exercise 92. What is the relation between t' 
and the data of Table 26 which refer to molaliti 
latent heats for water (MO =18), benzene (MO = 

65. Osmotic pressure. It was dis 

Pfeffer (1877) that solutes exercise 2 
membrane which is impenetrable to t! 
freely. This pressure is called osmoti 
Suppose that the liquid solution fills 
vessel (Fig. 21) closed above by the 


shall assume (at first) that the solutes are non-volatile so that the 
vapor, in the upper part of the vessel, is free from them; later we shall 
drop this restriction. Suppose that we have in the vessel a second 
piston consisting of a semipermeable membrane of the sort just men- 
tioned whose initial position coincides with the plane border surface 
MN between the liquid and the gaseous phases. We push this piston 
a little down, in a reversible isothermal way, to 
the new position M'N', the volume between the 
planes M'N' and MN being dV. As a result, 
there will remain only pure solvent above the 
membrane M'N': It is a process for recovering 
a certain amount of solvent, say dNo mols, from 
the solution. Since the volume V of the solution 
is a function of TVo, N^ . . . N ff , while only NQ 

changes, we have 
FlG. 21. -Calculation b 

of osmotic pressure. dV = (dV/dNo)dNo = VodN , 

where #o is the partial molal volume of the solvent (compare section 32). 
In pushing the membrane down, work must be done against the forces 
of osmotic pressure P to the extent 

DW = -PdV=- PVodNo, (9.44) 

>rding to section 7). 

re is, however, another reversible and isothermal way of sep- 

^'^Vo mols of the solvent from the solution. This can be done 

* wing two steps: (1) By moving the upper piston up, infi- 

and at constant temperature, we vaporize dNo mols of 

^he change of volume of the system is, then, AV-dNo, 

s the change corresponding to the vaporization of 

. In this expansion the system does the work 

r pn-ssure of the solution. (2) We cover 

ly impermeable membrane. Having lost 

the vapor is superheated at the pressure p 

'> p). As we push the piston down into 

done till the pressure assumes the value 

iccording to eq. (3.32), and considering 

^spond to godNv mols of the gas. From 

the pressure stays constant, and the 

ck is po&VodNo, where AFo again 

vent. If it is permissible to neglect 

id, compared with that of the vapor, 


and to use for the gaseous phase the laws of perfect gases, AF=oAFo: 
the total work done in both steps of the process becomes 

DW = - RTg Q dNo log (po/p). (9 AS) 

In the beginning and the end of this process the system is precisely 
in the same state as in the other method of separation carried out with 
the semipermeable membrane. Moreover, both processes are isother- 
mal, therefore, DW' = DW, since both elements of work are equal to 
the change of the function ^ between the initial and the final states 
(compare section 34). Equating the expressions (9.44) and (9.45), we 

g*. (9.46) 

We have assumed, for simplicity, that the solutes are non-volatile. 
However, this restriction is not essential and can easily be dropped: 
we need only imagine from the start a semipermeable membrane 
(opaque to the solutes) at the surface of the solution. This device 
permits us to have the pure vapor of the solvent in equilibrium with 
a solution containing volatile solutes, since the vapors of the solutes 
cannot penetrate the membrane. We can apply the reasoning leading 
to eq. (9.46) to the general case. This equation remains valid with 
the slight change that for p must be substituted the partial pressure 
of the solvent in the gaseous phase pxv (l) : 


The osmotic pressure depends on the ratio of the partial vapor pres- 
sures of the solvent, in its pure state and in the solution, being propor- 
tional to the logarithm of that ratio. 

Equation (9.44) permits another interpretation of the osmotic 
pressure: It has the form (2.02) of the element of work in the case of 
generalized forces. The differential dNo represents the change in mol 
number of the solution due to the removal of solvent. Pvo is, there- 
fore, the force with which the solution resists (because of the negative 
sign) such a removal. In other words, it can also be regarded as a 
tension or suction pulling the solvent back into the solution across the 
semipermeable membrane. Pfeffer's method of measuring the osmotic 
pressure is based on this interpretation (Fig. 22) : A glass tube, \xJiO8e 
lower end is closed by a semipermeable membrane MN (made of cop- 






H 2 

FIG. 22. Measure- 
ment of osmotic 

per ferrocyanide), is partially immersed in a beaker filled with pure 
water. When cane sugar is added to the water inside the tube, the 
solution in it rises because it sucks in water from the beaker. The rise 
stops when the pressure of the column AB is exactly equal to the force 
of suction, i.e. to the osmotic pressure P. This pressure is by no 
means small: In the case of more concentrated 
solutions, it is better to connect the tube with a 
manometer because the column becomes incon- 
veniently high. In fact, pressures up to 268 atm 
have been measured in sugar solutions. The fol- 
lowing familiar demonstration experiment shows 
how powerful the osmotic pressure can be. A part 
of an ox bladder is formed into a bag, filled with 
a strong aqueous solution of ethyl alcohol, and 
tightly sealed. The bladder is semipermeable in 
that it holds back alcohol but lets through water. 
When the bag is immersed in a water-filled vessel, 
the water is drawn into it with such force that the 
bladder cannot stand the strain and bursts within 
a short time. 

Let us turn to the general case when there are solutes on both sides 
of the membrane permeable only for the solvent. Both solutions 
exercise, then, a sucking action upon the solvent, and the resultant is 
a tension equal to the difference of the osmotic pressures, directed 
towards the side of higher concentration. Two solutions of the same 
osmotic pressure leave the solvent in equilibrium and are called isos- 
motic or isotonic. On these facts is based a method for determining 
the osmotic pressure within living cells, whose integuments are natural 
semipermeable membranes. When a living cell is placed in an aqueous 
salt solution of higher osmotic pressure than its own, it begins to lose 
water and to shrink. When the osmotic pressure of the medium is 
lower, it swells. Only in solutions approximately isosmotic with its 
interior does it retain, under the microscope, its size and general appear- 
ance. By this method of finding a solution isosmotic with them, the 
osmotic pressures in many physiological cells have been determined. 
In plant cells they are sometimes as high as 14 atm. In general, the 
vegetable and animal cells are nearly isosmotic with the natural saps 
and animal fluids in which they live. For instance the red corpuscles 
of the human blood and most other cells of the human body have the 
same osmotic pressure as the normal blood serum, which in turn is 
isosmotic with a solution consisting of 0.156 mol of NaCl per 1000 g of 
water. The harmful effect of certain diseases is believed to be due to 


the fact that they upset the osmotic equilibrium of the body fluids and 
cause a destruction of cells similar to the breaking of the ox bladder 
in the above-mentioned experiment. 

66. Van t'Hoff's equation for osmotic pressure. The expression 
for osmotic pressure (9.47) takes a simple form in the case of dilute 
and perfect solutions. On one hand, the partial molal volume then 
becomes identical with the ordinary molal volume of the solvent: 
t/o = VQ. On the other, we may substitute for o/p#o (1) from (9.33) 
and obtain 

p=-^log*o, (9.48) 


an expression which is valid both for volatile and non-volatile solutes. 
If the solution is dilute, log XQ takes the form (9.40) and the volume 
NQ VQ of .the solvent can be replaced by the volume V of the solution 
(V = NQVQ), whence 

PV (Ni + . . . + N ff )RT. (9.49) 

This is Van t'Hoff's equation for osmotic pressure, which is formally 
identical with eq. (1.18) for a mixture of perfect gases. By specializa- 
tion, we can obtain from it the partial osmotic pressures of the indi- 
vidual solutes: if only the component h is present in the solution, 

P h V - NiJRT, or P h x h RT/v Q . (9.50) 

It follows from (9.49) that the partial osmotic pressures are addi- 
tive P = PI + . . . + P . In other words, the solutes in a dilute 
solution behave like perfect gases. 

The experimental data are usually expressed in terms of the 
molality f i.e. mol number of the solute per 1000 g of the solvent, which 
is denoted by m h . For aqueous solutions, this reduces to NQ VQ 
1000 cm 3 and, together with eq. (9.50), gives P h - RTm h /lQQQ, or 
numerically for C, 

P k - 22.4m*atm. (9.51) 

A solution of the molality m = 1 is called normal. We see that 
the normal solution has a theoretical osmotic pressure of 22.4 atm. 
If the molality is computed from the gram-formula weight, eq. (9.51) 
can give a correct result only barring association, dissociation, and 
hydration. A few of the experimental values for dilute aqueous solu- 
tions (directly measured with copper ferrocyanide semipermeable mem- 
branes) are given in Table 27. The values of mi listed in this table 
are not corrected for association, etc. 







P (atm) 













20 5 









Mannitol . ... 


13 1 


When the solution is not dilute we have to fall back on the general 
eq. (9.48). For aqueous solutions, vo = 18.015 and the coefficient 
has the numerical value (at C) 

P = - 2865 logio * atm. (9.52) 

This formula can be tested for sucrose. 1 It is known that sucrose 
forms a hydrate, each molecule attaching five molecules of water. The 
molality of pure water is 1000/18.015 = 55.51; consequently, in the 
sucrose solution it is wo = 55.51 5wi and l/#o = l+wi/(55.51 5wi). 
The values thus obtained are listed in Table 28. 


P (calc.) . . . 
P (obs.) . . . 













The agreement is surprisingly good and shows that the sucrose solu- 
tion can be regarded as perfect, as far as this calculation is concerned. 
\ In a perfect solution the difference between solvent and solute dis- 
appears. It is better, therefore, to avoid the word " solvent " and 
to speak instead of the " component " without making any restric- 
tions as to its mol number. For a binary solution the interpretation 
of the preceding section can, then, be stated as follows: the partial 
osmotic pressure PI represents the force with which the other component 
is held in the solution (and vice versa). According to (9.48) it is 
proportional to the logarithm of the mol number of that component. 
If we gradually remove it from the solution, by the mechanism of 

1 Lord Berkeley and Hartley, Proc. Roy. Soc. (A) 92, p. 477, 1916; H. L. Cal- 
lendar, Proc. Roy. Soc. (A) 80, p. 466, 1908. 


Fig. 21 or by some other method, the difficulty of the removal con- 
tinually increases and becomes prohibitive in the end, because the last 
traces are held with an infinite force. 

This impossibility of removing the last vestiges of any component 
from a phase is quite general, as can be seen, in the simplest way, 
directly from the fundamental equation (9.17) which applies to this 
case as the solution is then certainly dilute with respect to this com- 
ponent. Log K is, by its definition (9.14), a finite number: therefore, 
Xh (l) cannot become zero. Otherwise the left side of eq. (9.17) would 
become infinite, unless J0 (2) were zero at the same time. This means 
that a component cannot be completely absent from one phase of a 
system if it has a finite concentration in another phase. All parts of 
our earth have been in direct or indirect interaction for immense 
periods of time. Consequently, every sample of terrestrial material 
must contain minute quantities of all known substances, and an abso- 
lutely complete purification of it is impossible. However, the second 
interpretation of the osmotic pressure here used should not obscure 
the fact that it is not an abstraction, but, in every way, a real pressure 
like that of a gas. After all, the only tangible thing about any pres- 
sure is that it is able to resist the motion of a piston or to set it in 
motion, and there is no difference between the osmotic and the ordi- 
nary pressure in this respect. We shall gain some insight into the 
deeper nature of the osmotic pressure in section 146 but shall bring out 
in this chapter the analogies between dilute solutions and perfect gases. 
We have already seen that the osmotic pressure has a tendency towards 
uniformity: if the concentration of a solution is non-uniform, the local 
differences of osmotic pressure set up diffusion currents which make 
it uniform (barring gravitational fields, compare section 108). In 
this process some mass of the solute (K) may be transferred from a 
place with the osmotic pressure Ph to one with P' h (in an isothermal 
way). The work done in this transfer must have in a dilute solution 
the same expression (3.32) as in a perfect gas, in view of the identity 
of the equations of state. Thus we obtain (per mol of the solute) 

w = RT log (Pk/P'd = RT log (x k /x' k ). (9.53) 

Exercise 93. What are the osmotic pressures of the following solutions, sup- 
posing that they satisfy the formula (9.51): (a) antipyrine (CiiHi 2 N 2 O, n 188.11) 
10 g in 1000 g of water; (b) saccharin (CyHfiNOaS, M = 183.1 1), 2 g in 1000 g of water; 
(c) glucose (CHi 2 O, /* = 180.09) 100 g in 1000 g of water? 

Exercise 94. What are the osmotic pressures of sucrose solutions (CuHtsOu, 
M = 342.17), according to the formula (9.52) and taking into account hydration aa 
in Table 28: (a) 250 g, (b) 500 g, (c) 1000 g in 1000 g of water? 




Exercise 95. Calculate numerically from eq. (9.53) the work (in calories) that 
must be expended to reduce the mol fraction of a solvent in the ratio 1:10, at 20 C. 

67. Saturated solutions. Solubility. In general, a solvent can 
take up only a limited amount of a solute at a given temperature. For 
instance, when we gradually add a salt or other solid solute to water, 
it dissolves at first, increasing the strength of the solution. However, 
when a certain limiting concentration is reached, further additions do 
not go into the solution but form a solid precipitate at the bottom of 
the vessel. Such a solution is called saturated, and the solute in it is 
in equilibrium with its solid crystalline phase. The behavior of liquid 
solutes is analogous, but we shall restrict ourselves in this section to 
solids, postponing the discussion of liquids until the next chapter. 

The mol fraction x, of the solute in the saturated state is sometimes 
called its solubility. More often, however, the solubility is defined as 
the molality m, of the solute in the saturated solution. If we denote by 
mo the molality of the solvent (mo = 1000/Mo), 

, = w,/(mo + m,). 


(For water mo = 55.51). The solubilities of some of the more common 
solutes in water, at C, are as follows: 













CuSO 4 . 






FeSO 4 

1 030 

NaOH ... 

10 50 

Ca(OH) 3 . 



5 36 



CaSO 4 


Lactose . . . 



17.3 * 

AgNO 8 

6 65 

Maltose . . 

1 70 

It should be mentioned that the table is meant to illustrate merely 
the general notion of saturation. The substances listed in it are for 
the most part " strong electrolytes ", they do not obey quantitatively 
the theory of dilute solutions of this section but need a correction which 
will be given in section 115, Moreover, the numbers m, listed in it 
present the " gram-formula weights " of the undissociated and unhy- 
drated solutes. 


Returning to the theory, we shall assume that the solvent is not 
soluble in the solute (as is the case with most solids) so that the excess 
of the solute represents a pure phase (xh (l) = 1). In applying to this 
case of equilibrium the fundamental eq. (9.17), we label the excess 
solute (1) and drop the superscript for the liquid solution 

-^log*M = logK. (9.5S) 

Both phases are here condensed, so that the change of volume in 
the process of solution is small and may be neglected in the first 
approximation (AF = 0). According to eq. (9.15), this means that K 
is independent of the pressure and is a function of the temperature 
alone: K = K(T). At the same time the heat, Q p = Qv, developed in 
the process is entirely due to the difference of the internal energies in 
the two states Q p = A 7 because, within this approximation, the work 
done by the system is negligible. If we denote by Aw* the difference 
of internal energies per 1 mol of solute, Q p = ^Aw*, and we find, from 
eq. (9.16) 

d log X* Aw* 

(9 ' 56) 

or substituting for Xk the expression (9.50), we obtain 

d log P h _ Aw* + RT _ l h 
dT ~ RT* ~~RT* 


This relation is formally identical with the approximation (7.07) of 
the Clapeyron-Clausius equation. The process of solution of a salt is 
completely analogous to the process of vaporization, and the osmotic 
pressure PK of the saturated solute corresponds to the boiling pressure 
ps of the vapor. In fact, the quantity k in eq. (9.57) has the same 
structure as the heat of vaporization: According to (3.21) and (3.24), 
/ = Aw + p&v, or, when the volume of the condensed phase is neg- 
lected and the vapor regarded as a perfect gas, / = Aw + RT. 

The heat of solution Aw* is almost invariably positive. Therefore, 
we conclude from eq. (9.56) that the solubility of a salt increases with 
temperature, unlike the solubility of a gas (compare section 62). Our 
assumption that the volume does not change is only a first approxima- 
tion, in reality AF has, usually, a very small but finite value. Conse- 
quently the effect of added pressure consists in a very slight increase or 
decrease of solubility. 


Exercise 96. The solubility of many aqueous solutes is well represented by the 

1 0.052234 
logio- +B. 


Range.. 7.Oto8.4C -27 to 10 -1.3 to 18 40 to 100 

A 12090 14980 17150 2142 

B -2.243 -2.70 -3.078 -2.102 

Calculate from (9.56) the approximate heat of solution AA within the range, assuming 
that there is no dissociation or hydration. 

68. Chemical equilibrium of solutes. We suppose that chemical 
reactions are possible between the components of a dilute solution, 
according to eq. (6.46) 

*>oGo + vid + . . . + v 9 G 9 = 0. (9.58) 

The general equation of equilibrium (9.12) takes then the form 

VQ log #o + v\ log x\ + . . . + v 9 log x ff = log K. (9.59) 

Since we restrict ourselves to dilute solutions, the volume is prac- 
tically determined by the amount of the solvent, and its change in the 
reaction (9.58) is very small. As in the preceding section, we conclude 
from this that K is very little dependent on the volume and with a 
good approximation can be regarded as a function of the temperature 
alone, K = K(T). Moreover, we can replace x h = N h /N by N h /No, 
because the difference would amount only to terms of the second order 
in the expression (9.59) which are systematically neglected in the 
theory of dilute solutions. In application to the solvent, this means 
that we can replace XQ by 1, obtaining from the last equation 

si-W... x." = K(T), (9.60) 

without regard to whether the solvent takes part in the reaction or 
not. Because of the relations (9.50) this is equivalent to 

P^Pa* 1 . . . P/ r = K P (T) = K[), (9.61) 

\ VQ / 

(v = v\ + V2 + . . + O- 

This equation is formally identical with eq. (8.17) for a mixture of 
perfect gases. However, the conditions under which the equilibrium 
takes place are different in the two cases: The mixture of perfect gases 
is usually observed at a given pressure p which is an important factor 


in determining the partial pressures ph, when only the mol numbers 
are known. On the other hand, the partial osmotic pressures P* of the 
solutes are completely determined by N h /No = WA/WO and have noth- 
ing to do with the pressure. This is brought out clearly, if we write 
the condition in the third form, resulting from (9.60) 

mi'W 1 . . . w/' K m = K(r).m v (9.62) 

(We recall that WQ is the molality of the solvent: wo = 1000 /Vo) 
The relation between the molalities depends only on the temperature 
and on the atomic weight of the solvent. 

As in the case of gases, the most interesting question is the deter- 
mination of the degree of dissociation. Let us take as an example the 
same reaction (8.22) which we discussed in section 52, the dissociation 
of nitrogen tetroxide (dissolved in chloroform). The expressions for 
the numbers VH and Nh remain the same as those given there. Sub- 
stituting them into (9.60), we obtain for the degrees of dissociation 

2 K**. (9.63) 

where m\ is the molality of N2CU originally dissolved. 

Measurements on the dissociation of N2O4 in chloroform were car- 
ried out by Cundall. 1 Lewis and Randall 2 give the following table 
calculated from his data (for C). 


#N204 #N02 10 & 

0.018 0.00010 49 

0.037 0.00012 35 

0.050 0.00015 43 

0.066 0.00019 52 

0.125 0.00029 66 

The constancy of K is not very good owing, perhaps, to the insuffi- 
cient accuracy of the colorimetric determinations of # N02 . Of course, 
the degree of dissociation is quite different from that which gaseous 
nitrogen tetroxide has under the same conditions: KP comes out here 
about 2.00 X 10" 4 atm, while (8.25) gives 0.020 atm for the gas 
reaction at C. This is not surprising since the heats of reaction are 

1 Cundall, J. Chem. Soc. 50, 1076, 1891; 67, p. 794, 1895. 

1 Lewis and Randall, Thermodynamics, p. 303, New York, 1923. 


very different in the two cases: in the chloroform solution there must 
be added the difference of the heats of solution of the two components. 

For very high dilutions (mi/wo 1), approaches unity. This 
is a general property of dilute solutions, as we can readily see from 
eq. (9.64): Whenever the dissociation leads to an increase in the 
number of molecules (v > 0), dilution (i. e. increasing NQ) has the same 
effect as increasing K (compare section 52). It advances the disso- 
ciation which becomes complete in the limiting case of extreme dilution. 

This influence of the relative amount of the solvent upon the degree 
of dissociation is the analogue of the effect of neutral gases upon a gas 
mixture discussed in section 54. The addition of neutral solutes has, 
according to eqs. (9.61) and (9.62), no appreciable effect upon the 
equilibrium. The reason for this is that in a dilute solution the quan- 
tity of a solute is, necessarily, small and its contribution to the avail- 
able volume negligible. On the other hand, the remarks made in the 
sections 53 and 55 about the depressing influence of the excess of one 
active component and about the mutual interdependence of two simul- 
taneous reactions remain fully valid in the case of solutions. 

Exercise 97. Calculate the factors RT/v Q and w entering into KP and K m 
for the following solvents: water (MO 18, density d = 1.00), chloroform 
(MO - 119.38, d - 1.526), ethyl ether (/*o - 74.08, d - 0.736), benzene (MO = 78.05 
d - 0.900). 

Exercise 98. Carbonic acid and acetic acid in aqueous solutions dissociate into 
ions according to the following equations: 

H+ + HCOa" - H 2 CO, (aq) =0, K m - 1.84 X lO" 6 , 

H+ + C 2 H 8 Or - C 2 H 4 O 2 (aq) - 0, ** 3.50 X 10' 7 . 

Calculate the degrees of dissociation for the molalities mi (of the acid originally 
dissolved) 0.001/0.01, 0.05, 0.10, 0.20. 

Exercise 98'. Show that the equation of equilibrium (9.62) can be also written 
in the form 

' (9.64) 


69. General remarks. The character of the equilibrium between 
different phases of a system with two independent components 
(binary systems) is of great importance in chemistry and metallurgy. 
A vast body of experimental data, pertaining to it, has been accumu- 
lated in these branches of science. The role of thermodynamics is to 
help in the explanation and classification of this material by enumerat- 
ing the theoretically possible types of equilibrium. Of necessity, much 
of the information derived from thermodynamics is qualitative: The 
problem consists in treating the equilibrium between solid, liquid, and 
gaseous binary solutions (in the sense of section 56), but we do not* 
possess the theoretical knowledge of the condensed phases which 
would be necessary for a quantitative treatment, viz. the equations of 
state and the energy expressions for the solid and liquid state. 
Fortunately there exist, if not real, at least ideal systems in which the 
theory can be completely carried through, namely, the simplified 
models which we called perfect solutions in section 56. 

The essence of this approximation is that the interactions of the 
components are taken into account only in so far as they affect the 
mutual solubility. Apart from this, each component is treated as if 
it were alone. We propose, therefore, to work out, for every type of 
equilibrium, the formulas and curves pertaining to the case of perfect 
solutions. Comparing them with the experimental curves, we shall 
see that this model is not oversimplified and gives a good account of 
the main features of real systems, giving examples of every type of 
diagram that occurs in experiments. 

According to the discussion of section 41, the maximum number 
of phases which can be simultaneously in equilibrium exceeds by two 
the number of independent components: for binary systems it is 4. 
Four phases can coexist only at the " quadruple point " for which all 
the variables of the system (viz. temperature, pressure, and composi- 
tion of the phases) are perfectly determinate. In the case of the 

1 This chapter may be skipped without loss of continuity. 


coexistence of three phases the system has one degree of freedom; i.e. 
one of the variables can be chosen arbitrarily. For experimental rea- 
sons, this is, usually, the pressure : when p is given, the whole system 
is determined, so that equilibrium can take place only at one definite 
temperature and for definite compositions of all the three phases. 
The most interesting case is, however, that of the coexistence of two 
phases, because it was shown in the general theory of Chapter VI that 
the other cases can be reduced to it. Then, two of the variables can 
be chosen at random: for instance, we can prescribe the temperature 
and the composition of one phase, so that equilibrium is possible at 
only one definite pressure. Or else, we can select arbitrarily the pres- 
sure and the composition of one of the phases, and so completely 
determine the equilibrium temperature. 

The parameter which determines the composition of a phase is the 
mol fraction of either component. If we indicate the components by 
the subscripts 1 and 2, and the phase by the superscript (/), we find 
from the definition (8.05) of the mol fraction 

Xl <* + * a < = 1. (10.01) 

In most of the applications of this chapter we shall suppose that 
the components do not interact chemically. In this case, there is no 
difference between the independent components and the actual 

A few remarks about ternary systems will be made in section 75. 

70. Vapor pressure of binary systems at constant temperature. 
The available experimental data have reference to the equilibrium of 
liquid binary solutions and their vapors, because the vapor pressures 
of solid solutions are too low to be conveniently measurable. There- 
fore, we shall now restrict ourselves to the liquid-gaseous equilibrium 
and we shall further assume that, in the liquid phase, the components 
are mutually soluble in all proportions. We shall indicate the mol 
numbers of the two components, in the liquid phase, by x\, #2 and, in 
the vapor, by x'i, #'2, and shall apply the relations derived in the pre- 
ceding chapter for perfect solutions. 

In a perfect solution either component can be regarded as the 
solvent and obeys Raoult's law. If the molecules remain unchanged 
in the process of vaporization, this law must be used in the form 
(9.33), with go = 1, 

px'i = PBI xi, px f 2 = pB2 x 2t (10.02) 

where PBI and pB2 denote the boiling pressures of the two components 




in their pure state. Adding these two equations and taking into 
account the relations (10.01) we find 

P = 



The plot of the pressure against the mol number x\ in the liquid 
phase is a straight line (Fig. 23). 

We can express the pressure also in terms of #'i, the mol number in 
the gaseous phase. Dividing the eqs. (10.02) by psi, pB2 respectively, 
and then adding them, 


P '- 

pBl(l - 


This corresponds to the curve of hyperbolic character labeled 
11 vapor " in Fig. 23. This figure represents the so-called px-diagram: 




FIG. 23. 

FIG. 24. 

FIG. 25. 

Boiling pressure as a function of composition. 

when the composition and pressure of the liquid phase are given by 
the point A (on the " liquidus " curve), the constitution of the vapor 
in equilibrium with it is obtained by drawing a horizontal to the point 
B of intersection with the "vapor" curve. 

When we turn to the experimental ^-diagrams of pairs of organic 
and inorganic liquids, such a large proportion of them is found to be 
of the general type of Fig. 23 that it is justifiable to call it the normal 
type. It is true, the representation is not quantitative in that the 
" liquidus " locus geometricus is, generally, not straight but also 
curved. But it is typical in that both curves are monotonic and the 
vapor pressure of the solution intermediate to the boiling pressures of 
the pure components. Especially, pairs of chemically closely related 
substances (benzene-toluene, hexane-octane, methyl and ethyl alcohol, 
etc.) belong to this type. We have mentioned in section 56 that such 
pairs must be expected to approximate the behavior of perfect solu- 


However, there exist two other types of (p, jc)-diagrams which are 
illustrated by the Figs. 24 and 25. We shall call them the mrtype 
and the mm-type. It is often somewhat uncertain what the molecular 
weight in the liquid phase is. Therefore, the pressure is usually plotted 
not against the mol fraction but against the mass fraction 

yi - Mi/(Mi + M 2 ) = *i/(*i + /*2*2/Mi), (10.05) 

Mi, M2 being the masses of the two components and /*i, /*2 their 
molecular weights. This way of plotting has also some theoretical 
advantages as we shall see in section 74. 

Our simplified model (the perfect solutions) permits us to predict 
a behavior of this kind in certain cases, viz. when the molecular 
weights of the components are different in the liquid and gaseous states. 
The generalization (9.33) of Raoult's equation must, then, be. used, 
and this leads, instead of (10.03), to the equation 

P = PBI xi l/ai + p B *(l - *i) 1/<7a , (10.06) 

where gi = MI/M'I and #2 = ^2/^2- It is to be expected that the case, 
when both components are subject to association or dissociation of 
molecules, is a rare one. It will be sufficient, therefore, to assume that 
only the exponent g\ is different from 1, while g% = 1. It is easy to 
see that, under this assumption, the slope dp/dxi will, in general, go 
to zero for a value of x\ between and 1, provided the boiling pressures 
PBI and pB2 are of the same order of magnitude. This means that the 
pressure reaches in this interval an extreme; whether it is a maximum 
or a minimum depends upon the sign of the second derivative 
d?p/dxi 2 (1 gi)gi~ 2 si#i (1 ~ 2ai)/w . In the case of association in 
the liquid phase gi > 1, and the sign is negative; in the case of 
dissociation, g\ < 1, and it is positive. We should, therefore, expect 
the max-type when the molecules of one of the liquid components are 
associated, and the />min-type when they are dissociated. 

On the whole, these expectations are borne out by the facts: It is 
known that hydroxylated liquids have a tendency towards association 
when mixed with another liquid which is free of the hydroxyl group. 
Upon looking through the list of systems showing the m*x-type, we 
find that it consists largely (although not exclusively) of such pairs of 
liquids. On the other hand, aqueous solutions of HI, HBr, HC1, HF 
which are considerably dissociated belong to the pmm-type. However, 
there is another cause which may produce these kinds of par-diagrams: 
the existence of chemical compounds of the two components. If the va- 
por pressure of the compound is higher than of either pure component, 


the px-curve must have a maximum, if it is lower, a minimum. The 
reason why the two curves in the Figs. 24 and 25 have a, joint maximum 
or minimum will become clear in section 74. 

This theory has an important practical application in the process of 
distillation. In the normal pressure curve (Fig. 26) there corresponds 
to the composition A of the liquid the composition B of the vapor; 
that is, the vapor is richer in the component (2) with the higher boiling 
pressure. Prolonged boiling at constant temperature, therefore, causes 
the liquid to become poorer in this component so that its representative 
point moves downward towards psi- If the purpose of distillation is 
to drive out the more volatile component, one has only to continue 
boiling and, eventually, the liquid will consist of the practically pure 
component (1). On the other hand, the pro- 
cedure is different when it is desired to prepare 
the pure component (2). Starting again from the 
state A, we let the solution boil for a time and 
collect its vapor which has the constitution B. 
Condensing this vapor we obtain a liquid of the 
same constitution represented in our diagram 
by the point C. Boiling this fraction (at the same 

temperature) gives a vapor of the composition Z>, w > 

etc. By such steps it is possible to approach FIG. 26. Distillation. 

closely the point pB2 representing the pure volatile 

component. It is clear that pressure curves of the min-type (Fig. 

25) preclude the success of the first kind of distillation, and curves 

of the max-type (Fig. 24) that of the second. The best-known 

example is the water-ethyl alcohol solution having a maximum at 

4.43% water by weight so that the purification stops with 95.57% 

alcohol (about 194 proof). Distillers have learned, in recent years, 

to get around this maximum by adding a third component to the 


71. Temperature of transformation at constant pressure. When 
the pressure is kept constant, the temperature of equilibrium of two 
binary phases is completely determined by the composition of one 
of them. To treat this case theoretically for perfect solutions, we have 
to fall back again upon the fundamental eqs. (9.17) or (9.22), but to 
evaluate the equilibrium constant K in a way somewhat different from 
that of the preceding section. We can apply to the first component 
the eq. (9.16) in the form 

, 10(m 
(10 ' 07) 


where l\ is the latent heat of transformation referred to 1 mol of the 
first component in the primed phase (v'i = 1, vi either integral or frac- 

We have pointed out in section 45 that the latent heat is a rather 
slowly varying function of T which can be regarded as constant in the 
first approximation. This approximation is amply sufficient for the 
purpose of the present investigation because we have already intro- 
duced an element of inaccuracy by supposing the phases to be perfect 
solutions. Integrating, we find therefore 

log Xi = - h/RT + Ci(/>), (10.08) 

and a similar relation with the subscripts 2, for the second 
component. In the particular case, when the second component 
is absent (xi = x'\ = 1), we find from (9.17) log K\ 0, and at the 
same time the temperature of equilibrium is the transformation tem- 
perature Tbi of the pure first substance. This serves to determine the 
constant as Ci(p) = h/RToi. There follows 

h 1 


On the other hand, the fundamental equations take the form (9.22), 
with v\ = j/2 = 1, 

Oc'i/^i) 17 " 1 = xi, (x'z/Kz) 11 ** = * 2 . (10.10) 

We shall treat in this section the case when the components in both 
phases ar^jnutually soluble in all proportions. Both components are 
then present in either phase, and eqs. (10.10) must be treated as simul- 
taneous. The relations (10.01) permit us to eliminate all the variables 
but xi. In particular, when there is no association in either component 

(vi = V2 = 1), 

*! = - (10.11) 

This equation and x'i = K\x\ resulting from (10.10) give the 
explicit expressions of the compositions of both phases (x\ and #'i) as 
functions of temperature. They apply to all cases of two-phase equi- 
librium, of which the more important (because of the accumulated 
experimental material) are liquid-vapor and solid-liquid. The graph- 
ical representation of these formulas leads to the so-called (Tx)- 






diagram (or Jy-diagram, if the temperature of transformation is 
plotted against the mass fraction y). As an example we give in Fig. 27 
the yT-curves computed with the help of our equations for the solid 
liquid equilibrium of an alloy of platinum (Toi = 2032, /i = 5270 
cal mol- 1 ) and gold (7o 2 = 1338, / 2 = 3140 cal mol" 1 ) as dashed 
lines. For comparison the experimental curves are drawn as solid 
lines. The agreement is by no means quantitative, but the general 
trend is well accounted for by the theory. It will be noticed that 
the order of the curves is inverted by comparison with Fig. 23, the curve 
giving the composition of the phase, stable at higher temperatures, 
lying above the other. Apart from this, there is a complete analogy 
between the cases of constant 
temperature (preceding section) 
and constant pressure inasmuch 
as there exist, in addition to the 
" normal" or monotonic type of 
Fig. 27, also r max - and r m i n -types 
of curves (of the same shape as 
those of Figs. 24 and 25). They 
are, especially, a frequent occur- 
rence in the liquid-vapor equi- 
librium of binary systems com- 
posed of organic substances. It 
would be easy to generalize our 
formulas for the case of any 
numbers v and to show that association or dissociation in the liquid 
phase leads to these types. However, this is unnecessary because 
in section 74 we shall give the general proof that conditions which 
produce the /w-type of the ^-diagram also cause the r m i n -type 
of the TVgraph. In the same relation to each other stand the 
mln -and r max -types. We can, therefore, refer, with respect to the 
influence of association and dissociation, to the discussion at the end 
of the preceding section. 

Curves of r m i n -type occur also in the equilibrium of binary metal 
alloys with their melts, especially, when one of the components belongs 
to the ferromagnetic group. We did not find in literature any discus- 
sion of the causes responsible for the minimum, but the prevailing 
tacit assumption seems to be that it is due to the existence of inter- 
metallic compounds. It should be mentioned, however, that in the 
case of Co-Mn alloys, a close investigation in Tammann's laboratory 
revealed that compounds of these metals do not exist. 1 In view of 
1 Hiege, Zs. anorgan. Chemie 83, p. 253, 1913. 

0.0 0.5 1,0 

FIG. 27. Temperature of fusion as a 
function of composition. 




this, we calculated from our formulas the Incurves (Fig. 29) which 
would result on the assumption that Co (T<>2 = 1762, /2 = 3950 
cal mol"" 1 ) has in the solid state a diatomic molecule when in alloy with 
Mn (Tbi = 1523, k = 3560 cal mol- 1 ). Comparing the theoretical 
curves with the experimental 7>diagram (Fig. 28) by Hiege, 1 we see 
that the minimum occurs at the right mass fraction (30% Co) but at a 
temperature which is some 80 lower than that measured. Although 
this discrepancy lies well within the limits of the theoretical inaccura- 
cies, we do not feel that the evidence is strong enough to prove posi- 
tively the existence of association, especially as it is not quite clear 
what this would mean in terms of the lattice structure of the alloy. 







FIG. 28. 
Temperature of fusion as a function of composition 

FIG. 29. 


But we do wish to point out that there lies here a problem which, pre- 
sumably, could be settled by experimental (X-ray) methods. 

72. Case of mutually insoluble components. Let us suppose that 
in one of the phases the components are mutually insoluble. This can 
be illustrated by many examples of which we mention only a few : (1) 
Aqueous solution of a salt which is insoluble in ice. When the liquid 
solution is cooled, either a part of the salt is precipitated or a part of 
the water congealed, conditional upon the concentration of the solu- 
tion. There are, therefore, two cases of two phase equilibriums: 
liquid solution -solid salt and liquid solution -pure ice. (2) Molten 
alloy of two metals which do not form mixed crystals. Upon cooling, 
either of the metals can freeze out partially, according to the composi- 
tion of the liquid phase. (3) Mixture of the vapors of two immiscible 
liquids. As the temperature is lowered, either one or the other liquid 

In all these cases we have equilibria of two kinds of pure substances 
with the (liquid or gaseous) solution. If we suppose that the pressure 
p is kept constant, eqs. (10.09) and (10.10) of the preceding section 

1 Hiege, loc. cit. 




still apply, but we must put in them x\ = 1 or 002 = 1, the condition 
that the unprimed phase is pure. Equations (10.10) become 

x'l = 

= (1 - 

= K 2 , 


and they are no longer simultaneous but refer separately to the two 
possible kinds of equilibrium. 

To make this quite clear, we shall consider a special example, the 
molten alloy of cadmium (Toi = 596, /i = 1245 cal) and bismuth 
(r 02 = 546, / 2 = 2110 cal). The first of eqs. (10.12) gives then the 
composition of the liquid phase when it is in equilibrium with solid 
bismuth; the second, when it is in contact with solid cadmium. Com- 







0.55 1.0 

FIG. 30. Solid lines experimental, 
dashed and dotted curves theoretical. 

* l.o 

FIG. 31. Experimental liquidus 

puting Ki and K% from eqs. (10.9), we find, for the first case, the theo- 
retical curve of A A' of Fig. 30; for the second, the curve BB'. Let us 
start from a composition (x'\) and temperature (T) of the liquid alloy 
represented by the point M. When this system is slowly cooled, the 
representative point first moves down vertically, until it hits the 
curve AA f . From then on, bismuth begins to freeze out; accordingly, 
the cadmium concentration of the liquid phase increases and, as the 
cooling proceeds, the representative point moves along the equilibrium 
curve toward the point C. In a similar way, when we start from the 
point N, the state of the cooled system first strikes the curve BB f and 
then begins to move along it to the left. In no case, however, does 
the representative point move beyond the intersection C of the two 
curves. The point C is a triple point in which the liquid phase is in 
equilibrium both with bismuth and with cadmium. As soon as it is 
reached both components begin to freeze out simultaneously, forming 
a microcrystalline mixture of the two solid phases (pure bismuth and 
pure cadmium). The point C corresponds, therefore, to the lowest 


temperature at which the alloy can be maintained in a liquid state at 
the given pressure p: for this reason, it is called the eutectic point 
(eutectic = Greek for readily melting). The dotted parts of the 
curves A A 1 and BB' have no physical reality as far as the system under 
consideration is concerned. For comparison we give the experimental 
Tjc-diagram of the cadmium-bismuth alloy 1 (solid curves in the same 
Fig. 30) : considering the crude approximation of the formulas (10.09), 
the agreement must be regarded as very good. The theoretical 
eutectic point lies at the same concentration as the experimental 
(x = 0.55) but about 25 too low. 

This example is quite characteristic. The microcrystalline mixture 
into which the liquid phase congeals at the eutectic point is called, 
in the case of metallic components, the eutectic alloy. It has, usually, 
the same content of the two components as the liquid, but this may 
be said to be accidental, in the sense that it is due to the kinematics of 
the freezing process. As we know from section 39, the equilibrium 
conditions do not depend on the extension of the phases. Therefore, 
the eutectic liquid should be in equilibrium with the solid phases taken 
in any proportion. This is, in fact, observed and indicated in Fig. 30 
by the horizontal line passing through the empirical eutectic point. 
In the case of aqueous solutions of salts, the eutectic congelations were 
formerly called cryohydrates. 

It is worth pointing out that, as long as the solution can be regarded 
as perfect, each of the two curves given by eqs. (10.12) depends on the 
properties of one component only and is quite independent of the other. 
In application to our example (Cd-Bi) this means that the curve AA' 
of cadmium has exactly the same shape and position, no matter what 
the other component is. In order to measure a large part of it and to 
bring out its curvature, we must choose, as the second component, a 
metal with a low boiling point, e.g. mercury. Unfortunately the 
7#-diagram for Cd-Hg was not available, and to illustrate this point 
we give in Fig. 31 the diagram for zinc-mercury. When we consider 
a mixture of vapors of two immiscible liquids in equilibrium with one 
of the pure liquid phases, the corresponding curve gives us the equi- 
librium of this pure phase with its own vapor. The partial vapor 
pressure of this component is, therefore, exactly the same as if the 
other component were absent. 

73. Case of partial mutual solubility. Other complications. In 

order to complete our discussion of binary equilibrium, we shall give 

here a brief schematic r6sum6 of the conditions which are to be expected 

in the case of partial solubility in some of the phases. To fix our ideas 

1 Petrenko and Fedorov, Zs. anorg. Chemie 6, p. 212, 1914. 


let us consider the case of two metals which form mixed crystals of 
two kinds. The bulk of the kind a consists of the first metal while the 
content of the second is smaller and can vary continuously from noth- 
ing to a certain limit. In the kind ft the conditions are reversed: the 
second metal dominates and the proportion of the first has a continuous 
range with a relatively small upper limit. Crystals of intermediate 
composition do not exist. There are, therefore, two cases of solid- 
liquid equilibrium: coexistence of the liquid alloy of the two metals 
with the mixed crystal a and with the mixed crystal ft. 

As long as the mol numbers of the second metal (#2, #'2) are small 
in both phases, they can change continuously and the conditions are 
exactly the same, in this range, as in the case of complete solubility 
(section 71). Assuming, by way of an approximation, that the solu- 
tion is perfect, we can describe the equilibrium between the liquid 
alloy and the crystal a mathematically by the formulas (10.09), 
(10.10), and (10.11). The graphical representation is, therefore, also 
analogous to the corresponding range of Fig. 27. Two curves diverge 
from the boiling point A of the pure first component (Fig. 32), the 
one giving the mol number x\ in the liquid phase, the other x\ in the 
crystal a; points of the two curves lying on the same horizontal are in 
equilibrium. However, these curves do not extend across the whole 
diagram but stop when x\ reaches its limiting value. This fact has a 
bearing on the interpretation of the constants TOI, To2 of eqs. (10.9): 
For the same reason as in section 71 we conclude that TOI is the 
melting point of the pure first metal. However, we cannot identify 
To2 with the melting point of the second because it is not permissible 
to extrapolate our curves until this metal is pure. It is better to denote 
this constant by r'o2, and to bear in mind that its numerical value 
cannot be predicted from any general considerations. It can be larger 
or smaller than TOI : When T'oz > TOI the two equilibrium curves go 
from the point A upwards; when T'o2 < TOI, they go downwards. 
Similar conditions prevail in the other case of equilibrium (the coex- 
istence of the liquid phase with the mixed crystal 0), when the mol 
numbers x\, x'\ are small. The two equilibrium curves diverge from 
the melting point B (temperature To2) of the pure second component 
and are again represented by eqs. (10.9), (10.10), and (10.11) with 
the new constants T'oi, To2- They go up or down from the point B 
depending upon the numerical value of the ratio T'oi/To2- We have, 
therefore, to distinguish three cases: (1) The equilibrium curves go 
downward on both sides, from the point A and from the point B. 
(2) They go downward on one side and upward on the other. (3) They 
go upward on both sides. 




The first case, illustrated by Fig. 32, is the most common. Suppose 
we slowly cool the system, starting from a state represented by the 
pair of points NN'. Some a crystal freezes out consisting mainly of 
the first component; the remaining liquid therefore becomes richer in 
the second component. The representative points move down along 
the branches AD and AC, so that later-freezing crystalline fractions 
are of lower XL This continues until the eutectic point C is reached 
marking the intersection between the " liquidus " curves AC and 
BC. In this state the liquid alloy is simultaneously in equilibrium 
with the mixed crystals a and ft. As in the case of the preceding 
section, the alloy freezes completely when the temperature is lowered 
further. The only difference is that it forms a microcrystalline mix- 


FIG. 32. FIG. 33. FIG. 34. 

Equilibrium in case of partial mutual solubility. 

ture, not of the two pure substances, but of the two mixed crystals 
a and 0, having the composition indicated by the points D and E. 
This is the end of the motion along the branches AD, AC, and their 
dotted continuation below the line DE has no physical reality. The 
mixed crystals represented by the points D and E are not only in 
equilibrium with the eutectic liquid alloy C but also with each other. 
The question may be put, therefore, as to the equilibrium of the two 
crystalline phases at temperatures below the eutetic. The answer is 
given by the two schematic lines going downward from D and E: 
Points of them lying on the same horizontals are in equilibrium. It is 
an empirical fact that these lines slope away from each other. 

The second case is represented by Fig. 33, which is self-explanatory 
because the notations in it are the same as in Fig. 32. We have used 
the example of two metals only to fix our ideas. All we have said 
applies, mutatis mutandis, to the equilibrium of two partially miscible 
liquids with their vapor, and to similar cases. The cases illustrated by 
the Figs. 32 and 33 are, in fact, as characteristic of mixed crystals as 
of the binary equilibrium of liquids and vapors. However, the third 
case represented by Fig. 34 applies only to a rather unusual system: 




Two substances form, in the solid phase, a mixed crystal in all pro- 
portions, but, upon being melted, they are partially miscible liquids. 
At temperatures lower than the eutectic (C) the crystal melts either 
into the solution a or the solution ft (depending upon its composition), 
at the eutectic point into a turbid emulsion of the two solutions. 

It goes without saying that the conditions considered here include 
the limiting case when the first component is partially soluble in the 
second but does not admit the second as a solute and crystallizes 
(condenses or melts) in its pure state. The point D in the above figures 
lies then on the vertical x\ = 1 passing through A. 

We have enumerated in this chapter the types of binary equilibrium 
which can be expected when the components do not form chemical 

FIG. 35. Liquid us curve. 

compounds. The presence of chemical compounds does not add much 
new to the discussion, from the conceptual point of view, but makes 
the curves look more complicated. One example will be sufficient to 
illustrate this: In Fig. 35 we give the 7*-diagram for the melting of 
Ca-Mg alloys. 1 These two metals form the compound CaaMg-r, 
there are three solid phases (pure Ca, pure Mg, CaaMg^ which happen 
to be insoluble in one another. At 721 C we have the equilibrium of 
the liquid and solid phase of the compound. The diagram is divided 
by this point into two halves: The left representing a binary system 
composed of the compound and magnesium, the right of the compound 
and calcium. 

74. Maxima and minima of the equilibrium curves. We have 
seen in the preceding sections that the treatment of binary systems, 
as if their phases were perfect solutions, gives a good qualitative 
account of all the observed phenomena. It is even probable that the 
not very large quantitative discrepancies are due, primarily, to the 
crudity of our approximation in putting / = const (section 71) and 
only in rare cases to the inadequacy of the description in terms of 
1 1.C.T. baaed on Baar, Za. anorg. Chemie 70, p. 352, 1911. 


perfect solutions. In particular, we could account for the appearance 
of maxima and minima in the equilibrium diagrams. However, it 
will be useful to give here a theorem (due to Gibbs) relating to these 
maxima and minima which is free from any assumption as to the 
nature of the system. 

Let us denote by <l>, Mi, M 2 the thermodynamical potential and 
the masses of the two components in the first phase, by <', M'i, M f 2 
the same quantities in the second. We shall use the equations of 
equilibrium between the two phases in the form (6.33) 

It was pointed out in sections 39 and 41 that $ is a homogeneous 
function of the first degree in the mol numbers Ni, N% and the masses 
Mi, M 2 . Consequently, the quotient </(Mi + M%) is a homogeneous 
function of the degree zero; in other words, it depends only on the 
ratio Mi/M2. In particular, we can represent it as a function f of the 
mass fraction (10.05), and a similar conclusion holds with respect 
to the other phase : 

<*> = (Mi + M 2 ) f(yi), *' = (M'i + M' 2 ) f'(y'i), (10.14) 
and by partial differentiation 

- toi) + (1 - *) , - toO - yi . (10.15) 

These expressions are quite general, no matter whether the compo- 
nents form chemical compounds or not. The variables entering into 
eqs. (10.13) are, therefore, yi, y\, p, T. We give to these variables 
increments dyi, dy'i, dp, dT in such a way that the two phases remain 
in equilibrium, i.e. eqs. (10.13) continue to hold. We can, therefore, 
take their total differentials 

3A<f> ^ 

"^T 1 + ^7 ^ l + ~^T d + ~^F dT = - etc ' 
3^i 3^ i 3^> 3T 

The partials with respect to p and T are given by the formulas 
(6.51) and (6.52), while those with respect to yi, y'i follow from 
eqs. (10.15) 


where AFi, AF 2 are the increases of the volume of the system and 
l g i, I g 2 the latent heats contingent upon the respective transfers of one 
gram of the first or second component from the unprimed to the 
primed phase. We can eliminate dy'i by multiplying the first equation 
by y ; i, the second by (1 y'i) and adding 

Ox'i - yi) y^ d y* +A'dp- B'dT = 0, 


with the abbreviations 

A' = y'i&Vi + / 2 AF 2 , B f = (y'il g i + y'd&llT. (10.17) 

We have now to distinguish two cases: (a) the ^-diagram, at 
constant temperature (dT = 0), 

& y=iiL (10 .i8) 

dyi A dyi* 

(b) the 7>graph, at constant pressure (dp = 0), 
dT y'i - yi 

dyi B' 


Obviously, the maximum or minimum of the py- and !Py-curves 
are at the places where the derivatives dp /dyi and dT/dyi vanish. 
Since A', B' do not become infinite and d 2 /dyi 2 is not known ever 
to be equal to zero, this condition reduces to 

y'i = yi. (10.20) 

The maxima and minima occur in the py- and Ty-curves at the same 
concentrations, namely, when the mass fractions in the two phases 
are equal. From the partial symmetry of eqs. (10.16) with respect to 
yit y'\ it is clear that the conditions dp/dy'\ = and dT/dy r \ = 
also reduce to the same eq. (10.20). This accounts for the fact that 
the two curves (liquidus and solidus, or liquid and vapor) have always a 
joint maximum or minimum. 

Whether the condition (10.20) leads to a maximum or to a min- 
imum depends on the sign of the second derivative. The second 
differentiation gives in the two cases (with y'\ y\ = 0) 

d 2 p 1 d*$ d*T 1 d*f 

^ dyi 2 A'dyi 2 ' x/ dyi 2 B' dyi 2 

The quantities A' and B f can be considered as positive. In fact, 
B' always is positive, since our notations are chosen so as to make the 


primed phase that of the higher heat a function (i.e. giving the positive 
sign to the latent heats /i, fe). The changes of volume A7i, and AF 2 
are always positive in the case of liquid-vapor equilibrium, and posi- 
tive with rare exceptions in the case of solid-liquid. The second 
derivatives d?p/dyi 2 and d?T/dyi 2 have, therefore, generally opposite 
signs. To a maximum in the py-curve there corresponds a minimum in 
the Ty-curve, and vice versa, as has already been mentioned in section 71. 
It is well to point out that all these conclusions apply, independently 
of the physical cause of the maximum or minimum type of curves, whether 
they are due to association, compound formation, or other effects. 
75. Remark on ternary systems. In a ternary system each phase 
consists of three (independent) components Gi, 62, G$ with the mol 
fractions #1, #2, #3 (or mass fractions yi, y^ ya) satisfying the condition 

XI + X2 + *3 - 1. (10.21) 

The scope of this book does not permit us to enter into the very 
extensive theory of these systems. All we wish to do here is to explain 

the type of diagrams used in connec- 
tion with them, in order to enable the 
reader, to understand the graphical 
material relating to ternary equi- 
librium. The triple of numbers xi, 
X2, #3 can be interpreted as a point 
within the area of an equilateral tri- 
angle. In fact, it is known from 
geometry that the sum of the dis- 
tances of any internal point P from 
the three sides (PAi + PA 2 + PA 3 , 

FIG. 36.-Graphical representation in Fi *' 36 > is e( * ual tO the ^\ f 

of the composition of ternary the triangle. If we choose the height 

systems. as equal to unity, we may identify 

the distances with the mass fractions 

(PA\ = xi, PA2 = X2, PA$ = #3) because the relation (10.21) is 
then satisfied. Every composition of the phase can, then, be repre- 
sented by a point of the triangle; and vice versa, every point within 
it corresponds to a possible composition. The mol fractions x\ = PA\ 
of the component G\ are, usually, marked on the side GiG2, so that 
one must draw the (dotted) line normal to PA\, in order to read yi 
in the point 61. For instance in our Fig. 36, x\ has the value of 
about 0.34. In a similar way the values of #2 (and #3) are listed on 
the sides G*Gz (and GsGi) and can be read by drawing (dotted) lines 
normal to PA* (and to PA*) to the points 62 (and 63). 




When two ternary phases (primed and unprimed) are in equilibrium, 
the condition (6.43) must be satisfied for each of the three components 

Vk?k(p,T, xi, * 2 ) + 'k?'k(p,T, *'i, *' 2 ) = 0, (* - 1, 2, 3) 

because the mol fraction #3 (resp. #'3) should not be regarded as a 
separate variable as it is completely determined by eq. (10.21). 

Two of the variables, x'i, #' 2 or x\, # 2 , can be eliminated from 
these three equations. If the pressure is, moreover, considered as con- 
stant, this leaves 

T - r(*i, * 2 ) or T - r(*'i, *' 2 ), (10.22) 

the equilibrium temperature expressed in terms of the mol numbers 
of either phase. These equations depend on three variables (T, x\ t # 2 


70 60 50 40 30 20 10 

FIG. 37. Isothermals of liquidus sur- 
face. (Temperature in C). 

FIG. 38. Eutectic lines. 
(Temperature in C). 

or r, x'i, #'2) and can be represented graphically only in a three- 
dimensional space. For instance, we could interpret the third dimen- 
sion, normal to the plane of the triangle in Fig. 37, as the temperature 
T. Equations (10.22) represent, then, two surfaces lying above the 
triangle. If the equilibrium is between a solid and a liquid phase, 
they are called, in analogy with the curves of Fig. 28, the soKdus 
and the liquidus surface. 

Naturally, one can represent in a plane drawing only the " levels " 
of these surfaces, i.e. their intersections with the planes of constant 
temperature, T = const. As an example we give in Fig. 37 1 the iso- 
thermal levels of the liquidus surface for the system Ni-Fe-Mn (dotted 
lines, the coordinates in this figure are not mol fractions but mass 
fractions /i, y 2 , 3/3). In the solid state, these three metals are 

1 1.C.T. based on Parravano, Gaz. Chim. Ital. 42 II, p. 367, 1912. 


mutually soluble in all proportions: therefore, there exists only one 
solid phase with which the melt is in equilibrium and the liquidus 
surface extends over the whole of the triangle. This is not the general 
case: if the solid components are mutually insoluble or partially solu- 
ble, there may exist at the same temperature three (or even more) 
different solid phases. This case is analogous to those of Figs. 30 and 
32 for binary systems. As we have there two separate liquidus curves 
corresponding to the equilibrium of the melt with the two solid phases, 
so here the area of the triangle (Fig. 38) * is divided into three regions and 
in each region we have a separate liquidus surface. The three liquidus 
surfaces are analytically represented by three different equations 
T = T'(y'\, y f 2) and each corresponds to the equilibrium of the melt 
with one of the three solid phases. The lines of intersection of two 
of these surfaces (eutectic lines) represent the states in which the melt 
is in equilibrium with two solid phases simultaneously (Fig. 38). In 
the point where all three intersect (ternary eutectic point) the melt is 
in equilibrium with all three solid phases. There may arise further 
complications in the number and shape of the equilibrium surfaces 
when the components are able to form chemical compounds. 

1 1.C.T. based on Goerens, Metallurgie 6, p. 537, 1909; Stead, Iron, Steel Inst. 91, 
p. 140, 1915. 


76. Definition of fugacity. It was shown in section 42 that chemi- 
cal and physical equilibrium depends on the partial thermodynamic 

We have reviewed, in Chapters VII to X, the cases in which some 
theoretical knowledge of the functions permits the making of com- 
plete or partial predictions about the details of the equilibrium. How- 
ever, in the large majority of cases such knowledge is lacking or in- 
sufficient, and the only resource is to combine the experimental and 
the theoretical methods: to measure and record the partial potentials 
from one group of observations and to use these data for the prediction 
of another set of phenomena. A large amount of experimental 
material has been collected and published in the chemical literature. 
However, the chemists usually record not the partial thermodynamic 
potential itself but another function/ which stands to it in the simple 

relation / = / (r) exp &/RT) (11.02) 

Of V = RT[logf - log/ (r)]. (11.03) 

The quantity / was introduced by G. N. Lewis 2 and is called the 
fugacity or absolute activity. fo(T) is an auxiliary function of tem- 
perature only, which is defined by the following requirement: in the 
gaseous phase, the fugacity approaches more and more the pressure, 
as the density is decreased, and becomes identical with it (/ = p) in 
the limit V = oo . 

The thermodynamic potentials of perfect and of Van der Waals 
gases were given by the formulas (5.41) and (5.44). In order 
to satisfy the above requirement we have to put in both cases 
f (T) = - w(T)/RT. For the perfect gas we obtain 

/ = P, (11.04) 

1 This chapter can be skipped without loss of continuity. 

1 G. N. Lewis, Proc. Am. Acad. 37, p. 49, 1901; Zs. physik. Chemie, 38, p. 205, 




XI 76 

at all temperatures; for the Van der Waals gas we find 

2a RTb 


Once/ is defined in the gaseous phase, it is definite also in the con- 
densed phases because (barring associations) the thermodynamic 
potentials (and, therefore, the fugacities) are equal in equilibrium. 
More generally, we have for two different conditions of the systems 
(1) and (2), at the same temperature, 

- /? - (11.06) 

In general, analytical expressions for / are lacking and, even in 
systems with one independent component, only the partial of log / 
with respect to p can be given explicitly from (5.37) 


The measurement of ^ can be easily carried out in pure 
substances (systems with one independent component): in this case 
s=^> = # Ts + pv, where each of the three terms can be directly 
measured. As an example we give in Table 31 the fugacity of liquid 
water at various temperatures and pressures. 1 


Activity a 

Fugacity / 

25 C 



25 C 


























































0. 19592 






















1 M. Randall and B. Sosnick, J. Am. Chem. Soc. 50, p. 967, 1928. 


On the other hand, in a phase (i) with many components the 
function <p h (f *, as is clear from its definition (5.39), is only then accessi- 
ble to direct measurement when the mol number JV* (i) can be changed 
independently from the other mol numbers. If this is impossible 
(owing to chemical reactions in the system), it is often still feasible 
to determine it in a roundabout way when this component can be 
isolated in another phase (j) which is in equilibrium with (i), making 
use of the equilibrium condition ^ ( ) = ^ A ( ^. After being determined 
from observations, these data can be used for theoretical purposes, for 
instance to test the relations (6.49) 

A*]ifctyk> =0, (11.08) 

which must obtain within the phase (i). 

77. Definition of activity. In some cases it is impossible or diffi- 
cult to determine the absolute value of the fugacity of a component 
in a complex phase (for instance, in a solution), while it is feasible to 
measure the ratio 

*-///. (11.09) 

of the fugacities of the same component for two states of the solution, 
different as to pressure and concentration, but of the same temperature. 
This permits choosing /, as the unit in which the fugacity is to be 
measured and regarding the state to which /. refers as the standard 
state for the component and for the temperature in question. The 
ratio a was also introduced by Lewis 1 and is called activity or relative 
fugacity. Its relation to the partial thermodynamic potential follows 
from the formula (11.03), taking into account that/and/, are measured 
at the same temperature so that/o(r) drops out of the ratio: 

?*-?* = RTlo%a h , (11.10) 

where v, h is the partial thermodynamic potential of the component h 
in the standard state. The* simplest example is the case of a pure 
substance: in Table 31 are given the activities of liquid water. As 
standard states at the temperatures 25, 37.5, 50 C, are chosen 
those corresponding to the pressure p = 1 atm (with/, equal to 0.03 125, 
0.06372, 0.1219, respectively). 

It must be distinctly understood that, in systems with several 
components, the standard states are chosen for each component inde- 
pendently and arbitrarily. For the definition of ?*, not only 

1 C. N. Lewis, Proc. Am. Acad. 43, p. 259, 1907; ZB. physik. Chemic 61 p. 129, 


but all the mol numbers of the other components, must be given. 
Therefore, the <p,& are mutually exclusive, in the sense that there does 
not exist any stable state of the system with these simultaneous values 
of the partial potentials. This implies, of course, that they do not 
satisfy the conditions (11.08). The expression 

log K., (11.11) 

1 h 

does not vanish but is a finite function of temperature and pressure. 
We find from (5.37) 


' ajai ) -&. <"' 

/ p Ai 


AF, =X) "A ^^* (? =]C vi^TldP* (11-14) 

On the other hand, the partial potentials^ in eqs. (11.10) refer to 
actual states of equilibrium and satisfy the condition A$ = 0. There- 
fore, there follows the equation 

]C ^ log a h u} = log X,, (11.15) 

which has the same analytical form as the mass law, the mol fractions 
being replaced by the activities. 

The physical meaning of the activity becomes a little clearer from 
the analogy which the formulas (11.10) and (11.15) bear to those valid 
for dilute solutions. In section 59 we found for the partial thermo- 
dynamic potential of a dilute solute 

Since log x* vanishes for x\ = 1, it may be said that the term ^ 
represents the molal thermodynamic potential of the fictitious state 
when the component is pure while having the same properties \\& in the 
solution. This means that the heat of solution is contained in <pn but 
the entropy only partially: the second term RT log XH represents its 
increase due to the entropy of mixing the solute (h) with the solvent. 
The interactions of the component (h) with the other solutes and the 
entropy of mixing it with them are negligible in dilute solutions. In 
the general case of non-dilute solutions, we have the analogous formula 

>gaA 0) . (11.17) 


The standard state for the component h is, practically, always 
chosen so that, in it, it is the only solute (xk = 0, k j& h). The term 
with log a represents, therefore, the increment of the thermodynamic 
potential due to interactions with the other solutes, to the entropy of 
mixing h with them, and (in a measure depending on the definition of 
the standard state) also to the interactions and entropy of mixing with 
the solvent. 

78. The activity coefficient. The simplest way of choosing the 
standard potentials ,/ is identifying them with the functions <f>k of 
eq. (11.16) used in the theory of dilute solutions 

V* - ** w . (11.18) 

The equilibrium constant K, becomes then identical with K in 
Van t'Hoff s eq. (9.13), whence (11.15) takes the form 

Yi "* a) log a h (i) -logJC. (11.19) 


The activities ah have then the meaning of effective mol numbers 
which one must substitute instead of XH in order to conserve the form 
of the mass law. The ratio between the effective and the actual mol 

*h = Ok/Xk (11.20) 

is called the activity coefficient. 1 

In dilute solutions the activity coefficient is usually defined in 
another way: as the ratio of activity and molality WH 

y h = a h /m h . (11.21) 

The relation of the two definitions (since in the dilute state 

Xh = WA/WO) is 

y h = CKA/WO, (11.22) 

where mo is the molality of the solvent (i.e. mol number per 1000 g 

It is obvious that the choice of the standard states defined by 
eq. (11.18) is permissible when the solution obeys (in the limit of low 
concentrations) the theory of dilute solutions treated in Chapter IX. 
The standard states are then, really, possible states of the system, 
and at low concentrations the activities approach the mol fractions: 

lim ah = x ht lim ah 1. (11.23) 

1 This term was first introduced by A. A. Noyes and W. C. Bray (Journ. Am. 
Chem. Soc. 33, p. 1643, 1911), but these authors defined it with respect to the 
molality as in eq. (11.21). 


However, even if this should not be so, if the values ,* = <PH cor- 
respond to no existing states of the system, they can still be used. 
They are, then, merely auxiliary mathematical quantities with respect 
to which the activities are defined. There is no reason why one 
should not use thermodynamic potentials defined with respect to 
fictitious states, provided there can be given an experimental way of 
measuring them, either separately or in combinations. Such an experi- 
mental procedure exists and will be described below. 

In the first place, it is simple to measure the activity of the solvent 
by the method of boiling pressures or boiling (and freezing) temper- 
atures. For the process of vaporizing >o (2) mols of solvent, to obtain 
FO (I) mols of vapor, we have from (11.19) the equation 


vo log a (1) - vo (2) log ao (2) - log K, (11.24) 

closely analogous to (9.17). We assume that the solutes in the liquid 
phase (2) are non-volatile so that the vapor phase (1) is pure and of 
sufficiently low pressure to be regarded as a perfect gas. For a perfect 
gas, eq. (11.16) or (9.09) is rigorous and the activity coincides with the 
mol fraction: a* (1) = #A (I) = 1. This gives for the liquid phase (drop- 
ping the superscript) 

vo log ao = - log K. (11.25) 

In the particular case when the liquid phase also consists of the 
pure solvent we can apply to it the theory of section 64 and find 
log K(p,T) = 0, where p, T are the pressure and temperature of the 
boiling point. When the solutes are present in a concentration that 
is not high, the solution boils at a lower pressure p + &p (provided T 
is the same), and we obtain from (11.25) 

*o log ao - - log K(p + A,!T) - AF- A/> (11.26) 

neglecting higher powers of A/>. 

On the other hand, if the theory of section 64 were applied to 
this case, we would have 

j>o log XQ = AF- 

Denoting the ratio of the actually measured pressure increase A 
to the theoretical Ao (if the solutes behaved like perfect gases) by 

we have 

log 00 = /3ok>g*o. (11.28) 


It is obvious, by analogy with section 64, that ft> can also be 
measured as the ratio of the actual (#) and the theoretical (tfo) lower- 
ing of the freezing point, 0o = #/#() Much of the experimental work 
has, in fact, been carried out by the freezing method. The above 
formulas imply, of course, concentrations sufficiently low for higher 
powers of A being neglected in the expansion (11.26). A more accu- 
rate formula for the activity of water as a solvent, taking in second and 
third powers of the lowering # of the freezing point, is 1 

logio ao - - (421.00 + 0.1640 2 - 0.0037tf 3 ) X 10~ 5 . (11.29) 

79. The activity function of electrolytes. Equations (11.28), 
(11.29) give a convenient way of determining the activity of the sol- 
vent. As to the solutes, their activities can be measured directly if 
they are volatile. This is, however, a comparatively rare case. A 
very important field of investigation are the electrolytes whose ions 
are mostly non-volatile. It was shown by Lewis that valuable knowl- 
edge about their activities can be obtained from eq. (11.28) by means 
of Duhem's relation (6.16). Let us consider the case that every 
molecule of an electrolytic solute is completely dissociated into ions 
according to the formula vid + . . . + v T G r G n = 0. If the mol 
number of the neutral electrolyte dissolved was N n , those of the ions 
are N* = VhN n , the sum being v e N n , (v 6 = v\ + . . . + v r ). Denoting 

* - V '?\T = 1 - *o, (11-30) 

(where NQ and XQ refer to the solvent), the mol fractions become 

x k = VKX/V., (ft - 1,2.. .r). (11.31) 

In Duhem's equation (6.16) the partial thermodynamic potentials 
<ph can be replaced by log a h because, in eq. (11.10) defining the activ- 
ity, the function ^. is independent of the mol fractions #*. As the 
variable of differentiation we take x 

3 log a h 

Now, from (11.20) and (11.31): log a* = log a* + log x + const. 
Substituting this for the activities of the ions, 

Abel, Redlich, and Lengyel, Zs. phys. Chem. 132, p. 201, 1928. 


The expression 

log 7 - ~ vh log a* (11.32) 

defines the quantity y called the activity function of the electrolyte 
(sometimes also referred to as activity coefficient or activation function). 

( i-,) + i+,-a. (n.33) 

In the limit of zero concentration (x = 0) the solution is reduced 
to the pure solvent (0o = 1) and it is experimentally ascertained that 
lim log ah = lim log 7 = 0. The integration of (11.33) gives, there- 

log 7 

_ /Ti + * -* 9l g' 

/o Lx jc 3^ 

In the particular case of considerable dilution, the expression 
(11.28) may be used for log a , and, moreover, it is then permissible 
to neglect terms of second order in the small quantities x and PQ. 
Hence the limiting law for low concentrations 

- go 3/3o 


Since in dilute solutions x cc m (molality) this can be also written 

The pioneer work of collecting data about activities of electrolytes 
by the freezing method was largely carried out by American chemists 
under the leadership of G. N. Lewis and A. A. Noyes. This material 
confirmed the division of these substances in two groups discovered 
earlier by other lines of study. 1 

(A) Weak electrolytes (mostly organic acids and their salts) are 
those which form aqueous solutions obeying, in their dilute state, the 
theory of Chapter IX. This implies /So = 1, 7 = 1 for all low concen- 
trations, so that eq. (11.34) is identically satisfied (0 = 0). 

(B) Strong electrolytes (the common strong acids and their salts) 
which exhibit the electrolytic properties in their most typical form 

1 There are two other methods of measuring the activities of electrolytes. 
(1) Electrical measurements (conductivity or electromotive force), as described in 
section 114. (2) Influence of other solutes on the solubility. This method is 
limited to electrolytes of low solubility. 


do not obey the laws of dilute solutions. While 0o tends towards 1 in 
the limit of infinite dilution (m = 0), it appreciably deviates from unity 
even for very low molalities. In fact, the curve representing 0o = 0o(w) 
has a vertical tangent at m = 0, i.e. QPo/dm = oo . Several empirical 
formulas have been proposed for this function, among them 

/So = 1 - ^W H , (11.35) 

which is supported theoretically (section 115) and represents the 
experimental material as well as any of the others. This formula 
permits one to carry out the integration of (11.34) and to find the 
limiting law for the activity function 

log 7 =- km*. (11.36) 

It will be better to postpone the discussion about the structure of 
the factor k and about the agreement of this formula with the experi- 
mental data until section 115. 

80. Activities in binary gas mixtures. The deviations from the 
laws of perfect gases in electrolytes (preceding section) are due, pri- 
marily, to electric forces. It will be instructive to mention, as another 
example, the deviations in gas mixtures which are caused by molecular 
cohesion. The activities of a number of gases in mixtures have been 
determined by several investigators from their chemical equilibrium 
with liquid or solid substances, and a summary of their results is con- 
tained in a paper by Randall and Sosnick. 1 The activity coefficient 
of a component is defined as a h = fh/fh (0) Xh where /* is the f ugacity of 
the gas h in the mixture, /A (O) its fugacity in the pure state, at the 
same pressure. 

Randall and Sosnick confirm for binary gas mixtures an equation 
which was found by Hildebrand 2 to hold for liquid solutions, namely, 

log ai//i (0) *i) = (Bx 2 2 + Cx 2 * + Dxz* + . . .)/RT. (11.37) 

The character of this function appears from the curves of Fig. 39 3 
relating to mixtures of ethylene and argon at 25 C. 

The only theoretical equation of state for gas mixtures is Lorentz' 
extension of the Van der Waals formula given in section 5. Although 
its validity is uncertain and, presumably, restricted to a narrow range, 
it is interesting to note that it leads to an expression similar to (11.37). 
Lorentz' equation is formally identical with the ordinary Van der 

i M. Randall and B. Sosnick, J. Am. Chem. Soc. 50, p. 967, 1928. 
'Hildebrand, Proc. Nat. Acad. Sci. 13, p. 267, 1927. 
8 Randall and Sosnick, loc. cit. 



XI 80 

Waals formula (1.21); only, the "molal volume" v has here a 
different meaning, namely, v = V/N (where N = ffi + N*), while 
the constants depend on the mol fractions as follows: 


+ 2ai2XiX2 + (122X2*, 
+ 2bi2XiX2 + 

The thermodynamic potential is obtained by an obvious generaliza- 
tion of (5.44) 

p + 

W1 (D 


c p idT -T 


Argon 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9 Ethylene 

Mol Fraction 

FIG. 39. Logarithms of activity coefficients (log o&) 

The entropy of the mixture must contain a term depending on the 
mol numbers of both components which represents the entropy of 
mixing the two gases. It can be evaluated by noticing that, in the 
limit of very large v, the components become perfect. For a binary 
mixture of perfect gases the thermodynamic potential is known to be 
(section SO) 

Ni*i(T) + N 2 <*2(T) + NRTlogp + RT(Ni 

+ N 2 log * 2 ), 


where the subscript i stands for " ideal.*' Comparing the two expres- 
sions, we conclude that 


It is our purpose to calculate the expression (11.37) to terms of 
second order in the coefficients a, 6. Expanding and making use of 
the equation of state (1.21), we find 

According to (11.04), the fugacity of a perfect gas is equal to its 
(partial) pressure, /u = pi = px\. Consequently (11.06) leads to 

This relation holds for all values of the mol fractions, including 
X2 = 0, xi = 1, when the left side is identical with log (/i (0) /)- Tak- 
ing the difference of the two expressions, we obtain, in fact, an equation 
of the type (11.37) with the following values of the coefficients. 

B = p[(an - 2ai2 + a 2 ^/RT - (6u - 2b i2 + 622)] 

- 612) + an(a 2 2/2RT - 6 22 ),] 


) [" 

(3an - 4ai2 + a 2 2)(ai2/2RT - 612) 
- & 22 )1 , 

3 . -(an - 

(611 2&i2 + 622)]- 

In the case of ethylene (1) and argon (2) at 25 C, the table values 
are an/RT = 135, in 42, a 2 2/RT 53, & 2 2 30. The order of 
magnitude of the data in Fig. 39 is consistent with this if a\2 and 612 


have commensurate values. At pressures of 25 atm the terms with f? 
are still unimportant and, neglecting them, our formula is symmetrical 
in the subscripts (1) and (2). This is borne out by the symmetrical 
character of the curves for argon and ethylene at this pressure. The 
constants can be adjusted so as to describe also the general character 
of the curves at the higher pressures of 50 and 75 atm, but it remains 
to be seen whether the expressions are fit for a quantitative descrip- 
tion of the phenomena. 


81. Surface layer and surface tension. The equilibrium of two 
adjacent phases was treated in Chapter VI as if each of them were 
uniform in its properties up to the very surface of discontinuity sep T 
arating them from each other. This, however, is not strictly true: 
the state of a substance at any point within it is influenced, not only 
by the molecules nearest to this point, but also by all the others in a 
small radius around it called the radius of molecular action. The sur- 
face layer of the thickness of this radius, between the phases or at the 
boundary, is, therefore, in a condition different from interior points, 
as it stands under the action of molecules of both media. It is called 
the capillary layer and exhibits peculiar properties depending on the 
nature of the two phases. When the phases are of large size, the 
energy and entropy of the capillary layer become negligible, and the 
conditions are those discussed in Chapter VI. But when the size of 
the phases is small, in one or several dimensions, the influence of the 
capillary layer is important. 

A general characteristic of the capillary layer is that it tends to 
contract. From the point of view of mechanics, it acts like a mem- 
brane in a state of uniform tension. By this we mean a condition of 
the membrane (Fig. 40) in which every line element dl of its boundary 
AB experiences a normal force F n = a dl to the inward, <r being the 
constant tension. If we draw an imaginary line CD in the interior of 
the membrane it is also under the action of two oppositely equal nor- 
mal forces of the magnitude a per unit length of the line. When the 
membrane has a chance to contract, it performs work in doing so. 
Suppose that the boundary line is displaced from the position AB to 
A'B', and the element dl to dl r , through the distance AL, as a result of 
the contraction. The work done against the element dl is, according 
to (2.01), F n AL cos (w,L) = a dl&L cos (n,L), or it is equal to a multi- 
plied by the area which is swept over by dl in its displacement. Conse- 
quently, the total work is equal to <r multiplied by the area swept over 

1 This chapter can be skipped without loss of continuity. 


by the whole boundary (which, in turn, equals the decrease of the 
area 2 of the membrane) : 

DW -- <r<G, (12.01) 

counting dS positive when it is an increase. It does not matter 
whether the membrane is plane or curved. 



B' 1 'B "2 (2) 

FIG. 40. Work done by surface tension. FIG. 41. Capillary layer. 

The tension <r of the capillary layer is called the surface tension. 
Its numerical values for the more important substances are contained 
in Table 32. A few additional data for solutions and for solids will 
be found in sections 86 and 87. 

The thickness of the capillary transition layer is not well defined 
as it represents a continuous change of properties from one phase to 
the other. In the interest of the next section we shall, therefore, pro- 
ceed as follows. We envisage a point in this layer and lay an imagi- 
nary geometrical surface AB through it and through all the other 
points which are similarly situated in relation to the two adjacent 
phases (Fig. 41). We lay a second imaginary surface, A\B\, parallel 
to it, in the upper phase (1), at a distance d\ where that phase is 
already uniform, and a third parallel surface, A^B^ in the lower 
medium (2) where this medium is uniform. In all the variations 
which we shall let the phases undergo, the distances di, dz from AB to 
AiBi and ^2-82 shall be considered as constant and not subject to 
variation. We shall restrict ourselves to the case when the distances 
d are small compared with the radius of curvature of the layer. 
Imagine now a wide l rigid and adiabatic envelope which contains 
parts of both phases and cuts through the capillary layer. By our 
device the space within it is divided into three regions, as labeled in 
Fig. 41, and the total internal energy and entropy can be divided into 
three parts pertaining to these three regions 

U - U+ t/ (2 >+ /< 3 >, S = 5 (1 >+ S+ S< 3 >. (12.02) 

1 Near the envelope the capillary layer is distorted because the walla add their 
own surface effects; but when the envelope is sufficiently wide, the distorted parts 
are negligible compared with the undistorted remainder. 

XII 81 





Adjacent medium 


9 (dyne/cm) 




75 49 


Water vapor 


73 21 






Chloroform. . . 


27 7 


Ethyl ether 


9 7 




471 6 




480 3 


Ethyl ether 


398 3 


Ethyl alcohol 


364 3 


Chloroform . 


356 6 



70 K 

18 3 


Vapor . 


10 53 


Vapor . . 



2 83 





13 2 

Carbon monoxide 



12 11 

Carbon dioxide 



9 13 

Ethyl ether 



16 49 




20 28 

Methyl alcohol 



23 02 

Acetic acid 



23 46 

Carbon tetrachloride .... 



25 68 

Ethyl alcohol 



23 03 



The energy differentials in the first and second region (first phase 
and second phase) have the usual expressions 



while that of the inhomogeneous intermediate layer is 

rfZ7 ( 3 ) = TdS-DW = TdS+ <rd2 - &4V*\ (12.04) 

where !T <3) and ps are the mean temperature and mean pressure in it. 
82. Conditions of equilibrium as modified by the surface layer. 
The fundamental conditions (5.03), in view of the relations (12.04), 
take the form 

0, 65 (1 >+ 6S+ 6S (3) = 0. (12.05) 

When treating equilibrium with neglect of the capillary layer 
(section 40) we found it convenient to divide the problem into two 
parts by suitable restrictions of the virtual variations (auxiliary con- 
straints). We shall now go a step farther and break up the present 
treatment into three partial problems. 

(1) We do not admit any changes in the composition and mass or 
in the volume and geometrical shape of the phases. The only per- 
missible variations are those of transfer of heat from one phase to 
another. Under these conditions 

6/o>= r<$S<, (j = 1,2,3). 

We substitute this into the first eq. (12.05) and add the second, 
multiplied by the Lagrangean factor X 



jxn = r (2) = r< 3) = r = - x. (12.06) 

In equilibrium the temperature is uniform throughout the system. 

(2) We consider only variations at constant temperatures and 
constant total volume. As was shown in section 36, the equilibrium 
is then determined by the minimum of the sum of the work functions 

whose differentials are (5.24) 

Moreover, we disallow any changes of composition or mass in the 
phases and permit only changes in volume and shape of the three 
regions at the expense of one another. Since 67* = 0, the variation 
d becomes, according to (12.03), (12.04) 

v S - 0. 

XII 82 



The variations of volume are subject to certain subsidiary condi- 
tions. In the first place the total volume is constant since we suppose 
it enclosed in a rigid envelope 


(pi - 

- * 52 = 0. 


As a further restriction, we specialize the geometrical shape of the 
phases in the following manner : let the capillary layer have the con- 
stant curvature 

C = - + 


r and r r being its main radii, and let the walls of the vessel be normal to 

the layer. We consider a transfer in which mass is added to the lower 

phase (1) and taken from the upper 

phase (2), so that all three surfaces 

defining the layer are displaced up- 

ward through the distance /. If we 

denote the respective areas of the 

surfaces AB, AiBi, A 2 B 2 by 2, Si, 

S 2 (Fig. 42) 

6 7<D = 2i $/, 5 7 (2) = - S 2 81. 

FIG. 42. Virtual displacement of 
capillary layer. 

On the other hand, we have from geometry 

S/rr' = 2i/(r - di)(r' - di), etc,, 
or neglecting squares of d/r, d/r', 

2i = (1 - Cdi) -2, 2 2 - (1 + Cd 2 ) 2. 
In the same way the increase of the area S due to the displacement 


62 = CS-6Z. 
Substituting all this into (12.07), we find 

pi - p2 = [<r - (ps ~ i)di - (#3 - p2)d 2 ] C. (12.09) 

The terms (3 p\)d\ and (3 pz)d 2 represent the difference 
between the actual force of pressure on the unit length of the cross- 
section through the region (3), and the force which would act if the 
phases (1) and (2) extended to the surface AB and changed discon- 
tinuously at it. Because of this subtraction, any indeterminateness, 


due to the arbitrary choice of the thickness di and </2, cancels out. 
The difference itself is an intrinsic property of the capillary layer of 
the same nature as the surface tension. In fact, there is no experiment 
which would permit its measurement separately from it. It is, there- 
fore, appropriate to throw it together with the surface tension and to 
denote by the symbol <r the whole bracket expression of (12.09). In 
this sense, we write 

p l - p 2 = Co- = * + - (12.10) 

It should be noted that in Fig. 42 the region (1) lies on the concave 
side of the surface AB. We therefore make the rule that C is to be 
counted as positive when (1) is on the concave side and as negative when 
it is on the convex. Since <r is essentially positive, the pressure is 
always higher on the concave side of the capillary layer. The relation 
(12.10) is known as Lord Kelvin s formula because it was deduced by 
that physicist from mechanical considerations (next section). 

(3) We maintain constant the temperature T and the local pres- 
sures p\ 9 pz, p3 in all the three regions. As we know from section 36, 
the equilibrium is then determined by the minimum of the thermo- 
dynamic potential : 6$ = 0. The only permissible variations are now 
those of transfer of mass from one phase into another. Besides, we 
restrict these variations still further by prohibiting any changes of 
mass or composition in the region (3) and we consider only the trans- 
fers of mass of any component h (through the capillary layer) from 
phase (1) into phase (2), or vice versa. These variations are precisely 
the same as those considered in section 40, and the presence of the 
capillary layer in no way influences the form of the conditions of 
equilibrium. We may, therefore, refer to that section and to 42 and 
take over the equations (6.43) and (6.45) 

^"W^+^W^O, (12.11) 


= ?W 2) . (12.12) 

In the second of these equations the partial thermodynamic poten- 
tial is referred to 1 g of the component h (and not to 1 mol). 

We may summarize the results of this section in the following way. 
We derived the conditions of equilibrium for two phases separated by 
a capillary layer. Comparing them with those obtained neglecting 
the surface layer, we find all the conditions the same, except that 
relating to the pressures of the phases. Instead of equality of pres- 

XII 83 




sure, we find a pressure difference (12.10) depending on the curvature 
of the capillary layer. When the layer is plane, C = 0, its influence 
on the equilibrium vanishes. 

83. Direct observation of the capillary pressure difference. It 
was known for a long time that the level of water is higher in narrow 
tubes (capillaries) than in large vessels (Fig. 43). The theoretical 
explanation of this phenomenon was given by Laplace, who showed 
that the column of liquid in the capillary tube is supported by the 
surface layer of the meniscus. In this way the names " capillary 
layer " and " capillarity " originated. A particularly simple vffcy of 
calculating the difference / of levels in the tube and 
in the wide vessel (" capillary rise ") was given by 
Lord Kelvin. Let us consider the case that a 
liquid rises in a tube of circular cross-section 
(with the radius R) and that the meniscus includes 
the angle # with the walls. The vertical com- 
ponent of the force of tension which acts on the 
rim of the capillary layer is then F = 2irR<r cos $. 
This force supports the liquid column of height / 
and must be equal to its weight in the surround- 
ing gaseous atmosphere. Denoting the densities 
of the liquid and gas by P2 and pi, respectively, 
and by g the acceleration of gravity, F = irR 2 l g(p% pi), whence 
2<rcos# = Rlg(p2 PI). For simplicity, let us assume the curvature 


FIG. 43. Capillary 

of the meniscus as constant over its whole surface, 
curvature is then r = R/cos &, and we find 

The radius of 

- Pi) 


For water and other wetting liquids # = and, roughly, R = r . 
On the other hand, for completely non-wetting liquids, like mercury, 
& = IT: i.e. r and / are negative. This means that the liquid is on the 
concave side of the meniscus and its level is lower in the capillary. 
The formula (12.13) offers one of the most accurate methods of measur- 
ing surface tensions. 

If p is the pressure at the plane surface in the vessel, the pressure 
in the gas at the meniscus is smaller by the pressure of the layer of 
gas of the thickness I, viz. pig/: 




Similarly the pressure pz in the liquid at the meniscus is decreased 

. . 2<r P2 

p 2 - p = 

r pi p2 

Pi- p2= (12.15) 

This is identical with the thermodynamical formula (12.10), in the 
special case r = r'. 

84. Influence of curvature on the pressure and temperature of 
transformations. Let the pressure and temperature of equilibrium 
between two phases be p and T when the capillary layer is flat (wide 
vessel). We first consider the case of a pure substance (or of a system 
with one independent component), <p = (p, and use the equation of 
equilibrium in the form (12.12), since we are not interested in the 
effects of association, p and T must then satisfy the relation 


When the capillary layer is curved, the conditions are slightly dif- 
ferent. We denote the temperature T + Ar and we have two pres- 
sures pi = p + &p and p 2 = p + A/> <rC, according to (12.10). 
The relation of equilibrium still has the same form 

<P M (P + A/> - <rC, r + Ar) - 9M ^(p + A/>, r + Ar) = o. 

Restricting ourselves to quantities of the first order in the small 
increments, we expand into a Taylor series and subtract (12.16) 

This formula can be specialized in two ways: 

(1) Transformation pressure at given temperature. Suppose that 
the flat and the curved boundaries are at the same temperature: 
Ar = 0, so that the third term of the equation vanishes. According 
to section 35, (9vW9/>)T ** ^M, the specific volume referred to Ig, which 
is the reciprocal of the density, V M = 1/p. Hence 

Pl (12.17) 


PI ~ P2 

In the special case of equilibrium between a liquid and its vapor 
this is identical with the relation (12.14) which was obtained by Lord 


Kelvin's method. In fact, the liquid boils at the meniscus in the tube 
at a slightly different pressure from that at the plane surface of the 
vessel, corresponding to the difference of altitudes. 

(2) Transformation temperature at given pressure. Suppose the 
pressure of the phase (1) is the same at its plane and at its curved 
boundaries: pi = p or Ap = 0. We know from section 42 that 

\ /W>\ _ I 

),~ \~&r) f - ~ 

\~&r,~ \~& 

where IM is the latent heat of transformation at the plane surface, i.e. 
the heat necessary to transform Ig of the substance from the phase (2) 
into the phase (1), 

AT = ~ (12.19) 


In addition to the case of a pure substance, we shall be interested 
in another which is but slightly more general. We assume that one 
of the phases, say (2), is condensed and pure, containing only the com- 
ponent h, while the other phase (1) is either a gas mixture or a dilute 
solution. We consider first the case of the gas mixture and apply to 
it eq. (9.24), since it can be considered an immediate consequence of 
(6.43) or (12.11), 

RT log p, = fok^tor)/** - *(r)J, (12.20) 

where g* = ^ (1) /^ (2) . At a flat surface of discontinuity, p means the 
pressure in phase (2) identical with that in phase (1) where it is com- 
posed of the partial pressure ph (l) of the component (h) and of the pres- 
sures of all the other components whose sum we shall denote by po, viz. 
p = p h w -f- pQ. The curvature of the surface of discontinuity cannot 
influence the pressure p^of those components which are absent in the 
pure condensed phase. It remains the same, and only the partial 
pressure of the component h changes into ph (l} + A/> A (1) . Consequently 
p 1 = p + A/> A (1) and p 2 = p +' A/> A (1) - aC, according to (12.10). At 
the temperature T, we have, therefore, from (12.20) 

>- aC, T)/g h - 
or, as above, 

When the first phase is a gas mixture (ph (1> = RT/vS 1 *), this is iden- 
tical with eq. (12.14) since obviously (&ifc (1) Vh^/Vk* = (P2 PI)/DI. 


In the case of a dilute solution, (1) labels the liquid phase and (2) the 
solid precipitate. ph (1) must be interpreted as the osmotic pressure of 
the component h which is proportional to the solubility x gh a \ according 
to the relation (9.50). Neglecting A/>* in the parentheses beside aC, 
(where C is negative, since the solid particles are convex) 


which expresses the influence of the curvature upon the solubility and 
is called the Ostwald-Freundlich formula. It is to be remembered that 
v h (2) is the molal volume in the condensed phase. If the substance is 
dissolved without change of molecule (g h = 1), the solubilities for the 
two negative curvatures C\ and 2 are related by the formula 

log*.u (1) - log*.A2 (1) = - ^ (2 V(Ci - C 2 )/RT. (12.22) 

8S. Applications. The effects listed in the preceding section have 
an appreciable magnitude only in the case of very large curvatures. 
Let us take as an example drops of water at room temperature. The 
constants for water at 20 C are v M (2) = 1, jr (1) 5.0 X 10 4 cm 3 /g, 
1 M = 2.45 X 10 l erg/g, <r = 72.3 dyne/cm. Thence and from (12.19), 
with C = - 2/r, 

(12 .23) 

Even for tiny droplets of the radius r = 10 ~ 5 cm the boiling point 
is only 0.174 degree below that at a plane surface. However, the sur- 
face layer is only two or three molecular layers thick, so that our for- 
mula is quantitative down to about r = 5 X 10~ 7 cm. 

Why then pay much attention to this apparently insignificant phe- 
nomenon? Because it plays a large role in the processes of trans- 
formation of state, as they actually occur in nature. Let us consider, 
for instance, the condensation of water vapor when its temperature 
falls slightly below the point of saturation. Granting that the boun- 
dary layers of the vapor can and do condense on the surface of the 
container, its main mass is spread through the volume. Here the 
condensation must first form a liquid nucleus before it can proceed 
further, a tiny droplet which has a low boiling point, according to 
eq. (12.23), and is, therefore, unstable. Instead of growing it is more 
likely to dissolve by vaporization. This is the reason why condensa- 
tion does not take place through the volume of a completely pure 
vapor. Only when dust particles are present, do they offer to the 
vapor a surface of reasonably low curvature and act as nuclei of con- 


densation. Very effective in this respect are electrically charged 
particles: their field produces an electrostatic tension at the surface of 
the condensed droplet which is inversely proportional to the fourth 
power of its radius 1 . When the radius is very small, it may overcom- 
pensate the effect of the curvature and make the condensation easier 
than at the plane surface. 

Quite similar conditions obtain in the case of vaporization. When 
water boils at 100 C, the outer pressure does not prevent small vapor 
bubbles arising in it. However, from the point of view of the water, 
the surface of the bubbles has a negative curvature, so that the boiling 
point in them is above 100 and they are unstable. Only when bub- 
bles of gases already exist in the water from the start, does the boiling 
take place through its volume. 

In the same way, surface tension has an effect on transformations 
in condensed phases. It is well known that crystallization does not 
begin without suitable nuclei, and the explanation is the same as for 
the condensation of gases. The case of fusion is also interesting, since 
most solids consist of very small microcrystals. The phenomena in 
crystals are somewhat complicated by the fact that the different 
crystallographic planes have different surface tensions, but, qualita- 
tively, they are similar to those treated above. We should, therefore, 
expect that, with the rise of temperature, the smallest units and the 
sharp corners and edges of the larger ones would melt first (because 
these parts have the lowest point of fusion) while the larger crystals 
swim loosely in this melt. This influence of microcrystalline structure 
upon the process of fusion has not yet been sufficiently appreciated. 
However, it seems to have a bearing on the observation of Errera that 
glacial acetic acid and a few other organic solids show an abnormally 
high dielectric constant two or three degrees below their melting 
points. 2 

1 The electrostatic tension or additional difference in pressure between the atmos- 
phere and the interior of the droplet, due to the charge e on it, is equal to e*/Sira 4 . 
This term must be added to aC in eq. (12.17). 

2 J. Errera (Trans. Far. Soc. 24, p. 162, 1928) determined the dispersion curve 
of the dielectric constant in the vicinity of the melting point and found that the effect 
in acetic acid has its maximum at 3 below 7> and for a critical frequency of electro- 
magnetic waves v = 10 3 . Assuming that the phenomenon is due to free solid particles 
capable of rotating in the liquid melt, my colleague Prof. G. W. Potapenko calculated 
from the critical frequency the mean radius of the particles and obtained r - 3 X 10 ~ 8 
cm. This result agrees well with the formula (12.19) which gives for the surface 
tension <r = Jrp 2 /jif &T/T. Substituting A r = 3 (as found by Errera), IM 1.9 
X 10* erg/g, P2 - 1.266 we obtain (at T - 298) a - 40 dyne/cm as the surface 
tension of solid acetic acid. This value lies in the range found experimentally and 
theoretically for other solids (section 86), 


Attempts to use eq. (12.22) as a means for determining the 
surface tension of solid salts against their saturated solutions were not 
very successful. The solubility of barium sulfate and calcium sulfate, 
in several states of dispersion, was measured by determining the elec- 
tric conductivity of the solutions 1 which increased with the dispersion. 
But the very high surface tensions calculated from these data 2 have 
been questioned since there are possible sources of error which would 
produce a similar rise of the conductivity. 3 

86. Formulas for surface tension. Only when the molecular 
forces holding a substance together are completely known, is the 
theoretical calculation of its surface tension possible. So far, this 
knowledge is available only in the case of crystalline solids and, even 
here, it is restricted to a few simple space lattices. For heteropolar 
halogen compounds of the type of NaCl (rock salt), KC1 (sylvine), etc. 
Born and Stern 4 calculated the following expression 

er = 4.022 X 10 3 . * . , (12.24) 

Ai + A 2 

where p is the density of the crystal and A\ t A% the atomic weights. 
This gives: 


Substance LiCl NaCl KC1 NaBr KI 

<r (dyne/cm)... 320 149 108 125 76 

Although the Born-Stern formula is only an approximation, the 
values derived from it are, presumably, as accurate as the results of 
the very difficult experiments with surface tensions of plastic solids. 
Experiments of this kind gave for pitch, at 20 C, about 50 dyne/cm 
(Ignatiew), and for lead glass, at 500 C, 70 dyne/cm (Berggren). 

Eotvos 5 proposed a formula for the tension of liquids against their 
own vapor which he obtained by a partially theoretical reasoning. He 
starts from the assumption that the extended law of corresponding 
states of section 27 applies also to the surface energy <r S which must 
be treated on the same footing as the volume energy pV or the internal 
energy U. In other words, for substances which obey the law of cor- 

1 Hulett, Zs. phys. Chcm. 37, p. 385, 1901; Dunadon and Mack, J. Am. Chem. 
Soc. 45, p. 2479, 1923. 

W. J. Jones, Zs. phys. Chem. 82, p. 448, 1913. 

Balarew, Zs. anorgan. Chem. 154, p. 170, 1926. 

* M. Born, Verh. Deutsch. phys. Ges. 21, pp. 13, 533, 1919, 

8 R. Edtvos, Ann. Phys. 27, p. 448, 1886. 

XII 86 



responding states, the ratio cr 2/r must be a universal function of the 
reduced variables TT, T. If we imagine 1 mol of the liquid substance 
expanding with the temperature in such a way as to retain its geometri- 
cal similitude, its free surface S will change proportionally to v*. 
Therefore, the extended law of corresponding states leads to 

av** _ 

~r~ ^ ' (7r ' T '* 

When the function / is found for one substance, it is the same for 
all others obeying the law of correspondence. From experiments with 
ethyl ether between and 190 C, Eotvos inferred that the function 
is/(7r, r) = 2.22(l/r - 1) = 2.22(r c /r - 1). Hence 

<rv" = 2.22 (T e - T) 


must be a universal relation between surface energy and temperature. 
In fact, this formula, due to Eotvos, holds with a fair approximation 
for numerous substances. 

Ramsey and Shields 1 secured a better fit with experiments by 

ov* = k(T c - T - d), (12.26) 

where k is slightly variable (but not greatly different from Eotvos' 
value) and d is a new constant (Table 34). The formula loses, of 
course, its theoretical significance and becomes empirical. 



t c C 



Range in C 

Ethyl ether 



8 5 


Methyl formate 



5 9 


Ethyl acetate 



6 7 


Carbon tetrachloride 


2 1052 






6 5 




2 077 

6 3 


A purely empirical formula, giving the surface tension as a function 
of temperature, is due to Van der Waals 2 


1 W. Ramsey and J. Shields, Zs. phys. Chem. 12, p. 433, 1893. 
2 J. D. Van der Waals, Proc. Amsterdam, 1893. 


It has been extensively tested by Verschaffelt l and represents the 
data for a number of substances fairly well. 


Ether logio A = 1.761, B = 1.270 

Benzene 1.839, 1.230 

Chlorobenzene 1.810, 1.214 

Carbon tetrachloride 1 . 445, 1 . 185 

87. Influence of temperature and of impurities. The experimen- 
tal data on surface tensions in solutions can be summarized as follows. 
All the solutes can be divided into two classes: 

(A) Inactive solutes produce a slight increase of o-, proportional to 
ttieir molality nth (provided the concentration is not high) 

ff = <r (l + km*), (12.28) 

where ox> is the surface tension of the pure solvent. Alkali salts are 
inactive in aqueous solutions and the coefficient k is of the same order 
of magnitude for all of them, namely, between 0.0174 and 0.0357. 
Similar conditions obtain for solutions of aromatic substances, like 
camphor, aniline, benzoic acid in organic solvents. 

(B) Active solutes have the opposite property of strongly lowering 
the surface tension. Such are many organic substances in aqueous 
solutions, for instance, alcohols, aldehydes, ethers, fatty acids, ter- 
penes, etc. Soaps are particularly effective: a solution of sodium 
oleate of the molality m^ = 0.002 has a surface tension <r = 25 dyne/cm, 
while that of pure water is <TO = 73. The cleaning properties of soap 
solutions rest on the low values of their surface tensions. 

In a qualitative way these facts were explained already by Gibbs. 
Let us write down the expression for the differential of internal energy 
of the surface layer. We consider the case that the mass of the system 
is variable and write dU in a form analogous to (5.06), taking into 
account (12.01) and (5.40) 

dt/< 3) = TdS+ <r dS + ?*W 3 >. (12.29) 


The term pdV does not appear in this expression because it is 
included in 0-dS (as far as the surface layer is concerned), as was 
explained in discussing the formula (12.09). Similarly, N h represents 
here the difference between the actual value which the mol number of 
the components h has in the whole system and that which it would 

1 J. E. Verschaffelt, Mitteil. Naturk. Labor. Gent No. 2, pp. 16, 19, 1925. 


have if the surface layer were replaced by a sharp discontinuity. In 
other words, Nh (3) is the quantity of the component h (in mols) adsorbed 
by the surface layer. Although not expressly stated in section 82, it 
follows from it that ^ in a surface layer of negligible curvature is the 
same as in the two adjacent volumes. For the further discussion it is 
convenient to introduce the generalized work function 

whose differential is 

^(3) = - sdT - W 3) <ty k + <* <&. (12.31) 


We now make the assumption (borne out by all observations) that 
the work function has the analytical form ^ (3) = fa S, where fa, the 
work function per unit area, is a specific thermodynamical characteristic 
of the surface layer independent of its area S. Substituting into 
(12.31), we obtain 

o- = fa. (12.32) 

The surface tension is identical with the specific work function per 
unit area of the capillary layer. Moreover, we find 

We apply the last formula to the special case when one of the 
phases, adjacent to the surface layer and containing the component 
(h), is either a perfect gas or a dilute solution in the sense of Chap- 
ter IX. According to (9.09) and (9.10), <p h = <p h + RT\ogx h , where 
<ph is a function of p and T only. The partial with respect to 53* in 
(12.33) is taken in the sense that all the other variables (including 
T and p) are to be kept constant: 3^ = RT'd log #&. In dilute solu- 
tions Xh can be replaced by WA/WO, so that 

This formula is the key to the behavior of solutes. (A) In the 
case of inactive solutes: fiff/dm*, > 0, whence n^ h < 0, i.e. the adsorp- 
tion (per unit area) of the capillary layer is negative. This means 
that the concentration of the component (h) is less in the capillary 
layer than in the bulk of the solution. As we deal here with dilute 
solutions, the solute can exercise but a small effect upon the surface 
tension, in agreement with the observed facts. (B) In the case of 


active solutes there follows n^h > 0, the adsorption is positive. The 
surface layer is very much smaller in volume than the adjacent phases 
and may build up to a high concentration in the solute (A), by adsorb- 
ing it even from a very dilute solution. This is the explanation why 
the increase of the surface tension by a small amount of solute is 
always slight, while the decrease may be considerable. 

It will be well to say here a few words about the temperature 
dependence of the surface tension. As the data of Table 32 indicate, <r 
decreases with rising T. Moreover, it may be concluded from the 
good fit of the formulas (12.25) and (12.26) that this drop is linear 
over wide ranges of temperature. In fact, the molal heat v appearing 
in these expressions changes but little with T and can be regarded as 
constant in the first approximation. The underlying cause of the 
linear drop can be inferred from the following considerations. Let us 
apply eq. (12.30) to a pure substance: the question of adsorption has, 
then, no interest and we may leave out the last term, writing it in the 
form (dividing by S) fa = w 2 - 7s s or from (12.32), (12.33) 

Within the range where a can be represented as a linear function 
(<r = cri aT) this gives 

u% = <7i = const. 

The internal energy (per unit area) of the capillary layer is constant 
and independent of temperature. This ceases to be true in the vicinity 
of the critical temperature T where the linear laws (12.25) and (12.26) 
are no longer valid. 

The understanding of the physical nature of certain surface films 
(oil films) has been considerably advanced by investigations of Harkins 
(and collaborators), Langmuir, and Adam, 1 but this subject lies out- 
side the scope of the present book. 

1 W. D. Harkins, F. H. Brown, and E. C. H. Davies, J. Am. Chem. Soc. 39, 
p. 354, 1917; I. Langmuir, ibidem 39, p. 1848, 1917; N. K. Adam, Proc. Roy. Soc. 
(A) 99, p. 336, 1921; 101, pp. 452, 516, 1922; 103, pp. 676, 687, 1923. 



88. The principle of Thomsen and Berthelot. It was emphasized 
in Chapter IV that the fundamental eq. (4.07) by means of which the 
concept of the entropy is derived from the second law is a differential 
equation. Because of this, the definition (4.09) of S contains an 
additive constant of integration so that the absolute value of the entropy 
cannot be determined from the first and second law. The indeter- 
minateness of the entropy constant has serious implications for the 
general character of the thermodynamical laws, in that they contain 
coefficients of indeterminate numerical value. As an example, we can 
point to the law (8.17), (8.16) of chemical equilibrium of perfect gases: 
The expression of the equilibrium constant K contains a factor /, 
which depends, according to the formula (8.14), upon the entropy 
constants of the gases taking part in the reaction. It is clear from 
this that the knowledge of the absolute entropy would be of inesti- 
mable value, as it would greatly enhance the usefulness of many 
thermodynamical formulas. To obtain this knowledge one must 
have recourse either to experiment, or to new theoretical principles 
not contained in the first and second law. Of course, it is quite suf- 
ficient to determine the entropy of a substance in one special case, in 
order to deduce the entropy constant, and it was pointed out by 
Nernst (1906) that experimental data were available to make a 
general conclusion about the behavior of the entropy of condensed 
systems in the vicinity of the absolute zero point of temperature. 

Let us start our considerations from the Gibbs-Helmholtz equa- 
tion (5.48) relating to an isothermal process 

We have seen in section 36 that, under ordinary conditions, the 
process (reaction) takes place spontaneously when A$>0. The dif- 
ference of thermodynamic potentials A* m $ 2 $1 can be regarded, 
therefore, as a measure of the chemical affinity. Suppose now that 


the partial (d&$/dT) p does not become infinite; when T goes to zero. 
In this case we find, putting T = 0, 

AX) = 0, (13.02) 

or lim Tm0 (&$ Q p ) = 0, if the process is conducted also at constant 

In the vicinity of the absolute zero point the heat of reaction 
becomes identical with A< and the criterion of the reaction taking 
place can be replaced by AX > or Q p > 0. For a long time it 
was thought that this form of the criterion holds good at any tem- 
perature. In fact, Thomsen and, independently, Berthelot stated the 
principle that out of a number of possible reactions the one will take place 
which evolves the largest amount of heat. According to our present 
knowledge, this would be true if the right side of eq. (13.01) were 
always negligible so that we could write at all temperatures 

A$ - AX = 0. (13.03) 

As we have already mentioned in section 37, this equation is 
erroneous. Modern investigations have shown that the term 
T(d&$/dT) p must be taken into consideration. Nevertheless, the 
approximation given by the principle of Thomsen-Berthelot, and by 
eq. (13.03), is surprisingly good even at room temperature, provided 
the system is condensed and the affinity of the reaction not very small. 
A comprehensive set of measurements of A<i> = $2 3>i for reactions 
in condensed systems was carried out by T. W. Richards 1 with the 
result that the difference A< AX was very small, indeed. 

Exercise 99. Check that at room temperatures r(DAf>/3D p is small compared 
with A* for the reactions in galvanic cells given in formula (5.52) and in exercises 
56, 57. 

89. Nernst's own formulation of his postulate. This last fact led 
Nernst to assume that in condensed systems the right side of eq. (13.01) 
decreases more rapidly, when the temperature goes to zero, than is 
accounted for by the factor T. According to him, the remaining 
factor also decreases to naught in the limit T = : for any isothermal 

The assumption expressed in this formula is known as Nernst's 
postulate or the third law of thermodynamics. That it is natural and 

* T. W. Richards, Zs. phys. Chemie 38, p. 293, 1902, 


convincing is best brought out by the following considerations. Equa- 
tion (13.01) can be written in the following form 

If the partial on the left side does not become infinite, there follows 
from (13.01) that both AS> AX and T become zero in the limit 
!T = 0. It must be well understood what a partial with respect to T 
at constant pressure means in this case. The pressure in the two terms 
of A$ = $2 $1 and AX = X2 Xi is not necessarily the same: 
A<i> = <S>2(r, 2 ) - 3>i(r,i), etc. But the partial is taken so that it 
remains equal to p% in the first term and to p\ in the second. Let us 
now take the limiting values of both sides of eq. (13.05) for T 0, 
keeping the pressure constant (i.e. equal to p% in <i>2 and X2 and to 
pi in <J>i and Xi). The right side is indeterminate, being equal to 0/0, 
but the indeterminacy can be readily resolved by means of the well- 
known mathematical rule : instead of the numerator and the denomina- 
tor, one has to substitute, respectively, their partials with respect to T. 

/9A$\ /3AX 

mr - w- hmr - Var /," hmr - 


0. (13.06) 

If we plot the change of the heat function AX in any isothermal, 
chemical or physical, process against the temperature T at which it is 
conducted, the curve has a horizontal tangent at 
the limit T = (Fig. 44). We repeat that this 
conclusion is independent of Nernsfs postulate'. 
the only assumption on which it rests is that 
OAS/Q^Op does not become infinite for T = 0. 

The difference A<J> is equal to AX in the limit 
T = 0, and is very little different from AX over 
a considerable range of temperatures, according FlG 44. 

to the experimental data mentioned in the preced- postulate. 

ing section. Nernst's heat postulate (13.04) has 
the purpose of insuring such behavior. The curves for A< and for 
AX have, if it is accepted, a common horizontal tangent, at T = 0, 
and remain for some distance close to each other. 

Owing to eq. (5.37), (d$/dT) p S, an alternative form of 
Nernst's postulate (13.04) is 

lim r .o(5 2 - Si) - 0. (13.07) 


The entropy remains unchanged in any isothermal process taking place 
in a condensed system in the vicinity of the absolute zero. 

Although Nernst enunciated his postulate only for condensed 
systems, this restriction may now be dropped for the following reasons: 
(1) In sufficient proximity to the zero point of temperature every known 
system of nature is condensed, if the pressure is finite. (2) We shall 
see in Chapter XVI that even the ideal equation of perfect gases has 
been changed by the modern theory of statistics so as to satisfy 
Nernst's law. 

Exercise 100. Starting from eq. (5.58) prove: 

(a) If limr-oteA^/aTV ^ , then lim^.o (&&U/dT)v - 0. 
Use the same argument as that used in the text for proving (13.06). 

(b) From (a) and from (13.06) there follows 

limr-o (CV2 - CVO - 0, Hmy.o (C P2 - C p i) = 0. 

These results are independent of Nernst's postulate. 

(c) Equation (13.07) requires 


90. Thermal expansion in the vicinity of the absolute zero. There 
can be deduced from the formula (13.07) some immediate consequences 
about the properties which all substances have in their condensed state 
at temperatures close to T = 0. Let the process to which we apply 
this equation consist, merely, in an infinitesimal increase of pres- 
sure (from pi = p to p2 = p + dp) at constant temperature. The 
entropy of the system is changed, accordingly, from Si = S to 
S 2 = S + (dS/dp) T dp, and Nernst's postulate (13.07) takes the form 

lim r .oO5/3/>)r = 0. (13.08) 

Because of the relation (4.46) of section 28 this is identical with 

lim r . (||) =0. (13.09) 

The coefficient of thermal expansion vanishes in the vicinity of T = 0. 
This fact is important in itself and leads to some interesting inferences. 
If the pressure is kept constant, the volume is sure to remain constant 
automatically: in the limit T = 0, a partial with respect to T at 
constant p is, at the same time, a partial at constant v. In the formulas 
of the last sections we could replace the subscript p by v or drop it 
altogether. According to the definition (3.08) of the specific heat 
this applies also to the quantities c p ,c v , 

lim r _ C P =* lini r .o d Hm r . c. (13.10) 


We can readily obtain the generalization of the formula (13.08) 
for the case when the system is not simple but is described by the 
variables Xi, J^2, X n and the generalized forces y\, 3^2, ... ^n, in 
addition to T (compare section 7). In fact, let us consider processes 
which consist in changing only one parameter, while the rest of them 
remain constant: for instance, Xh changes to Xh + dX^ while all the 
other Xj (j j h) and all the yj (including yh) stay constant; or else, 
y h changes to yh + dyn and y, (j 7* h) and all the Xj are constant. We 
find as above, respectively, 

oC oC 

lim r .o ^T - O t lim r . f- - 0. (13.11) 

dXh Qyh 

Because of (5.30) and (5.45) this is equivalent to 

Hm,.o - 0. fonr-o ? - 0. (13.12) 

The temperature gradients of all the parameters X^ yh describing the 
em vanish in the vicinity of T = 0. In particular, we conclude that 


91. Specific heat in the vicinity of the absolute zero. Nernst's 
postulate implies that the entropy is finite at all (finite) temperatures, 
including the limiting case T = 0. We can choose, therefore, T = 
as the lower limit of the integrals (4.33), (4.34) representing the 
entropy of a simple system 

*(7>) = f 



r 1 , JT 


These integrals must be convergent, a requirement which could 
not be satisfied, if c p and c v had a finite value for T = (because this 
would make the integrals logarithmically infinite). We must con- 
clude, therefore, that the limiting values of c p , c v vanish 

lim r- o c p = Hm r . c* * 0. (13.15) 

The specific heat vanishes in the vicinity of T 0: It should be 
noted that Nernst's postulate is not needed for this conclusion in its 


entirety: it is derived merely from the assumption that the entropy 
is not infinite in the limit T ~ (lim r . S ^ ) 

As far as the second law is concerned, the terms /i(fl),/2(/>) in 
(13.14) may be any functions of volume and pressure (section 26). 
However, if we apply to them the conditions (13.08) and (13.13) 
flowing from Nernst's postulate we find dfi/dv = 3/2/3/> = 0. These 
terms turn out to be independent of v and p and are, therefore, constant 

T fT 

%dT + s , *(P,T) = -* 

* '0 * 

dT + s , *(P,T) = -*dr + s . (13.16) 

SQ, the limiting value of the entropy at the absolute zero point, is a 
constant independent of the process by which this point is reached, as 
may be inferred from the requirement (13.07). From these expressions 
can be obtained the molal thermodynamic potential, for a condensed 
simple system, 

PV-T f 


c p dlog r- TSQ. (13.17) 

Exercise 101. Prove eq. (13.15) in a different way. The condition lim^-oS 5^ <*> 
can also be written in the forms limr-o (W/dT)v ^ and limr-o (d&/dT) p 
Derive from the eqs. (5.56) and (5.54) by the method used in obtaining (13.06) 

r n 

) - limr_o I - 


Hmr-o Cv = Hmr-o C p = 0. 

92. Planck's formulation of Nernst's postulate. It is obvious 
that the last result can be generalized for the case of systems which 
are not simple but depend on any number of variables. Let us sup- 
pose, in fact, that the zero point of temperature is attained at definite 
values of all the other variables and that the entropy is found to be 
equal to 5 . According to the postulate (13.07) or to eqs. (13.11), 
further changes in the variables (leaving the temperature constant) 
will fail to have any effect upon SQ. The state T = is, therefore, a 
state of constant entropy. 

Planck goes beyond Nernst in that he makes an additional assump- 
tion about this limiting value of the entropy: he postulates that the 
entropy in the state T = is equal to zero, in every system occurring in 

lim r . 5 = 0. (13.19) 


In the special case of simple systems, the entropy expressions 
(13.14) are reduced to 

/T /*T 

^dT, s(T,p) = / C -*dT. (13.20) 

1 A) * 

The modern statistical theory has borne out Planck's contention. 
When we refer to Nernst's postulate in the following chapters, we 
shall always mean Planck's formulation of it, unless expressly stated 

Equation (13.19) helps to determine the constants or arbitrary 
functions of integration in many thermodynamical relations. Thus, 
the presence of the functions /(/>), f\(v) in eqs. (5.55) and (5.57) is but 
a consequence of the indeterminacy of the entropy inherent in the 
second law. This is brought out, for instance, by partial integration 
(taking into account the formulas 3.12 and 3.26) 

The parentheses are equivalent to the entropy expressions (13.14). 
Replacing them by (13.20) we obtain 

* = T / U v d(l/T), * - T I X p d(l/T). (13.21) 
JQ ^o 

Since (3^/3 F)r = p, the first of these equations gives 

Knowing the internal energy as a function of T and 7, down to 
T = 0, we can deduce the equation of state. 

Exercise 102. The molal heats c p (in cal mol~ l , at 1 atm) of diamond, graphite, 
and copper are given in the following table: 

T ....... 20 K 30 40 50 60 70 80 90 100 150 200 250 300 

Diamond ............................ 0.01 0.06 0.33 0.60 1.00 1.50 

Graphite 0.02 0.06 0.10 0.14 0.18 0.24 0.30 0.33 0.38 0.77 1.2 1.4 2.05 

Copper.. 0.1 0.4 0.8 1.4 1.94 2.5 3.0 3.5 3.9 5.0 5.4 5.7 5.9 

Calculate in a rough, graphical way the absolute entropies of these substances 
at 300 K. 

93. Einstein's formula for the specific heat of solid bodies. In 
order to establish his postulate it became necessary for Nernst to test 
whether its consequences are borne out by experiment. In particular 


he carried out with his pupils l an extensive program of measurements 
on specific heats of solid bodies at low temperatures. In all cases the 
law expressed in formula (13.15) was confirmed: as the substances 
were cooled, their specific heat decreased, and, at the lowest point 
attained (temperature of liquid air, later liquid hydrogen and helium), 
it amounted to only a small fraction of its value at room temperature. 

The previous knowledge of this question had been as follows. In 
1819 Dulong and Petit deduced from their measurements the law that 
elementary substances in the solid state have all the same molal (or 
atomic) heat of approximately c = 6 cal/deg (section 16). How- 
ever, it was realized for a long time prior to Nernst's work that the 
law of Dulong and Petit is not accurate and that c varies with the 
temperature. A well-known example of this was the diamond, which 
exhibited, in a striking way, the type of behavior later found to be 
general by Nernst and his pupils. The specific heat of diamond has 
a value approaching the 6 cal/deg of Dulong and Petit only at 
temperatures which are comparatively high. It decreases rapidly upon 
cooling and is already pretty small (1.5 cal/deg) at room temperature, 
a fact which was the source of much embarrassment for the kinetic 
theory of matter. 

Boltzmann had given the following simple approach to the problems 
of the internal energy of elementary solid bodies: In such a body every 
atom has a position of rest about which it carries out thermokinetic 
oscillations. As a first approximation, one can regard the forces with 
which the atom is attached to its normal position as proportional to 
the distance from it. It is well known that forces of this kind (called 
" elastic forces ") cause the atoms to oscillate harmonically, with a 
constant frequency v. Since all atoms of an elementary body are 
equivalent, each of them is attached by the same force and has the 
same frequency v. What is the energy of such a model of the solid 
body, according to statistical mechanics? According to (4.71), the 
classical statistics of Boltzmann-Gibbs leads to the so-called principle 
of equipartition of energy. The mean kinetic energy of every particle 
subject to heat motion is kT/2 per each degree of freedom (where k 
denotes Boltzmann's constant of section 4). It was also stated in 
section 30 that, in the case of elastic forces, the mean potential energy 

1 Monographs on Nernst 's postulate: W. Nernst, Thermodynamics and 
Chemistry. New York 1907; W. Nernst, Die theoretischen und praktischen 
Grundlagen des neuen Warmesatzes. Halle 1918; F. Politzer, Berechnung der 
chemise hen Gleichgewichte. Stuttgart 1912. A very useful critical study of the 
subject is due to W. H. Van de Sande Bakhuyzen, Het Warrate-Theorema van 
Nernst, Thesis. Leiden 1921. 


is equal to the mean kinetic and the total energy becomes kT per 
degree of freedom. An atom is supposed to have only three degrees 
of freedom and, therefore, the mean energy 3 kT. Correspondingly, 
the internal energy of one mol is 

u = 3n A kT + UQ = 3RT + u Qt 
since WA* = R, according to eq. (1.14), and the specific heat 

c = du/dT = 3R. (13.23) 

Substituting R = 1.986 cal/deg, we obtain from this theory the 
numerical value c = 5.95, in good agreement with the law of Dulong 
and Petit. However, it does not offer any explanation for the devi- 
ations at low temperatures. 

This difficulty was resolved by Einstein l in one of his most impor- 
tant papers. He pointed out that the quantum theory provides a 
modification of statistics sufficient to account for these phenomena. 
In fact, Planck had found that, in the case of the harmonic oscillations 
of an electron of frequency v, the mean energy is not the equipartition 
value kT but kTx/(e x 1), where x is an abbreviation for x = hv/kT, 
and h denotes Planck's quantum of action (h = 6.55 X 10~ 27 erg sec). 
Einstein made the assumption that what is true for an oscillating 
electron must also apply to an oscillating atom. It was the first appli- 
cation of the quantum theory to atoms and molecules, and led directly 
to the result 


The entropy follows from the expression (13.20) of Nernst's theorem 
in Planck s formulation 


Y - log (1 - e-*)J. (13.26) 

When the temperature T is high compared with 9 = hv/k (i.e. 
x <3C 1), we find, neglecting terms of the order x 

u = 3RT + UQ, c - 3-R, 1 

5 = 35(1 -logic), .j 

1 A. Einstein, Ann. Physik 22, pp. 180, 800, 1907. 


The limiting values, for high temperatures, are the same as in 
classical statistics and in agreement with the law of Dulong and Petit. 
On the other hand, in the vicinity of the absolute zero point (r 0, 
#-*oo) the formulas (13.24), (13.25) reduce to u = UQ, c = 0, 
satisfying the requirements of Nernst's postulate. 

Einstein's formula agrees with the general trend shown by the 
experimental values of the specific heats but the numerical agreement 
is not very close. Later work on the subject, which resulted in more 
accurate formulas, will be treated in section 123. For theoretical uses, 
however (when a fictitious solid body is as good as real one), Einstein's 
model is very convenient because of its simplicity. 

Exercise 103. Find the work function (^ = Ts) of the Einstein model 
and calculate its approximate expressions for very high and for very low tempera- 
tures (x 1 and x 1). 



94. Chemical equilibrium and Nernst's postulate. The bearing of 
the first and second laws of thermodynamics upon the problem of the 
chemical equilibrium of perfect gases was discussed in Chapter VIII. 
It was stated in section 51 that the first step in solving this problem 
was the mass law of Guldberg and Waage (1867) 

> log p h = log K p , (14.01) 

which gives a relation between the partial pressures pi, . . . p<r of the 
gases taking part in a reaction 

^jV h G h = 0, (14.02) 


conducted at a given temperature. 

The next big advance was due to Gibbs (1871) who determined 
the dependence (8.16) of the mass constant K p on temperature. 
Within the range where the molal heats can be regarded as constant 

log K p =^ log T - % + log I. (14.03) 

Provided the heat of the reaction Q p and the change of the heat 
capacity AC in it are known, the constant of integration / can be 
determined from the measurement of K p at one single temperature. 
The formula (14.03) serves, then, to calculate the mass constant for 
any other temperature. We have seen that the constant / depends 
upon the entropy constant of the gases in a way expressed by eq. (8.14) 


This shows that a theoretical calculation of the constant / is beyond 
the scope of the first and second law of thermodynamics, which provide 



no way of determining the absolute value of the entropy. The formula 
(14.03) with an empirical constant / represents, therefore, all the 
information which can be obtained from the older thermodynamics, 
with respect to the chemical equilibrium of gases. It was pointed 
out in section 59 how Van t'Hoff (1878) succeeded in generalizing this 
formula (still with the empirical constant /) for the case of the chem- 
ical equilibrium of solutes in a dilute solution. 

The problem remained in this state until 1907 when Nernst brought 
his postulate to bear on it. It is possible to establish a relation 
between the entropy constants of a substance (ft) in its gaseous state 
soh and its condensed state W c) . In fact, let us consider the equilibrium 
between the vapor and the solid phase. According to eq. (7.02), the 
condition of equilibrium is * = gh<Ph (c \ where <p h without upper index 
represents the molal thermodynamic potential of the gas and ^ (c) that 
of the condensed phase while g h = M*/W C) = v h (c) /v h is the ratio of the 
molecular weights in the two phases. We suppose that the temper- 
ature and pressure are such that the vapor can be considered as a 
perfect gas. In this case we can substitute for ^ the expression 
(8.08), (8.10) where we consider c ph as constant, and for <(>h c) the eq. 
(13.17) and obtain a relation which may be written in the form 

<*r + iog4, (14.05) 


), (14.06) 

denotes the latent heat of vaporization per 1 mol of the vapor, and 


is a constant called by Nernst the chemical constant of the gas. 

Equation (14.05) is, of course, only the integrated form of eq. (7.05) 
of section 44, It is, however, important for us to know that the 
constant of integration ** is connected with the entropy constant by 
the relation (14.07) which permits us to express $o* in the form 

SOH = Ri h + ft** 1 *. (14.08) 

Substituting this into (14.04) we find 

R log I = #2>A*A -Z'A^W* (14.09) 

The second sum on the right side is of a structure familiar to us: 
It represents the limiting value (for T = 0) of the entropy change 
A5 in the reaction SJ>A (C) (?A (C> =0; in other words, it is the entropy 


change which would result if all the components taking part in the 
gas reaction (14.02) were condensed and the reaction conducted in the 
condensed state in the vicinity of T = 0. We know from Nernst's 
postulate in the form (13.07) that 

Z)* W W* = lim r-o AS = 0. 


Therefore the relation (14.09) is reduced to 

log /*]*. (14-10) 


This formula represents the third important advance in the 
problem of the chemical equilibrium of gases and is due to Nernst. 
While before 1907 the constant / had to be determined experimentally 
for every separate gas reaction, eq. (14.10) made it possible to compute 
it from the chemical constants i h of the components if they were known. 
It was sufficient to determine by measurement the constants 4 of the 
individual gases by investigating their equilibrium with the condensed 
phases in order to know the constants / for all possible reactions 
between them. 

We have derived the relation (14.10) using the more restricted form 
(13.07) of Nernst's postulate. If we accept Planck's formulation of 
it (13.19), we have to put W c) = 0, and this reduces eq. (14.07) to 

4 = SM/R. (14.11) 

From this point of view the chemical constant of a perfect gas is, 
simply, proportional to its entropy constant. Nernst's relation 
(14.10) is then an immediate consequence of the definition of the 
constant / given by eq. (14.04). 

95. The Sackur-Tetrode formula for the chemical constant. The 
experimental determination of the chemical constant i involves 
very difficult measurements. It was, therefore, fortunate that in 
1912 Sackur l and Tetrode 2 succeeded in making an important step 
beyond Nernst. Working independently they gave theoretical expres- 
sions for the chemical constant of monatomic gases which differed but 
slightly. Tetrode's formula proved to be the correct one 

1 0. Sackur, Nernst-Festschrift, p. 405, 1912; Ann. Physik 40, p. 67, 1913. 
1 H. Tetrode, Ann. Phyaik 38, p. 434; 39, p. 255, 1912. 


Of course, this result could not have been obtained by thermo- 
dynamical reasoning: the resources of thermodynamics, including 
Nernst's theorem, are exhausted with the formula (14.10). Sackur 
and Tetrode used statistical methods about which we shall say more 
in the next section. 

The argument of the logarithm in eq. (14.12) is not a pure number, 
so that the numerical value of i depends on the system of units used 
for the description of the gas. The reason of this lies in the structure 
of eq. (14.05), where in addition to two terms of dimension zero, we 
have two logarithmic terms. Taking into consideration that, for a 
monatomic gas c p /R = f, the combination of the two logarithmic 
terms is log (p/T^). This term depends on the units in which the 
pressure and the temperature of the gas are measured and, therefore, 
the same is true with respect to the chemical constant i. If we 
substitute into (14.12) the numerical values in the G.C.S. sys- 
tem R = 8.315 X 10 7 erg/deg, n A = 6.061 X 10 23 mol" 1 , h = 
6.554 X 10~ 27 erg sec, we obtain 

j = i - f = 10.171 + f log M , (14.13) 

an expression which corresponds to pressures measured in dyne/cm. 
For use with decimal logarithms, this expression must be divided by 
log 10 = 1/0.4343 

jio - no ~ 1.086 - 4.417 + f logio /* (14.14) 

If we wish to measure the pressure in atmospheres instead of 
dyne/cm and to use decimal logarithms, we have to subtract 
logio 1 013 249 = 6.0057, obtaining the constant 

jio, atm = iio, *tm ~ 1-086 = - 1.589 + f lo glo M- (14.15) 

It was later found that the Sackur-Tetrode expression gives the 
main part of the chemical constant but is incomplete: there are 
additional terms depending on the electronic and nuclear spins. The 
agreement of the theoretical values with the experimental is, on the 
whole, very good but it will be better to postpone the comparison 
until section 120 where we shall derive the missing terms. 

When accuracy is not required and the question is only as to the 
order of magnitude, it is permissible to neglect the refinements and to 
use the above expressions without correction. Therefore, it will be 
useful to write out here the explicit form which the equation (8.16), 


(8.17) of equilibrium of perfect gases assumes when the numerical 
values of the constants are substituted in the form (14.15) 

logio K p = ]C fc logio ph = 0.503 v f 2J vh logi 

lo glo 

The pressure p is here expressed in atmospheres, the heat of reac- 
tion Q p in cal/degree. 

96. Theoretical derivation of the Sackur-Tetrode formula. 
Although Sackur and Tetrode obtained the correct expression (14.12) 
of the chemical constant, the method of reasoning they used did not 
stand the test of later criticism, and cannot be considered as altogether 
valid. However, a rigorous derivation of this expression was given by 
O. Stern 1 in 1913. As this problem lies beyond the pale of pure 
thermodynamics, he had to appeal to the principles of statistics and 
of the quantum theory. But the statistical elements in Stern's 
argumentation are so simple that it can be given in its completeness 
even in a textbook of thermodynamics. The fundamental idea is to 
consider a vapor obeying the law of monatomic perfect gases in 
equilibrium with its condensed phase, and to calculate its pressures, 
on one hand, thermodynamically from eq. (14.05), on the other hand, 
statistically. The comparison of the two results leads to the expression 
for the chemical constant. This constant determines the entropy of 
the perfect gas and, therefore, it can depend only on the properties 
of the gaseous phase of our system remaining the same no matter 
what kind of condensed phase is in equilibrium with it. It is, there- 
fore, permissible to consider the equilibrium even with an imaginary 
condensed phase which does not really exist but could exist, inas- 
much as it is constructed in agreement with the laws of nature. Stern 
takes as the condensed phase Einstein's model of the solid body 
described in section 93 and assumes that the temperature T of the 
system is sufficiently high compared with = hv/k to use the approx- 
imations (13.27). 2 We substitute them into eq. (14.05), taking the solid 
body also as monatomic (g = 1) and its volume as negligible (v 0) 
and recalling that for a monatomic perfect gas u 3.RZV2 + wo, 
c p = 5.R/2. If we drop the subscript 

log = f + f log T - 3[1 - log (hp/kT)] - "~^ + * 

. - | - log T + 3 log (hv/k) - U ^ (C) + i. (14-17) 

* O. Stern, Phys. Zs. 14, p. 629, 1913; Zs. Electrochemie 25, p. 99, 1919. 

pifferently from the preceding section, v is here the freqnency. 


With this result the thermodynamic part of the investigation is 
concluded and we proceed to the statistical part. We have specified 
that the temperature is high, and this permits us to use the principles 
of statistics in their traditional form since, in this case, the differences 
between classical and quantum statistics disappear. All we need of 
the quantum theory is already contained in eq. (14.17), which embodies 
the properties of a quantized solid. In Einstein's model every atom 
of the solid body is regarded as a harmonic oscillator attached to the 
position of rest by a force F proportional to the distance r from it. 
The equation of motion of an atom is, then, 

where M is its mass and co = 2irv its angular frequency. The right side 
of the equation represents the force F, so that the potential energy of 
an atom at the distance r from its position of equilibrium is 

e r = - f F dr = Mu 2 r 2 /2. (14.18) 


These assumptions were sufficient for the theory of specific heats 
(section 93) but, in order to calculate the equilibrium of the solid with 
its vapor, we must specify them a little farther. How far does the 
field of the force F extend? The atoms are all considered as indepen- 
dent in their oscillations so that it cannot reach as far as the next atom. 
To satisfy this requirement we imagine, around every position of rest 
as a center, a sphere of the radius a. Inside each sphere there exists a 
radial force F = u?Mr while the space outside the spheres is field- 
free. The model of the solid body is reduced to a number of little 
spheres with fields of forces in them. The atoms which are inside the 
spheres belong to the condensed phase, while the free atoms outside 
form the vapor. 

Let dr be an element of space in one of the spheres situated at the 
distance r from the center, and let Zrdr be the mean number of atoms 
in this element (i.e. the mean of many observations taken at different 
times). The only proposition from statistical mechanics which we 
need is Boltzmann's principle (4.69) which tells us that the mean 
numerical density ZT is proportional to an exponential function of the 
potential energy e. 


C being a constant of proportionality. 


As the space outside the spheres is field-free, the potential energy 
eo of the free atoms is the same as that of an atom at the border of the 
sphere eo = Mw 2 a?/2. Denoting the numerical density (i.e. number 
of atoms per unit volume) of the free atoms by ZQ we can apply 
Boltzmann's principle also to them 

0o = Cexp (-eo/^r), (14.20) 

whence eliminating the constant C from the two equations 

z r = 2 exp 

If we integrate z r over the whole volume of the sphere, we obtain 
the mean number of atoms in it. It is obvious that this number must 
be equal to 1. In fact, in a real solid the number of atoms is equal to 
the number of positions of rest. If a part of the solid is vaporized the 
positions of rest disappear with the atoms. We can write, therefore, 

/FO / 
Zrdr = 4irzo0f I 


In the thermodynamic eq. (14.17) small quantities of the order 
x = hv/kT were neglected. To be consistent we must neglect terms 
of the same order also in the expressions derived from statistics. 
This remark shows the way of getting rid of the radius a. Since the 
Einstein model is not a real but an imaginary solid, the quantities v 
and a can be selected at our discretion and we can choose them so that 
y = Mu?a?/2kT will be still large at the high temperature at which 
x = hv/kT is very small, so that exp( y) can be made smaller than 
x. The integral from to a in (14.21) can then be replaced by the 
integral from to oo , since the two differ only by terms of that order. 
We obtain thus 

1 = so(27r*r/Mco 2 )* exp (eo/r). 

The potential energy of 1 mol of the gaseous phase is n A eo. It 
is the energy which would have to be expended to make n A atoms free, 
if all the bound atoms were at rest in the centers of their respective 
spheres. This state of affairs prevails at the absolute zero of temper- 
ature when the atoms have no kinetic energy. The physical meaning 
of n^eo is, therefore, the difference in the zero point energies of the 
gaseous and the solid phases: w^eo = wo W c) . Therefore, 

exp [(W c) - u )/RT] t (14.22) 


since 1? = kn A and o> = 2irv. The number ZQ defines the vapor pres- 
sure according to eq. (1.15), p = zokT. Substituting ZQ from (14.22) 
and taking the logarithm, 

log p = f log 2wM + | log k - 3 log h 2 - | log T 

~ (C) . (14.23) 

Comparing this expression with (14.17), we find that in forming 
the difference the quantities p, T, v, UQ o (c) are eliminated and we 
obtain for the chemical constant i precisely the expression (14.12) of 

In the preceding discussion we have not made use of the electric 
neutrality of atoms. All our considerations and formulas would apply, 
therefore, just as well to electrified particles attached to their respec- 
tive spheres by electric forces, provided a gas consisting of such parti- 
cles is subject to the law of perfect gases. The slight corrections 
which are necessitated by the existence of electronic and nuclear spins 
as well as the theory of chemical constants of diatomic and polyatomic 
gases will be dealt with in section 120. 


97. Unattainability of the absolute zero. In the years following 
the discovery of Nernst's theorem the question was much discussed 
whether it really represented an independent and new principle of 
science or was contained, in some way, in the first and second laws of 
thermodynamics. The relation of the isothermal !T = to the system 
of adiabatics, S = const, played an important role in these discussions. 
The second law not only states that the entropy is a function of the 
state but implies also that it is a unique function of the state. No 
system can have two different values of the entropy at the same time. 
In fact, if this were possible in some state of the system, this state 
would be common to two adiabatics (say, S = S\ and S = 2, where 
Si > S%) representing their intersection. This leads to a contradiction 
with the postulate contained in the second law that in an adiabatic 
process the entropy cannot decrease: starting from the entropy Si 
one could lead the process to the state of intersection and thence along 
the adiabatic 52. It follows that the system of adiabatics S = const 
is a family of surfaces which do not possess intersections or envelopes. 
However, it must be remembered that the adiabatic process is defined 
as one in which no heat is imparted to the system, its equation being 

dQ = TdS - 0. (15.01) 

In addition to the solution dS = 0, or S = const, this equation 
has also the singular solution T = which represents at the same 
time an isothermal and an adiabatic process. Taking the (/>, V)- 
diagram of a simple system as an example, one is inclined, at first 
sight, to expect conditions as in Fig. 45: the lines of constant entropy 
form a non-intersecting family (of which we give only the two members 
5 = Si and 5 = 2), while the curve T = cuts across. It seems 
possible to lead the system by a succession of reversible adiabatic 

1 With the kind permission of Yale University Press parts of this Chapter were 
patterned after the exposition by P. S. Epstein, Commentary on the Scientific Writ* 
ings of J. W. Gibbs, Article 0, Sections 6, 7. 



processes from the state A to the state D of a different entropy, and 
this would be a contradiction with the second law of thermodynamics. 
On the other hand, this contradiction would not exist if the curve 
T = were identical with one of the lines S = const, instead of being 
a stranger in their midst and cutting across their system. 

There are, however, two reasons why it is not permissible to 
conclude from this argument that Nernst's postulate is a consequence 
of the second law. The first of them was realized early, being brought 
home forcibly by the example of the perfect gas which obeys the 

second law but contradicts Nernst's theo- 
rem. In the (p, v) -diagram of the perfect 
gas the line T = is represented by the 
axes p = and v = which serve as 
asymptotes both of the system of isother- 
mals pv = const and of the system of 
adiabatics pv y = const. For this reason 
the line T = cannot be reached by any 
FIG. 45. State T - in finite and reversible adiabatic process so 
relation to adiabatics. that the conditions illustrated by Fig. 45 
do not apply. It is true that, in the case of 

the perfect gas, the isothermal T = also belongs, mathematically 
speaking, to the system of lines of constant entropy, S = const, being 
its extreme member S = <*> in view of the entropy expression (4.18) 

5 = c p log T - R log p + *o, (15.02) 

(where we suppose p j& 0, v = 0). Nevertheless, this system does not 
satisfy Nernst's theorem which requires that no change of entropy be 
possible for T = 0. By changing the pressure from pi to p2 we can 
change s in formula (15.02) by the finite amount As r-0 = R log (pi/p2), 
while it retains its singular value oo . 

The example of the perfect gas is a conclusive proof that the second 
law of thermodynamics does not contain Nernst's theorem in its en- 
tirety. But attempts continued to show, at least, that certain conse- 
quences deduced from it could be also obtained from the second law. 
In this connection there was advanced by Nernst the principle of the 
unattainability of the absolute zero of temperature. 1 If one of the states 
T = could be reached by a reversible adiabatic process, the diagram 
of the Fig. 45 would be an adequate illustration of the actual conditions. 
Pointing out that the succession of adiabatics AB, BC, CD could be 

1 W. Nernst, Die theoretischen und praktischen Grundlagen des neuen WSrme- 
eatzes. Halle 1918. 


used to decrease the entropy of the system, Nernst concludes that 
the assumption of attainability is in contradiction with the second law. 

This conclusion was challenged by Einstein, 1 who questions that the 
part BC of the process (corresponding to T = 0) can be carried out. 
A reversible process is only an ideal, and there is in reality always a 
certain degree of frictional waste of work, developing heat. However, 
the slightest amount of heat production would throw the system off the 
curve T = 0. To bring out the gist of Einstein's objection, we may 
amplify it as follows. From the point of view of physics, a process can 
only then be considered as defined and meaningful when an experimen- 
tal procedure is given by which it can be carried out. In general, the 
adiabatics and isothermals satisfy this requirement. In the case of a 
simple system, for instance, the adiabatic process is carried out by 
compressing (or expanding) the system while it is enclosed in a heat- 
insulating envelope, the isothermal process by the same procedure 
while the system is in thermal contact with a heat reservoir. The 
particular process T = has the peculiarity that, while it is isothermal, 
no heat is transferred in it to the reservoir. Nernst treats it, therefore, 
as an adiabatic process. This is equivalent to dropping the heat 
reservoir altogether and imagining the system adiabatically insulated 
in all three branches of the Fig. 45. However, the lack of experimental 
differentiation between the compressions along the lines BA and BC 
makes the branch BC meaningless. In fact, no experimental direction 
is given (or, indeed, possible) to insure that a compression, starting 
from the point B, should cause the system to move in the curve BC 
and not in BA. This is the second reason why the argument making 
use of the diagram of Fig. 45 is fallacious. 

These objections refer, however, only to the attempt to derive the 
principle of unattainability from the second law. If Nernst's postulate 
is accepted, unattainability becomes a matter of course since it states 
(in Planck's formulation) that T = coincides with 5 = 0. The 
process T = belongs, therefore, to the family S = const and is 
identical with its extreme member. As the adiabatics S = const do 
not intersect, it is obvious that 5 = cannot be reached by any 
reversible process represented by one of the other adiabatics. 

98. Solutions, supercooled liquids, and the statistical interpreta- 
tion of Nernst's postulate. Of great practical and theoretical import- 
tance is the question whether Nernst's postulate is a general law or 
admits exceptions for certain condensed systems. Some controversy 

1 A. Einstein, Structure de la matiere (Second Solvay Congress of 1913). Pub- 
lished 1921. 


arose with respect to solutions : Planck 1 maintains that his formula- 
tion of the theorem applies only to chemically pure substances while 
in the case of a solution or mixture one should add to the entropy the 

so = - R(Ni + . . . + #/)/, log *,, (15.03) 


where xi, #2, represent the mol fractions of the components of the 
system and N\, N2> their mol numbers. This term represents the 
entropy of mixing the components in the case of a mixture of perfect 
gases (compare section 50). According to Planck, a condensed solu- 
tion or mixture should have the same value of the entropy constant 
since it can be converted into the gaseous phase by vaporization. 

In this country Planck's views were advocated by Lewis and 
Gibson. 2 These authors went beyond Planck and questioned, on 
theoretical and experimental grounds, the applicability of Nernst's 
theorem to supercooled liquids even of a chemically homogeneous 
constitution. On the other hand, Nernst 3 had always claimed the 
general validity of his theorem and was supported by other authors. 4 
The contradiction was resolved, and partly reconciled, by an investiga- 
tion of O. Stern's 6 concerning the entropy of mixed crystals. 

It will be best to say here a few words about the statistical aspect 
of the problem, as both the nature of the difficulty and its ultimate 
resolution are reduced in it to its simplest terms. We have seen (in 
section 30) that in statistical mechanics the entropy of a state is inter- 
preted as proportional to the logarithm of its probability 5 = k log P f 
which, in turn, is measured by the number of ways in which the state 
can be realized. If we consider a chemically homogeneous crystal at 
T = 0, every one of its Z atoms has a perfectly definite position of rest. 
We have discussed this case in section 30 and seen that two probability 
definitions are possible. If we regard permutations of identical atoms 
as independent realizations of the crystal (specific definition), we obtain 
P. = Z! and So = k log ZL On the other hand, if (denying the possi- 
bility of telling whether the atoms were permuted or not) we consider 
all the permutations as one single realization (generic definition), we 
find P = 1 and So = 0. Nernsfs -theorem has, therefore, the generic 
probability definition as its statistical equivalent. 

1 M. Planck, Thermodynamik (6th ed), p. 285, 1921. 
* Lewis and Gibson, J. Am. Chem. Soc. 42, p. 1542, 1920. 
1 W. Nernst, Sitzungsberichte Berlin, p. 972, 1913. 
E. g. W. H. Keesom, Phys. Zs. 14, p. 665, 1913. 
1 0. Stern, Ann. Phyaik, 49, p. 813, 1916. 


For a mixed crystal consisting of atoms of two kinds (Zi and Z* in 
number) the generic definition gives the probability by formula (4.66). 
By the use of the Stirling approximation (log Z! = Z log Z) and of 
mol fractions, xi = Z\/(Z\ + Z*), etc., it can be written as 

Pi 2 - xr Zl x 2 ~ z *, (15.04) 

so that the zero entropy of the mixed crystal appears to be of Planck's 

5 W ~ R (Ni + N 2 )(xi log xi + x 2 log * 2 ), (15.05) 

since kZi = RNi, according to (1.14). If we call each of the realiza- 
tions of the mixed crystal (differing by the arrangement of at least 
one pair of atoms of the two kinds) a modification of it, the number of 
modifications is identical with Pi2 and we meet, apparently, with the 
same difficulty which was encountered by Planck. However, the 
difficulty is removed by the following remark. The number of possible 
modifications only then measures the probability in a meaningful way 
if all these modifications belong to the same thermodynamical state. 
It was pointed out by Stern that this is not always the case. The 
rearrangement of atoms changes in a slight degree the internal energy 
and the other thermodynamical functions of the mixed crystal. Every 
modification has, therefore, a slightly different thermodynamical 

Suppose now that we have a very large number Z of copies of the 
same mixed crystal at the temperature T. Owing to the thermokinetic 
motions, the atoms will be continually rearranging themselves, chang- 
ing the modifications to which the crystals belong. We may ask, 
therefore, how many of these copies are, at any given time, in the 
modification j having the thermodynamic potential $/, and we shall 
show in the next section that their number is 

Z y = const-exp (-*,/W). (15.06) 

Correspondingly the ratio Z//Z represents the probability of find- 
ing a given crystal in the modification j. Since the differences in the 
function $, are very minute, all the numbers Z/ are practically equal at 
all temperatures which are not too low. This means that all the 
modifications are equally probable representing statistically equiva- 
lent realizations of our crystal. It is appropriate to count, as the 
probability of the crystal, the number of these realizations, as was 
done in formula (15.04), and this leads to the expression (15.05) for 
the entropy of mixing. On the other hand, at very low temperatures 
even very small differences in */ begin to tell because of the factor. 


1/r in the exponent. The nurfibers Z, become unequal and, in the 
extreme case of the immediate vicinity of T = 0, the modification j m 
with the lowest thermodynamical potential $ m i n dominates to such an 
extent that all the others are impossible. In the vicinity of the abso- 
lute zero the mixed crystal is, therefore, in the modification j m . Since 
every modification can be realized in only one way, the probability 
of the crystal, instead of (15.04), becomes Pi 2 = 1 and the zero 
entropy So = 1, in agreement with Nernst's postulate. Offhand it is 
conceivable that several modifications could have the same thermo- 
dynamic potential, and its lowest value $ mln might be shared by n of 
them. In this case, the probability at T = would be Pi 2 = n and 
the entropy So = k log n. 'The purport of Nernst's postulate is, there- 
fore (as far as mixed crystals are concerned), that there is one and only 
one modification of lowest thermodynamic potential. 

Similar considerations apply to the entropies of supercooled 
liquids. In a liquid the positions of the atoms and molecules are 
irregular and the possibilities of arranging them more numerous than 
in a crystal. This makes their entropy, at high temperatures, larger 
than that of a crystal. At low temperatures, however, the regular 
crystalline modification is the most probable and, in the vicinity of 
r = 0, even the only possible arrangement of atoms. 

From the theoretical point of view Nernst's postulate is, therefore, 
completely vindicated since there is no evidence that it admits of any 
exceptions. However, the differences in $/ on which the above argu- 
ment depends are so very small that one can expect a perfectly regular 
arrangement only at extremely low temperatures which are, in many 
cases, below the range accessible to our experimental technique. 
Another important consideration is the time element since, under these 
conditions, all processes are extremely slow. For this reason one can- 
not be certain that the substances under investigation have reached 
the state of true thermodynamic equilibrium. In this sense, we may 
say that the view of Planck and of Lewis and Gibson is justifiable from 
the practical standpoint of experimental measurements. This is, in 
fact, borne out by the recent and very accurate observations l of 
Giauque and his collaborators. The method used by these authors 
is based on the fact that the absolute entropies of many diatomic and 
polyatomic gases can be calculated with great accuracy from spec- 
troscopic data (compare section 119). They take the same sub- 
stances in the solid state at very low temperatures and measure the 

1 Older work pointing in the same direction was due to Gibson, Parks, and 
Latimer (J. Am. Chem. Soc. 42, p. 1542, 1920) and Gibson and Giauque (ibidem 
45, p. 93, 1923). 


entropy difference attending their gradual transformation into gases 
by heating and subsequent vaporization. In this way it was found 
that hydrogen, 1 carbon monoxide, 2 nitrous oxide, 3 and ice 4 possess 
still appreciable entropies as solids at about 15 K. The lattice 
theory of solid hydrogen was treated by Pauling, 5 who showed 
that the crystals are built up of para- and ortho-molecules (com- 
pare section 118) whose distribution is entirely irregular at tem- 
peratures above 5 K. The interpretation which Giauque and 
co-workers give with respect to the other substances just mentioned is 
that their molecules are not uniquely oriented but even at temperatures 
as low as 15 K have still the choice between several orientations. 
Pauling 6 investigated the case of ice and finds the entropy R log (3/2) 
= 0.805 cal deg" 1 mol" 1 , due to the statistical weight of the indeter- 
minacy of orientation. This is in excellent agreement with the 
experimental value 0.82 cal deg 1 mol"" 1 found by Giauque and Stout. 

99. Equilibrium of modifications of a mixed crystal. There remains 
to prove the formula (15.06) on which the conclusions of the preceding 
section rest. We shall do this by an argument which is thermo- 
dynamical in its main points. Its essential idea (but not its mathe- 
matical form) is due to Stern. The formula in question refers to the 
equilibrium of a very large number of copies of the same mixed 
crystal. In order to deduce it, we must provide an ideal experimental 
arrangement by which such an equilibrium can be secured. Let all 
these crystal copies be contained in an enormously large vessel where 
they are floating in a neutral gas medium removed from the action of 
gravity. Through their Brownian movements they will set themselves 
into thermal equilibrium with the medium and with one another. 
Provided the size of the crystals is small compared with their mean 
distance, such a suspension can be regarded as a perfect gas with 
extremely large molecules obeying all the laws valid for gases (compare 
section 146). In our particular case every modification of the crystal 
may be considered as a separate gas, so that we have a mixture of as 
many gases as there are modifications. 

Any modification (1) can be converted through the internal 
rearrangement of its atoms into any other modification (2), and we can 
apply to this process the laws of chemical equilibrium of perfect gases. 

1 W. F. Giauque and H. L. Johnston, J. Am. Chem. Soc. 50, p. 3221, 1928. 
1 J. O. Clayton and Giauque, ibidem 54, p. 2610, 1932. 
R. W. Blue and Giauque, ibidem 57, p. 991, 1935. 

4 Giauque and Ashley, Phys. Rev. 43, p. 91, 1933; Giauque and Stout, J. Am. 
Chem. Soc. 58, p. 1144, 1936. 

L. Pauling, Phys. Rev. 36, p. 430, 1930. 

L. Pauling, J. Am. Chem. Soc. 57, p. 2680, 1935. 


In the law of reaction (6.40) we have to substitute v\ = 1*2 = 1, 

Gi - G 2 - 0, (15.07) 

while the equation of equilibrium (6.43) is reduced to 

<f>i - <p 2 = 0. (15.08) 

The difference between the system under consideration and the 
ordinary perfect gases of Chapter VIII lies in the fact that in the 
present case the thermodynamic potential consists of two parts 

9 = n + v (15.09) 

<pi is the " internal " potential of the crystals themselves, <p e the 
" external " part due to the Brownian motions. For <p e we have to 
use the expression (8.08) of the thermodynamic potential in perfect 
gases, omitting the terms UQ TSQ which refer to the intrinsic proper- 
ties of the " molecules " (here crystals) and are, therefore, included in 
<pi. The total potential is, therefore, 

V = T(R log p - c p log T + c p ) + Vi. (15.10) 

At this stage of the reasoning we have to invoke another result of 
statistical mechanics, the fact that the " external " molal heat c p (due 
to the movements of the crystals) depends only on the number of 
degrees of freedom of the " molecules " regarded as rigid bodies. It is, 
therefore, the same for all modifications: c p \ = C P 2. In substituting 
(15.10) into the condition (15.08), the terms with c p will cancel out 

RT log (Pi/fr) = - (<pn - ^2). (15.11) 

<pn is the molal thermodynamic potential, i.e. it is referred to n A 
crystals (n A being the Avogadro number). It stands, therefore, in the 
following relation with the thermodynamic potential <>,- of an individual 
crystal used in the preceding section <pn = n A $,-. Noting that R = n A k, 
we obtain 

log (Pi/fr) --(*i- *2)/*r. (15.12) 

pi and p2 are the partial pressures in our vessel of the two gases con- 
sisting of the modifications (1) and (2). These pressures are propor- 
tional with the numbers Zi and 2 in which these modifications are 
represented in the vessel, according to the formula (1.15). The last 
equation becomes, therefore, identical with (15.06), and this is what 
we set out to prove. 


We have obtained this result by applying to our imaginary sus- 
pension the laws of the classical perfect gas. It may be objected that 
these laws have doubtful validity at very low temperatures. How- 
ever, we shall see in the next chapter that the approximation given by 
them is the better, the larger the weight of the molecules. Since, in 
our ideal experiments, we may attribute to the crystals any size, the 
accuracy may be considered as of any desired degree even in the 
immediate vicinity of the zero point. 


100. Equation of state of the monatomic degenerate gas. Nernst's 
theorem was enunciated only for condensed systems, and it was 
realized, from the beginning, that the classical perfect gas fails to 
fulfill it. In fact, the entropy of the perfect gas has the form (4.18) 
which would need an infinite entropy constant in order to meet the 
requirement s r _ The coefficient of expansion of the perfect 
gas is, according to (1.19), a. = 1/T, while the specific heats c v and c p 
are constants independent of temperature (section 15), in contradic- 
tion with the two immediate consequences (13.09) and (13.15) of the 
theorem which would require lim r . a = and lim r . c = 0. Begin- 
ning with Sackur and Tetrode (section 95) there were many attempts 
to subject the perfect gas to quantum conditions and to obtain a cor- 
rected equation of state. As long as these endeavors were based on 
the classical statistics of Boltzmann-Planck they failed, but the 
advent of new types of statistics led to a great success, the discovery 
of the correct laws of perfect gases which happen to be in agreement 
with Nernst's theory. 

A textbook of thermodynamics is not the place to enlarge upon the 
statistical principles underlying the new theory of the perfect gases. 
We shall give only a brief reference to questions of statistics in section 
104 of this chapter and shall take the equation of state as given, in the 
same way as we did not inquire into the statistical justification of the 
classical perfect gas. If we restrict ourselves to monatomic gases, the 
corrected equation of state can be written in the form 


It differs from the old equation of perfect gases (1.13) only by the 
factor P(6)/9 appearing on the right side. By P(G) is meant a certain 
function of the argument 




where the letters have the same meaning as before. In particular, 
/x is the atomic weight, h Planck's quantum of action (6.554 X 10 ~ 27 
erg sec), and n A the Avogadro number. The function P(9) is so com- 
plicated that it cannot be given by an explicit formula but only 
implicitly. In order to define it we introduce two auxiliary functions 
F(A) and G(A) of a new argument A by the equations 

r ^' 




1 d-Ae~* 

We have to distinguish three types of perfect gases which differ, 
according to the value assigned to the constant 6 and according to 
the range of variability of the variable A : 

(1) The classical or Boltzmann-Planck gas 

6 = 0, 0^-4^oo. 

(2) The Fermi degenerate gas 

d =- 1, ^ A ^ oo. 

(3) The Einstein-Bose degenerate gas 

+!, O^A^l. 

When 5 =+ 1, A cannot be larger than 1 because otherwise the 
integrals (16.03) would be divergent. 

In all three cases the connection between F(A), G(A) t on one 
hand, and P(0), 0, on the other, is the same, namely: 

F(A) = 8-*. (16.04) 

This relation expresses A in terms of 9, so that G(A) can now 
also be regarded as a function of 6. Therefore, the second relation 
which we impose, 

P(0) = 0* G(A), (16.05) 

amounts to defining P(0) as a function of 0. 

It should be noted that the functions F(A) and G(A) are not inde- 
pendent but there exists between them the relation 

. (16.06) 

dA A 

which is readily obtained by multiplying the numerator and denom- 



XVI 100 

inator of the integral for G(A) by e*/A, then differentiating the result 
with respect to A and partially integrating with respect to x. 

In the range A ^ 1 the functions (16.03) can be represented by 
the following series 


The opposite case of very large values of A, (A ~2> 1) is of interest 
only in connection with the Fermi gas, 5 = 1. The approximations 
for the two functions are, then, as follows 


(log ^ 


In the case (d = 1) of the Einstein-Bose gas, A cannot be larger 
than 1 and, for the classical gas (5 = 0), no approximation is needed 
since F(A) = G(A) = A are, then, the rigorous expressions for the 

It is clear from the formulas (16.07) and (16.08) that F(A) is 
parallel with A in the sense of being small or large when A is small or 
large. On the other hand the equation (16.04) shows that 9 is in an 
inverse relation to F(A) and, therefore, to A: when A is small, is 
large, and vice versa. From the four equations (16.07), (16.04), and 
(16.05), we can eliminate by the method of successive approximations 
the three quantities A, F(A), G(A) and obtain the following approx- 
imate relation between P(0) and 6: 

(0 1) 

The same procedure carried out with the formulas (16.08), (16.04), 
and (16.05) applies only to the case d = - 1 of the Fermi gas 

(0 1) 


We see from (16.09) that, for very large values of the variable 9, 
the function P(9)/9 becomes closely equal to 1 so that eq. (16.01) takes 
the classical form pv = RT. By its definition (16.02) the quantity 6 
is proportional to v*T, it is large when the temperature is high or 
the density low. Under these conditions the gases of Einstein-Bose 
and Fermi do not deviate in their properties from the classical perfect 
gas: they are non-degenerate. As 9 decreases the deviation from the 
classical laws becomes more and more marked : the degree of degener- 
ation increases. However, we shall postpone the quantitative discus- 
sion of the degree of degeneration until section 105. 

101. The thermodynamic characteristics of the degenerate gas. 
From the equation of state (16.01) the internal energy can be obtained 
by means of the relation (4.23) 


or integrating with respect to dv, 

The function f(T) is determined by the condition that the gas is 
non-degenerate at high temperatures. The internal energy must take, 
for T >oo, the classical value (3.18) without becoming infinite for 
T = 0. It is convenient to change the variable of integration from 
v to 9, given by the relation (16.02) which can be also written 

f log 9 = f log T + log v + const. (16.13) 

Since T is to be considered as constant with respect to the inte- 


On the other hand, T(dp/dT) v - 9(3/>/99) t ,. After these substi- 
tutions the integration can be readily carried out, giving, with a suit- 
able disposal of the f unction /(r), 


This is, in fact, identical with (3.18) in the limit 9 -oo, since for 
a classical monatomic gas c v = ^ R. 


Knowing the internal energy u and the pressure p, we can calculate 
the entropy differential (4.15), ds = (du + pdv)/T. According to the 
expressions (16.01), (16.13), and (16.15) for p, v, and u, we can choose 
as variables T and G. The differential dv can be expressed in terms 
of dT and dQ: it follows from (16.13) 

3dQ ^$dT dv 

2 9 ~2 r + v 9 

This change of variables reduces ds to the form 

depending only on 9. 

Nernst's postulate in the form (13.19) can be directly applied only 
in the case of the Fermi gas. In fact, in the case of the Einstein-Bose 
gas the variable A cannot be larger than 1, and the relation (16.04) 
shows that 9 cannot sink below a certain finite limit and is not defined 
in the vicinity of T = 0. We pointed out in section 1 that the postu- 
late does not apply to the classical perfect gas. Therefore the validity 
of the simple entropy expression 

-i R J. -r (16 - 17) 

is restricted to the Fermi gas. The integration can be carried 
out by transforming back to the variable A with the help of 
the relations (16.04) and (16.05). The first of them can be written as 
log 9 = f log F(A) t or differentiating 

d9 2dF 

9 ~~3T' 

In this way the integrand is transformed into 

dP(Q) __$ G W5dG_2dG 
9 ~~ 3 F 2 + 3 F 3 F* 

The first two terms give $d(G/F) and we can write instead of the 
last -%dA/A, because of the relation (16.06). Hence 

The condition 5 * 0, at the limit 9 = 0, or A = oo is taken care 
of because, for very large values of A, the formulas (16.08) give 


log A. We shall show by a special investigation in section 
103 that the form (16.18) of the entropy expression holds also in the 
case of the Einstein- Bose gas. 

Finally, the formulas (16.01), (16.15), and (16.18) give for the molal 
thermodynamic potential <p = u Ts + pv the simple expression 

<p = RT log A + u Q , (16.19) 

which is true for both kinds of degenerate gases. 

102. Chemical constant of degenerate gases. Their relation to 
Nernst's postulate. We have seen in section 1 that for high tempera- 
tures or low densities (6 <3C 1 and A ^> 1) the gases of Fermi and 
Einstein-Bose are non -degenerate, approximating in their properties 
the classical perfect gas. When the variable A is extremely large, the 
functions (16.07) are reduced to F(A) = G(A) = A, whence eq. (16.04) 
gives A 9"^. The entropy expression (16.18), therefore, takes 
the form 


or because of (16.02) and R = n A k, 

s = R log v + %R log T + R i 1 
= %R log T - R log p + Ri, 


The entropy has the classical form (4.18), specialized for the monatomic 
gas (c v = !?, c p = -j^R), while the constant i exactly coincides, with 
the Sackur-Tetrode expression (14.12). Putting the question as to 
the validity of Nernst's postulate we must remember that we so far 
obtained complete expressions for the energy and equation of state 
(valid down to the vicinity of T = 0) only in the special case of the 
Fermi gas.- In deriving them we made explicit use of Nernst's theorem 
through eq. (16.17) so that there is no question but that it is fulfilled. 
It will be well, however, to check here the two immediate consequences 
from the theorem which we have derived in sections 90 and 91, that 
the specific heat and the coefficient of thermal expansion must vanish 
for T = 0. 

For the specific heat we have from (16.15) and (16.10) the value 


which is proportional to T, because of formula (16.02), and vanishes 
with T. 

For the coefficient of thermal expansion, eqs. (1.02) and (1.05) give 

Since p differs from u only by the factor 2/3v, 


(dp\ _ 2* 
\dT/ v 3 v ' 

so that a also vanishes for T = in a linear way. 

To elucidate the conditions in the case of the Einstein-Bose gas we 
need a separate investigation to which we now turn. 

103. Condensation of the Einstein-Bose gas. It was pointed out 
in section 100 that in the Einstein-Bose case (5 = 1) the variable A 
cannot be larger than 1. In view of this, 9 has a lower limit 9 m m 
determined by eq. (16.04) with 4 = 1: 

F(l) = 6-. (16.25) 

Now F(l) = 2.6123, and the numerical value of the limit is, there- 
fore, 6 m i n = 0.528. Since the parameter 9 depends on v and T 9 accord- 
ing to eq. (16.02), we may ask what will happen if, keeping the temper- 
ature of the gas constant, we should compress it sufficiently to make 9 
fall below the limit 9 mln (or else decrease the temperature keeping the 
volume constant). To this question Einstein gives the answer that a 
part of the system will cease to be gaseous and will be precipitated out, 
forming a condensed phase in equilibrium with the remaining part of 
the gas. The condition of equilibrium is given by the equation (7.03) 

9 - W, (16.26) 

where <p and <ppr represent the thermodynamic potentials of the 
gaseous phase and of the precipitate. Since the expressions (16.18) 
and (16.19), derived for the entropy and thermodynamic potential of 
the Fermi gas, give the correct value of the chemical constant, it is 
desirable to conserve them for the Einstein-Bose gas. Under the 
conditions of condensation (saturated state) 9 = 9 mln and A = 1, so 
that eq. (16.19) gives <p = UQ. Consequently, <ppr = u pr Ts pr + pVpr 
must be also equal to UQ, for all temperatures and pressures compatible 
with 9 9 m in. This requires u^ = o s pr = 0, v pr = and means 
that in the solid phase the material is much in the same state which 
prevails at T 0. It does not possess any thermokinetic energy, and 

XVI 104 



it has a perfectly regular arrangement of atoms and a negligible 

In support of Einstein's explanation we can point to the peculiar 
shape of the isothermals of the Einstein-Bose gas. 1 By means of (16.02) 
and (16.05) we obtain from (16.01) the expression for the pressure 

\ n A ) A 3 



and calculate from it the partial derivative 

/aA _ dp t dA 

\dv/T~ dA ' dF 



Because of (16.06) and (16.04), this gives 

dp\ _ RT 1 dA 
at; A" v 2 Q 3 A dF 



It is easy to see from the definition (16.03) of the function F(A) 
that (in the case 6 = 1) the derivative 
dF/dA becomes infinite for A = 1. Con- 
sequently, (dp/dv)T = 0, for A = 1, 6 = 9 m i n . 
In the state of saturation the gas is, there- 
fore, characterized by an infinite compress- 
ibility and its isothermals by horizontal 
tangents. These conditions are illustrated 
in Fig. 46, where the solid lines represent 
the isothermals and the dotted curve cor- 
responds to 8 = m i n . We have seen in the 
theory of the Van der Waals equation that 
the points (dp/d^T = are those where a 
single phase of the gas begins to be completely 
unstable. The condensation of the ideal gas 
of Einstein-Bose is, therefore, closely analo- 
gous to the conditions in real gases. 

104. Statistics underlying the theory of degenerate gases. Rule 
of alternation. 2 We have introduced and treated eq. (16.01) of 
degenerate gases in a purely formal manner. It will be well to say now 
a few words about its origin and about the question of which gases of 

1 Compare P. S. Epstein, Commentary on the Scientific Writings of J. W. Gibbs, 
Article 0, Section 10. 

1 For a detailed exposition of quantum statistical principles, see: P. S. Epstein, 
ibidem, Article V. 

FIG. 46. Isothermals in 

Einstein-Bose degenerate 



nature are subject to degeneration of the types of Fermi and of 
Einstein-Bose. Classical statistics and the statistics of the older 
quantum theory were dominated by Boltzmann's principle (4.70): 
If we have a single particle (or other system) which can assume dif- 
ferent dynamical states, the probability of finding it in a state char- 
acterized by the energy e, is 

P i = Cexp(-e,/*r), (16.29) 

C being a constant. Nothing is changed if we consider Z particles of 
one and the same kind under joint external conditions (field of forces 
or vessel), determining a set of energy levels ei, 62, . . . e/, . . . which 
any of the particles can assume. The mean number Z/ of particles 
occupying the energy level e/ is again given by 

Zy/Z = C exp (- e,-/*r), (16.30) 

because in the classical theory the particles are regarded as statistically 
independent: the probability of a particle assuming any state is not 
influenced by the presence of the other particles and remains the 
same as if it were alone. 

With the advent of wave mechanics the point of view had to be 
changed. In the case of a single particle, Boltzmann's principle (16.29) 
remains valid but Z particles forming a system of the kind mentioned 
above can no longer be regarded as statistically independent. 1 The 
primary particles of matter (the proton, neutron, and electron) fulfill 
the Pauli exclusion principle : No more than one of Z identical particles 
can occupy any given quantum state. (In the language of wave mechan- 

1 It was pointed out in section 30 that the fundamental difference between 
classical and quantum statistics lies in the question whether identical particles can 
be told apart. This question has an intimate relation to Heisenberg's principle of 
indetermination which states that the uncertainties A* (of a coordinate of position) 
and A/> x (of the conjugate momentum) are restricted by the inequality Ax- &p x <h/4T. 
In fact, when the uncertainty of position of each particle in a gas is small compared 
with the mean distance between the particles (v/n A )^, then there exists the con- 
ceptual possibility of keeping track of any particle and mentally identifying it as 
distinct from the others. Therefore, classical statistics must give a correct result 
when the condition (/n 4 )**Ax is satisfied. On the other hand, when the two 
quantities are of the same order of magnitude, there is no way of telling which 
particle is which, the concept of permutation loses its sense, and classical statistics 
can no longer be applied. The uncertainty of the momentum is, obviously, the 
mean momentum itself A/> x J, since the particlejnay have any velocity. As long 
as classical statistics (equipartition) is valid p x * M*v x * = MkT. We obtain, 
therefore the condition (MkT)^(v/n A )^ A/4r, or using the notation (16.02), 
6l/8r. This is the deeper reason why degeneration is absent when 6 is large 
and sets in when this quantity becomes small. 


ics the same thing is stated by saying that the primary particles have 
antisymmetric wave functions.) We cannot enter here into the details 
of the statistical calculations based on the exclusion principle and 
must be satisfied with their results: they lead to the modified formula 

(5 a l), which represents the fundamental law of Fermi's statistics. 
The quantity A gives a measure of the deviation from the classical 
conditions of the formula (16.29). It is, therefore, obvious that A must 
be a function both of the particle density in the system and of its 
temperature. In fact, the sum of all Zj is the total number of particles, 
Z, leading to the condition 

Another condition follows from the energy relations. The total 
energy of the Z, particles in the state j is e,Z, and, therefore, the total 
internal energy of the system U = 2e,Z, + UQ or 

U - U = ZC 


It is possible to eliminate C from these two equations and to obtain 
A as a function of U, Z, and T. 

Let us now turn from primary particles to composite ones, for 
instance, to atomic nuclei built up of protons and neutrons. If anti- 
symmetric wave functions are used for the protons and neutrons 
within the necleus, the translational motions of it, as a whole, are 
described as follows: (a) by antisymmetric wave functions, if the 
number of primary particles in the nucleus is odd, (6) by symmetric 
wave functions, if this number is even. Nuclei of the first kind have 
the same statistical properties as the primary particles themselves, 
they obey the exclusion principle and the formula (16.31) of Fermi's 
statistics (with 5 = 1). On the other hand, nuclei of the second 
kind possess entirely different qualities: the statistical calculation 
shows that in their case the number Z, can be expressed by the formula 
(16.31) with d =+ 1 which characterizes the statistics of Einstein-Bose. 
The law that the statistics of Fermi applies to composite particles 
with an odd number of primary elements, and Einstein-Bose's to those 
with an even number, is known as the rule of alternation. Of course, 
this rule is theoretically valid also in the case of neutral atoms which 


contain primary particles of three kinds: electrons in addition to 
protons and neutrons. The experimental test of the rule could not be 
carried out by observing the degeneration of gases (compare next 
section) but by study of molecular band spectra whose structure also 
depends on the statistical properties of the nuclei composing the 
molecule. In the present state of our knowledge the rule of alternation 
seems to hold without exception. Doubts exist only with respect to 
the sulfur nucleus, but the experimental data are insufficient to decide 
whether they are well founded. There was introduced into the theory 
of nuclear structure the hypothesis of the existence of another primary 
particle, the neutrino, which is supposed to have a very minute mass. 
This assumption proved very useful in the theory /3-ray emission and 
of nuclear transformations, but it seems that the neutrino could be 
dispensed with as far as the rule of alternation is concerned. 

The conditions (16.32) and (16.33) determining the quantity A are 
closely related to the integrals (16.03) of section 100. If the system is 
a gas, contained in a vessel of the volume F, the energy levels are 
determined by the vessel. It is shown in the quantum theory that in 
the interval of energy values between e and e + Ae there lie 
2irF(2/i/w^^ 2 )^e^Ae levels, and, if V is sufficiently large, their dis- 
tribution is practically continuous. By introducing the variable 
x = e/kT the sums (16.32) and (16.33) can be thrown into the form 
of integrals proportional to (16.03). The comparison shows 


U - Uo = %ZkTG(A}/F(A). 


Thus, the probability definition (16.31) leads directly to an energy 
expression equivalent with the formula (16.15). 

The gases here discussed are perfect in the sense that their molecules 
(or atoms) have no extension and there are no forces of interaction 
between them. However, the statistical limitations imposed on identi- 
cal particles by Pauli's exclusion principle or the requirements of 
Einstein-Bose have an effect upon the thermal and caloric equations 
of state which resembles the effect of force interactions (compare next 
section). It must be emphasized that no such limitations exist for 
non-identical particles. In a mixture of several degenerate perfect 
gases, there is no mutual influence between them, and each gas behaves 
as if it were alone and the others absent. This implies, in particular, 
that the formula 

P - Pi + p2 + ...+P., (16.35) 


representing the total pressure as the sum of the partial pressures of 
the component gases (each calculated as if the rest were not present) 
remains valid. 

105. Chances of observing the degeneration experimentally. 
Some of the laws of perfect gases have the same mathematical form 
for all three kinds of them (classical, Fermi, and Einstein-Bose). 
Such, for instance, is the equation of the adiabatic process: it follows 
from the expression of the entropy differential (16.16) that s = const, 
when 9 = const. According to (16.02) this can be written 

const, (16.36) 

or multiplying by pv from eq. (16.01) 

pv* = const. (16.37) 

The relation between pressure and energy density resulting from 
(16.01) and (16.15) 

is also valid for the classical and both degenerate gases. 

On the other hand, the pressure expressed in terms of the molal 
volume and temperature depends on the nature of the gas. Written 
out explicitly the expressions are 

p = * (i - 5/2*6") 


h?n A ** 1 \ 7 

^Ukf^ V + '")' 

(9 <C 1 occurs only in the Fermi gas) 


Substituting into the expression (16.02) for 9 the numerical values 
of the constants k,n A , h, and VQ (molal volume at normal conditions), 
we find 

9 = 3.67 M (-) T. (16.41) 


This gives for helium (/* - 4.00) at C and 1 atm, 9 - 4033, so 
that the condition 9 ^> 1 is fulfilled for the permanent gases of 


nature. On the other hand, electrons in metals also form a gas (com- 
pare next chapter) of the atomic weight /z = 1/1821 = 0.000549 
and with a remarkably small molal volume, namely of the order 
V/VQ 1/2000. In such an electron gas the quantity 9 has at room 
temperature a value of the order of 9 = 0.004, satisfying the opposite 
condition 9 <C 1. The two extreme cases represented, respectively, 
by the formulas (16.39) and (16.40) happen, therefore, to be sufficient 
for the treatment of all the degenerate systems occurring in nature. 

For the permanent gases, eq. (16.39) must be used, and its form is 
analogous to the Van der Waals equation of state (1.21) which, in the 
case of not too small molal volume (v J> b), can be written 

Perfect gases have, of course, no extension of molecules (6 = 0), 
but the Einstein-Bose gas (6=1) acts as if its molecules were attract- 
ing one another, the Fermi gas (5 = 1), as if they exercised repulsive 
forces. This behavior is in harmony with the statistical meaning of 
degeneration in the two cases (section 104). 

The most promising gas for observations is helium: it is the 
monatomic gas of lowest atomic weight, and, because of this, has 
the highest degeneration. Moreover, it can be cooled to lower 
temperatures than any other gas. But even in helium the condi- 
tions are not favorable, the Van der Waals terms of eq. (16.40) are 
in it (under normal conditions) b/v = 1.1 X lQ-*,a/RTv = 6.8 X 10~ 5 
while the degeneration term of (16.39) has the much smaller value 
1/2 H 9** = - 6.9 X 10 ~ 7 . It is true that the ratio becomes more fav- 
orable at lower temperatures, but barring the immediate vicinity of 
the critical state the constant 9 remains fairly large even then. The 
effect is, therefore, completely obscured by the Van der Waals 

The velocity of sound can be measured with considerable accuracy 
even at very low temperatures and in a small volume of gas. The de- 
terminations of Keesom in helium at 4 K and 0.08 atm gave the value 
to be expected in a non-degenerate gas. It is easy to see the reason 
for this. Since the equation of the adiabatic (16.37) is the same as 
for the classical perfect gas, the expression (3.40) for the velocity 
of sound also remains valid. This gives (with y = ^) 

- -'RT (1 - 6/2 M 6). (16.43) 

3 /* 3/*' 


Under the conditions of Keesom 9 = 24 and - 1 /2 W 9* = - 1 .5 X H)- 3 . 
The effect of the degeneration is only 0.15% and entirely within the 
errors of measurement. 

Summarizing we can say that the observation of degeneration in 
atomic gases is hopeless. The only system in which the phenomenon 
is accessible to the experimental test is the electron gas which is 
treated in the next chapter. 


106. Thermodynamic properties of charged gases. As long ago 
as 1888 the pioneers of thermodynamics, Arrhenius and Ostwald, 1 
applied the mass law to electrified particles (ions). We are indebted 
to them and their pupils for extensive experimental data which show 
conclusively that the thermodynamical theory of chemical equilibrium 
applies to ions just as well as to neutral atoms. The idea that a cloud 
of electrons can be also described by the laws of perfect gases proved 
useful in the theory of metallic conduction and associated phenomena. 
It was introduced by Drude 2 and Richardson, 3 who dealt with it from 
the statistical point of view, while the first thermodynamical treatment 
was due to H. A. Wilson. 4 

A part of the internal energy of such a gas must be of electric 
origin. Let us focus our attention on a portion of the gas so small 
that we can consider the outer electric potential 12 as constant over its 
volume r. The potential 2 is produced, partly, by causes extraneous 
to the gas, partly by charges on the gas particles outside the volume r. 
The contribution to the total electric energy of the portion within r 
consists of two items: the mutual energy of this portion and of the 
rest of the system, and the electrostatic energy of the volume r itself. 
The first item is proportional to the first power of the volume r ; the 
second goes mainly with r 2 and only rarely contains a portion linear 
in r. The latter case may arise when the charges within r are of both 
signs so that there is a possibility of pairing, i.e. of positive charges 
hovering, by preference, in the vicinity of negative ones, and vice 
versa, a phenomenon which we shall discuss in section 115. At present 
we shall treat the case where pairing is either absent or negligible, so 
that the inner electric energy of the volume r is quadratic in r. If we 
select r sufficiently small, the quadratic term becomes negligible com- 

1 W. Ostwald, Zs. phys. Chemie 2, pp. 36, 270, 1888. 
* P. Drude, Ann. Physik 1, p. 566; 3, p. 369, 1901. 

O. W. Richardson, Proc. Camb. Phil. Soc. 11, p. 286, 1901; Philosophical 
Transactions (A) 201, p. 497, 1903. 

H. A. Wilson, Philosophical Transactions (A) 202, p. 258, 1903. 



pared with the mutual electric (mentioned above) and the caloric 
terms which are both proportional to r. Let us denote the charge per 
mol by / = en A (e being the charge on each particle). 1 The electric 
energy of the gas in the volume r (due to the outer potential 8) will 
then be/12 per mol. This is the only part of the electric energy which 
has to be considered and added to the internal energy of caloric origin. 
For instance, if we assume that the caloric part has the expression 
(3.18), valid for a perfect gas, the internal energy becomes 

u = c v T + uo+fto. (17.01) 

The additional term /12 is independent of the condition of the gas 
in the volume r. It is a constant with respect to it and not one of the 
parameters describing its state. The parameters, subject to change, 
defined within r, are temperature, pressure, and specific volume, as in 
every simple system. It has been established by statistical reasoning 
and by experiment that the conditions concerning the equation of 
state and the caloric internal energy are in no way different in a 
charged gas from those in the uncharged. The equation 

pv = RT, (17.02) 

together with (17.01) define the classical (or non-degenerate) perfect 
charged gas and describe the phenomena in clouds of ions and electrons 
to the same degree of approximation as the equations lacking the 
electric term describe the ordinary permanent gases. 

Since /ft must be considered as constant, with respect to changes 
in the volume r, it may be regarded as forming part of the energy 
constant UQ: in fact, we have seen in section 98 that UQ itself repre- 
sents, in part, potential energy. Therefore, the entropy differential 
ds does not contain the electric potential so that we find for s precisely 
the same expression (4.18) as in uncharged gases. On the other hand, 
the thermodynamic potential <p becomes according to (5.41), (5.43) 

<p = RT log p - CpT log T + (c p - 5 )r + u + /0. (17.03) 

107. Equilibrium in charged gases. The problem of equilibrium 
was treated in section 40 in its generality, but the conditions in charged 
systems are somewhat peculiar, and it will be necessary to say a word 
how they fit into the picture there given. The phases in a system 
consisting of charged gases may be determined by external conditions 
and separated by surfaces of discontinuity, for instance, when we 

1 f becomes identical with the Faraday F when the charge is monovalent and 
positive. In general, / = <rF, if <r is the valency; for the electron (whose charge 
ia negative)/ - F. 



XVII 107 

consider the free electrons inside a heated metal, as one phase, and the 
electron cloud outside it, as another. However, when the electric 
potential Q changes continuously with the position in the gas each 
small volume r, considered in the preceding section, may be also 
regarded as a separate phase of the system because the electric condi- 
tions in it are different from those of the rest of the gas. In either case 
the potential is one of the essential characteristics of the phase, and 
this fact has an important bearing on the equilibrium. In section 40, 
we mentioned two possible cases, represented by eqs. (6.26) and (6.29), 
arising when there is no transfer of matter from one phase to another. 
Which of these applies to charged gases? The first equation is valid 
when the subsidiary condition (6.24) holds: when it is possible to 
change the volume of the individual phases at the expense of one 
another, while keeping the total volume of the system constant. 
Obviously this is not feasible in the system we are now considering. 
The expansion of any phase would push a part of it into a region of a 
different electric potential or, in other words, out of the phase. On 
the other hand, it is clear that the equilibrium will not be disturbed 
if we imagine each phase surrounded by a rigid heat-conducting 
envelope, i.e. if we assume the subsidiary conditions underlying the 
formulas (6.27) and (6.29). We must conclude, therefore, that a 

system composed of charged gases in 
an electric field possesses, in its state 
of equilibrium, a temperature T uni- 
form for all its parts while the pressure 
is not uniform but " local," changing 
with the potential from phase to phase. 
The remaining condition of equilib- 
rium (6.31) is derived from the analysis 
of processes of transfer of matter, from 
one phase to another and from one 
component to another, while the tem- 
perature of the system T and all the 
local pressures of the phases are kept 
constant. It applies to a charged system 
to the extent to which such processes 

are possible in it. Let us consider, for simplicity, a chemically pure gas 
(i.e. having molecules of one kind only) in equilibrium at the temper- 
ature r, regarding the small volumes r and r r (Fig. 47) as two of its 
phases with respective electric potentials 12 and Q'. The equilibrium 
will not be disturbed if we imagine the small spherical surfaces L and 
L' within the phases hardened and transformed into rigid adiabatic 

FIG. 47. Equilibrium of charg- 
ed gas in electric field. 


shells. If the spheres are taken so small that the electric potential 
due to the gas within them is negligible compared with the rest, we may 
imagine this part of the gas removed without prejudice to the equilib- 
rium. Of course, we could have selected the spherical hole in r larger 
by an infinitesimal amount and having the surface M (dotted line), 
and that in r' slightly smaller with the surface M f . Let us adjust the 
radii in such a way that the spherical layer between L and M contain 
dN mols of the gas (in its original distribution), and that between 
L 1 and M 1 an equal number dN' (i.e. dN = dN'), while the tem- 
perature and the local pressures remain the same. The same effect can 
be obtained by expanding the sphere in r from L to M and contracting 
that in r' from M 1 to L'. This last procedure provides the mechanism 
we were looking for, a process for transferring dN mols of the gas from 
the phase T' to the phase r without change of temperature or local 
pressures in any part of the system. We can, therefore, apply the 
formula (6.31) to it. Let the mol numbers and molal thermodynamic 
potentials in r and r' be N,<p and N',<p', respectively. Changes of $ 
occur only in them, and the formula is reduced to <p dN + p'dN' = 0, 
or because of the relation dN + dN' = 0, 

<? - j. (17.04) 

It is well to say here a word about the phase rule. We found in 
section 43 that only three phases of a chemically pure substance can 
coexist. However, that applies only to uncharged systems in which 
temperature and pressure must be uniform throughout. On the 
other hand, let us envisage a phases of the charged gas we are dis- 
cussing now. They are described by a + 1 thermodynamical variables 
r, p (l \ . . . p (a \ and there exist between them a - 1 relations (17.04). 
Even if the electric potential differences are prescribed, two of the 
variables can be chosen at random (no matter how large the number a 
is), for instance, the temperature and the pressure in one of the phases. 
If the electric potentials are not prescribed, there may exist still 
further degrees of freedom. 

The relation (17.04) is valid not only for a chemically pure gas: by 
suitable use of semipermeable membranes it can be shown to hold for 
any component of a gas mixture. However, we shall be interested 
only in mixtures of perfect gases and, then, the following proposition 
is obvious: if the relation 

*/ = f r , (17.05) 

holds for the component j, when it is pure, it will continue to hold, 
when the component forms a part of a mixture. This follows from 


the fact that perfect gases do not exercise forces on one another but 
behave as if the others were not present. 

In the particular case of a non-degenerate perfect gas, eqs. (17.03) 
and (17.05) give 

RT log W,/pd = ~ /(&'- 0), (17.06) 


This special .form of the relation could have been obtained also in 
simpler ways, for instance, from Boltzmann's principle (4.69). How- 
ever, we shall have to fall back upon the general form (17.05) in treating 
degenerate gases (section 111). 

The other type of processes included in the equation of equilibrium 
(6.31) are chemical reactions taking place within the individual phases. 
In the interior of a phase the electric potential does not vary and merely 
plays the role of an additional energy constant. The conditions are, 
therefore, precisely the same in a charged gas as in an uncharged and 
eq. (6.31) remains valid. In particular, when we consider the case of a 
reaction of the type (6.46) 

+ . . . + v0G0 = (17.08) 

between perfect gases, the equation of equilibrium has the form (6.49) 
of section 42 : 

+ . . + r&t = 0. (17.09) 

Comparing it with (17.05), we see that the equation is satisfied in 
all the phases simultaneously so that we have equilibrium in every 
part of the system. For non-degenerate perfect gases we have to use 
for <f> the expression (17.03) obtaining the following explicit form of the 
equilibrium condition 

log pi = logK p - (Q/RT) },*&. (17.10) 

K p is here exactly the same equilibrium constant as that defined by 
eq. (8.15) in the theory of ordinary gas mixtures (section 51). The 
sum Sv// = nj&vfij (where e$ is the charge on a molecule of the 
component j) represents the change of electric charge in the reaction. 
However, there is the law of conservation of electric charge, which 
cannot be created or destroyed, so that the sum S^,- = 0. The law 
of equilibrium has, therefore, precisely the same form (8.17) in a mixture 


of charged as of uncharged gases and can also be expressed in the 
equivalent formulations (8.18) and (9.60). 

108. Remark on heavy gases in a gravitational field. The con- 
siderations of the preceding section can be transferred in their entirety 
to a system of heavy gases in a Newtonian gravitational field produced, 
partly, by external causes, partly, by their own masses. The only 
necessary change is to interpret 8 as a gravitational potential and to 
write, instead of /, = n A e it the molecular weights M? = n A Mj, (Mj being 
the masses of the molecules). The analogue to (17.06) is then 

fifa - exp [-M/W- Q.)/RT\, (17.11) 

which is simply a slight generalization of the well-known barometric 
law of Laplace's. In a gravitational field the equation of chemical 
equilibrium takes the form 

log pt = log K p - (0,/RT) T\/M/. ( 17 - 12 > 


In most cases the sum S *>//*/, representing the change of mass in 
the reaction, may be taken as zero. However, the law of conservation 
of mass does not hold for matter alone, and there are phenomena for 
which this sum is significant and may not be neglected (compare 
section 133). 

109. Thermoelectric potential differences in the classical electron 
theory. As stated in section 106, the theory of electrons in metals 
achieved some signal successes by assuming 

that the free electrons form a perfect gas. In ^^^4^3 <2> I 
the older (classical) theory this gas was re- j 

garded as non-degenerate, with an osmotic FIG. 48. Thermoelectric 
pressure obeying the law (17.02) and eq. couple. 

(17.07) resulting from that law. As an 

application of these ideas, let us consider two bars (1) and (2), of two 
different metals (Fig. 48), in contact at the junction J and forming 
an open chain. Suppose that, in their electrically neutral state at 
the temperature T, the two metals contain, respectively, z\ and $2 
free electrons per unit volume. At the moment the two bars are 
brought in contact, there exists on both sides of the junction a 
difference of concentrations and, consequently, of osmotic pressures 
of the free electrons. As we know from section 65, the osmotic pres- 
sure tends to equalize the numbers z\ and 212 and drives the electrons 
from (1) to (2), if z\ > z 2 . How long will this so-called thermoelectric 


action continue? As the electrons are driven toward bar (2), there 
is set up an electric field: the potential of this bar will fall 
(because of the negative electronic charge) and that of bar (1) will 
rise. Equilibrium will be reached when the forces of the electric 
field just compensate the force of the thermoelectric (osmotic) action. 
Supposing that the system retains the uniform temperature !T, we can 
calculate, for the state of equilibrium, the difference of electric poten- 
tials Oi fa between the two bars. We have only to apply the for- 
mula (17.07) of section 108, taking into account that pi/pz = zi/Z2i 
according to (1.15), and R/f = - R/F = - k/e l 

kT z\ 

$21 = 81 fo = log (17.13) 

e 22 

The assumption underlying this formula is as follows: the electron 
density z of each metal is completely determined by its nature and, 
possibly, by the temperature T, while the loss or gain of electrons 
necessary for charging up the system is entirely negligible at a small 
distance from the junction. In the immediate vicinity of the junction, 
metal (1) carries a positive charge, through loss of electrons, and metal 
(2) a negative one, through excess of them. The two opposite charges 
form an electric double layer accounting for the drop of potential. 2 

As long as the whole system is kept at the same temperature, the 
thermoelectric potentials are not accessible to measurement. In fact, 
their experimental determination would require a pair of wires going 
from the free ends of bars (1) and (2) to a suitable measuring instru- 
ment. If the two wires are of the same material (3) we can denote the 
thermoelectric differences with respect to the two bars by 831 and ^32. 
We see, then, from the expression (17.13) that 

832 = Osi + Oi2, (17.14) 

so that both wires are at the same potential. It is obvious that taking 
the wires of different materials will not remove the difficulty. 

1 See footnote on p. 267. 

1 Compare, however, footnote on p. 274. It was pointed out by some authors 
that the energy necessary to form the double layer must be taken into account in 
setting up the conditions of equilibrium (see: P. W. Bridgman, Phys. Rev. 27, p. 173, 
1926). Theoretically this is true, but the terms arising from this cause are very small 
numerically and share with the capillary surface effects the property of decreasing 
in importance as the size of the system is increased (compare sections 39 and 81). 
Since capillary forces are neglected in this and the following chapters, consistency 
demands that the electric surface effects be treated on the same footing. 


In the present connection, we are interested in the thermoelectric 
actions only in so far as they offer an interesting example of potential 
differences in electron clouds. We shall see in section 139 that it can 
be indirectly measured from the electromotive force of a thermo- 
couple with two junctions kept at different temperatures. Without 
any doubt, the above theory explains the gist of the thermoelectric 
phenomena, but it is somewhat oversimplified. This appears from 
the experimental fact that thermoelectricity is structure sensitive: a 
slight distortion in the lattice structure of a conductor, due to mechan- 
ical strain, produces an appreciable change of its thermoelectric 
properties. As the theory stands today, it cannot account for the 
observations, quantitatively, but does give the correct order of 

It must be borne in mind that the formulas of this section apply 
(within the approximation of the theory) only to conductors in which 
the free electrons can be considered as forming a non-degenerate gas. 
As was shown in section 105, this situation prevails in semi-conductors. 
Experimentally, it is known only that such materials possess unusually 
large thermoelectric potential differences (compare Table 56 in 
section 139). This is unquestionably due to the wide range of the 
numbers z occurring in them. The thermoelectric conditions in 
metallic conductors will be considered in section 113. 

110. Contact potentials and thermionic emission in the classical 
theory. Let us consider a conducting plate in vacuo: at its surface 
the density of free electrons changes abruptly. The forces of osmotic 
pressure (mentioned in the preceding section) tend, therefore, to 
drive the electrons out of the conductor into the empty space beyond, 
producing there a so-called thermionic cloud. We call the potential 
of the conductor fli, the number of electrons in it (per unit volume) 
01, and the same quantities in the thermionic cloud at the surface of 
the conductor fl'i and z'l, and we apply the formula (17.13) 

flj- G'i = log^- (17.15) 

e z i 

The number z\ is determined by the nature and physical state of 
the conductor, but the density z'\ of the cloud may assume any value. 
What condition, then, makes the problem definite and fixes the poten- 
tial difference Oi 8'i and, through it, the number z'i? The action 
of the osmotic pressure in producing the outer electron cloud is limited 
by the fact that a definite amount of work W\ is required in order to 
liberate one electron from the conductor , which is performed by the electron 


while it passes through the surface. Therefore, the left side of 
eq. (17.15) is determined by the relation 

Qi - fl'i = Wi/e, (17.16) 

whence ^ = ^ ^ (-n A Wi/RT) - zi exp (-bi/T), (17.17) 

where the quantity bl = n A w 1/R (17 . 18) 

is called the thermionic work function. 1 

It is generally accepted that the force resisting the loss of electrons 
and pulling them back into the conductor is the image force. As known 
from the theory of electricity, a perfectly conducting plane acts upon 
a point charge with a force equivalent to the attraction of the reflected 
image of the charge. If the distance of the point from the plane is x 
(its charge being e), the distance to its reflected image becomes 2x 
and the image force is e?/4x 2 . The work necessary to remove the 
electron from the position x = a to x = oo is, then, e?/4a. The actual 
surface of the conductor is, of course, imperfect since it is constituted 
of atoms with interstices through which the electrons can pass into the 
outer space. Nevertheless, the above expressions for the image force 
and the work done against it remain approximately valid as long as 
x (or a) is not too small. To obtain the total work W\ that must be 
supplied to an electron in order to get it free, one must attribute to a 
a value of the order of the atomic distances (10~ 8 cm). The equations 

Wi = e 2 /4a, fli - fl'i e/4a (17.19) 

give then, in fact, the correct order of magnitude for the potential 
difference, namely, a few volts. 

Sometimes the question is asked how the electron cloud builds up 
this difference of potentials. This question is illegitimate and based 
on a misconception resulting from the fact that the term " potential 
difference " is used in the above formulas in a sense slightly dif- 
ferent from that given it in most textbooks on electricity. 2 As under- 

1 Not to be confused with the "work function" denned in section 34. " n 
1 The difference of electric potentials fij Q! between the points (2) and (1) 
is determined by the work W n which must be supplied to a test body (of the charge e) 
in order to move it from (1) to (2). Two different definitions are possible: either, 
8a - 81 - lim (Wi2/e), when e approaches 0; or fti fli = Wu/e, when e is 
finite. Most textbooks on electricity do not distinguish between the two definitions. 
The second, obviously, takes in the work of any field induced by the test body and is, 
therefore, unsuited for many theoretical purposes. However, it is the definition 
which must be used in connection with Boltzmann's principle and thermodynamics, 
since Cpot in eq. (4.69) is the energy imparted to a particle under the actual conditions. 
In this sense (17.19) represents the actual difference of potential between the con- 
ductor and the outer space. 


stood here, the potential difference (17.19) exists from the start and is 
not, to any material extent, built up by the electrons leaving the 
conductor: the potential added by the electron cloud (so called, space 
charge potential) is usually so small as to be negligible. 

If a second conducting plate is placed at a short distance from the 
first facing it in vacuo, it will also set itself in equilibrium with the 
electron cloud and through it with the first plate. Two potential 
differences relating to the plates are of interest. On the one hand, the 
difference fli ^2 between the potentials in the interiors of the two 
conductors depends (according to the theory of section 107) only on 
their internal properties. It has, therefore, the same expression as 
when the plates are in direct contact, and is identical with the thermo- 
electric potential difference discussed in the preceding section. On 
the other hand, there is the so-called mutual contact potential 8'i - Q' a 
of the two substances. It is defined as the difference of potentials 
between two points in the electronic cloud which are, respectively, close 
to the surfaces of the two plates. The potential 12' i at the surface of 
the first plate is given by the eq. (17.16), that at the second plate by 
the analogous formula $2 ^'2 = W^/e. Subtracting the two equa- 
tions, we find 

tti - fl' 2 = Oi - &2 ~ (Wi - W 2 )/e. (17.20) 

Substituting for &i 82 and W\ W% from the formulas (17.13) 
and (17.18), we obtain for the contact potential in the classical theory 

ffi - / 2 = - * [b, - 6 2 - r log (21/22)]. (17.21) 

The second term in the brackets is usually very small compared 
with the first. 

The work function b can be measured either photoelectrically or by 
thermionic currents. If we let light of a high frequency v fall upon a 
metal plate, it emits photoelectrons having a kinetic energy JE^m. 
Einstein's photoelectric equation 

hv = Ek* + W (17.22) 

expresses the hypothesis that the energy of an absorbed photon hv 
is imparted to the photoelectron. The part W of it is spent in getting 
through the surface of the metal, the remaining part appears as 
kinetic energy. As the incident frequency v is decreased, Ewn becomes 
smaller and smaller and vanishes altogether for a certain limiting value 

hv<> - W. (17.23) 


Light of lower frequency cannot produce any photoelectric effect, 
and this permits the experimental determination of W. If we identify 
W'with Wioi eq.(17.16), we find from (17.20) for the contact potential 1 

fi'i - o' a - - - (y i - w) + * T log fa/**). (17.24) 
e e 

The method of thermionic emission for measuring b is based on the 
formula from the kinetic theory of gases 

n = z (RT/2*p)* 9 (17.25) 

giving the connection between the number z of molecules per unit vol- 
ume in a perfect gas and the number n of those which pass from one 
side to the other of any plane per unit time and unit area, n means 
the molecular weight of the gas, in our case that of the electron 
Me = 1/1821. This expression holds also for the electrons going in 
either direction at the surface of the metal. However, of those going 
outward the number rn are reflected at the surface, if r is the coefficient 
of reflection. Only the remaining (1 r)n electrons come out of the 
metal itself. The thermionic current is observed by making the heated 
metal plate the cathode of a vacuum tube and applying a strong field 
between the cathode and the anode. The electron cloud in the vacuum 
is swept away by the field, but the conditions within the metal are not 
appreciably changed and the electrons leaving it are still the same in 
number, carrying with them the current 

J = e(l - r), (17.26) 

or substituting from (17.25) and (17.17) 

J = A'T"e-" r , (17.27) 


A' - (R/2*n)*zie(l - r). 

This is Richardson's formula for the thermionic current. It was 
deduced by its author 2 from kinetic considerations and by H. A. 
Wilson 3 from thermodynamics. For the reasons pointed out at the 
end of the preceding section its validity cannot be claimed for metals 
but is restricted to semi-conductors. 

1 It can be made plausible by statistical reasoning that Wi W is, in fact, the cor- 
rect identification. The data on semi-conductors are not precise enough to test the 
accuracy of formula (17.24). 

* See footnote on p. 266. 

See footnote on p. 266. 


111. The degenerate electron gas. We have seen in section 105 
that the free electrons in metals must be regarded as a completely 
degenerate gas. This fact was first pointed out by Pauli, 1 while 
extensive study of metallic electrons, on the basis of Fermi's statistics, 
is due to Sommerfeld. 2 The formulas of Chapter XVI, however, need 
a small correction before they can be applied to electrons. It is caused 
by the electron spins which obey the following rule: the spins of any 
two electrons in the cloud are either parallel or anti-parallel. We have, 
therefore, two types of electrons with respect to the spin which we 
shall call, for short, the first kind and the second kind. The statistical 
weight and the intrinsic energy of either kind of electron are the same, 
so that our gas is a mixture of equal parts of two components. Accord- 
ing to the remarks made at the end of section 105, the formulas of 
Chapter XVI apply to each component separately, and it is easy to 
infer from them the equations for the mixture. The specific volume 
of each component is 2v, twice the specific volume of the mixture, 
while the partial pressure of each kind of electrons is p/2, one-half 
of the total pressure. All we have to do is to substitute in all equations 
2v, instead of v, and p/2, instead of p. As this leaves the product pv 
unchanged, the only correction which the laws of degenerate gases 
require in the case of electrons is a redefinition of the quantity 9 
(replacing v by 2v) 

instead of (16.02), where /u is the atomic weight of the electron 
/LI, = 1/1821. Substituting the numerical values of the constants 

e = 0.403 x io- 5 v*r. (17.29) 

There is reason to assume that in monovalent metals there are as 
many free electrons as atoms. The molal volume of the free electrons 
coincides then with that of the metallic material itself. For instance 
itisv = 10.26 cm 3 mol^in the case of silver, giving = 1.91 X 10~ 6 r. 
In other metals the values of 6 are of the same order of magnitude and 
must be considered as small even at the highest temperatures (about 
2300 C) accessible to thermionic measurement. It is, therefore, per- 
missible to use the expansion (16.10) and to obtain from (16.01) 

W. Pauli, Zs. Physik 41, p. 81, 1927. 

* A. Sommerfeld, Zs. Physik 47, pp. 1, 43, 1928. 


When O 2 is negligible, the gas is completely degenerate. Its pressure 
then becomes independent of temperature, being determined by the 
specific volume alone which, in turn, is fixed by the nature of the 
metal. We shall introduce the notation 


where !2< is called the inner potential. Hence its numerical value is 

!2< = - 25.85 tT H electron-volt. (17.32) 

For instance, in the case of silver Q- = 5.48 e.-v. With the 
abbreviation (17.31), eq. (17.30) takes the form 

while the expression for the energy follows from (16.15) and (17.01): 
- %pv +/Q + UQ. The entropy is determined by (16.17) and 
becomes for small values of 9, according to (16.10), 

-2 / \2 T 

-'7 5- 


while the molal heat has the expression 


Finally, the thermodynamic potential is expressed by 

IT 2 

^ +0 ~ + *' (17 - 35) 

The electronic spin has an important bearing also on the entropy 
expression for high temperatures. In of eq. (16.20), 2v must be sub- 
stituted for v, and this is equivalent to adding the term R log 2 to 
the entropy constant, or log 2 to the chemical constant. The entropy 
(16.21) and the thermodynamic potential (17.03) of the non-degenerate 
electron gas retain, therefore, the form of their expressions, but the 
constant JQ has an increased value. The chemical constant (14.15) 
of the electron gas becomes (with /* = 1/1821): 

jio, oi =-6.180 (17.36) 

or jio 0.826 - 1. 

A similar increase of the chemical constant occurs also in the case 
of gaseous atoms having an angular momentum and can be calculated 


in a quite analogous way. It will be better, however, to postpone this 
calculation until section 120, where it will be treated together with the 
chemical constant of diatomic and polyatomic gases. 

112. Thermionic emission and thermoelectricity from the modern 
point of view. We are now going to apply the results of the preceding 
section to the equilibrium of a metal with the thermionic cloud outside 
it. While the free electrons within the metal are in a degenerate state, 
those forming the outer cloud are completely non-degenerate. In 
fact, after deriving the conditions of equilibrium, under this assump- 
tion, we shall justify it by showing that the density of the cloud is 
extremely low and the value of the quantity 9 in it very high. We 
substitute into the condition (17.04), <p = <p f , the expressions (17.03) 
for the outer cloud, and (17.35) for the metal. Noticing that 
c p * | R, we find (with / = - en A ) 

log P = j + f !og r b/T, (17.360 

with the abbreviation 

- r _2 1.21-21 


where the potentials 0' and ft refer to the outer cloud and to the 
metal, respectively. Neglecting second order terms, 

b - | (Q, + - a')- (17.37') 


With the help of (1.15) we throw (17.360 into the form 

z - 7^ exp (j - b/T), (17.38) 

and obtain from (17.25) and (17.26) the thermionic current 

- r)7V" i/p , 


A e (n A /2wpk)* exp j 
120.2 amp cm~ 2 deg~ 2 . 

This formula was first derived by Dushman l from a semi-classical 
hypothesis of thermionic emission. Its derivation in terms of the 
degenerate electron gas was due to R. H. Fowler. 2 

The work function b receives, according to eq. (17.370, an interpre- 
tation different from that of section 110. The work which the electron 

1 S. Dushman, Phys. Rev. 20, p. 109, 1922; 21, p. 623, 1923. 
* R. H. Fowler, Proc. Roy. Soc. (A) 122, p. 36, 1929, 



XVII 112 

has to do in getting through the metal surface is still determined by 
the image force w which, in turn, fixes the outer potential difference 
Q 12' as in eq. (17.17). However, the electrons bring with them the 
energy eQ* depending on the inner potential (which has the nature 
of a quantized kinetic energy, as is shown in the statistical version of 
the theory). Only the difference bk = e(Q 0' + 0<) has to be 
supplied by thermokinetic or photoelectric forces. The connection of 
the thermionic work function with the photoelectric threshold fre- 
quency remains, therefore the same as in the classical theory, namely, 

hvo - Kb. 

Our knowledge of the inner potential fi, is not exclusively theoreti- 
cal: its existence and order of magnitude can be inferred from other 
phenomena, for instance, from experiments on the diffraction of 
electron beams falling on the surface of the metal. The values of ft< 
found by such measurements are not at all accurate, as they are 
affected by an uncertainty of d= 5 volts, but theoretical results obtained 
from formula (17.32) do not lie far outside these limits. A few nu- 
merical data are given in Table 36. 1 






t C 



o o' 




If V 









18 5 









18 " 








17 " 











17 " 










18 " 





11 " 



















1 Taken from Hughes and Du Bridge (Photoelectric Phenomena, pp. 76, 235, 
New York, 1932) and slightly revised. A more complete list will be found in the 
article by J. A. Becker, Rev. of Mod. Physics 7, p. 123, 1935. 


The inner potentials Q< in Table 36 are calculated on the assump- 
tion that there is one free electron per atom. This is sound in the case 
of metals of the first group (Ag, Au, Cu) but less certain in other cases. 

From the tabulated values of b one can determine z by means of 
eq. (17.38) and find the molal volume and the quantity 9 in the ther- 
mionic cloud at different temperatures. For instance, in the case of 
tungsten (neglecting the change of b with temperature) these data are 
as follows : 


/C. 20 300* 600 1000 1500 2000 2300 3000 3800 
v.... 1.1X10" 4.6X10" 3.4X10* 5.2X10" 1.6X10" 3.4X10" 2.5X10" 4.8X10* 5.1X10 
6... 6.0X1078 7.6X10i 1.9X10" 7.8X10" 1.0X10* 2.1X10* 4.3X10* 3.7X10* 1.0X10* 

We see from this table that 9 has exceedingly high values in the 
electron gas outside the metal, so that the assumption of its being 
non-degenerate is entirely justified. 

In comparing Dushman'* formula (17.39) with experiments, it 
must be remembered that it has its theoretical limitations. The 
thermionic work function is not strictly a constant but has a slight 
temperature dependence. This is due, in the first place, to the 
neglected terms of the expression (17.35) and to the thermal expansion 
of the metal which changes the molal volume v in it and also may 
affect the image force, and through it, the potential difference 8 0'. 
In the second place, the picture of electrons being completely free is 
only an approximation; in reality, there are some mutual forces 
between them and the metal ions which also could account for small 
deviations from the constancy of b. Finally, the factor (1 r), which 
is supposed 1 to be of the magnitude 0.9S, is again slightly temperature 
dependent. On the other hand, the measurement of thermionic cur- 
rents involves an extremely difficult experimental technique whose 
accuracy cannot be pushed beyond a certain limit. In view of this 
situation, the agreement must be considered as quite satisfactory. 
The results for A, as obtained experimentally, 2 do not agree with the 
theoretical value 120.2, as shown in Table 38. This is probably due 
to the work function b depending on T and containing a linear term 
(b = b' + aT.) Therefore, the formula (17.39) can be rewritten in the 
same form but with the new constants V = b aT, A' A exp a. 
From the analysis of the measurements one obtains directly V, A' and 
not 6, A, but, knowing the theoretical value of A, one can calculate a 
and the true work function b. 

L. Nordheim, Proc. Roy. Soc. (A) 117, p. 626, 1928. 
L. A. DuBridge, Proc. Nat. Acad. 14, p. 788, 1928. 


TABLE 38* 

XVII 112 



(in volt) 



0X 10* 


Bu Bridge 



0.99 X 10 



i < 



1.05 X 10-' 



1 1 



0.57 X 10* 



1 1 



1.37 X 10" 


Cu (liqu) 



1.65 X 10~ 4 


Zr, Hf 




0.57 X 10 a 






1.89 X 10-* 


* See: S. Dushman, Rev. Modern Physics, 2, p. 381, 1930. 

A curious and not entirely explained result of DuBridge's is that, 
in the same metal but for different conditions of its surface, the 
observed log A' is a linear function of the observed V (i.e. 
log A' = &b' + const). The coefficients are listed in Table 38. 

With respect to thermoelectric forces between two metals (1) and (2), 
the qualitative considerations of section 4 remain valid. The equi- 
librium of the electrons in the two metals is determined by the same 
condition <pi = ^2- The only difference is that we have to substitute 
for the thermodynamic potential in either metal the expression (17.35) 
since we consider both clouds as degenerate. Thus the potential dif- 
ference becomes 

V 2 k 2 I 1 

- 2 - (Q - Da) --- T 2 ( - -f- 
Iz e? \Sm S 

< 1M > 

The expression fli $2<2 is independent of the temperature. It 
represents a potential difference existing between the two metals under 
all circumstances. It cancels out of every closed circuit and cannot be 
observed by galvanometric or electrometric methods. Therefore, only 
the second term need to be taken into account for the thermoelectric 

821 - 0.611 

x lo-sr 2 ( -f- - 





where Q-i, 12<2 are expressed in volts. This formula was first given by 
Sommerfeld. 1 For the reasons outlined in section 109, we can hardly 
expect from it more than the correct order of magnitude. Experi- 

1 A. Sommerfeld, note on p. 277. This formula does not represent the e.m.f. 
in a thermoelectric circuit, which is given at the end of section 139. 


mental data and a thermodynamical discussion of thermoelectricity 
will be given in section 139. 

As to the contact potential of metals, it can be obtained directly 
from the formula (17.37), substituting for 12i Ife the expression 

This relation was tested by Millikan, Lukirsky and Prilezaev, and 
Olpin * and was confirmed in the case of metals with very clean 

113. lonization in the solar atmosphere. In their systematic 
study of solar radiation the astronomers succeeded in analyzing spec- 
troscopically the light coming from the different altitude layers of the 
chromosphere. They found that the emission of the lower levels mostly 
consists of arc lines (characteristic of the neutral atoms), while the 
spark lines (belonging to the ionized atoms) appear higher up and 
grow in intensity with altitude at the expense of the spark lines. 
This result seemed puzzling because the lower layers of the sun are 
hotter and, in the laboratory, the spark lines are enhanced by raising 
the temperature. The reason for this discrepancy was explained by 
M. N. Saha, 2 who pointed out the influence of the pressure on the 
degree of ionization and applied the thermodynamical equations of 
chemical equilibrium to the problem. There are charged particles of 
both signs in the solar chromosphere, but at the temperatures and 
pressures prevailing there the phenomenon of pairing (mentioned in 
section 106) is quite negligible. Therefore, the form of the equations 
of equilibrium is not appreciably changed either by the charges on the 
electrons and ions or by the gravitational field of the sun (section 108). 
These conditions also exclude degeneration so that we can apply the 
equation of chemical equilibrium (14.16) for classical perfect gases. 
The reactions which we have to consider as taking place are those 
of the single and double ionization of atoms by splitting off one or two 
electrons. In the symbols of section 42 

#++GL-G n = and C++ + 2GL - G n - 0, (17.42) 

where" 6+ and C++ refer to the positive ions, GL to the electron and 
G n to the neutral atom. The formula (14.16) needs a slight correction; 

1 R. A. Millikan, Phys. Rev. 7, p. 355, 1916; P. Lukirsky and S. Prilezaev, 
Za. Physik 49, p. 236, 1928; Olpin, Phys. Rev. 36, p. 251, 1930. For a discussion 
from the point of view of the statistical theory see: Mitchell, Proc. Roy. Soc. (A) 
146, p. 442, 1934; 153, p. 513, 1936. 

* M. N. Saha, Phil. Mag. 40, pp. 472, 809, 1920. 



XVII 113 

there must be added log 2 to the chemical constant of the electron 
(section 111). However, this is not all : it will be shown in section 120 
that a similar term log g belongs to the chemical constants of the 
atoms and ions (because of the statistical weight of the angular 
momenta of these particles. Since the atomic weights of the atom 
and the ion are practically the same, we can write f S> A logio MA = 
f v logio M ~ 4.892v, where p e means the atomic weight of the elec- 
tron. For monatomic gases c p = %R, and the heat of reaction be- 
comes, according to (3.28), Q = 2^x* = At/o + fyRT. If we con- 
tinue to measure the pressure in atmospheres but take as the unit of 
heat of reaction not the erg but the electron-volt (1 erg/mol = 
0.9646 X 10 12 e.-v.), the equations of equilibrium become for the 
two reactions (17.42) 

log (P+P-/P*) - ~ 6.180 + logio 

- f ^ 
f logio T 

- 12.360 + 

+ 5 logio T 

5037 AC/o 

5037 AC/o 


The values of AJ7o and g for the most important ingredients of the 
solar atmosphere are given in Table 39. The data of this table refer to 
the normal states of the particles', the statistical weights of the nuclear 
spins are not included in g because they are the same in the neutral 
atom and the ion and cancel out of the reaction. 




(in e.-v.) 

Particle g ( .^ v } 



(in e.-v.) 

H + 



Ca 1 
Ca + 2 


Zn + 


9 36 





Sr** 1 
Sr + 2 
Sr ++ 1 


Cd + 

1 J 






Ba 1 
Ba+ 2 








Ba++ 1 

rig +4. 






We see from eqs. (17.43) that the ratio P+/p not only is an increas- 
ing function of the temperature but also depends on the pressure of 
the free electrons, being inversely proportional to it. This pressure is 
an appreciable fraction of the total pressure and rapidly decreases with 
the general attenuation of the chromosphere as the level is raised. The 
equation, therefore, gives a qualitative explanation of the observations 
described above. Moreover, it accounts for the observed fact that, 
the lower the ionization potential, the lower also the level at which 
the appearance of the spark lines sets in. On the other hand, it is 
very difficult to calculate p+/p n quantitatively because the pressure 
of the electrons />_ depends on many factors. All the elements of the 
mixture which constitutes the solar atmosphere contribute to it, but 
even if we had atoms of one kind only the degree of ionization would 
depend on the excited states and its computation would be a difficult 
problem. 1 However, if our object is only to obtain a check of the 
thermodynamical formulas (17.43), we can get around this difficulty 
by choosing one of the elements of the solar chromosphere as a stand- 
ard. Suppose we have measured, for different levels of the sun, the 
intensities of the arc lines and the spark line of this standard substance 
and inferred from these measurements the relative abundance 
/>+<> //><> of its ions and neutral atoms. Writing eq. (17.43) for the 
standard element and for any other element, we eliminate />- and 

The theory was tested by similar devices and received a fair con- 
firmation, but it was pointed out by H. N. Russel 2 that great accuracy 
in the agreement cannot be expected. The gases of the solar chromo- 
sphere are not strictly in a condition of thermodynamical equilibrium 
since they are exposed to the one-sided radiation coming from the 
lower and hotter regions of the photosphere. 

114. Ion concentrations in electrolytes. Galvanic cells. The 
theory of dilute solutions also offers an important field of applications 
for the results of sections 106 and 107. A large class of solutes, the 
electrolytes, undergo a partial or complete dissociation into electrically 
charged components (ions). This is particularly characteristic of 
acids and salts in aqueous solutions: when their molecules dissociate, 

1 It was treated by Milne, R. H. Fowler, and C. G. Darwin. The most compre- 
hensive review of the whole field of ionization in the solar atmosphere is due to 
A. Pannekoek (Handbuch der Astrophysik, Vol. Ill 1, pp. 257-350. Berlin 1930). 

* H. N. Russel, Astrophys. J. 79, p. 317, 1934. 


the hydrogen or the metal acquires a (single or multiple) positive 
charge and is called the cathion, or positive ion, the acid radical gets 
a negative charge and constitutes the anion or negative ion. As an 
example we take cadmium sulfate which, dissolved in water, dissoci- 
ates according to the formula 

Cd++ + SO 4 ' - CdSO 4 = 0, 

where the superscript ++ indicates a bivalent positive ion (i.e. one 
carrying two elementary charges). 

We know from section 68 that non-electrolytic dilute solutions 
behave like perfect gases. If the conditions of equilibrium are not 
changed by the electric charges of the ions, we should expect the laws 
derived in that section, especially the mass law (9.60), to apply also 
to electrolytic solutions. For instance, the above reaction would give 

X++X /Xn = K(T), 

where x++, x , x n are the respective mol fractions of the positive and 

negative ions and of the neutral (undissociated) salt. In the more 
general case, when the molecule is ionized into vi, i>2, ... V T positive 
and negative ions, their total number being v t ~ v\ + . . . + v rt the 
equation takes the form 

vid + P2G* + ... + v T G r - G n = (17.45) 

and the mass law has the expression 

X) ^ log Xj - log x n = log K. (17.46) 


This equation is of the same type as those considered in section 52, 
and we can draw from it the conclusion that a certain fraction of the 
salt molecules is ionized (dissociated into ions). In fact, experiments 
show that the equilibrium constant K is fairly large in these reactions, 
so that solutions of low concentration are almost completely ionized. 

The hypothesis that dilute electrolytes are completely dissociated 
into ions was first advanced by Arrhenius, who adduced in its support 
their high osmotic pressures and low freezing points. Moreover, he 
showed that the electric conductivity is in agreement with this assump- 
tion, being nearly proportional to the molality of the dissolved salt. 
The work of Arrhenius and his successors had a great influence on the 
development of thermodynamics, on one hand, and laid the founda- 
tions of the science of electrochemistry, on the other. The laws of 
electrolytic conduction, as developed by Kohlrausch, afforded a means 


of determining the mol fractions Xj of the individual ions and gave a 
rough confirmation of the formula (17.46). However, as the technique 
of these measurements grew more and more accurate, it was found 
that the law (17.46) is rigorously valid only for a limited class of sub- 
stances, the so-called weak electrolytes. To this group belong a number 
of organic acids and their salts. On the other hand, the common 
inorganic acids and their salts form the class of strong electrolytes which 
exhibit appreciable deviations from the formula (17.46) even in very 
low concentrations. 

At first, it was doubtful whether these discrepancies were due to 
the inadequacy of the formula itself or to the failure of the conduction 
measurements to give the correct mol fractions. The investigations 
based on the freezing method, described in section 78, were under- 
taken to decide this question. They led to the result that eq. (17.46), 
actually, is not valid for strong electrolytes, or in other words, the ionic 
clouds of strong electrolytes do not obey the laws of perfect gases. 
As was shown in section 77, the form of the mass law can always be 
retained if the numbers Xj are replaced by the effective mol fractions 
or activities a,: 

JjiV log a,- log a n = log K. (17.47) 

Putting for the neutral salt a n = x n and making use of the notations 
of section 79, especially of the activity function 7 defined by (11.32), 
we can write (for dilute solutions) 

vj log Xi log x n = log K v, log 7. (17.48) 

The data from conduction measurements turn out to be in excellent 
agreement with those obtained by the freezing method: when the 
proper values of 7 are substituted, the formula (17.48) is confirmed with 
great accuracy. The theoretical reasons for the abnormal behavior 
of strong electrolytes will be discussed in the next section. 

The notion of electrolytic ionization affords an insight into the 
mechanism producing the e.m.f. (electromotive force) of galvanic cells 
which were treated in section 37 from the more formal point of view 
of energy relations. In all the constructions mentioned there the 
chemical reaction is between two metallic salts, MR and M'R (with 
the same radical R) which ionize in aqueous solutions according to 
equations similar to 

MR - M + + -, M'R = M'+ + R-, (17.49) 

and in which the reaction M'R + M > MR + M' is exothermic. 



XVII 114 

The current is, in part, due to the negative radical ion R- traveling 
through the cell from the vicinity of the positive to the negative plate. 1 
We are now able to say what force drives the ions and produces the 
current: it is the force of osmotic pressure. In fact, the -R_ ions react 
at the negative plate with the metal M of which it consists. Therefore, 
their concentration and partial osmotic pressure P_ sinks below the 
value P'_ which it has at the positive plate. When the resistance of 
the outer circuit is very large, an electric difference of potentials 
fl' 12 builds up which almost completely compensates the e.m.f. 
(electromotive force) of osmotic pressure. 

According to (17.06) we have then (since/ = F) 


- - f- log (P'/P), 

provided the cloud of R~ ions in the solution can be regarded as a 
perfect gas. The explanation for the movement of the M'+ ions to 
the positive plate is also osmosis (since in equilibrium the total osmotic 
pressure must be everywhere the same), but the calculation of their 
concentrations would lead us too far into electrochemistry. 

It is essential for the operation of the cells described in section 37 
that its electrodes consist of two different metals; they are, therefore, 
called electrode cells. There exist cells of a different kind, known as 
concentration cells, an example of a modern concentration cell is as 
follows: 2 





solution (m) 



solution (m f ) 



The first half of this chain (from Ag to the amalgam K-Hg) repre- 
sents an electrode cell with the reaction AgCl + K = KCl + Ag, 
which is strongly exothermic. If this half chain were left to itself, the 
molality m of KCl in the chamber filled with potassium chloride solu- 
tion would increase producing a current in the direction to the left 
(i.e. from K-Hg to Ag). In fact, in this case, the K+ ions move to the 
left and replace some of the silver in the AgCl paste (the reduced 
silver being deposited on the Ag electrode). At the same time the 
Cl_ ions move to the right and react with the potassium metal in the 
amalgam. In both processes KCl is produced and added to the solu- 
tion. On the other hand, if an electric current is sent through the half 

1 In the example of the Weston cell (Fig. 11, p. 95) the radical (R-.) is SO 4 , 
the negative plate (M) consists of Hg and the positive (M') of Cd. 
* Maclnnea and Parker. J. Am. Chem. Soc. 37, p. 1445, 1915. 


chain in the direction to the right all processes are reversed and the 
molality of potassium chloride solution must decrease. Now, the 
second half of the concentration cell is of the same construction and of 
opposite orientation: the resultant current will be, therefore, the 
difference of the currents produced in the two half chains and will flow 
through both of them. If we assume m' > m (see below), it will have 
the direction from K-Hg to Ag, in the first half, and from Ag to 
K-Hg, in the second. Therefore, the molality in the first KC1 cham- 
ber will increase at the expense of that in the second. There is no cur- 
rent when m r = m : the driving power is due to the difference of osmotic 
pressures in the two chambers. 

All this is also characteristic of the general case when the reaction 
is given by eq. (17.45). The essential feature of every concentration 
cell are two chambers C and C filled with the same solution in different 
concentrations. In the beginning, when the chambers are freshly 
filled, the solutions are electrically neutral (as a whole, containing 
equal positive and negative charges) so that the osmotic forces are 
entirely uncompensated by electric potentials. Under the ordinary 
conditions of construction, there is no tendency for potential differences 
to build up, and the solutions can be regarded as neutral during the 
whole length of their operation. Owing to these forces, the ions pass 
from the chamber of higher osmotic pressure into that of lower in 
numbers proportional to the coefficients v$ of eq. (17.45) as is implied 
in the electric neutrality of the solutions. We shall calculate , the 
e.m.f. of such a cell, only for the case of low concentrations when the 
electrolyte (in both chambers) can be considered as completely ionized. 
The change of thermodynamic potential which the ions undergo in 
being transferred from the chamber C to the chamber C is then 
(referred to 1 mol of the neutral electrolyte) A* = %v h fo\ ? h ). 
The left side, according to (5.35), is the (negative) non-mechanical 
work done by the system APT - W - EJ, where / is the total 
charge of the ions pertaining to 1 mol. On the right side, we substi- 
tute the activities from formula (11.17) 

JE - - RT y>* log (aV**). (17.51) 

If we denote the valency of the ion h by <r h (taking it positive or 
negative, according to the sign of the charge) 


Further, making use of the abbreviation (11.32), we can write 
E - - (RT/J){v 9 log ( 7 '/7) +Z* log (x'k/tk)}- (17.52) 

The measurement of electromotive forces of concentration cells offers 
a method for the determination of relative activity functions of electrolytes. 
Numerical data for the activity function will be given in the next 

115. Strong electrolytes. All the results so far obtained in this 
chapter rest on the following hypothesis made in section 106: the 
mutual electric energy of the particles in a small volume r is propor- 
tional to the square of the number of particles in it and, therefore, 
negligible. However, this argument is sound only as long as the 
electric forces do not affect the uniformity of distribution of the 
particles, and it ceases to be strictly applicable in the case of mixtures 
of positive and negative ions. Owing to the mutual attraction, each 
ion has the tendency to hover in the neighborhood of other ions of the 
opposite sign. The mean distance between unlike ions is smaller than 
that between like ones. This gives rise to energy terms, proportional 
to the number of particles, which may become significant under 
certain circumstances. It had been suspected for a long time that 
these additional electric terms may account for the abnormalities in 
strong electrolytes mentioned in sections 79 and 114. If this point 
of view is accepted, the sharp line of division between dissociated and 
undissociated molecules disappears: the distances between the ions 
can assume all values and are distributed according to the laws of 
probability. The transition from association to dissociation is gradual, 
and all ions can be considered as free, in this sense. 

The problem was formulated with complete clarity and partially 
treated by Milner. 1 It was made better amenable to quantitative 
evaluation by Debye and Hiickel, 2 who gave a simple method for the 
approximate calculation of the effect at low concentrations. 

Let us consider a neutral solution containing ions of several kinds, 
the type (j) being represented by Ni mols. We denote their valency 
by <r/, which is positive or negative, according to the sign of the charge 
on this ion. The total charge in the solution is then 

0, (17.53) 

Milner, Phil. Mag. 23, p. 551, 1912; 25, p. 742, 1913. 

P. Debye and E. HUckel, Phys. Zs. 24, pp. 185, 305, 1923. 


where F = en A is the Faraday constant or the charge per 1 mol of 
monovalent ions. We denote farther by 

zt-NtnJV, (17.54) 

the mean density, i.e. the number of ions of the type j per unit volume. 
Let us now focus our attention on a " preferred ion " of the kind (1) 
and let us ask how the other ions are distributed around it. The 
answer is given by the Boltzmann principle (4.69). Suppose that, at 
a point P, its electric potential is 81 ; in the time average, the density 
of j-ions at this point will be C exp (-e<rjQi/kT). Let the potential 
81 be counted "from infinity", i.e. from a distant point where the 
field of the preferred ion does not make itself felt (Qi = 0) and the 
density has the normal value /; then the constant of proportionality 
has the value C = */. The total density of electric charge at the point 
P, produced by ions of all kinds, is, therefore, 

exp ( eat Qi/kT). (17.55) 


The potential d is produced by the preferred ion, in part directly, 
in part indirectly through the action of the " atmosphere " of other 
ions which hover about it. Assuming the validity of the Coulomb law 
and, on the average, a spherically symmetrical distribution of the 
atmosphere, fij. must satisfy Poisson's equation in the form 


r 2 dr \ dr / D 

where D is the dielectric constant of the solvent. Eliminating p from 
(17.55) and (17.56), we obtain a differential equation for the determina- 
tion of QI. We restrict ourselves to the case when the exponent 
<TJ e$li/kT is so small that its square can be neglected. Because of 
(17.53), the term of zero order vanishes, leaving 

2 m , 

H --- = *r 81, (17.57) 
r dr 

*-%*%* <"- 58 > 

The integral of eq. (17.57) is 

! = -". (17.59) 


The coefficient A\ must be determined from the "boundary 
conditions." Let a\ be the mean distance of nearest approach of the 
other ions to the preferred ion: within the little sphere of the radius a\ 
the field is not influenced by the ionic atmosphere and the potential in 
this region may be taken as obeying the simple Coulomb law 

tf x - ? + ODI, (17.60) 

while outside this region it follows the law (17.59). At the surface of 
the sphere r = ai, there must be continuity of potential (&'i = fli) 
and of dielectric displacement (dQ'i/dr = dtti/dr). This leads to 

ffl e exp 

Without the ionic cloud about the preferred ion, its potential 
(against infinity) would be represented merely by the first term of the 
expression (17.60). Therefore, QOI gives us the potential difference, 
built up through mutual interaction, of the ion with respect to the mean 
potential in the solution. In a similar way the other ions assume the 

w TT (17 - 62) 

D 1 + *&/ 

In arriving at this formula, a dual capacity was attributed to 
every ion. On the one hand, it acted as a " preferred ion " collecting 
an atmosphere around it; on the other, it formed part of the atmos- 
pheres of all the other ions. It is clear that it is not always permissible 
to separate and superpose these two functions of the ion, as is done in 
Debye-Huckel's theory. The superposition implies, ultimately, a 
linearity of the equations controlling the electric field and comes back 
to the same restriction which was made in eq. (17.57). It seems, 
therefore, risky trying to improve the theory by taking into account 
terms of higher order in the exponent of (17.56), as has been done by 
some authors. 1 The question needs a closer investigation whether 
the theory is not restricted by its very method to cases in which 
(ffj-eQi/kT) 2 can be neglected. Another idealization of the theory 
lies in the schematic character of the " boundary conditions " which 
lead to the formula (17.62). In view of them, the dependence of the 
potential Ooy on the ionic radius a,- can hardly be considered as entirely 

1 Gronwall, LaMer, and Sandved, Phys. Zs. 29, p. 358, 1928. 


We shall show in section 138 that the change of internal energy of 
the solution produced by the potentials (17.62) is a rather complicated 
one. The work function SF, however, can be calculated without much 
trouble. In order to do so, it will be well to bring out the thermo- 
dynamical significance of our results. An ion of the charge cr,- 6 acquires 
the potential flo/' let us consider an imaginary, ideal way of producing 
such a distribution of charges and potentials. Suppose that e is not 
constant but that the charge of every ion can be changed and built 
up from to the normal value. 1 The work which must be done in 
order to increase all the charges from <r,-e to <r,( + de) (e being the 
same for every ion) is known from the theory of electricity to be 
DW* = Qo/ffi <fct where the summation is extended over all the 

ions. Since there are n A N, ions of the kindj, DW 9 = n A / .ffjN&oide. 


The total work of building up the distribution at constant temperature 
and constant volume is, therefore, 

W. - - n A I y\yQ ,-tf/ de. (17.63) 

It was shown in section 36 that the non-mechanical work done in a 
reversible process, at T = const, V = const, is oppositely equal to 
the increase of the work function. This increase, due to the electric 
charges and potentials of the ions, we denote by ^, = W 9 . The 
expression (17.63) is somewhat simplified if we use, instead of the 
individual ionic radii a/ of eq. (17.62), a mean radius a. In fact, the 
accuracy of the theory is hardly sufficient to justify such distinctions 
and, moreover, the term *a/ = *a in the denominator is altogether 
neglected in our ultimate applications. According to the definition 
(17.58) of K we can write, then, 


The partial thermodynarnic potential can be obtained directly 
from the work function by means of eq. (5.39) 

_ _. kTV "* ( d "\ . 

T 4ir l + Ka\dN h / VtT 2D 1 + *a " ' 

This is the increase of the thermodynamic potential due to the 
electric fields of the ions. If we define the activity coefficient of the 
ions h as log a h = ?eh/RT (section 78), this means that we compare the 

1 R. H. Fowler, Statistical Mechanics, p. 318, Cambridge, 1927. 


activities in strong electrolytes with the theoretical ones in dilute 
solutions at the same volume, whereas experimentally they are compared 
at the same pressure. This difference, however, is of slight significance 
in view of the very small compressibility of water and other condensed 
solvents. 1 

In very dilute solutions K is comparatively small, and Ka in the 
denominator can be neglected. From (17.58) 

or in terms of molalities (V = 1000, AT,- = 

log a* = - ,^* H ( ^- > V- 2 ,- ) (17.66) 

In water at the temperature of 25 C the dielectric constant is 
D = 78.8 leading to the expression 

logio <* h = - 0.356 fft? 

if we denote by m the molality of the neutral electrolyte originally dis- 
solved (mj = Vj-m). Hence the activity function, defined by eq. (11.32), 
is given by 


lo 7 . _ 

while the coefficient becomes 0.345 for the temperature C. This is 
the so-called limiting law for very dilute electrolytes due to Debye and 

The agreement of the limiting law with the measured activities is 
very good for ions of low valency as appears from Table 40. 

The very accurate measurements by Neumann, 2 on silver chloride 
and barium sulfate, show, however, that the limiting law is not rigorous 
even for low-valency salts. In plotting the measured y against m** 
this author finds curious oscillations of the curves around the theo- 
retical value (17.67). The deviations, although very small, seem 
outside the limit of experimental error. High-valency electrolytes 

1 The pressure increase due to the effect here considered is only a small fraction 
of 1 mm Hg. According to (9.42), (11.35), and (17.67), in water of 25 C it is: 
A - A - 0.006v e m log y - 0.12(2<r,yym)^ mm Hg. 

E. W. Neumann, J. Am. Chem. Soc. 54, p. 2195, 1932; 55, p. 879, 


were investigated by La Mer and co-workers, 1 who found large dis- 
crepancies with the limiting law. 


Type <TI = 1, <r a = 1, n 1, v* 1 



- 2 


Type o\ = 2, 0-2 = 2, v\ 1, v\ 1 

m .............. 0.001 0.002 0.003 0.005 

7 ......... (theor) 0.746 0.661 0.601 0.519 

BeSO 4 . . . . (exp) 0.754 0.670 ..... 0.534 

CdSOi.... " 0.754 0.671 0.621 0.540 

CaS 2 O 8 ... " 0.754 0.674 ..... 0.540 

When the solutions are not very dilute (presumably, m > 0.005, 
for low- valency salts), the term KO, in (17.65) can no longer be neglected. 
In fact, the measured activity functions usually fall, at the higher 
concentrations, below the values of formula (17.67). However, in 
the present state of the theory the dependence on a is not accurately 
enough known to make its quantitative discussion worth while. 2 

i La Mer and Mason, J. Am. Chem. Soc. 49, p. 410, 1927; La Mer and Cook, 
ibidem 51, p. 2622, 1929; La Mer and Goldman, ibidem 51, p. 2632, 1929. 

* A review of the work done on the theory of strong electrolytes by statistical 
methods will be found in an article by O. Halpern (J. Chem. Physics 2, p. 85, 1934). 
A critical discussion of the foundations of the theory was given by R. H. Fowler, 
(loc. cU.) 





. . (theor) 




HC1 . . . . 

.. (exp) 






AgNO 8 . 



AgC10 3 . 


. . . 


m . 




= 2, a a = l, 

v\ 1, 

CaCl 2 








H*SO*. . 





116. General considerations. We had occasion to give the expres- 
sions for the specific heats of a few ideal systems, such as perfect gases 
(non-degenerate and degenerate) and the Einstein model of a solid. 
Purely thermodynamical reasoning does not enable us to say much 
about the heat capacities of the real substances of nature except that 
they must vanish at T = 0. The internal energy of a system com- 
pletely determines its equation of state through eq. (13.22), but the 
reverse is not true: the knowledge of the equation of state is not 
sufficient for calculating the energy since eq. (16.12) 

still contains an unknown function of the temperature which usually 
gives the major contribution to the specific heat. However, heat 
capacities are of such fundamental importance as material for the 
applications of thermodynamics that a textbook of this branch of 
science would be incomplete if it did not contain some information 
on the status of our theoretical knowledge about them, even if this 
knowledge is obtained by non-thermodynamical methods. It seems 
appropriate, therefore, to include a chapter partially devoted to the 
way in which the kinetic theory of matter approaches the problem of 
specific heat. 

The fundamental tool of the kinetic treatment is eq. (4.70) of the 
Boltzmann principle. It is true that there are other types of statistics 
which we have described in section 104. However, as it was stated 
there, they apply only when many identical particles are distributed 
over energy states determined largely by joint external conditions. 
On the other hand, in such systems as an assembly of rotating mole- 
cules, each particle rotates around its own center of gravity and its 
quantum states are determined by conditions peculiar to itself as if it 
were alone and not part of a system. The same thing applies to the 
other systems occurring in the theory of specific heats, for instance, 



linear oscillators. There is, therefore, little occasion to use the statistics 
of Fermi or Einstein-Bose, and the Boltzmann principle reigns supreme 
as far as these investigations are concerned. If a particle can assume 
the quantum states 0,1,2, ... with the respective energies eo, ei, 82, ... 
the principle postulates that the number of particles in the state / be 
Zi = Cexp ( Bi/kT). The sum of the numbers Zj represents, obvi- 
ously, the total number Z of particles in the system 

Z = c exp (-ci/*r) - CY. (18.02) 

The expression Y = S exp ( i/kT) is usually called the sum of 
states. Every quantum state of the particle must be represented in it by a 
term. If the same energy level e< belongs to several (gi) different 
quantum states, the corresponding term must be repeated g\ times. 
It is, therefore, more convenient to write it in the form 

V = Z ft exp (- ei/*r). (18.03) 

We call the integer gi, as in section 111, the statistical weight of the 
level /. 

The energy of all the particles in the state / is 
X exp ( ei/kT), and the total energy of the system 

U = C]C gi*i exp (- ei/*r). (18.04) 

The sum in this expression can be obtained by differentiating Y 
with respect to -(l/*r,) i.e. U = - CkdY/d(l/T). Eliminating C 
with the help of (18.02) 

If we refer our considerations to 1 mol of matter, the number of 
particles is Z = n A (Avogadro's number n A = R/k), and the molal 
internal energy becomes 

u = - R a log F/a(i/r>. (IBM) 

We may consider this as the fundamental formula of the statistical 
theory of specific (molal) heats, since they can be derived from u by 
the formula c = du/$T or 

R a 2 log Y 

< 18 - 07 > 


Comparing (18.06) with the relation (5.S6), we find as the thermo- 
dynamical interpretation of the sum of states 

* = - RT log Y, (18.08) 

where \l/ is the molal work function defined as ^ = u Ts. There- 
fore, the entropy has the expression 

s -^A-r + *"* r < IM 

When the temperature T is very low, all the terms of the sum of 
states (18.03) become negligible compared with those of the lowest 
energy eo. The sum is reduced to Y = goesp ( eo/Jfer), whence 
RT log Y = n A o + RT log go and s = R log go- It may seem, 
at first sight, that this result is in contradiction with Planck's formula- 
tion (13.19) of Nernst's postulate. However, these formulas are no 
longer valid in the vicinity of T = 0. The expression (18.03) refers 
to the rotational and oscillatory motions of the molecules, and its use 
implies the assumption that the probabilities of these motions can be 
evaluated apart from the translational movements. This ceases to be 
true under conditions of temperature and density in which the statis- 
tics of Fermi or Einstein- Bose begin to play a role for the translational 
degrees of freedom. We refer, in this connection, to the remarks made 
in section 111 with respect to the entropy of the electron gas. In that 
example the statistical weight is go = 2, as there exist two kinds of 
spin. In the non-degenerate state the entropy contains, therefore, the 
term R log 2 (entropy of mixing), but in the degenerate gas this term 
disappears because the two kinds of electrons are regularly and 
uniquely arranged over the quantum states. 

117. Diatomic gases. The classical kinetic theory inevitably 
leads to the principle of equipartition of energy (4.71) : each degree of 
freedom takes up the kinetic energy %RT per mol. It b'assumed that 
the molecule of a monatomic gas has only 3 (translational) degrees of 
freedom, the molecule of a diatomic gas 5 (3 translational, 2 rota- 
tional). The corresponding molal heats are c v = $R and c p = -|J?, 
respectively. If the gas does not strictly obey the equation pv = RT, 
the resulting correction is taken care of by eq. (18.01) or by its deriva- 
tive with respect to T 


dv+f'Cn, (18.10) 

while c p can be obtained from (4.27). The main success which the 
classical theory has to its credit lies in the sphere of monatomic gases, 


whose specific heats it represents accurately. It is not so satisfactory 
in application to diatomic substances (compare section 15). For the 
more perfect permanent gases of nature the agreement is fair in the 
region of room temperatures, but there are large deviations both for 
low and very high temperatures. 

The following treatment is based on the theorem of mechanics that 
the kinetic energy of any system (molecule) can be represented as a 
sum of the two terms : the kinetic energy of the center of gravity and 
the energy of the motion within the system (molecule) relatively to the 
center of gravity. Correspondingly, the internal energy (as also the 
entropy and the specific heat) may be divided into two parts. It 
happens, moreover, that the contributions of the rotations and vibra- 
tions (though not rigorously additive) are separable with a sufficient 
degree of approximation 1 

u = u t + u r + u vi . (18.11) 

The first belongs to the translational motions and is calculated 
by the classical formula 

wo, (18.12) 

since the center of gravity is completely determined by the three 
coordinates of translation, assuming that degeneration of the type of 
Chapter XVI is negligible (compare preceding section). The second 
and third parts are due to the rotational motion of the molecule as a 
whole or to the vibrations of the atoms inside it. They are computed 
with the help of the quantum eqs. (18.03) and (18.06). 

As to the rotational heat c r , a good approximation is often afforded 
by the so-called dumb-bell model. The diatomic molecule is considered 
as consisting of two material points at an invariable distance from each 
other rotating around their center of gravity in a joint plane. Quantum 
dynamics gives for the energy levels of the dumb-bell model 

j + 1), (18-13) 

where K is the moment of inertia of the molecule. At the same time 
the statistical weight of the level e, is g/ = 2j + 1 (provided the 
nuclei have no spins). 

1 Occasionally, more rigorous expressions are used in which the rotational and 
vibrational energies cannot be separated (compare section 119). This is, however, 
unnecessary as far as the accuracy of caloric measurements is concerned. In addi- 
tion to the items of eq. (18.11), sometimes the energy of electronic configurations 
(excited states) is also of importance. 


Two cases must be distinguished with respect to the values of 
the quantum integers j. (A) When the molecule is asymmetrical, con- 
sisting of two different atoms, it can assume all integral values 
(j as 0,1,2, . . .). (B) When the molecule is symmetrical, built up of 
two identical atoms, only even integral values are permissible 1 
(j = 0,2,4, . . .) This distinction was recognized even in the old 
quantum theory and was there interpreted as follows: the asym- 
metrical molecule must be rotated through the angle 2ir until it returns 
to the initial position; the cycle of the symmetrical molecule is only 
half as large, for after being turned by the angle TT it is in a position 
which cannot be told from the initial. The interpretation of quantum 
dynamics is somewhat different: when we have two identical atoms 
(without nuclear spins) they obey the statistics of Einstein-Bose 2 and 
admit only quantum states with symmetrical wave functions; these 
happen to be those with even numbers j. No such restriction exists 
for the asymmetrical molecule. 

In either case, we can express the sum of states (18.03) by the 
single formula (j = 1,2,3. . .) 

(2aj + V exp l- Tfl tffo' + *>]' < 18 - 14 ) 

r = h 2 /Sir*KkT, (18.15) 

if we introduce the symmetry number <r, which is, respectively, equal to 
1 and 2, for asymmetrical and symmetrical molecules. 

For high temperatures, r is very small and (18.14) yields the equipar- 
tition value. In fact, putting n = in eq. (18.22), below, we find 
in this case 

Y r - 1/rcr 

or log Y r = - log (1/r) + const. Hence (18.07) gives c r = R, 
corresponding to the two degrees of freedom of the dumb-bell model. 
On the other hand, for very low temperatures, only the first two terms 
of the sum are significant, log Y r = (2<r + 1) exp [ TO-(<T + 1)], 
whence c r - R(2a + IJo^cr + 1)V exp [-T<T(<T + 1)]. Therefore, the 
formula (18.14) represents a gradual rise (with temperature) of the 
rotational molal heat from zero to the equipartition value, a behavior 
which is in qualitative agreement with the observed facts. 

1 We consider here only nuclei without spins. The case of atoms with nuclear 
spins will be treated in the next section. 

1 Nuclei without spins are, necessarily, built up of an even number of primary 


When quantitative accuracy is desired, the dumb-bell model often 
turns out to be oversimplified. For one thing, nuclear spins of the 
two atoms will introduce changes in the statistical weights, as we 
shall see in the next section. For another, the molecule may be 
capable of several electronic configurations (excited states) causing a 
multiplication of energy levels. The spectroscopy of the band spectra 
makes available a material of large and rapidly increasing volume 
relating to the rotational and vibrational energy states. For a great 
many gases the levels e/ have been accurately measured and tabulated. 
If a simple formula is not available, the procedure usually followed is 
to substitute into the sum of states (18.03) the spectroscopically 
measured levels (or exact theoretical expressions where they are known) 
and to obtain the specific heats by numerical calculation. 

In view of this, we shall mention here specifically only a particu- 
larly simple case, that of the doublet structure of the energy levels. It is 
not restricted to the rotational energy of molecules but has a quite 
general application and was first treated by Schottky. 1 Suppose that 
the energy levels occur in pairs ej and ej + Ae (the increment Ae being 
constant and independent of /) with the relative statistical weights 
go : gi. Every term of Y (18.03) will then appear to be multiplied by 
the same factor Y d = go + gi exp ( Ae/jfer). Writing for short 

9 = Ae/fc, 7 = \ log (gi/go), (18.16) 

we find 

Y d = go + gi exp (-8/r) = 2go exp (jy - j cosh ^ - T j. 
Correspondingly, log Y will contain the additional term 

log Yd = - + log cosh "" 7 + * log ( 4 S<*i),(18.17) 

whence the doublet heat is calculated from (18.07) 

(6/2D 2 

cosh 2 rz; - 

This expression is valid, of course, even when no other energy 
spectrum exists, i.e. when each particle is capable only of the two 
energy states eo and eo + Ae. In the special case of the rotational 
doublets, Ae is not strictly constant. Its formula was worked out by 

1 W. Schottky, Phys. Zs. 23, p. 448, 1922. 


Hill and Van Vleck 1 and shows a slight dependence on j. It seems, 
however, that in most cases the simple formula (18.18) gives an 
approximation, sufficient within the accuracy of heat measurements 
(section 119). The presence of the term c d produces a characteristic 
hump in the specific heat curve as exemplified in Fig. 54. The condi- 
tions are not very different when each energy level, instead of splitting 
into two components, splits into several or many closely spaced sub- 
levels (multiplet structure): the analytical expressions are more 
complicated but the result is a hump in the curve of specific heat not 
unlike the doublet hump. 

It is well known that the energy levels of the linear oscillators are 
given by the formula 

e,i = hv Q (v + J), v = 0,1,2, . . . 

VQ being its characteristic frequency. The vibrational levels of 
diatomic molecules are not quite so simple as that but in most cases 
they can be sufficiently well represented by an expression of the form 

+ i) - *(v + ) 2 L (18.19) 

(anharmonic oscillator) where x is a constant. A theoretical founda- 
tion for this formula was given by Kratzer 2 and in a different way by 
Morse. 3 We repeat that, strictly speaking, the vibrational energy is 
not entirely independent from the rotational but there exist mixed 
roto-vibrational terms, which are, however, negligible as far as the 
computation of specific heats is concerned. The fact that e/ and e v 
are simply additive means that, by taking into account the levels 
(18.19), each term of the sum F r (18.14) is multiplied by the same 
expression F,<. Since the statistical weight of each vibrational energy 
level is 1, we can write 

F. -53 exp J - || [(v + i) - *(o + *)] J . (18.20) 

In other words, the rotational and vibrational sums of states are 
multiplicative, Y = 7 r - F v *, whence log Y = log Y T + log Y vi and 
c = c r + c vi . The rotational and vibrational heats are additive, 
within our approximation, and can be evaluated separately. 

It must be noted that the sum of states with the energy levels 

* E. L. Hill and J. H. Van Vleck, Phys. Rev. 32, p. 250, 1928. On the basis of 
thia work a correction to (18.18) was calculated by G. Gregory (Zs. Physik 78, p. 789, 

A. Kratzer, Zs. Physik 3, p. 289, 1920. 

Ph. M. Morse, Phys. Rev. 34, p. 57, 1929. 


(18.19) would be divergent if v could increase to infinity. This means 
either that the number of states is finite, being cut short by the disso- 
ciation of the molecule, or that the expression (18.19) represents an 
approximation valid only for low order terms. There is no practical 
difficulty connected with this because the terms decrease rapidly 
even at high temperatures and no ambiguities arise in the numerical 
calculations. For a few gases the expression (18.19) does not hold 
with sufficient precision; recourse is then taken to the empirical 
energy levels tabulated by spectroscopists. On the other hand, when 
the coefficient x is small compared with 1 , it is possible to obtain a fair 
approximation neglecting it altogether, and extending the summation 
to infinity. 1 In this case it can be readily carried out and gives 
the result 

Y vi = 1/2 sinh (0/2r), 1 

f (18.21) 

c vi = *0 2 /4r 2 sinh 2 (e/2r), J 

where the "characteristic temperature" 9 is an abbreviation for 
6 = hvo/k. At any event the law of equipartition of energy does not 
apply to expressions of the type (18.19) or (18.20). The vibrational 
specific heats, therefore, rise steadily without reaching a limiting 
value even at the highest attainable temperatures. 

The constant hvo in (18.20) is always considerably larger numeri- 
cally than rkT of (18.15). Therefore, the vibrational heats come into 
play at much higher temperatures than the rotational. The gases 
mentioned in the beginning of this section as apparently confirming 
the classical theory are those for which the rotational degrees of 
freedom have reached complete equipartition at room temperature 
while the vibrational degrees are not yet appreciably excited. 

Exercise 104. Take a sum slightly more general than (18.14) 

fa + n + $) exp [-rfo* + n)(*j + n +1)] A/. 

A/ = 1 is the increment which j receives from term to term. Write \/r <rj #/; 
if T is very small, Ax/ = \fc <rbj is also very small, and the summation can be replaced 
by an integration with respect to */. Show that the result of the integration is 
Y = (1/ar) exp [ n(n + l)<r], or when a is extremely small, 

Y - 1/crr. (18.22) 

118. Hydrogen (Protium, E^H 1 )- The decline of rotational 
specific heats at low temperatures was first observed in hydrogen. 2 Its 

1 W. Nernst and K. Wohl, Zs. techn. Physik 10, p. 611, 1929. 
* A. Eucken, Preuss. Akad. d, Wise., p. 141, 1912. 


complete theory was developed in slow steps which were highly 
instructive. The simple dumb-bell model is here inadequate because 
of the nuclear spin of the hydrogen atom. There exist two modifica- 
tions of the molecule: para-hydrogen in which the spins of the two 
nuclei are anti-parallel, and ortho-hydrogen in which they are parallel. 
The total spin of the para-hydrogen is zero and it is indifferent to a 
magnetic field, while ortho-hydrogen is capable of assuming in such a 
field three quantized orientations. Therefore, the factors (2j + 1) in 
the sum of states (18.14) represent correctly the statistical weights of 
those energy levels which belong to para-hydrogen while those corre- 
sponding to ortho-hydrogen must be multiplied by three. It happens 
that the levels of the two modifications divide neatly, para-hydrogen 
being capable only of even quantum numbers j, and ortho-hydrogen 
only of odd. The corrected sum of states as first given by Hund l is, 

F= F, + 3F , (18.23) 

F, - (4j + 1)<T* 2 ' (2 ' +1) , F a - W + 3)<T' (2 ' +1)(2 > +2) . (18.24) 

This is in complete agreement with the band spectrum data where 
the lines corresponding to odd levels are three times as strong as those 
corresponding to even levels. Hund gave also the theoretical explana- 
tion of the selectivity of the two modifications with respect to even 
and odd levels. It lies in the requirement of wave mechanics that the 
wave functions representing the states of the molecule must be anti- 
symmetric with respect to the two nuclei (compare section 104). 
These wave functions consist of two factors which characterize, 
respectively, the spins and the rotational motions. It is clear, there- 
fore, that the arrangement of the spins will influence the selection of 
the rotational states. When the spins are anti-parallel, the interchange 
of the two nuclei produces the same effect as a reversal of the spins: 
para-hydrogen is, therefore, anti-symmetric in the spins and must 
have a symmetric rotational function (which happens to be the one 
with even j). On the other hand, when the spins are parallel an inter- 
change of the nuclei does not produce any effect: ortho-hydrogen is 
symmetric with respect to the spins and has an anti-symmetric rota- 
tional function (the one with odd j). We shall, therefore, drop the 
prefixes para- and ortho- and speak instead of symmetrical and anti- 
symmetrical hydrogen (meaning symmetry with respect to the rota- 

F. Hund, Zs. Physik 42, p. 93, 1927. 


Yet the specific heats calculated from the sum of states (18.23) do 
not agree with those obtained from measurements. 1 The reason for 
this has been cleared up by Dennison 2 and lies in the fact that the 
two modifications of hydrogen are very slow in reaching equilibrium. 
The theory of wave mechanics permits one to foresee that transitions 
between the two modifications of hydrogen are extremely unlikely, 
and this is borne out by the complete absence of band lines correspond- 
ing to such transitions. When heat is added to hydrogen, it imme- 
diately distributes itself over the rotational levels within each modifi- 
cation separately, but it takes a very long time for the equilibrium ratio 
of the two kinds of molecules to be established. The number of mole- 
cules of symmetrical hydrogen is proportional to F t , that of antisym- 
metrical to 3F ; we give the equilibrium ratio of these numbers for 
several temperatures using the value <rT = 82.6 deg. which follows 
from the spectroscopical value 3 K = 4.80 X 

Temperature F, : 3Y a 

T = oo 1.00:3.00 

293. IK 1.00:2.98 

78 1.00:1.07 

20.4 1.00:0.0145 

While at high temperatures symmetrical hydrogen accounts for 
only one-quarter of the gas, it is practically pure in equilibrium at the 
temperature of boiling hydrogen. Under usual conditions, the deter- 
mination of specific heats takes only a fraction of an hour, and the 
actual ratio F, : 3 F in the gas is that corresponding to room tempera- 
ture and not to the temperature of measurement. In other words, 
symmetrical and anti-symmetrical hydrogen behave like two inde- 
pendent gases in the permanent ratio of about 1:3. The appropriate 
procedure is, therefore, to calculate separately, by means of eq. (18.07), 
the specific heats of the two modifications: c from F,, and c a from F a . 
The rotational heat of the actual gas is then 

c r - \c. + \c a . (18.25) 

The result is indeed in excellent agreement with experiment, as will 

1 Experimental determinations were due to: F. A. Giacomini, Phil. Mag. 50, 
p. 146, 1925; J. H. Brinkworth, Proc. Roy. Soc. (A) 107, p. 510, 1925; Partington 
and Howe, ibidem, 109, p. 286, 1925; Cornish and Eastman, J. Am. Chem. Soc. 50, 
p. 627, 1928; Scheel and Reuse, Ann. Physik 40, p. 473, 1913; A. Eucken, Sitzungs- 
ber. Berlin, p. 141, 1912. 

* D. M. Dennison, Proc. Roy. Soc. (A) 115, p. 483, 1927. 

Compare: R. T. Birge, I.C.T. V, p. 409, 1929. 



XVIII 119 

be seen from Fig. 49, where the dashed curve is calculated from eq. 

An investigation by Bonhoeffer and Harteck l showed that, under 
ordinary conditions, the time necessary for reaching equilibrium of 
para- and ortho-hydrogen is about ten months. However, it can be 
enormously shortened, in fact, reduced to a few minutes, by letting the 
gas be absorbed in carbon. In this way it is possible to prepare almost 

*- 6 

100' K 200 300 

FIG. 49. Specific heats of hydrogen and deuterium. 

pure symmetrical hydrogen which keeps for several months when stored 
in a glass container. 

119. Deuterium, N, O2> Ck, NO, Cl. Limitations of space do 
not permit us to give an exhaustive account of the work on specific 
heats of gases, and we restrict ourselves to a few typical examples. 

As to rotational heats, the situation in hydrogen repeats itself in 
many other gases. Most of them possess a symmetric and an anti- 
symmetric modification of the molecule, so that the complete sum of 
states is 

Y r = g.Y. + gaY a . (18.26) 

The ratio of statistical weights g : g is, in general, not the same 
as in hydrogen (where it is 3 : 1). The two modifications interact but 
little and behave, under the conditions of measurement, like two 

1 Bonhoeffer and Harteck, Sitzungsber. Berlin, p. 103, 1929; Zs. phys. Chem. 
(B) 4, p. 113, 1929. 


independent gases. The experimental specific heats are, therefore, 
represented by the formula 

cr - g ' C t gaC *- (18.27) 

The vibrational heats are usually calculated from expressions of the 
sum of states of the form (18.20). 

(1) Deuterium (H 2 H 2 ). 

The nucleus of deuterium (deuteron or the heavy isotope of hydro- 
gen) possesses a double spin (with the quantum number 1, as compared 
to ^ of the proton). As shown in the quantum theory, the molecule is 
then capable of 6 symmetrical and 3 antisymmetrical spin configura- 
tions. Moreover, the deuteron obeys the Einstein-Bose statistics, 
being built up of an even number (2) of protons. Therefore, the 
symmetrical spin states combine with the symmetrical rotational 
states, etc., and we have g = 6, g a = 3. The specific heats of deuterium 
have been calculated with great accuracy (even taking into account 
mixed vibro-rotational terms) by Johnston and Long. 1 Their results 
are represented by the solid curve of Fig. 49. The decline takes place 
at lower temperatures than in the case of protium, caused by the, 
roughly, two times larger moment of inertia. The dotted curve of 
Fig. 49 refers to the mixed proton-deuteron molecule rPH 1 as given 
by the same authors. 

The data given below for several gases are mostly taken from 
Trautz and Ader, 2 in whose paper can be found the references to the 
older work on the subject. 

(2) Nitrogen, N 2 (Fig. SO). 

The rotational heat (determined by K = 39.65 X lO" 40 , <rT 
= 2.873, g a : g. = 2 : 1) reaches the full equipartition value at 29 abs. 
The vibrational levels are represented by 2t 3 

0,< = fc- 1 ** - 3374.24(0 + J) - 20.66(t> + |) 2 . (18.28) 

(3) Oxygen, O 2 (Fig. 51). 

1 H. L. Johnston and E. A. Long, J. Chem. Phys. 2, p. 389, 1934. 

2 M. Trautz and H. Ader, Zs. Physik 89, pp. 1, 12, 15, 1934. The experimental 
data included in Figs. 50, 51, 52 are taken from the following papers: Eucken and 
v. Liide, Zs. phys. Chem. 5, p. 413, 1929; Eucken and Miicke, ibidem 18, p. 167, 
1932; P. S. Henry, Proc. Roy. Soc. 133, p. 492, 1931; Shillings and Partington, 
Phil. Mag. 9, p. 1020, 1930. 

1 Compare also: Giauque and Clayton, J. Am. Chem. Soc. 55, p. 4875, 1933. 



XVIII 119 

The equipartition of the rotational heat is reached at temperatures 
still lower than in the case of nitrogen. The vibrational levels are 1 


- 16.26(o 



+ ( 

"" x i 

calc.) F< 
calc.) Fc 
obs.) H< 
obs.) Sh 

jrmula < 
>r mula ( 
ckn am 
illings ai 

J Miickt 








c x 

| e vl in Nitrogtn | 

200 400 600 800 1000 1200 1400 1600 1800 
FIG. 50. Vibrational specific heat of nitrogen. 

The specific heat of O 2 has been calculated also by Johnston and 
Walker, 2 who used the rigorous spectroscopic expressions for the 
energy levels (including the mixed vibro-rotational terms). Their 

o 15 












+ ( 

c vl in 2 

calc. ) Formula ( 18.29 
calc.) Formula (18.21 
calc.) Johnston and W 
obs.) Euckenand Muc 
obs. > Henry 





200 400 600 800 1000 1200 1400 1600 1800 
Tabs * 

FIG. 51. Vibrational specific heat of oxygen. 

data are represented by the dashed curve in Fig. 51. It will be seen 
that the difference is appreciable only at very high temperatures. 

(4) Air (Fig. 52). 1 

The constitution of air is 78.06% N 2 , 21.0% O 2 , 0.94% Ar. 

(5) Chlorine, C1 2 (Fig. S3). 1 

1 Trautz and Ader, he. rit. 

1 Johnston and Walker, J. Am. Chem. Soc. 55, pp. 172, 187, 1933. 

XVIII 119 



The conditions in Cfo are complicated by the existence of the two 
isotopes Clasi Cls7 which form three kinds of molecules. The band 
spectra, however, show that the amount of ClsyCls? is small and 
that chlorine can be taken as consisting of 60% ClasCbs and 40% 

1 _ 



200 400 600 800 1000 1200 1400 1600 
Tabs >- 

FIG. 52. Vibrational specific heat of air. 


c vj in i 

and Mt 








The rotational heat of chlorine is of only academic interest 
as it reaches equipartition at 1.5 abs. The vibrational levels of 
are given by 2 

= 807.81(t; 


c v , in CI 2 
<cak.) Formula (18.30) 
(obs.) Eucken and Hoffmann 
+ (calc.) Formula (18.21) 

500 1000 1500 2000 

FIG. 53. Vibrational specific heat of chlorine. 

In the case of ClasCla? the two coefficients must be multiplied 
(36/37)* and (36/37), respectively: 2 


- 5.57(* 


1 A. Elliott, Proc. Roy. Soc. (A) 127, p. 638, 1930. 
* Trautz and Ader, loc. tit. 



XVIII 119 

(6) Nitric oxide (NO) does not offer much new with respect to the 
rotational and vibrational heats. In fact, c r reaches equipartition 
below 30 abs. and comes into play only above room temperature. 
In the intermediate range NO is interesting because it offers an example 
of doublet heat c* as given by the formula (18.18). From spectro- 
scopic data the doublet difference is Aeo = 354 cal/mol, whence 
9 178 deg. As mentioned in section 117, it is not strictly constant. 
The measurements were carried out by Eucken and d'Or, 1 who also 
calculated the theoretical values from the approximate formula (18.18), 
with 7 = 0. Accurate calculations of the specific heat of NO were 
carried out by Johnston and Chapman, 2 who used the rigorous 
spectroscopic expressions of the energy levels. We give the values of 
c d inferred from their results in the last column of Table 41. It will 
be seen that the approximation given by formula (18.18) is pretty 
good and certainly sufficient in view of the not very accurate experi- 
mental data. 





ca/R (calc.) 

Eucken and 

Johnston and 

50. OC 







































(7) Atomic chlorine (Cl) offers another example of doublet heat 
(18.10). From spectroscopical data Ae = 2 X 580 cal/mol, 9 = 1299, 
7 = 2 log. The calculation 3 leads to the curve shown in Fig. 54. 

There are more gases whose complete curves of specific heats 
have been accurately calculated, but limitations of space do not 

1 A. Eucken and L. d'Or, Nachrichten Gdttingen, p. 107, 1932. 

* Johnston and Chapman, J. Am. Chem. Soc. 55, p. 159, 1933. 

* W. Nernst and K. Wohl, Zs. techn. Physik 10, p. 608, 1929. 

XVIII 120 



permit us to extend the number of examples, and we can only give 
here an incomplete list of references: HgO has been treated by Trautz 
and Ader (loc. cit.) ; CO and N2 by Johnston and Davis (J. Am. Chem. 
Soc. 56, p. 271, 1934); OH by Johnston and Dawson (ibidem 55, 
p. 2744, 1933). The work previous to 1930 can be found in I. C. T. and 
in other Tables of Constants. 

Figures 50-53 contain also the values calculated by Nernst and 
Wohl (loc. tit.) with the help of the approximate formula (18.21). 

500 1000 1500 2000 2500 3000 3500 
T e abs > 

FIG. 54. Doublet specific heat of atomic chlorine (theoretical). 

The very good agreement shows that this simple way of computation 
can be used to great advantage when the coefficient x of the formula 
(18.19) is small compared with unity (and, presumably, T not too 
high). The values of the characteristic temperature 0, for the common 
diatomic gases, as used by them, are given in Table 42. 













o a 




Li 2 




S 2 




Na 2 








K 2 




Te 2 . . . . 














120. The chemical constant of gases. The notion of the chemical 
constant was introduced in section 95. It is that part of the entropy 
of a perfect gas which is left undetermined by the first and second laws 


of thermodynamics. Its determination completes the definition of the 
thermodynamic functions, especially of the thermodynamic potential 
<p = u Ts + pv t and so makes definite the equations of equilibrium 
(6.50). In section 95 we showed that it has the Sackur-Tetrode 
expression (14.12) in the case of molecules having only translational 
degrees of freedom. Generally, the thermodynamic potential of a non- 
degenerate perfect gas can be divided into two parts 

where <p t refers to the translational and <p* to the inner (rotational, 
vibrational, etc.) degrees of freedom of the molecules. To <p t applies the 
expression which was derived for a monatomic gas consisting of mate- 
rial points, i.e. (5.41), (5.43) with c p = SR/2. On the other hand, the 
inner motions do not contribute to the term pv and consequently <?,- is 
identical with the work function ^ whose connection with the sum of 
states is given by (18.08): 

^ = u Qt + RT[lo% p + f log T - log Y - j t ], (18.32) 

where jt is the expression (14.13). 

(A) Monatomic gases do not possess rotational or vibrational 
degrees of freedom. The sum of states Y = Y e refers here only to the 
energy levels of electronic configurations. The term jRriog Y e 
contains, however, a contribution to the zero point energy. In fact, 
factoring off the first exponential in the sum (18.03) we may write 
Y. = exp (- eo/kT) g, or log Y. = - (eo/kT) + log g e . The quantity 
gt can be thrown into the form 

< exp (-GOT, (18.33) 


where 6f = (e, eo)/. Consequently, 

V - uo + RT(lo%p - flog T - j), 
jio - - 1.589 + f logio M + logio g.> (18.34) 

(provided the pressure is expressed in atmospheres). o represents 
the statistical weight of the fundamental state due to the moment of 
momentum of the atom. The quantities Qf are usually so large that 
the higher terms of (18.33) are negligible even at fairly high tempera- 
tures. Only in rare cases are the terms with / = 1,1 2 appreciable. 
Data relating to a number of atoms are contained in Table 43. 1 

^ Taken from K. Wohl, Landolt-Bornstein, Second Supplement, p. 1254, 1930. 

XVIII 120 




of level 




of level 



Noble gases 




5 1 

H, Cu, Ag, Au, 

1 2 5i 



3 (9) 


alkali metals 

J M 




Zn, Cd, Hg, 

I 'So 





earth alkalis 

f ** 

2 Pj4 







2 P& 





2 Pn 




3 (9) 



2 P% 


'P a 



2 P^ 







2 P^ 





2 P^ 







6 I>4 





5 #, 






6 Z?2 






6 Z?i 









(B) Diatomic gases. The concept of the chemical constant can be 
extended to diatomic and polyatomic gases only within the approxi- 
mation with which log F r , pertaining to the rotational states, can be 
regarded as additive to log Y due to the other degrees of freedom. 
We have seen in the preceding section that this approximation is, 
indeed, quite sufficient for thermodynamical purposes. The range 
of temperatures is restricted to those above equipartition of the 
rotational energies when Y r = (l/or), according to (18.22) or 

Y r = - 

+ log K - log <r + log T. 

In practice, moreover, it is assumed that the molecule is in its 
fundamental electronic state, so that g e reduces to go and the remainder 
of log Y becomes log go + log Y vi = log go - t*t/RT. Eq. (18.32), 
therefore, takes the form 

* - o< + *.< + RT[logp - |log T - j], 

where ^* can be calculated either from the approximate equations 
(18.21), with the data of Table 42, or from the more accurate formula 
(18.20). Since the numerical value of -logio(h?/&ir*k) is 38.402, we 
find for the chemical constant the expression 

jio 36.813 + | logic M + logio K - logic * + logic go- (18.35) 



XVIII 120 



10" K 





lO 40 ^ 




H a 










F a 




















Br 8 

330 ? 



2.49 ? 














2(0 1 e = 173) 



575 ? 



2.89 ? 














2(6^ = 182) 


S 2 

67 ? 



1.08 ? 

Na 2 






860 ? 



3.09 ? 

K 2 















The fractional symmetry numbers (for C\2 and Br2 in Table 44) 
represent the mean values for the isotopes. 

(C) Polyatomic gases. Without entering into the derivation we 
only state here the final result for the expression of the thermodynamic 
potential * 

9 = o< + fc< + RT(\ogp - 41og T - j], 
jio = 56.563 + | log p + | log 'K - log <r + log go, (18.36) 

where ~R = (^1X2^3)^ is the geometrical mean of the three moments 
of inertia possessed by the molecule. 

A few numerical values are given in Table 45. 










N 2 




C 3 H 


CH 4 


In conclusion it must be pointed out that the above expressions 
are still incomplete in so far as they do not contain the statistical 
weights g n of the nuclear spins. Strictly speaking, one should add to 

A. Eucken, Phys. Zs. 30, p. 118, 1929; 31, p. 361, 1930. 


the expressions (18.34), (18.35), and (18.36) the term log g n . However, 
at temperatures at which the symmetrical and antisymmetrical 
modifications of the molecules (sections 118, 119) have practically 
reached the ratio g a : g a , corresponding to T = oo , this term is the 
same in the molecules and in an equivalent number of free atoms so 
that it drops out of the equations of chemical equilibrium. At room 
temperature this condition is satisfied for all gases except hydrogen, 
for which it is true from about 70 C on. In Chapter XIX we shall 
discuss problems of transmutation of matter in which the term log g n 
may play a role. We mention, therefore, that in the atom of ordinary 
hydrogen (protium) g n = 2, in the deuteron g n = 3, in the helium 
atom g n = 0. 

121. Heat capacity of a reacting gas mixture. The diagrams of 
section 119 give the specific heats of several gases up to temperatures 
of 2000 abs. Under these circumstances dissociation may be already 
appreciable and must be taken into account. It will, therefore, be 
well to discuss how chemical reactions, taking place in a gas mixture, 
affect its heat capacity. We shall restrict ourselves to conditions 
under which the equation of perfect gases pv = RT gives a sufficient 
approximation, permitting us to use the equilibrium theory of Chap- 
ter VIII. 

Let us consider a mixture composed of the gases 1, 2, ... in the 
respective mol numbers Ni, N2, . . . N ft , and let us suppose the 
following reactions to be possible in the mixture 

v\G\ + . . . + vpGp = 0, 




The condition of equilibrium for the first reaction is represented by 
the mass law (8.18) 

X) "i log xi log K(p,T) (18.38) 


Substituting the definition of the mol fraction xi = Ni/N (where 
N = NI + . . . + Np), we can write 


i log Ni-vlogN* log K, (18.39) 

with the abbreviation v v\ + . . . + v ft . According to eqs. (8.19) 
and (8.21) 

-~, (18.40) 


Q being the heat of the reaction 

In the same way we find for the other reactions 

2>, log Ni - v' log N = log K'> (18.42) 

1 m-^ 

and so on. 

We are now ready to calculate the heat capacity at constant pres- 
sure. The total heat function of the gas mixture is 

, fi Xl, 
and according to (3.26) 

The mol numbers change only inasmuch as the molecules of the 
mixture take part in the reactions (18.37). Each of the variations 
dNi, therefore, must be expressible in the form 

dNi = vi da + v'i da' + . . . , (18.45) 

where the quantities a, a', ... are independent of the coefficients v. 

Noting (18.46) and the relation (dxi/d^) p = c p i and writing for 

2Nic plt (18.47) 

we have 

In the case of a mixture of neutral, non-reacting gases this reduces 
simply to 

C p =",Nic P i. 


The process of change of the mol numbers is supposed to take place 
in such a way that the equations of equilibrium (18.39) and (18.42) 
are always fulfilled. We can, therefore, obtain (da/dT) p by differ- 
entiating the first of them and taking into account that the contribu- 
tion of this reaction to (dNi/QT) p is the first term of (18.46), namely, 


With the abbreviations 

etc., we obtain as the final expression 

c p = N [r p + F( X ) ^ + TO ~z + 

The mol fractions x refer here, of course, to the state of chemical 
equilibrium. They can be obtained either theoretically, by the solution 
of the simultaneous eqs. (18.38), (18.42), etc., or empirically from 
measurements of the density and other properties of the gas mixture. 

For the heat capacity at constant volume, we must start from the 
internal energy of the mixture 

U = ^,NiU h (18.51) 



For dNi we have again to use the expression (18.45). Noting that 
(dui/dT) v = Cii, 2viUi = AC/, and introducing the abbreviation 

i^ (18-53) 


we obtain 

In view of the relation (8.19), log K = log K p (T) - vlogp, the 
equation of equilibrium (18.39) can be transformed by substituting 
for the pressure its expression from the equation of state, p = NRT/V: 

Y,n log Ni - log K p - v log T + v log (V/R). (18.54) 



Introducing the abbreviations 

/(*) = GE'iV*!)" 1 , /'(*) = (XV' 2 /**)' 1 . ( 18 - 55 ) 

and taking into account the relation xi = i + RT, we obtain by 
differentiating (18.54) with respect to T 


/(*) VaTVV \RT 2 T 
The expression for the heat capacity (18.52) is, therefore, 

An interesting application of this theory is due to McCollum, 1 
who investigated the heat capacity of nitrogen tetroxide (N2O4). 
On the one hand, he made calorimetric determinations of C p at temper- 
tures between 30 and 100 C. On the other hand, he calculated C p 
from a formula equivalent to (18.50). The equilibrium in N2O4 was 
treated in section 52: the only reaction in it (8.22) has the coefficients 
vi 2, V2 = 1 v = 1. As was shown there, the mol fractions can 
be expressed in terms of the degree of dissociation as follows: 
*i - 2{/(l + ), * 2 - (1 - *)/(! + *) Hence we find for the 
(single) functions F(x) and f(x) 

D, /(*) - 1(1 - 0/(l + Q(2 - Q. 

McCollum used empirical values of . The excellent correspon- 
deiice between measurement and calculation which he obtained is 
shown in Fig. 55. 

122. Velocity of sound in gases and gas mixtures. Determina- 
tions of the velocity of sound in gases have been often used as a 
simple and, apparently, accurate method of measuring their specific 
heats. As was mentioned in section 18, the square of the velocity of 
sound in a pure gas is supposed to be given by the formula 

> McCollum, J. Am. Chem. Soc. 49, p. 28, 1927. 

XVIII 122 



p being the mass density and M the molecular weight. If the gas fol- 
lows the equation pv = RT, this expression takes the form (3.40) 

a 2 = yRT/n, 

1 = 

c v 



FIG. 55. Specific heat of nitrogen 

It was pointed out that these relations apply without regard to 
whether c p , c vt y are constants or functions of temperature. 

However, the specific heats c v 
measured by the sonic method in 
oxygen and in air 1 at high temper- 
atures (from 100 to 1000 C) are 
considerably below those determined 
by other methods. In a similar way, 
the sound velocities in NO give the 
same specific heat at low tempera- 
tures as at C, 2 instead of the 
increase tabulated in Table 41 of 
section 119. This fact seemed to 
indicate that the changes of state, 
produced by the sonic waves, are 
too rapid to allow the equilibrium of 
the molecular degrees of freedom to 
be established. In fact, the disper- 
sion of sound waves in gases (CC>2) had already been observed a few 
years earlier 3 and was interpreted as a lag in the molecular adjust- 
ment. 4 Extensive work on the subject was done by Kneser, 6 who was 
able to show that it is the vibrational energy which requires periods up 
to 10~~ 2 sec to assume its equilibrium value, while the rotational 
degrees of freedom respond much more quickly. At low frequen- 
cies the ratio 7 of the formula (18.58) was experimentally equal 
to 1 + R/(c t + c r + c*i), at high frequencies, to 1 + R/(ct + c r ). 

These investigations were greatly influenced by a paper of Ein- 
stein's 6 which suggested sonic measurements as a method of deter- 

1 W. C. Shillings and J. R. Partington, Phil. Mag. 9, p. 1020, 1930. 
1 Partington and Shillings, Phil. Mag. 45, p. 416, 1923. 
1 G. W. Pierce, Proc. Am. Acad. 60, p. 271, 1925. 
4 K. F. Herzfeld and F. O. Rice, Phys. Rev. 31, p. 691, 1928. 
*H. O. Kneser, Ann. Physik 11, pp. 761, 778, 1931; 16, p. 337, 1933; H. O. 
Kneser and V. O. Knudsen, Ann. Physik 21, p. 682, 1935. 
6 A. Einstein, Sitzungsber. Berlin, p. 380, 1920. 


mining the rates of reaction in gas mixtures. While this paper was 
indirectly responsible for the research on dispersion in pure gases, 
the results of this research defeated its original purpose. The obser- 
vations in mixtures are too difficult to interpret when possible abnor- 
malities of the components themselves must be taken into account. 1 
For this reason, we shall calculate here the velocity of sound in mix- 
tures only for low frequencies when the question of lag does not 

The velocity of sound in a gas mixture of the total mass M can 
be written 

a 2 = (dp/dp) a = ~ V 2 (dp/dV) s /M, (18.59) 

since the mass density is p = M/V. Let us envisage a mixture of a 
given initial constitution. If its reacting components are always in 
equilibrium, the total volume is completely determined by the tempera- 
ture and pressure. The system is, therefore, simple in the sense of 
section 2: its entropy differential, dS = (dU + pdV)/T, can be 
regarded as a function of pressure and volume, and the equation of 
the adiabatic can be written 

since OX/9F) P = (dU/dV) p + p, in view of the definition of the heat 
function X = U + pV. Because of (3.12) and (3.26) 


V 1 = *"V ITI / \ 

\dp/v \dp/v 



(dp/dV)a BB y(dp/dT) v /(dV/dT) p = 7(9/9tOrl 

| (18.60) 

7 = Cp/Cv ] 

This equation holds generally for any simple system. When we 
apply it to a gas mixture, the heat capacities in the ratio y are, of course, 

1 C. E. Teeter, J. Chem. Physics 1, p. 251, 1933. This paper contains a good 
review of the experimental and theoretical work about measuring rates of dissocia- 
tion in nitrogen tetroxide by the sonic method. 


those calculated in the preceding section. From the equation of state, 
pV = NRT, 

We transform this with the help of the relations (18.43), (18.46), 
and (18.54), obtaining 

1 + tf ) + //'(*) 

(18 ' 62) 

It will be noted that the last factor is much less affected by the 
heats of reaction than 7: it is linear in Q/RT, etc., while Cv, C p 
contain the squares of these large numbers. 

Experimental work concerning the velocity of sound has been 
published by several authors. 1 While the results are not quite con- 
sistent in their bearing on the rate of the reaction and the speed 
with which equilibrium is established, it seems that indications of a 
lag in reacting gas mixtures are to^be found only in the measurements 
of absorption of sound. There is no convincing proof of the existence 
of dispersion, and the formula (18.62) appears to represent the observed 
velocities fairly well. This would mean that chemical equilibrium is, 
practically, reached after the lapse of only 10~ 4 sec. 

123. Solids. Einstein's model of a solid body was described in 
section 93. It treats the atoms of the solid as harmonic oscillators, 
all independent and having the same frequency v. For the energy 
levels of the three-dimensional harmonic oscillator the quantum theory 

e ni n 2 n, = kv(m + W2 + W3 + f), 

where n\ t n%, #3 are the quantum numbers of the three degrees of 
freedom. The sum of states is 

exp [-*(i +H2 + H3+ f)] = C (18.63) 

\ l ~- e ) 

(x = hv/k). The energy and entropy can be derived from it by eqs. 
1 Compare the review in the paper of Teeter's (he. cto.) 


(18.06) and (18.09), and are those given in section 93. In particular, 
the mean energy of the oscillator is, per degree of freedom, 

- "* 

' e - l ' 2 

As was mentioned before, the agreement of Einstein's formulas 
with the measured specific heats is not very close. The reason for this 
discrepancy was pointed out in a later paper 1 of his: the model does 
not represent the actual conditions accurately enough because, in 
reality, the atoms of a crystal do not oscillate independently of one 
another with the same frequency but form a coupled system. The 
forces of interaction influence the characteristic frequencies so that 
they all become different and are drawn out into a spectrum. More 
accurate theories were given from this point of view by Debye 2 and 
independently by Born and Von Karman. 3 Debye's theory is par- 
ticularly simple: he assumes that the 3Z frequencies of a body con- 
taining Z atoms can be calculated as if it were a homogeneous elastic 
continuum. In other words, these frequencies correspond to the 
fundamental tone and to the first 3Z 1 overtones of its elastic 
spectrum. The following considerations support this assumption. 
In the region of high temperatures prevails equipartition and the 
energy depends only on the number of frequencies (or degrees of 
freedom) and not on their values. On the other hand, at low tempera- 
tures, the high order overtones are not appreciably excited ; according 
to eq. (18.64) their mean energy -becomes very small. All depends 
then on the low order frequencies, and for these the approximation by 
a continuum is quite adequate because their wave length is large 
compared with the atomic distances. Therefore, we should expect 
accurate results both at low and at high temperatures. 

The calculations are easily carried through in the case of isotropic 
elementary materials. If we take the solid in the shape of a cube with 
the edge /, 4 the theory of elasticity gives the following expressions for 
the frequencies of the standing longitudinal and transverse waves 

v - ai(m 2 + 2 2 + w 3 2 ) H /2/, v = a 2 (ni 2 + n 2 2 + w 3 2 ) H /2/, (18.65) 

where a\ and a* are the velocities of the longitudinal and transverse 
wave systems. Every triple of integral numbers wi, w 2 , m determines 
a state of the body (standing wave) in each of the two wave systems, 

1 A. Einstein, Ann. Physik 35, p. 679, 1911. 

P. Dcbye, Ann. Physik 39, p. 789, 1912. 

*M. Born and Th, Von Karman, Phys. Zs. 13, p. 297, 1912; 14, pp. 15, 65, 1913. 

4 R. Ortvay, Ann. Physik 42, p. 745, 1913. 


the corresponding frequencies being given by eqs. (18.65). It is easy 
to see that the number of all longitudinal states with frequencies 
smaller than a given value v is 




where V = P is the volume of the cube. In the transverse system the 
number must be doubled: every transverse wave has two degrees of 
freedom because of the two possible states of polarization 

(18 ' 67) 

Exercise 105. If we imagine the numbers ni, n 3 , n 9 plotted in three orthogonal 
directions, their integral values will represent a cubic lattice. The number Zi(v) is the 
number of lattice points within the surface given by the first eq. (18.65). Show 
that the expression (18.66) holds when / is very large. i, n^ n\ are positive. 

If the solid were really a continuum, there would be no upper limit 
to its characteristic frequencies. But it has, actually, atomic structure, 
and the total number of its possible states must be equal to the number 
of its degrees of freedom, namely 3Z. Debye, therefore, makes the 
assumption that the elastic spectrum is cut off at an upper limit VQ 
determined by the condition 


Substituting (18.66) and (18.67) we find 

The number of characteristic states in the frequency interval from 
v to v + dv is 


dZi(v) + dZ*(v) - Mv. 

Each of these states is a harmonic oscillation whose mean energy 
is given in the quantum theory by the expression (18.64). The con- 
tribution of the frequency interval dv to the internal energy is, there- 
fore, QZpZiPdv/vo 3 * The total internal energy is obtained by integrat- 
ing this expression from to VQ and is, per mol, 

" *dv , /40 ^ 

- + " (18 ' 68) 



XVIII 123 

The specific heat results from differentiation with respect to T 
(contained in x). If we introduce the abbreviation hvo/k = 9, 

c = 


This formula shares with Einstein's (18.64) the property that 
the individual characteristics of the substance are all contained in 
the single constant 9, the characteristic temperature. If we plot c 
against T/9 (the reduced temperature) we obtain for all bodies one 
and the same universal curve. In the limiting case of very low tem- 
peratures (r/9 1) the integral can be regarded as constant and 
equal to 47r 4 /15. The formula becomes then 

127T 4 

c = 

The agreement with experiments can be judged from Fig. 56 l 
giving the measured c v . The value of 9 for each substance is here 

2 -3 -4 -5 -6 -7 -8 -9 2 -1 

Pb, Ag, KCI, Zn, NaCI, Cu, Al, CaF 2 C 

FIG. 56. Specific heat of solids. 

chosen so as to give the best fit for that substance with the continuous 
curve representing eq. (18.69). Curve II is in the normal position; 
the others are displaced because the experimental material was too 
large to be accommodated in a single curve. The key to the sub- 
stances is contained in Table 46. 

1 Taken from E. Schrodinger, Phys. Zs. 20, pp. 420, 450, 474, 497, 523, 1919. 

XVIII 123 








Points in Fig. 





CaF 2 
FeS 2 



















Potassium bromide. . . 



32-95 * 
22-57 * 








Iron pyrites . . 


* Rises above the curve after these temperatures. 

The formula (18.69) was derived on the assumption that the sub- 
stance is isotropic and elementary. Within this scope, it represents 
the facts on the whole very well although a few exceptions are known. 
But Fig. 56 shows that it applies, in some measure, also to substances 
of a more complicated structure, presumably because they do not 
deviate greatly from isotropy. 

The approach of Born and Von Karman permits to treat the more 
general cases, at least, by approximation. If the substance is an 
element in an anisotropic system of crystallization, the elastic waves 
propagate in it with three different velocities and form three sets of 
overtones. Born l finds that the specific heat is then the arithmetical 
mean of three terms of the Debye type (18.69), each with a different 
characteristic temperature. If the substance is a chemical compound 
containing r different kinds of atoms, there still exist, in addition to 
the spectrum of overtones, r 1 characteristic frequencies represent- 

1 M. Born, Dynamik der Krystallgitter, 1915. 



XVIII 123 

ing the oscillations of these atoms with respect to one another. The 
molal heat is, therefore, expressed by three Debye terms and r 1 
terms of the Einstein type (13.25). Unfortunately, these expressions 
are so unwieldy that their comparison with experimental results is 

The third power law at very low temperatures remains valid also 
in the general case because the Debye terms satisfy it and the Einstein 
terms become negligible. While a great many substances fulfill the 
third power law down to the very lowest attainable temperatures, 1 
other metallic and non-metallic solids show marked deviations in the 

region of the temperature of 
liquid hydrogen and below. 
The measured specific heats lie, 
in this case, above those pre- 
dicted by Debye's formula. It 
was suggested that in metallic 
substances the discrepancy 
may be due to the specific 
heats of the free conduction 
electrons (compare sections 102 
and 111). In fact, in the case 
of silver the influence of the 
free electrons seems well 
authenticated by recent 
measurements in the laboratory 
of Leiden. 2 In Fig. 57 is plot- 
ted the difference between the 


















-T 2 4 6K 

FIG. 57. Electronic specific heat in silver. 

measured c and the value 

predicted by the third power law (18.70). It shows both the linear 
dependence required by the Sommerfeld formula for free electrons 
(17.34) and the correct absolute value on the assumption that there 
is one electron to every atom of silver. 

The only other (non-supraconductive) metal investigated in the 
same temperature region is zinc, and here the free electrons do not 
give a sufficient explanation of the trend of specific heats. Even 
after correcting for them, there remains a hump-like residual curve 
with a maximum at 4 abs., not unlike in its shape to that of Fig. 54. 
Indeed, it was suggested that they are due to the same cause, doubling 
of the lowest quantum level of the zinc atom. However, the relation 

1 Compare in this connection, the work of Simon and collaborators (Ann. Physik 
68, p. 241, 1922; Zs. phys. Chemie 123, p. 383, 1926; Zs. Physik 38, p. 227, 1926). 
W. H. Keesom and J. A. Kok, Physica 1, p. 770, 1934. 

XVI II 123 








1.6 2.0 

of the position of the hump and its height is such that only a small 
fraction of the zinc atoms could possess the double level, a circumstance 
which makes the explanation doubtful. There are some solids (such 
as gadolinium sulfate, 1 samarium sulfate, 2 orthohydrogen 3 ), in which 
the doubling or tripling of the atomic and molecular levels gives a 
complete explanation of the anomalies of specific heats according to 
(18.18) or to similar formulas. We reproduce here the very instructive 
curve measured by Giauque and McDougall in gadolinium sulfate 
(Fig. 58). 

In other cases (e.g. tin and silicon 4 ) there were observed humps 
in the curves of specific heats lying at somewhat higher temperatures 
whose detailed explanation is still 
lacking. Simon suggested that they 
may be due to a doubling of the 
energy levels according to the 
formula (18.18). It should be men- 
tioned, however, that electrons 
which are not completely free and 
cannot cause electric conduction 
may nevertheless contribute to the 
specific heats, sometimes even in 
a larger measure than the free 
electrons. At extremely low tem- 
peratures the specific heat due to 

them is proportional to r, in non-magnetic materials, and to T*, in 
ferromagnetic ones. 6 At somewhat higher temperature this cause, 
being essentially a multiplet action (compare section 117), should 
produce a hump similar to that of formula (18.18). 

Recent work by Jaeger and co-workers (on metals at high tempera- 
tures) drives it home that our theoretical knowledge of specific heats is 
still incomplete. Measuring very accurately, 6 they found that c p is 
structure sensitive and varies appreciably with the mechanical treatment 
of the specimens. Another series of investigations 7 was devoted to 
testing Neumann's law (section 16) with the result that it was found 
satisfied, within the experimental accuracy of 1%, in mixed crystals 

1 Giauque and McDougall, Phys. Rev. 44, p. 235, 1933. 

2 Ahlberg and Freed, Phys. Rev. 39, p. 540, 1932. 

1 Mendelssohn, Ruhemann, and Simon, Zs. phys. Chem. (B) 15, p. 121, 1931. 
4 F. Simon, Sitzungsber. Berlin, p. 477, 1926. 

F. Bloch, Zs. Physik 52, p. 555, 1928; Leipziger Vortrage, p. 67, 1930; P. & 
Epstein, Phys. Rev. 41, p. 91, 1932. 

6 Jaeger, Rosenbohm, and Bottema, Proc. Amsterdam 35, pp. 763, 772, 1932* 

7 Bottema and Jaeger, ibidem, 35, pp. 352, 916, 929, 1932. 

0.8 1.2 

FIG. 58. Anomaly of specific heat in 
gadolinium sulfate. 



XVIII 124 

(Ag-Au). On the other hand, in homopolar intermetallic compounds 
(AuSn, PtSn) there were found appreciable deviations from Neumann's 
law which, moreover, increased with rising temperature (between 
and 200 C). 

124. Liquids. The liquid state of matter is the most complicated 
and has not yet been made amenable to theoretical treatment. There 
exists a number of elements and compounds having in the liquid state 
approximately the same molal heats which are characteristic of solids, 
according to the laws of Dulong and Petit and of Neumann (section 
16), namely, about 6 cal/mol per atom (Table 47). 














H 2 O 








S 2 C1 2 








SiCl 4 























in cal /mol 

This would indicate that the conditions are similar to those in 

solids in that one must take into 
account both the kinetic and the 
potential energy of the atoms. 
However, many liquids have atom- 
ic heats much larger than 6 cal. 
Because of its practical im- 
portance, much attention was 
devoted to the specific heat of 
water. Of particular interest is 
the dependence of c p on pressure 
represented graphically in Fig. 
59. These curves were not de- 
termined by direct measurement 
but inferred from the thermal 
expansion at different pressures 1 
by means of the relation p (4.59), 

or Cp = /r(aV3^ 2 )p dp. We 
shall see that these data have an important bearing on the question 
of specific heats of aqueous solutions. 

*Bridgman, Proc. Am, Acad. 48, p. 309, 1913; F. Zwicky, Phys. Zs. 27, p. 271,1926, 


p-10T 3 inkgXcm 2 

59. Specific heat of water as a 
function of pressure. 

XVIII 124 



As was shown by Zwicky, the specific heats of non-electrolytic 
solvents in water are additive: i.e. the total heat capacity of the 
solution can be expressed by the formula 

No c p o + Ni c p i 


where C P Q = 18.01 is the molal heat of water (at 15 C) and c p i of the 
solute. It is interesting to note that the values of c p \ observed in the 
solution are pretty close to those observed in the solid state of the 
solute, as appears from Table 48. 



C 6 H 12 6 


OCN 2 H 4 

C,H 8 0, 

Tartaric acid 
C 4 H 6 6 

c p i from eq. (18.71) .. 
Cpi in solid state 

51 7 

98 5 

19 3 



When we turn to aqueous solutions of electrolytes, we meet with 
the unexpected fact that the heat capacity is not increased but lowered 
by the solutes. A qualitative explanation of this behavior was given by 
Tammann, 1 who pointed out that the effect is analogous to a rise of 
pressure to about 500 atm: the electrolyte must increase, in some 
way, the pressure under which the particles of water stand. Accurate 
modern measurements in solutions of alkali chlorides 2 showed that 
at low molalities (up to m = 0.5) the heat capacity is still well repre- 
sented by the linear formula (18.71). However, the coefficients c p \ 
here take negative values of the order 40. 

Taking up Tammann's suggestion, Zwicky (loc. tit.) developed it 
into a detailed theory. The field of the ions exercises large ponderomo- 
tive forces on the electric dipoles of the water molecules which actually 
produce a strong compression of the aqueous medium in their vicinity. 
Each ion is, therefore, surrounded by a little sphere within which the 
specific heat of the water is reduced by pressure. As long as the con- 
centration is low, the spheres do not overlap and the reduction of the 
heat capacity is proportional to their number, i.e. to the molality. 
Hence the validity of the linear expression (18.71). The quantitative 
calculation of this effect gave the right order of magnitude. 

1 G. Tammann, Innere Krafte und Eigenschaften der Losungen, 1907. 

1 Richards and Rowe, J. Am. Chem. Soc. 43, p. 770, 1921; 44, p. 684, 1922. 


125. Equivalence of mass and energy. We devoted Chapters 
XIII to XV to questions pertaining to the determination of the abso- 
lute value of the entropy S. It should not be forgotten, however, that 
there is another fundamental function, the internal energy U, whose 
absolute value is also indeterminate. As far as the first and second 
laws are concerned, S and U are on the same footing since only the 
differentials of these two functions are defined in thermodynamics 
(compare sections 13 and 22). Both contain, therefore, additive inte- 
gration constants. That the indeterminacy of the energy constant 
was much less emphasized was due to historical and practical reasons. 
It was taken as a matter of course, because the thermochemical 
methods of measuring the differences of these constants in a reaction 
had already been developed before thermodynamics was established 
as a separate science. 

It became possible to determine the absolute value of the energy 
after Einstein l had established the law of proportionality of mass and 
energy as an important result of his theory of relativity. The fact 
that the inertial mass of a particle depends on its kinetic energy was 
not an exclusive feature of relativity but was proved long before its 
advent. But only the theory of relativity gave a sharp definition of 
the concepts here involved, and abolished the difference between 
inertial and gravitational masses. Let M be the mass of a system, 
E its total energy, and c the velocity of light. Einstein's law is then 
expressed by the simple relation 

E Me 2 . (19.01) 

If we consider 1 mol of a substance at the temperature T = 0, its 
internal energy will be UQ, while its mass will be equal to its molecular 
(or atomic) weight ju Einstein's law gives, therefore, 

*o => 2 . (19.02) 

1 A. Einstein, Ann. Physik 18, p. 639, 1905. 


Strictly speaking, the mass depends on the temperature since the 
energy changes with it. To be exact we should have specified that M 
is the molecular weight at T = 0. Chemistry proper deals with dif- 
ferences Ao (for instance in dissociation of molecules) so small that 
the attendant change of mass AM = Awo/c 2 cannot be detected by the 
finest measurements. In this chapter, however, we shall treat proc- 
esses in which an appreciable part of the energy of the reacting sub- 
stances is converted into radiation. To the same extent their mass 
will undergo a loss: conservation of energy and conservation of mass 
hold only for matter and radiation together and not for the material 
part of the system separately. Compared with the huge changes of 
mass involved in these processes the difference between /u at T = 
and T = 300 abs is irrelevant and we shall understand under the 
symbol AC the molecular weight as tabulated by international agreement. 

As an example we write out the explicit expression for the thermo- 
dynamic potential of a monatomic perfect gas 

V = RT(log p - | log T - j) + nc*. (19.03) 

For some applications it will be more convenient to express the 
pressure in terms of z (the number of molecules or atoms per unit 
volume) by the relation (1.15), p = zkT, 

V = RT(log z - | log T - j + log *) + c? (19.04) 

126. Energy and entropy of black radiation. The theory of radi- 
ation is so extensive a subject that it is best treated in separate books, 1 
and it is not advisable to discuss it at length in a textbook devoted to 
thermodynamics. We limit ourselves to a brief review of some of its 
concepts and laws which shall be needed in the remaining part of this 

Let us consider a closed system part of which is free of matter 
(vacuum) and filled only with radiation, while matter is present in 
other parts. Left to itself such a system reaches a state of equilibrium 
in which the temperature T is uniform throughout and the radiation 
in the evacuated part has quite definite properties. It is called black 
radiation, and its spectral composition is given by a universal law, 
independent of the nature of the material part of the system in equilib- 
rium with it: the energy density u/lv (i.e. energy per unit volume) 
belonging to the interval of frequencies between v and v + dv is a 
universal function of v and T only 

M" - F(v,T)dv. (19.05) 

1 For instance, M. Planck, The Theory of Heat Radiation, 1914, 


This implies, of course, that the total energy density 

/CO /*00 

u& - / F(v,T)dv = u(T) (19.06) 


is a function of the temperature alone. 

If black radiation is enclosed in an evacuated container, it exerts 
a pressure p on its walls and through it resists compression. Assuming 
the walls to be movable, the radiation does the work p dV when the 
volume is increased by dV (compare section 7). If an element of heat 
dQ is imparted to the radiation, the first law of thermodynamics 
requires that it be converted, in part, into an increase of its energy U, 
in part, into work done by it 

dQ = dU + pdV. 

The second law can be also applied to radiation and includes the 
familiar expression (4.07) for the entropy differential 

<ZS= (dU + pdV)/T. (19.07) 

In particular, when the volume of the system remains constant 
(V = const, dV = 0) the increment of the entropy is 

dS - dU/T. (19.08) 

The same results hold, of course, if the vessel is not quite evacuated 
but contains, in addition to the radiation, gases in equilibrium with it, 
of so low a density that their refraction is negligible, so that the prop- 
agation of light is appreciably the same as in vacuo. This is the 
case we shall consider in the following sections. The restriction to 
low densities, though not absolutely necessary, greatly simplifies mat- 
ters and saves lengthy discussions. 

The general relation (19.08) which contains practically no restrict- 
ing assumptions about the function U is all we shall need. As far as 
our applications are concerned, the rest of this section has a purely 
academic interest. The quantum theory of radiation leads, in its 
present form, to the result that empty space contains a certain con- 
stant energy density o even at T = 0. It is usually thought that 
this is merely a formal defect and that this zero point energy (which is, 
by the way, infinite) has no physical reality. There were, however, 
those who argued that a part of it may have physical significance, 
that it may influence the absolute entropy constant of the radiation 
and through it the equilibrium with matter. We shall see in the 
^t section that the conditions of equilibrium are independent of the 


absolute entropy of radiation. Nevertheless the question how it is 
affected by UQ is worth looking into. We shall write, therefore, for 
the energy density, u + Wo, meaning by u that part which is due to 
temperature radiation. The total radiation is then U = (u + u$)V\ 
as to pressure, it is known to be p = w/3, since it is certain from 
empirical and theoretical considerations that the zero point energy 
does not contribute to it. The entropy differential (19.08) takes the 

V du /4 \ d V 

dS - -dT + (j u + no ) (19.09) 

The reciprocity relation (2.10) is easily integrated and gives 

u = ar 4 - f wo, (19.10) 

a being a constant. We see that the inclusion of a constant term UQ 
(not exerting pressure) into the energy density leads to a contradiction. 
The result gives for p = \u \aT* Jwo, so that UQ does, after all, 
produce a negative pressure. It follows that the zero point energy, 
whether it is in other ways real or not, must be left out in computing 
the entropy. We arrive thus at the usual form of the Stefan-Boltz- 
mann law 

u = ar 4 , (19.11) 

and substituting this into (19.09) 

5 = |ar 3 F+5 . 

So is an integration constant and as such must be independent of V 
and T. It is obvious that it cannot be anything but So = 0, other- 
wise we would have still a finite entropy when the volume is reduced to 
zero, i.e. when we have no system. The entropy density is, therefore, 

* = | = i r3 - (19tl2) 

Comparing this with (19.11), we verify 

ds - du/T. (19.13) 

This equation is compatible with the following interpretation form- 
ing an important part of the theory of black radiation. The entropy 
density 5 is considered as the sum of the densities sjiv belonging to 
the different frequency intervals 




where $ is defined by 

ds v - du,/T. (19.14) 

In fact, the integration of this relation with respect to dv (at con- 
stant T) gives eq. (19.13). The relation shows that the frequency 
intervals of the radiation can be regarded as statistically independent 
systems in equilibrium with one another at the temperature T (com- 
pare section 22). In the particular case of constant volume (dV = 0), 
we can write 


ds T 9 (19 * 15) 

if we denote U v Vu,, S, = Vs . 

127. General conditions of equilibrium. Let us envisage a system 
enclosed by rigid, perfectly reflecting, adiabatic walls and consisting 
of a sufficiently thin mixture of perfect gases (compare preceding sec- 
tion) and radiation. Let N\, N%, . . . N ft be the mol numbers of the 
gases; the total energy and total entropy of the system are then 

if UR, SR stand for the energy and entropy of the radiation. The total 
volume of the system is expressed in terms of the molal volume vi of 
each gas as 

The conditions of equilibrium (compare section 31) are given by S 
having its maximum while U and V stay constant, in view of our 
assumptions about the walls. In variational form they are 



+ 8U* = 0, (19.16) 

+ 8S* - 0, (19.17) 

0, (/ = 1, 2, . . . 0) (19.18) 

We can regard the first equation as the main condition, the others 
as subsidiary ones. The difference between this system and that of 
section 40 lies only in the presence of radiational terms. Apart from 


this, the procedure is exactly the same as in treating the ordinary gas 
mixture. We first eliminate the variations of the internal energies by 
means of (4.15) and (19.13) 

dU R = T R dS R , 
whence (19.16) takes the form 

+ uitNi) + T R dS R 0. (19.19) 

Now we apply the method of Lagrangean multipliers (section 38) : 
we multiply the first subsidiary condition (19.17) by Xo and the others 
(19.18) by X/, (/ SB 1, 2, . . . 0), and add them to the main condition 

#i(- Pi 
X, vi)*Ni] + (T R + Xo)SS* - 0. (19.20) 

As was explained in section 38, the variations foj, dS R , dvi can now 
be considered as independent, whence their coefficients must vanish 

- Xo - Ti = TR - T 

(i.e. the temperature is uniform) and 

Xz - pi. 

The last equation supplies only the physical interpretation of the 
multipliers Xj. Recalling the definition (5.33) of the molal thermo- 
dynamic potential ?i = u\ Tsi + pi vi, 1 we can write what remains 
of (19.20) in the form 

T] PI#I - 0, 

which is identical with (6.41) derived without considering radiation. 
If the reaction taking place in the mixture is expressed by the equation 

0, (19.21) 

the condition of equilibrium is 

T] n Vi - 0. (19.22) 

1 As in (8.08) we use the notation "Qi (and not <PI) to indicate that this function is 
calculated as if the other gases were not present, i.e. in terms of the partial pressure. 


Though formally identical with (6.49), these conditions have a 
wider scope of applications. In deriving them we explicitly admitted 
that the energy and mass of the material part of the system need not 
remain constant but may be converted, wholly or in part, into radia- 
tion. We are, therefore, free to apply them also to such reactions 
(19.21) in which substances disappear altogether under emission of 
radiation (annihilation of matter) or in which one kind of atoms dis- 
appears and another is created (permutation of elements). We may 
also mention that, because of eq. (19.15), we would have obtained the 
same result writing in (19.16), (19.17), dU v , 5S,, instead of dU Rt &S Rt 
where v is the frequency of the photons emitted in the reaction. In 
other words, the reacting substances set themselves in direct equilib- 
rium with the frequency interval corresponding to their emission and 
through it, indirectly, with the rest of the radiation. 

Using the expression (19.04) for the thermodynamic potential of 
monatomic gases, we obtain from (19.22) 

"i log zi = log / + I v log r - A-c 2 /RT, (19.23) 


where v = v\ + v^ + . . + v$\ the change of mass in the reaction is 


log / = Z"tf'i - v log k. (19.24) 


According to section 120, the chemical constant of a monatomic 
gas is jio = 4.417 + logio M + log g, where g must include also the 
nuclear statistical weight, and logio k = 15.8630. Therefore, 

! log Zi = log K n , 

T- - ' ' - - * (19 2S) 

logio K. = 20.280? +vi (f logio M< + logio gi) 

4.695 X 10 12 A M 

+ f v logio T - 

128. Negative and positive electrons. Equilibrium of matter and 
radiation as a problem of thermodynamics was first treated by 
O. Stern. 1 He envisaged a state of affairs in which atoms disappear, 
being converted into radiant energy, and others suddenly appear in 
the field of radiation, absorbing the amount of energy equivalent to 

1 0. Stern, Zs. Elektrochemie 31, p. 448, 1925. 


their mass. As science progressed processes of this sort were regarded 
as less and less speculative. With the discovery of the positive elec- 
tron (positron) by C. D. Anderson 1 the creation and annihilation of 
electron pairs became an established fact. The tracks of the two 
electrons (one positive, one negative) produced by a cosmic ray or 
7-ray can be observed in the Wilson chamber. There is also direct 
evidence that they recombine, emitting their whole energy in the form 
of two photons. 

If we characterize the electron and positron by the subscripts e and 
p, eq. (19.21) of the reaction becomes 

G e + G p - 0, (19.26) 

whence ? = v p = 1, v = 2, and 2 

K. (19.27) 

For the electron of either sign f log /z + log g = 4.59 (compare 
section 113), A/* = 2/x = 1.096 X 10"* 

5 146 X 10 9 
logio K = 31.38 + 3 logic T - - - jr -- (19.28) 

In the empty space, in the absence of all other types of matter, 
positive and negative electrons would exist in equal numbers, z p = z et so 
that the density of positive electrons would be 

z p = K*. (19.29) 

This number is exceedingly small at ordinary temperatures and 
becomes appreciable only when the temperature approaches 10 8 
degrees. In fact, for T = 8 X 10 7 deg, we find z p = 2.5 X 10~ 6 cm" 3 , 
or 1 positron per 40000 cm 3 . From then on the rise is rapid: for 
T = 2 X 10 8 , we already have z p = 1.9 X 10 15 cm- 3 . 

Of course, temperatures of this order do not prevail in interstellar 
space. The only place where they may possibly exist is the interior 
of stars. Here the conditions are different from those in empty space, 
in that there is a large excess of free negative electrons, which depress 
the density of the positrons in the way explained in section 53. Let 
us denote by ZQ the density of the free electrons which are in the star 
from the start, being stripped by ionization off the atoms of the stellar 
material. In addition to this, there are the positive-negative pairs 

1 C. D. Anderson, Science 76, p. 238, 1932. 

* Since there is no clanger of misapprehension, we drop the subscript % of K in 
the remainder of this chapter. 



XIX 129 

created by the reaction (19.26). The strong electric attraction will 
assure that the stellar matter remain electrically neutral. If the 
density of positrons is z p , the total density of negatives will be 
z e = z p + ZQ, the excess negative charge being neutralized by the 
atomic ions. Equation (19.27) gives, therefore, 


Z P (Z P + ZQ) = K, 

z p = ((K + so 2 /4) H - 20/2]. 


To fix our ideas, let us assume ZQ 5 X 10 21 cm" 3 , which is, pre- 
sumably, of the right order of magnitude for stars of the G and K 
types. Table 49 gives the density of positrons z p for a large range of 
temperatures which may or may not exist in the stars. The last col- 
umn lists the heat function stored up in the pairs per 1 cm 3 . From 
about 5 X 10 8 degrees on, z p and x become independent of ZQ and 
would be the same for all values of this density smaller than 5 X 20 21 . 
At the same time the kinetic energy of the electrons becomes more 
important than the intrinsic. 







8 X 10 7 

6.0 X 10- 10 

1.2 X 10-" 

4.5 X 10~ 45 


1.2 X 10~ 2 

2.4 X 10- 24 

9.3 X 10- 38 

1 X 108 

8.5 X 10 3 

1.7 X 10~ 18 

6.1 X 10- 32 


9.1 X 10 29 

1.8 X 10 8 

7.3 X 10-' 


5.0 X 10 38 

1.0 X 10 17 

4.2 X 10 3 


6.2 X 10" 

7.9 X 10 22 

3.5 X 10 9 

1 X 10" 

1 . 7 X 10" 

4.1 X 10 26 

2.2 X 10 13 

1 X 10 10 

7.4 X 10' 

2.7 X 10 30 

5.5 X 10 17 

At first sight, one is inclined to doubt if the assumption made in 
section 127 is fulfilled, namely, if the index of refraction of the radia- 
tion is approximately 1 in matter as dense as that. However, a check 
shows that such is indeed the case because the wave length which must 
be here considered is that emitted in the reaction, and it is extremely 
short (1.2 X 10 " 10 cm) even at low temperatures. 

129. Neutrons and protons. The neutron has a mass slightly 
larger than the proton. There is, therefore, the possibility of its being 


converted into a proton under ejection of a negative electron and per- 
haps of a quantum of radiation. In fact, there is much reason to 
believe than an intranuclear process of this type plays a role in the 
emission of radioactive 0-rays. Equation (19.21) takes here the form 

G n G H + G e = 0, 

whence v n = 1, J> H + = v e = 1, v = 1 if we characterize the neutron 
and proton by the subscripts n and H +. The equation of equilibrium 
(19.23) is then 

The atomic weights of the neutron and proton are 1 n n = 1.0090, 
1.0076, A/z = 0.0009, the (nuclear) statistical weights gn = gH+ = 2, 

3 4 2 v 10 9 

logio K = - 15.688 - - log 10 T - y -- (19.31) 

The process of dissociation is here exothermic, and, therefore, the 
conditions are peculiarly inverted. The presence of excess electrons 
does not depress the degree of dissociation but favors it. The equilib- 
rium constant does not rise monotonically with temperature but only 
rises to a maximum (at T = 6.5 X 10 9 ) and then drops again. This 
maximum value of K is equal to 

K = 7 X 10- 32 . 

Of course, the last decimal figure in the atomic weights /-in and 
/IH+ is not quite certain and the heat of reaction is affected with a con- 
siderable probable error. But even if Aju were only half as large 
(0.00045) the maximum value of K would be ttut little increased, 
namely K = 2 X 10 ~ 30 . As the density of electrons z e in ordinary stars 
(not white dwarfs) is only of the order of magnitude 10 21 (referred to 
in the preceding section), we conclude that the ratio of free neutrons to 
protons in them must be negligible. 

This conclusion would not be safe with respect to white dwarfs. 
The density in them is so enormously high that the electrons may form 
a completely degenerate gas even at temperatures as high as 10 8 de- 
grees. The thermodynamic potential of these electrons is then (1 7.35), 
<p e = fQt + jji e c 2 , and the equation of equilibrium, <p n ^>H+ <? = 0, 
gives simply 

1 Bonner and Brubaker, Phys. Rev. 50, p. 308, 1936. 


since /B< is negligible compared with AM- The molal volume of the 
electrons in the white dwarfs may be as low as v = 3 X 10 ~ 6 cm 3 , 
whence, by the formula (17.29), 9 = 0.7 X 10~ 8 T. Therefore, the 
electrons may be regarded as degenerate at temperatures of the order 
10 s but not of the order 10 9 . In other words, the temperature within 
the white dwarfs hardly rises higher than 10 9 degrees. We are, there- 
fore, justified in concluding that free neutrons do not play any impor- 
tant role in stars, including the white dwarfs. 

130. Hydrogen and deuterium. As early as 1815 the English 
chemist, William Prout, formulated the hypothesis that the atoms of 
heavier elements are formed by the association of hydrogen atoms. 
This view is confirmed by modern science and found its expression in 
the very name "proton," which is derived from the Greek "TrpSros" 
meaning " first " or " primary". A scientific theory of nuclear struc- 
ture became possible after the discovery of the neutron : every nucleus 
is supposed to consist of neutrons and protons. 1 There is a consider- 
able body of experimental facts to support this theory. In the so- 
called experiments on permutation of matter it is observed how fast- 
moving particles knock protons and neutrons from or into atoms, con- 
verting them into other elements. Inasmuch as a neutron can be 
formed by the association of a proton and an electron (preceding sec- 
tion), we can also say that any nucleus may be produced out of a 
suitable number of protons and electrons, although the electrons do 
not retain in it their individuality but merge with the protons, forming 

The simplest examples of the building up of heavier elements is the 
formation of deuterium and helitim, which we shall discuss in this and 
the next section. There is direct experimental evidence for these 
processes. When a fast-moving deuteron hits a target, phenomena 
are observed which offer evidence of its being split sometimes into a 
proton and a neutron. On the other hand, we have seen in the pre- 
ceding section that a neutron dissociates into a proton and an electron. 
It must be possible, therefore, for two protons and one electron to 
unite, forming a deuteron 

2Gn+ + G e GD+ = 0, 
PH+ = 2, v, = VD+ =1, v = 2 and 

* W. Heisenberg, Zs. Phys. 77, p. 1932; 78, p. 156, 1933. 


The atomic weight of the deuteron 1 is MD+ = 2.0142, the statistical 
weight 2 is #D+ = 3. Hence A/* = 0.0015, 

7 04 v 10 9 
lo glo K = 35.63 + 3 logio T - ^^ (19.33) 




T K K* 

10 8 1.6 X 10- 11 4 X 10~ 

2 X 10 8 2 X 10 26 4 X 10 12 

4 X 10 8 7 X 10 48 8 X 10 21 

It is clear from this equation that in the state of equilibrium there 
should be very little hydrogen below 10 8 degrees. The density of H+ 
cannot be larger than K^ since zn+/z e is certain to be smaller than 1. 

At low temperatures the conditions are somewhat different because 
one cannot treat the gases as completely ionized but must regard them 
as atomic or even as molecular. For atomic hydrogen and atomic 
deuterium the equation would be 

2G U - CD = 0, 
J>H ==: 2, VD == 1 v = 1. 

ZH = CKz D ) M or 2 D = Z H 2 K, 

3 7 04 V 10 9 
logio K = 20.260 + - lo glo T - -^-~ (19.34) 

Again we see that in the state of equilibrium there should be an 
enormous excess of deuterium. There is little doubt that positive and 
negative electrons, in our universe, are really in equilibrium in the 
sense of section 127. The recombination of pairs progresses with 
great rapidity, and under ordinary conditions there are no more of 
them than calculated from our equations. Also, there is no direct 
evidence that the free neutrons are more numerous than would be 
expected theoretically. But when it comes to the formation of deu- 
terium from hydrogen and to the analogous processes considered in the 
next two sections, we must conclude that the world is very far from 

1 Bonner and Brubaker (footnote on p. 339). 

2 At the high temperatures here in question there may exist and be stable another 
modification of the deuteron (with zero nuclear spin). The joint statistical weight 
would then be #D+ = 4 and the constant 35.50. 


equilibrium, indeed. It appears that the protons, unless they are in 
some unusual state of activation, are unable to combine and form 
heavier nuclei. 

131. Hydrogen and helium. The case of formation of helium by 
the association of four hydrogen atoms is of historical interest because 
it was the first example of permutation of elements to which the theory 
of thermodynamical equilibrium was applied. 1 The helium nucleus or 
a-particle can be formed of four protons and two electrons: 

4G H + + 2G. - G a = 0, 

whence *>H+ =4, v e = 2, v a = 1, v = 5, and 
z H + 4 Ze 2 /z a = K or SH+ = (Kzjz^z"* . 

The atomic weight of helium is HH e = 4.0040, and of the a-particle 
H a = 4.0029, the statistical weight in both cases gn e = g = 1- This 
gives AM = 0.0284, and 

i 333 y inn 
logio K = 92.44 + 7.5 lo glo T - ~ -- (19.35) 



T K K y * 

7 X 10 2 X 10- 1.2X10-' 

8 X 10 8 4X10- 8 1.4X10-* 

10 4 X 10 2 4.6X10' 

5 X 10 3 X 10 1 ' 8 4.3 X 10" 

Seeing that z a < z e and that, in ordinary stars, zj 4 is considerable 
(up to 10 6 ), hydrogen should be practically absent at temperatures 
below 10 9 . At low temperature, the gases must be taken as atomic 
because they are no longer ionized 

4G H - G H = 0, 
or Z H 

It is hardly necessary to calculate K in this case, because it is imme- 
diately obvious that the positive terms in its expression are diminished 
while the negative remain the same as in (19.35) so that logio K has 
an enormously large negative value. If equilibrium prevailed, with 
respect to this reaction, no appreciable amount of hydrogen could 
exist outside the stellar cores. 

1 R. C. Tolman, J. Am. Chem. Soc. 44, p. 1902, 1922. 


The formation of deuterium and helium are merely the beginning 
of the process of association of protons into heavier elements. The 
further combination of a-particles, protons, and electrons into stable 
nuclei is also exothermic, taking place under a reduction of mass 
(mass defect). The heavier atoms are, thermodynamically, still more 
probable than helium. Although the nuclei with the largest mass 
defects are the most abundant, it does not seem that their abundance 
relatively to helium and to one another is in quantitative agreement 
with the thermodynamical expectations. It is not certain that the 
present data on atomic weights are sufficiently accurate to decide this 
question. Be this as it may, it is quite certain that equilibrium does 
not exist with respect to hydrogen, as has already been emphasized at 
the end of the preceding section. 

132. Annihilation of hydrogen. The only experimentally estab- 
lished case of annihilation of matter is the mutual destruction of a posi- 
tive and a negative electron. The annihilation of protons, hydrogen 
atoms, or heavier atoms has not as yet been directly observed. How- 
ever, in the present state of theoretical knowledge no valid reason can 
be advanced why such a particle could not completely disappear, con- 
verting its energy into radiation. The equations for the annihilation of 
a neutron, on one hand, and of a proton-electron system, on the other, 
are very little different, and we shall consider the latter because of the 
greater abundance of protons. The reaction is given by 

G H + + G e = 0, 
"H+ = " = 1, v = 2, A/x = 1.0081, whence and from (19.25), 

ZH+ = K/z*> \ 

i r o*c.7i r 473 X 10 12 (19.36) 

logio K = 36.275 + 3 logio T 



T = 5 X 10 10 K - 2 X 10-" 

7 X IQio 2 X 10* 

8 X 10 10 7 X 10' 

10 11 1 X 10" 

If this process is possible, the universe is extremely far from the 
state of equilibrium: no matter should exist at temperatures below 
7 X 10 10 degrees. 

The view was expressed that there may exist also negative protons, 
unobserved, so far, because of their scarcity. If this is true, there is 


the possibility of the creation and mutual annihilation of positive- 
negative proton pairs. The reaction is the same as in the case of pos- 
itive-negative electron pairs only the heat is much larger (A/* = 2.015). 
It follows 

logio K = 39.948 + 3 logio T - 

9.48 X 10 12 


This means that temperatures of the order of 10 12 degrees are 
necessary to create such pairs. Expressed in electron-volts, the energy 
of such a pair is 1.88 X 10 9 e.-v. 

How exceedingly small the numbers in (19.36) and (19.37) are, 
appears from the following fact. A few years ago the astronomers 
calculated the size of the universe as 10 87 cm 3 (in the meantime it has 
become doubtful whether the universe is finite). The reciprocal of this 
is still very large compared with K at T = 10 10 or at lower tempera- 
tures. It follows that there could not be a single proton, either isolated 
or as a positive-negative pair, in the whole of the universe. The last 
atom of matter should have been converted into radiation if equilib- 
rium prevailed. 

133. Influence of gravitational fields. The interest of the proc- 
esses discussed in this chapter lies in their applications to problems of 
cosmology. However, in treating them we have not taken into con- 
sideration the gravitational fields existing in and around the stars. It 
will be well to say a few words of explanation why the effects of the 
gravitational fields can be neglected on account of their smallness. 
We have treated the problem of equilibrium of heavy gases in a gravi- 
tational field in section 108 and have seen there that the only correc- 
tion which the equations need is a term 

added to the equilibrium constant log K. This constant, as given by 
eq. (19.23), already contains the term A/i-c 2 /jRr, which, combined 
with the correction for gravitation, becomes 

-(1 + Q g /c 2 )&'C 2 /Rr. 

In other words, the heat of reaction or the energy available in the 
process is changed in the proportion (1 + & g /c 2 ) owing to gravitational 

V = (1 + fl,/c 2 )A/u. (19.38) 


The gravitational potential Q g vanishes at a great distance from a 
star and is negative in its vicinity, having its minimum at the surface. 
The factor is, therefore, smaller than 1, meaning that the heat of 
reaction is decreased. But even the largest known gravitational 
potentials, those of the white dwarfs, are materially smaller (in abso- 
lute value) than 0.002 X c 2 , whereas within the accuracy of the preced- 
ing chapter a correction of 0.2% is negligible. 

Equation (19.38) is, of course, quite general and applies to any 
molecular or atomic process in which energy is converted into radia- 
tion. According to the quantum theory, the frequency v of the 
emitted radiation is connected with the available energy per atom or 
molecule AE = c*&n/n A by the relation AE = hv, whence 

v' = (1 + tt g /c 2 )v. (19.39) 

The gravitational field changes the frequency of radiation emitted 
in an elementary process, decreasing it (gravitational red shift), a 
phenomenon predicted by Einstein and since confirmed experimentally. 

We have based these remarks on the Newtonian theory of gravita- 
tion. On general grounds we are justified in saying that the condi- 
tions cannot be materially different in the general relativity of Ein- 
stein as long as tt g /c 2 is small. In fact, chemical equilibrium was 
investigated from the standpoint of the general relativity by Tolman, 1 
who found the same relations as in the absence of any gravitational 
fields. Einstein's formula of the gravitational red shift 

/ = [(1 + 212,A 2 ) 
is, in practice (!2 g /c 2 1), identical with (19.39). 

1 R. C. Tolman, Proc. Nat. Acad. Sci. 17, p. 159, 1931. 


134. Langevin's theory of magnetization. The systems treated in 
the preceding chapters were completely described by the thermo- 
dynamical variables p, V, T, apart from data about their composition. 
Even in the case of charged gases, discussed in sections 106-115, the 
electric potential was regarded as only a part of the energy constant 
and not as an additional parameter of the system. It is instructive 
to consider also variables of a non-mechanical nature, and we are going 
to do so in this chapter, beginning with the theory of magnetics or 
substances capable of magnetization. It will be sufficient to treat the 
case of isotropic substances because the generalization is obvious and 
would only encumber our expressions without adding anything essen- 
tial to them. Suppose that a system of this sort is placed in a homo- 
geneous magnetic field whose strength is H in the absence of the system 
(i.e. before the system is brought in). Let M denote the component 
of the total magnetic moment (or total magnetization) in the direction 
of the field. 

We know from the theory of electromagnetism that the work which 
the field does, in raising the magnetization from M to M + dM, is 
DW' = HdM. In accordance with the convention of section 7, we 
count the work done by outer forces against the system as negative. 
The total element of work (2.02), including the mechanical and mag- 
netic parts, is then 

pdV - HdM 

and eq. (3.04) of the first law has the form 

DQ - dU + DW = dU + pdV - HdM. (20.01) 

The state of the magnetized system depends on the two thermal 
parameters T, V we used all along and, in addition, on the magnetic 
parameter M. In order to have a complete description of the system, 
we must know the three functions 

U = C7(r, F, M), p p(T, F, M), (20.02) 

H - H(T, V, M) (20.03) 



In the sense of section 10, we may call these three relations the 
caloric, the thermal, and the magnetic equations of state. Of course, 
it is arbitrary which of the six parameters we regard as independent. 
For most of the applications it will be convenient to describe the sys- 
tem in terms of T, p, H, but occasionally we shall use also T, p, M. 

The first application of the second law of thermodynamics to such 
a system was due to Langevin. 1 As he pointed out, in the expression 
(4.07) of the entropy differential, dS = DQ/T, the magnetic work 
gives rise to the additional term 

dS' = = dM, (20.04) 

and he postulated that dS' be an exact differential. This is possible 
only when the factor H/T is a function of M (compare section 8) and, 
vice versa, M is a function of H/T, 



This is the form of the magnetic equation of state (20.03), accord- 
ing to Langevin. It is quite obvious that this law is not general, a 
case in point being the behavior of diamagnetic substances whose mag- 
netization is known to be independent of temperature: 

M = - AH, (20.06) 

where A is a constant. 2 What is, then, the special assumption made 
in this theory and leading to the formula (20.05)? It lies in the sep- 
arate treatment of the magnetic part of the entropy (20.04) as an 
exact differential, independently of the remaining part 

(dU + pdV)/T. (20.07) 

This implies, of course, that the expression (20.07) is also an exact 
differential which cannot depend on the variable M (or H) since its 
differential is absent in it. Consequently the functions U and p must 
be also independent of M. Instead of (20.02), 

U = U(T, V), p = p(T, V). (20.08) 

We met with a similar case in the theory of perfect gases. In the 
expression (4.16) of dS the two terms are separately exact differentials, 

1 P. Langevin, Ann. Chim. Physique 5, p. 70, 1905. 

8 We speak here of the ordinary or atomic diamagnetism. There exist a few 
substances (as bismuth, antimony, etc.) with so-called crystal diamagnetism which 
has a temperature coefficient. 


and this is due to the internal energy depending only on the thermal 
variable T. By analogy we shall call a substance whose equations of 
state follow the simple laws (20.05), (20.08) a perfect magnetic. 

Langevin initiated also the statistical theory of magnetization and 
gave an explicit expression of the law (20.05) for paramagnetic mate- 
rials. As corrected to take into account the restrictions of the quan- 
tum theory, it can be written 

M - 


and is sometimes called the Langevin-Brillouin formula. Z represents 
here the total number of atoms, and the product jgfo the total magnetic 
moment of one atom. The factors have the following meaning: 

o the Bohr magneton 

A) = ^- = (0.9174 db 0.0013) X 10- 20 erg gauss- 1 , (20.10) 
g the Lande factor 

* 2+ 2/C7+1) ' ( } 

j is the quantum number of the total angular momentum of the atom, 
compounded of the resultant spin momentum s and the resultant 
angular momentum /. At 20 C the numerical value of fo/kT is 
2.28 X 10~ 7 : since jg is rarely larger than 7, the argument of the 
function F is small (< 0.050) at room temperature, even for fields of 
the order of 30 000 gauss. Under these circumstances all terms of the 
expansion of F(a) into a power series are negligible, excepting the first 
linear term, and the magnetization is well represented by Curie's law 

*, CH 

M = , (20.12) 

where C is a constant. For large values of the argument, the function 
F asymptotically approaches the value 1 (magnetic saturation), but its 
indications appear only at very low temperatures. 

The formula (20.09) was derived for gases and solutions of salts 
in which the atoms or ions can freely rotate. Even for these sub- 
stances its validity is not general, being often impaired by doublet or 
multiplet structure of the energy levels. Only when the width of 


these multiplets is very small or very large compared with kT is the 
formula applicable. It is remarkable, however, that it represents 
with great accuracy the magnetization of solid paramagnetic salts of 
rare earths, especially those in which the paramagnetic atoms are not 
very close together but separated by non-magnetic atoms (as, for 
instance, in hydrates). The classical example, in this respect, is 
hydrated gadolinium sulfate investigated in Leiden. 1 The magnetic 
moment of this substance was measured down to 1?3 K and found to 
obey strictly the law (20.09) with j = -J and g = 2. At the lowest 
temperature and highest field the saturation was far advanced, the 
magnetization being 95% of the possible maximum. 

Exercise 106. Calculate the entropy term S' from (20.04) for substances whose 
magnetic moments are expressed by (20.09). Show that 

S' = kZ\ log sinh ( a ~\ ) /sinh -- aF(a) \ -f- const. 

I L \ 2j/f 2/J J 

Exercise 107. In his original formula Langcvin assumed that the orientations 
of the molecular magnetic moments are not restricted by quantum conditions. 
This formula can be obtained from (20.09) by the following transition to the limit: 
let j go to oo, while goes to zero in such a way that jg&o = /3 remains constant 
where j3 represents the molecular magnetic moment). Show that the result is 

M = Z/5 f(0H/kT), f(a) coth a - I/a. 

135. Magnetothermal and magnetocaloric relations. It will be 
well to derive a few relations which will elucidate still further the 
thermodynamical status of perfect magnetics and diamagnetics. We 
start from the generalized thermodynamic potential 

* = U - TS + pV - HM, (20.13) 

whose differential is, according to (20.01) and (4.07), 

d& = - SdT + Vdp - MAIL (20.14) 


, V = ( J , M = - P^j (20.15) 
Substituting this into (20.13) 

-* - < 20 - 16) 

\dP/T,H \d 

1 H. R. Woltjer and H. Kamerlingh Onnes, Leiden Comm. 167 b, c. 


The three partials (20.15) give reciprocity relations when differ* 
entiated a second time. Thus we obtain from the second and third 

p.T T.H 

This is a relation between measurable quantities. The left side 
is the change of volume due to the field, its relative value (divided 
by V) is called volume magnetostriction or Barrett effect. The right 
side represents the change of the total magnetic moment due to pres- 
sure. Such an effect was foreseen and first measured by Nagaoka and 
Honda 1 and is named after them. Although accurate modern mea- 
surements of both effects are available, they were not carried out under 
comparable conditions and are not suitable for a quantitative test of 
the relation (20.17). It should be noted that in perfect magnetics 
the volume does not depend on the magnetic parameters (preceding 
section). The experimental existence of magnetostriction in ferro- 
magnetic materials shows, therefore, that these substances do not 
strictly follow Langevin's law (20.05). However, the effect is very 
small and the deviation, presumably, not large. 

Exercise 108. In a particular specimen of iron (at / = 20 C, H = 1 gauss) 
the partial (dM/dp)T,H was found to be 3 X 10 ~~ 3 gauss per unit volume and per 
1 atm. Calculate the volume magnetostriction from formula (20.17). (Use 
absolute units). 

The other two reciprocity relations following from (20.15) are 


97 / ptH 

We shall use them for transforming eq. (20.16) as follows. We 
differentiate (20.13) partially with respect to p, and with respect to H, 
and take into account the relations (20.17) and (20.18): 



In the case of perfect magnetics we have (dU/QH) TtP 
0, and from (20.17) and (20.20) 

1 Nagaoka and Honda, Phil. Mag. 46, p. 261, 1898; see also S. R. Williams, 
Int. Crit. Tables VI, p. 439, 1929. 


This is only another proof of Langevin's result because the general 
integral of this equation is the expression (20.05). On the other hand, 
the law (20.06) of diamagnetism gives OJ7/9#)r, P = ~~ AH- = -W> so 
that the internal energy contains a magnetic term whose differential is 

dU' = (dU/QH) Ttp dH = MdH = HdM. (20.21) 

A look at the structure of the element of heat (20.01) shows that 
the additional term in dU just cancels the work of the magnetic field 
DW' = HdM, so that DQ is independent of magnetic parameters. 
We see from this that diamagnetic and perfect magnetic substances are 
complete opposites with respect to the caloric action of putting them 
into a magnetic field. In diamagnetics, the (positive) work against 
the field is done at the expense of the internal energy without any 
contribution from external heat sources. It is a process at the same 
time adiabatic and isothermal: the loss is sustained, as it were, by the 
constant of the internal energy and does not affect the temperature of 
the body or its equilibrium with the environment. On the contrary, 
the (negative) work of magnetizing perfect magnetics is completely 
supplied by outer sources, when conducted isothermally. When the 
process is adiabatic it leads to a rise of temperature in the sub- 

In the interest of the following section we shall say here a few 
words about the heat capacity of magnetics. In treating simple sys- 
tems, we distinguished in section 14 the specific heats at constant vol- 
ume c v and at constant pressure c p . Similarly, we have to speak here 
about the following two heat capacities. The heat capacity at constant 
magnetization (and, say, constant pressure) is the heat that must be 
imparted to the magnetic when its temperature is raised by 1 degree 
and, at the same time, the magnetic field is changed in such a way as 
to keep its magnetization constant: C P M = lim(AQ/Ar) p Af. According 
to the expression (20.01), no part of this heat is used for magnetic 
work. On the other hand, the heat capacity at constant field strength 
C P H = Hm (&Q/&T) P H refers to a process accompanied by a change of 
M and involving magnetic work. In view of the relation AQ = T&S, 
we can also write 

The relation between these quantities analogous to (4.27) is found, 
in the simplest way, by considering 5 a function of T and M, while M , 
in its turn, depends on T and H, i.e. S = S[T, M(T, H)]. (We need 


not bother about the pressure p, since it is assumed to be always con- 
stant.) The rules of partial differentiation give us 


dT/ P> M \dM/ PtT \d ptH 

In the partial (dS/dM) PtT the variables p and T are considered as 
constant so that M is a function of H only: therefore, 

Substituting this and making use of eq. (20.18), we find the desired 

r c r 
C vn - C PM = T - 

136. Cooling by adiabatic demagnetization. Adiabatic demag- 
netization of a paramagnetic substance is analogous to the adiabatic 
expansion of a simple system. In either case, the work against external 
forces is done at the expense of the internal energy of the system and 
leads, in general, to a decrease of its temperature. The use of this 
" magnetocaloric effect " for the production of very low temperatures 
was proposed independently by Giauque J and Debye. 2 The usual 
method of lowering the boiling pressure of liquid helium does not per- 
mit in practice 3 to reach temperatures below 0.7 K. Work with 
the magnetocaloric method has been under way in recent years, at 
Berkeley (California) under the direction of Giauque, 4 and at Leiden 
(Holland), under the direction of deHaas. 5 It has been very suc- 
cessful in bringing the range of accessible temperatures considerably 
nearer to the absolute zero point. 

The equation of the adiabatic dS = dQ/T = is, according to 

dS = (dU + pdV - HdM)/T = 0, (20.24) 

when S is expressed as a function of T, p t H, also 



<3 T/ VtH \ OP/ H t T ^O** / T.p 

i W. F. Giauque, J. Am. Chem. Soc. 49, pp. 1864, 1870, 1927. 

* P. Debye, Ann. Physik 81, p. 1154, 1926. 
Keesom, Proc. Amsterdam 35, 136, 1932. 

W. F. Giauque and C. W. Clarke, J. Am. Chem. Soc. 54, p. 3135, 1932; 
W. F. Giauque and D. P. McDougall, Phys. Rev. 43, p. 768; 44, p. 235, 1933. 

W. J. deHaas, E. C. Wiersma, and H. A. Kramers, Physica 13, p. 171; 1, p. 1, 
1933; W. J. deHaas and E. C. Wiersma, Physica 1, p. 1107, 1933. 


Substituting from (20.22) and (20.18), 

c f-(f-r) *+(!!?) w -- < 20 - 25 > 

i \0-l/p,H \0* / Pt ff 

The procedure consists in precooling the specimen in vacuo, in a 
strong magnetic field, and then turning off the field. 

With p = 0, eq. (20.25) gives for the rate of the adiabatic cooling 

or from (12.23) 


As was pointed out in the preceding section, the heat capacity C P M 
does not contain any magnetic work. Therefore, it should not be 
materially different from the ordinary capacity C p as measured in the 
absence of a magnetic field. Of particular interest are the perfect 
magnetics: from the kinetic point of view, they are the substances in 
which every atom is free to adjust itself in the field without being 
hindered by its neighbors. Other things being equal, they should 
show the highest magnetization and the highest caloric action. For 
this reason, both Giauque and Debye suggested the use of gadolinium 
sulfate as a testing substance because it was known to follow the law 
(20.09) down to the lowest temperatures. If we denote x = H/T, 
this law implies 

dM\ __x_dM /aM\ 1 dM 
dT/ Pt a~ ~~ T doc ' \9/// Pl r"" T dx ' 

\dT/ Pi H J. ax \ou/p,r 

so that (20.27) becomes 

dT _ x(dM/dx) 

dH~ C pM +x 2 (dM/dx)' 


Of course, in the case of perfect magnetics there is no need to use a 
differential formula because the two terms (with p = 0), in the formula 
(20.24), are separately integrable. At zero pressure there is no differ- 
ence between c p and c v , and the energy differential is dU CvdT. 



If the specific heat follows the third power law (18.70), the first 
integral is equal to 6.45 X 10 9 (r/9) 3 erg/deg mol. Suppose we start 
the demagnetization at the temperature To and the field HQ. When 
the field is reduced to H = 0, the temperature has the value 

s*XQ J 

To 3 - 1.55X10- 10 8 3 / x- 


where XQ = Ho/To and m is the molal magnetic moment. 

It was found, however, that the specific heat of gadolinium sulfate 
does not follow the third power law but exhibits an anomaly due to 
multiplet structure of the lowest energy level (compare section 117). 
The curve reproduced on p. 327 was obtained by Giauque and McDou- 
gall by measuring the magnetocaloric effect dT/dH and by calculating 
C pM from a formula equivalent to (20.27). Although they reached a 
temperature of 0?287 K, this substance is not particularly suitable for 
magnetocaloric cooling. At the time of the writing, the greatest 
success was obtained by deHaas with a mixture of two alums, 
K 2 SO 4 Cr 2 (SO4)3 24H 2 O + 14.4K 2 SO4A1 2 (SO4)3 24H 2 O. Demagnet- 
izing from 24 075 gauss to 1 gauss, he reached 0?0044 K. In all this 
work the measurements themselves were used to establish the scale 
of absolute temperatures by the method explained in section 29. 
However, the last figure (0?0044 K) rests, in part, on an extrapolation 
and has only approximate validity. 

Exercise 109. Consider a substance satisfying eq. (20.30) whose magnetization 
obeys the law (20.09), with j = , g = 2. Start with T 3 K and H = 25 000 
gauss and demagnetize to H = 0. What will be the final temperature if = 200? 

137. Supraconductivity and thermodynamics. It was discovered 
by Kamerlingh Onnes that the electric resistance of certain metals 
suddenly drops to practically nothing when they are cooled below tem- 
peratures called their transition points. We give the transition points 
of the principal supraconductive substances in Table 53. 

The transition point is displaced when the system is put into a 
magnetic field H. As the magnetic field is increased, the transition 
point moves continuously to lower temperatures, and at a certain 
strength of it (threshold value) the supraconductive state ceases to 
exist altogether. From the point of view of electromagnetism, the 
immediate effect of switching on a field consists in setting up currents 
in the supraconductor, in cpnformity with Faraday's law of induction. 
But the conductivity of these substances is so large that the currents 
continue to flow, without apparent loss, as long as the state of supra- 

XX 137 










70 K 

V 4 3 K 

Au a Bi 1 84 K 

ZrB 2 82 K 



Nb 9.2 

CuS... 1.6 

TaSi . . . . 42 


0.6 (ca) 

Ta 4.4 

VN.... 1.3 

PbS .... 41 


4 22 

La 47 

WC 2 8 


1 14. 

WaC 2 05 


. . 1 05 


MoC. 7 7 



9 37 

A/fnP 9 1 


1 75 

TIN 1 4 

Pb-Sn-Bi 8 5 


1 5 

BiflTlj. 6 5 

TiC. 1 1 

Pb-As. 8 4 



Sb 2 Tl 7 .... 5.5 

TaC... 9.2 

Pb-As-Bi 9.0 



Na 2 Pb 6 ... 7.2 

NbC... 10.1 

Pb-Bi-Sb 8.9 



Hg 6 Tl 7 ... 3.8 

conductivity is maintained: they are therefore called persistent cur- 
rents. Another result also follows from the theory of electromagnet- 
ism: when the magnetic field is switched on, after the supraconductive 
state is established, neither the field nor the currents can penetrate to 
any appreciable depth of the metal. The persistent currents flow at 
the surface and produce a magnetic field which exactly compensates 
within the metal the external field. Experimental work has shown 
that the same conditions prevail when the metal is cooled below the 
transition point in an already existing magnetic field. The lines of 
magnetic induction are pushed out of the metal, as it becomes supra- 
conductive, until the induction in the interior vanishes. This result 
was first announced by Meissner and Ochsenfeld 1 and has been con- 
firmed since by many investigators. Although there is still some 
doubt whether it holds rigorously, it describes the phenomena with an 
accuracy sufficient for thermodynamical purposes. In short, a supra- 
conductor behaves in a magnetic field like a substance of the perme- 
ability /i = 0. Since /* is connected with the susceptibility K by the 
relation /* = 1 + 4?r/c, the supraconductors can be formally described 
as diamagnetics with the susceptibility K = 1/47T. 

Let us consider a long stretched (needle-shaped) supraconductive 
body placed with its axis in the direction of the magnetic field. Accord- 
ing to the electromagnetic theory, the outer magnetic field produced 
by a body of this shape is negligible . Its j(negative) magnetic energy 

1 W. Meiasner and R. Oc'senfeld, Die Naturwiseenachaften 21, p. 787, 1933. 


is due to the absence of any field in the interior and is equal to 
per unit volume, or to 


per mol. The magnetization (per unit volume) is an expression of the 
type (20.06) characteristic of diamagnetics. We can apply, therefore, 
to supraconductors the results established in the preceding section for 
diamagnetic substances: the internal energy contains U H as an additive 
term, while the entropy is independent of the magnetic parameters. 
Consequently the thermodynamic potential (<p = u Ts + pv) has, 
in the case of supraconductive materials, the form 

vH 2 
*-*>- , (20.32) 


where w is the potential in the absence of a field. At the very low 
temperatures we are here considering, the pressure effects are 
negligible: there is no observable change of specific volume, so that 
it is permissible to ignore the pressure as a variable and to regard 
<po(T) as a function of temperature only, and v as a constant. The 
normal (non-supraconductive) modification of the metal is, as a rule, 
non-magnetic, so that its thermodynamic potential does not contain 
any magnetic term but is, simply, <p n (T). The equation of equilibrium 
between the two phases becomes, therefore, A3> = ^> n <p = or 

A* = Vn (T) - ,(D + ~ = 0. (20.33) 


Of course, it is not a priori certain that the supraconductive state 
is in true equilibrium and that the second law can be applied to it. 
The hypothesis that it represents a phase in the thermodynamical 
sense was first made by Langevin. 1 It was corroborated by the great 
sharpness of the transition point in good crystalline specimens. The 
experimental test of the equation (20.33) is also reassuring: its conse- 
quences hold with good accuracy. This relation gives the dependence 
of the transition point on the strength of field H and can be tested in 
several ways. The first and second partials of A$ with respect to T 
are, according to (5.37) and (7.22), 

1 P. Langevin, Rapports du 1* Conseil & vay, p. 301, 1911. 

XX 137 



The differentiation of (20.33) gives the equation (l/T)dT 
(vH/v)dH = or 

analogous to the Clausius-Clapeyron equation. It was tested by 
Keesom and Kok l and found to be in agreement with experiments. 
Of great interest is the observation of these authors that the latent 
heat decreased as the field diminished and could not be measured at 
all in the case of field free transitions (i.e. dH/dT 7* oo for H = 0). If 
this observation is confirmed, it would mean that the equilibrium, in 
the case H = 0, is of the second order (section 49). Let us consider 
this case: we denote the transition point in the absence of a field by 
To and express A$, for T TQ + dT, H = dll, to terms of the second 
order. From (20.34) 


2\r / 

(dry - - 


AO _ v (dlf^ 
TQ 4ir\a 


This equation was obtained by Rutgers 2 and in a different way by 
Gorter and Casimir. 3 All these authors regarded the equilibrium in 
question as one of the second order. The equation (20.36) was con- 
firmed with remarkable accuracy by the beautiful measurements of 
Keesom and Kok on tin and thallium. 4 



cm l mol" 1 

gauss deg~ l 

Aco (calc) 
cal deg^mol"" 1 

Aco (obs) 



14 2 

151 2 





16 9 

137 4 



Nevertheless we do not agree with Rutgers, Gorton, and Casimir. 

1 W. H. Keesom and J. A. Kok, Physica 1, pp. 503, 595, 1934. 

1 Appendix to Ehrenfest's paper, note on p. 128. 

* C. J. Gorter and H. Casimir, Physica 1, p. 306, 1934. 

4 W. H. Keesom and J. A. Kok, Physica 1, p. 175, 1933. 



XX 137 

The equilibrium cannot be of the second order since an unsurmount- 
able difficulty is introduced by the double sign of 

dH ^ ATT A^oV 
dT " \v To/ 

Represented graphically (Fig. 60), the transformation is of the 
general type of Fig. 18 on p. 132. With any plausible choice of the 
functions <f>o(T), <p n (T), the analysis shows that the lines representing 
the equilibrium continue beyond the point TV Therefore, both 
regions labeled (1) in Fig. 60 correspond to the supraconductive state 
so that cooling along the line // = does not involve any transition. 
We maintain, therefore, that the equilibrium in question is, in reality, 

FIG. 60 FIG. 61 

Supraconductive transition point in its dependence on magnetic field. 

of the first order, there being a small but finite latent heat. According 
to the general discussion of section 49, this would imply a transition 
curve of the shape given in Fig. 61. That Rutgers' formula is satisfied 
with such a remarkable accuracy is due to the following coincidence. 
Equation (20.35) holds for the whole curve and can be differentiated 
along it. Since // T = As, its differential is (Ac/T)dT, and we find 



Where the line of Fig. 61 is straight (d 2 H/dT 2 = 0), the slope is given 
exactly by Rutgers* expression, whether the equilibrium in TO is of the 
first or of the second order. 1 

There were attempts to make other aspects of supraconductivity 
amenable to a thermodynamical treatment. However, they failed to 

1 In a later paper (Phys. Zs. 35, p. 963, 1934) Gorter and Casimir admit that the 
supraconductive state would be thermodynamically stable above the transition 
point "if it existed." They postulate, therefore, that it cannot exist. This seems 
an adhoc hypothesis, and the difficulty is much better disposed of by the simple 
assumption made in the text, that the equilibrium is of the first order. 


give a complete picture of the phenomena for which they were intended. 
Limitations of space do not permit us to enter into their discussion. 
138. Electrostatic phenomena. Dielectrics or substances capable 
of electric polarization P form, in many respects, an analogue to the 
magnetics treated in section 134. The work which must be done on 
a dielectric body in order to increase its total electric moment (or 
polarization), in the direction of the field , from P to P+ d P, is 

-E dP, 
so that the complete element of work becomes 

DW = pdV - E dP, (20.38) 

in analogy with the magnetic expression of section 134. There are 
two types of dielectric polarization which, from a thermodynamical 
point of view, correspond closely to the types of magnetization in 
atomic diamagnetics and in perfect magnetics. 

(A) The Lorentz type of polarization. If the molecules of a sub- 
stance have no electric moments in their normal state, the application 
of an electric field creates such moments in them by displacing the 
electrons from their normal position. The expression for this sort of 
polarization (taking account of the field of the electric moments) is 

P AE', (20.39) 

where A is a constant and E = E + P is the effective field strength. 


The phenomenon is mainly intramolecular, and A is independent of 
temperature. This has the thermodynamical implications discussed 
in connection with diamagnetism. 

(B) Dipole polarization. It was pointed out by Debye l that some 
dielectric substances may possess permanent molecular dipoles. He 
suggested, therefore, taking over Langevin's formula (developed for 
magnetic dipoles) in order to account for empirically observed changes 
of electric polarization with temperature. The case of polarization is 
simpler in two respects. In the first place, the electric dipoles have 
no tendency to orient themselves in discrete quantized directions. 
The original Langevin formula derived on lines of classical statistics 
holds for them: this means that one has to take in eq. (20.09) j = oo 
and jgPo = p e = const. 2 In the second place, the conditions in dielec- 

1 P. Debye, Phys. Zs. 13, p. 97, 1912. 

2 The transition to the limit is the same as in exercise 107. When the argument 
a is small, there holds the expansion coth a I/a -f -J-a; therefore, the function of 
exercise 107 becomes /(a) = ^a, leading to formula (20.40). 


tries hardly ever approach saturation so that the term of first order in 
E! IT represents an excellent approximation 


where fa is the electric dipole of a molecule and Z the total number of 
molecules. Of course, one should expect this expression to hold only 
in substances where the dipoles can freely rotate, i.e. in gases and 
liquids. In analogy with the perfect magnetics of section 134 they may 
be called perfect dielectrics. 

In general the dipole dielectrics possess also an appreciable polariza- 
tion of the Lorentz type so that very few of them are, even approxi- 
mately, perfect. The dielectric constant D (which is accessible to 
direct measurement) is connected with the polarization by the rela- 
tion (D 1)/(D + 2) = $irP/VE' and has the general expression 

F+I" iv + i z kT' (20 - 41) 

with z = Z/F. By plotting D against \/T one can separate out the 
part due to dipoles and determine the moment. This proved a very 
helpful method for investigating the properties of many organic 
molecules. 1 

The electric analogue to magnetostriction is the phenomenon of 
electrostriction. Because of the close parallelism of the thermody- 
namical properties of dielectrics and magnetics we can directly take 
over eq. (20.17), writing 


We see that electrostriction (represented by the left side) is asso- 
ciated with another effect called piezoelectricity, which consists in the 
dependence of the polarization P on pressure. Our formula refers to 
the volume effect, but piezoelectricity is mostly observed in crystals, 
where it is particularly strong in certain crystallographic directions. 
In turmaline the piezoelectric effect (increase of a component of the 
polarization per unit volume and per unit increase of a strain com- 
ponent) is 5.78 X 10~ 8 in e.-s. cgs units, in quartz -6.9 X lO" 8 , but 
it is particularly large in Rochelle salt, where it reaches 8100 X 10"" 8 . 
An electromagnetic wave, acting upon a plate cut from a piezoelectric 
crystal, periodically changes its size, because of the electrostriction 

' See P. Debye, Polare Molekeln, Leipzig, 1929. 


associated with piezoelectricity, and sets up in it elastic oscillations. 
The amplitude is particularly large when the frequency of the wave is 
in resonance with the characteristic period of the elastic oscillations 
in the plate. Such piezoelectric resonators have received important 
technical applications in the last decades. According to eqs. (20.02), 
the deeper reason for the existence of piezoelectricity is the dependence 
of the pressure (or strain) on the polarization P. It is to be expected 
that the internal energy and entropy should then also depend on P 
and H. From (20.18) we can, therefore, conclude that piezoelectric 
substances are at the same time pyroelectric, i.e. their electric moment 
is influenced by a change of temperature. 1 

It is here the place to make a few supplementary remarks about 
the theory of strong electrolytes (section 115) and to explain why we 
could not calculate the internal energy U directly but had to take 
the indirect way over the work function \I>. The reason is that the 
electric or magnetic energy of a system cannot always be regarded as 
a part of its internal energy. We have seen that such simple condi- 
tions obtain only in the case of atomic diamagnetism or of polarization 
of the Lorentz type, when the electromagnetic work does not depend on 
the thermal parameters. The opposite extreme are the perfect mag- 
netics and dielectrics in which U remains completely unaffected by 
the electromagnetic energy of the system. The strong electrolytes 
occupy an intermediate position. The addition U to the internal 
energy due to the ionic interaction should be calculated from the 
expression (17.64) of the work function V e by means of the relation 
. = U e - TS e or 

TJ ty _ T\ - - 

u * ~~ ^* l \ IT 

\9* / v 

The difficulty is, however, that * e depends on T not only explicitly 
but also implicitly through the dielectric constant D. In a similar 
way, the thermodynamic potential $ is connected with *, by the 
formula $ ^$% + P*V, or 

and here again the dependence of D on the volume is not sufficiently 
well known for a quantitative evaluation. 

139. Thermoelectric phenomena. In section 109 we introduced 
the concept of the potential difference built up by thermoelectric action 

1 For a discussion of these phenomena see: Geiger and Scheel, Handbuch der 
Physik, Vol. XIII. Berlin 1928. 


across a junction of two conductors. The discovery of the e.m.f. 
(electromotive force) of thermoelectricity was made by Seebeck in 
1822. It was supplemented in 1834 by observations of Peltier's on a 
peculiar heat development due to electric currents flowing through 
thermoelectric junctions. Let / denote the current flowing from con- 
ductor (1) to conductor (2): in order to maintain the junction at con- 
stant temperature, it is necessary to impart to it, in unit time, the 



where Hi2 is the so-called Peltier coefficient, depending on the nature of 
the conductors and on the temperature of the junction. IIi2 is positive 
(i.e. heat must be imparted to the system) when the current / has the 
same direction as that produced by the thermoelectric action of the 
junction. The sign of the Peltier coefficient is reversed when the cur- 
rent flows in the opposite direction (1X21= IIi2). This fact gives 
the experimental possibility of separating it from Joule's heat, which 
does not depend on the direction of the current. From the theoretical 
point of view, the same fact is taken as an indication that the develop- 
ment of Peltier heat is a reversible process, amenable to thermodynami- 
cal treatment. 

The final step was the discovery (1854) by William Thomson, later 
Lord Kelvin, of the following phenomenon: when the current / flows 
through a wire of homogeneous material and cross-section, but of non- 
uniform temperature, heat must be supplied in order to maintain the 
temperature gradient. To an element of the wire with the tempera- 
ture rise dT, there must be imparted in unit time the heat 

dQ = <rdTJ. (20.43) 

a is called the Thomson coefficient, and it is positive when the current 
flows in the direction of rising temperature. The Thomson effect is 
reversible, in the sense that dQ changes its sign when the direction of 
the current is reversed. 

The thermodynamical theory of thermoelectricity is due to Lord 
Kelvin. Let us consider the circuit of Fig. 62, consisting of two 
wires (1) and (2) whose junctions are kept at the respective tempera- 
tures T and T'. When the current J is sent through the (closed) cir- 
cuit, the total heat which must be imparted to it in unit time to keep 
the conditions stationary is, according to (20.42) and (20.43), 



From the point of view of the first law of thermodynamics, Q is the 
energy which maintains the circulation of the current. In the theory 
of electricity this energy has the expression 

Q - /, (20.45) 

where E is the e.m.f. (electromotive force) of the circuit, J being the 
electric charge flowing through any cross-section in unit time. Equa- 
tion (20.44) indicates how this energy is spent. There is a profound 
analogy between the conduction electrons streaming across a thermo- 
j unction and a gas forced through a cotton plug in the Joule-Thomson 
process (section 15): the energy which the electron gas gains or loses 
is equal to the difference of its heat functions in the two conductors, 
Q = AX. The same is true for the Thomson effect (20.43): dQ = dX. 
If the molal heat function is denoted by x and the molal electron charge 
by/ = F = n A e, eqs. (20.42) and (20.43) can be written 

_ (X2 - Xl) Idx /IA,HC\ 

H 12 = - j , *-- (20.46) 

Combining (20.44) and (20.45), we can write 

((72 - ai)dT, (20.47) 


as the expression of the first law of thermodynamics. In particular, 
when the temperature difference of the junctions is infinitesimal, 
T - r = dT 

g-f? + -* (20.48, 

Further relations can be obtained from the second law, assuming 
that the heat items of eq. (20.44) are imparted to the system in a 
reversible way and that the thermoelectric effect satisfies the entropy 
principle separately. Let us follow a portion of the electron gas, 
responsible for the current, in its motion around the circuit back to 
the initial position. The process is cyclic in that the final state is 
identical with the initial and the total entropy change in it must be 
zero: 2 AS = SAQ/r. The items of heat A() imparted to the electron 
gas are the same as in eq. (20.44). We find, therefore, 



or taking again T T' - dT, 



The ratio Uvt/T is called the thermoelectric power of the pair of 
conductors. There follows from (20.48) 


""" rp 9 


* 2 ffl T dT\Tj * dT 2 ' 

Finally, the third law of thermodynamics has also a bearing on 
thermoelectricity. Since Iliz/T represents the change of entropy 
which the electrons (or other carriers) undergo in passing across the 
junction, Nernst's postulate (13.07) gives directly 

lim r . (H/r) - 0. (20.51) 

Hence the integrated form of (20.50) becomes 

/ T (7 ~ (T 
2 ^ * dT. (20.52) 


This equation implies, of course, that the integral must be con- 
vergent, so that lim r . (^2 0*1) = 0. But not only the difference 
must vanish (in the limit T = 0), but also <n and a* individually. In 
fact, we see from (20.46) that <r has the physical meaning of the specific 
heat of the carriers: eq. (13.15) requires, therefore, 

lim r ..o <r *= 0. (20.53) 

The empirical representation of the thermoelectric e.m.f. is usually 
given by the equation 

E - (at + 10- 2 JW + 10-* $d] X 10- volt, (20.54) 

supposing that one of the junctions is at C, the other at f C. A 
few data are given in Table 55. 

From (20.54) we find (neglecting c) 

a - (II/TVo, ** - n - bT. 

It is interesting to note that the coefficients a and b in some cases 
have opposite signs. Because of the parabolic character of the 
formula (20.54), as the temperature is increased the e.m.f. rises to a 

XX 139 



maximum, then declines again, and finally reverses its sign. For 
instance, in the case of Cd-Pb the reversal takes place at about 
300 C. 



Range in C 





















































- 1.79 


















Very large thermoelectric e.m.f.'s are observed in circuits composed 
of semi-conductors (Table 56). 


Range in C 







Bi 2 0, 






Co,0 4 





- 22.1 

Cr 2 0a 




- 704 






- 1029 












- 408 

- 47 





- 735 


As to the Thomson coefficients, their temperature dependence can 
be represented by the empirical formula 

<r - [a + 10- 2 # + 10-V] X 10-* volt deg- 1 . (20.55) 



XX 139 

It is apparent from Table 57 that the Thomson effect is positive 
in some metals and negative in others. 


Range in C 













- 13 










+ 25 





- 60 





- 51 











- 72 










Only the order of magnitude of the constants in Tables 55, 56, and 
57 is significant: the thermoelectric power is highly sensitive to the 
purity and mechanical treatment of the substances and shows con- 
siderable variation in different samples of the same material. Never- 
theless, it is possible to test the thermodynamical relation (20.50) by 
carrying out on the same sample the measurement of the e.m.f. and 
the calorimetric determinations of II and a. The agreement is very 
good, as appears from the examples in Table 58. 




(T2 <TI 







dT 2 


In 10- 

" fl volt 

In 10- 

* volt 

Cu Pt ... 

3 66 




3 6 


Cu Fe 





2 7 


Cu nickeline 







Cu German silver . 







Borelius, Ann. Physik 56, p. 388, 1918. 
* Berg, Ann. Physik 32, p. 477, 1910. 


Recent measurements of the Thomson coefficient at very low tem- 
temperatures l showed that <r vanishes in supraconductors and con- 
forms to eq. (20.53) in non-supraconductive metals. 

Let us now find the connection between the Peltier coefficient Hi2 
(or the thermoelectric power Ui2/T) and the difference of potentials 
12 1 122 which we calculated in sections 109 and 112 from the electron 
theory of thermoelectricity. We must bear in mind, however, that in 
its [present form this theory cannot yet 
account for the detailed phenomena and 
gives correctly only the order of magnitude 
of the thermoelectric power. The results 
of Chapter XVII refer to open chains, but, 
if we take the case of a closed circuit (Fig. 
62) with a very large resistance, the current 
will be very weak and potentials com- 
pensating the thermoelectric forces will be FIG. 62. Thermoelectric 
built up, practically, to the same extent as couple, 

in the open circuit. Starting from eq. 

(20.46), we can apply to the heat function the expressions it has in 
equilibrium. We begin with the case when the electron gas is non- 
degenerate obeying eqs. (17.01) and (17.02) of the classical perfect 
gas. The heat function has then the expression x = c p T + Wo + /ft, 
so that we obtain, taking into consideration (17.13), 

H 12 = 12 2 -fli = log-- (20.56) 

e z\ 

It was mentioned in section 109 that this expression applies only to 
semi-conductors. As k/e = 86 X 10 ~ 6 volt deg~ ! , this would imply 
that log (z 2 /zi) for some of the substances of the Table 56 is of the 
order of magnitude 20, whence z 2 /zi would be about 5 X 10 8 . While 
it is within reason that a semi-conductor should have an electronic 
density 10 8 times smaller than that of lead, the signs of a in Table 56 
offer a greater difficulty. They seem to indicate that in some of the 
semi-conductors the carriers of the electric current have a positive 
charge. As we shall see in the next section this difficulty is not quite 
so baffling as it was a few years ago. 

Turning to the case of the degenerate electron gas, we shall make 
use of the equation ^2 = ^1, from which the results of the section 112 
were obtained. In view of the relation v = x Ts, we can write 

1 Borelius, Keesom, and Johansson, Comm. Leiden, 196a; Borelius, Keesom, 
Johansson and Linde, ibidem, 2l7a, b, c. 


/Hi2 X2 ~ xi = T($2 si). Hence with the help of eq. (17.33) we 
obtain the formula due to Sommerfeld 


1112 " 2 

^ e ) Lft,2 fltiJ ' 

~ = 3.67 X 10~ 8 T\ -^- - -^- ) volt deg- 1 . (20.57) 

1 \\li2 A*l/ 

For the combination lead-silver the formula (with the data of 
Table 55) gives U 12 /T = 0.86 X 10~ 8 volt deg- 1 , which is of the right 
order of magnitude. However, the electron theory, in its present 
form, is incomplete as it does not account for the phenomena quanti- 

According to eq. (20.50), the e.m.f. is obtained from the thermo- 
electric power by the relation 


~f ' 

Intimately connected with thermoelectricity are the so-called 
galvanomagnetic and thermomagnetic effects describing the influence of 
a magnetic field on the phenomenon of electric conduction. The 
scope of this book does not permit us to enter into their discussion, 
and we refer the reader interested in the subject to its analysis in a 
paper by Professor Bridgman. 1 

140. Semi-conductors. The difference between metallic con- 
ductors and semi-conductors is not only quantitative but also qualita- 
tive. The conductivity of metals (in the range about C) has numeri- 
cal values between 10 15 and 10 17 <abs e-s and is inversely proportional 
to the absolute temperature 


so that it has a negative temperature coefficient. On the other hand, 
in semi-conductors \ is only of the order 10 4 to 10 13 and its temperature 
dependence can be best represented by the law 2 

X = Xoexp(-a/r) (20.58) 

with a positive temperature coefficient. 

Recent work on the quantum theory of electrons in metals 3 has 

1 P. W. Bridgman, Phys. Rev. 24, p. 644, 1924. 

1 W. Voigt, Krystallphysik, 1910; E. Engelhard, Ann. Physik 17, p. 501, 1933. 

* A. H. Wilson, Proc. Roy. Soc, 133, p. 458; 134, p. 277, 1931. 


greatly contributed to our understanding of the peculiarities of semi- 
conductors. As a part of the argumentation can be given a thermo- 
dynamical guise, 1 we shall outline here its main ideas, even though 
the experimental data for testing the results are extremely meager. 

While treating electron clouds in metals in Chapter VII, we 
assumed that the electrons are free, actually implying in these words 
two different assumptions. In the first place, the free electron does 
not belong to any particular atom; it is shared by all atoms and can 
travel from one end of the conductor to the other. In the second 
place, we assumed that the energy levels of the free electrons can be 
calculated, as if the atoms were absent and as if the electron gas were 
contained in an empty box. It is of particular importance that the 
energy levels in our calculation were distributed practically continu- 
ously: any small energy increment could raise a few of the electrons 
to a higher level. We shall show now that electrons, which are free 
in the first sense, need not necessarily be free in the second, and that 
in this case they do not contribute to the conductivity, they are not 
conduction electrons. 

To fix our ideas let us consider two elements, the one divalent, the 
other monovalent. The free divalent atom has two valency electrons 
which, in their normal or 5-states, have the same energy and differ 
only by the orientation of their spins. There are no other states of 
the same, or nearly the same, energy available: just as many 5-states 
as electrons. The next higher electron levels belong to the first 
excited or -state and have a considerably larger energy. It is differ- 
ent with the monovalent atom] its single valency electron can assume 
a new quantum state without change of energy, simply reversing its 
spin: there are twice as many 5-states as electrons. When Z atoms 
are brought together to form a crystal lattice, we know from general 
mechanical principles the following two facts. (1) The number of 
quantum states remains the same. The Z atoms had in their free 
condition 2Z 5-states; therefore, the number of 5-states in the crystal 
is also 2Z. (2) While the 5-states of the free atoms had all the same 
energy level, this level is split up in the crystal owing to the forces of 
interaction between the atoms. In general it is replaced by 2Z 
slightly different levels lying close together. In short, to the 5-states 
of the free atoms there corresponds in the crystal an 5-band of closely 
crowded levels (Fig. 63a). 

The distribution of electrons over the states of the band is very 
different in the cases of monovalent and divalent elements. In 

J R. H. Fowler, Pfays. Zs. Sovjetunion 3, p. 507, 1933. 


monovalent elements there are only half as many electrons as quantum 
states. Because of the exclusion principle, there can be only one 
electron in each level so that, at T = 0, the electrons would fill the 
lower half of the band. At finite temperatures, they can spread out 
into the upper half. These are the conditions of freedom, in the 
second sense mentioned above, which formed the basis of the electron 
theory of section 111. It is immaterial that the band has an upper 
limit because the electron gas is under ordinary conditions largely 
degenerate and the overflow above the lower half is but small. On 
the other hand, in divalent elements, there are in the s-band exactly as 
many levels as electrons. Supposing that the excited states (forming 

FIG. 63. Position of s- and -bands in (a) insulators, (6) metallic conductors. 

the />-band) lie so high that they are unattainable at ordinary tem- 
peratures, each state is occupied by an electron: the electrons are 
restricted to their positions in the band, they are " bound electrons 1 ' 
in the second sense. 1 

What bearing have these facts on the problem of electric conduc- 
tion? From the point of view of the quantum theory, the mechanism 
by which an electromotive force produces a current is redistributing 
the electrons over the quantum states. According to the above analysis, 
such a redistribution is possible in the case of a monovalent element 
because it possesses plenty of unoccupied levels of accessible energy. 
A monovalent element is, therefore, always a conductor (provided its 
electrons are "free in the first sense"). On the other hand, a divalent 
crystal may be either an insulator or a conductor, even when its elec- 
trons are shared by the whole system. When the p-band of excited 
levels lies so high as to be inaccessible (Fig. 63a) the electrons com- 
pletely fill the s-band, they are " bound electrons" and their redistri- 
bution is impossible: the crystal is then an insulator. On the con- 

1 Interchanges of electrons need not be considered since they do not lead to a 
new state of the system. 

XX 140 



impurity level 

trary, when the p-band lies low and partially overlaps with the 5-band 
(Fig. 63&), it can take the overflow of electrons so that they can be 
redistributed: the crystal is a metallic conductor. 

In this way, the quantum theory gives us a simple explanation of 
why some substances are metals, others dielectrics. It has, more- 
over, the great advantage that the several kinds of semi-conductors 
can be easily fitted into the picture. 

(A) Intrinsic semi-conductors. Suppose that the p-band of excited 
states lies neither very high nor so low as to overlap with the 5-band, 
but has an intermediate position (Fig. 64a). We call Ae the energy 
difference between the lowest />-levels and the highest 5-levels. We 
consider the case that, at T = 0, 
the 5-band is completely filled with 
electrons (as in the preceding ex- 
ample of divalent elements). At 
higher temperatures, there is the 
possibility of an electron spontane- 
ously leaving the 5-band and rising 
to one of the ^-levels. We assume 
that Aeis small enough for the prob- 
ability of such a rise being appre- 
ciable at room temperature, so that 
the number of electrons belonging 
to the p-band is z_ per unit volume 
of the crystal. These Z- electrons 

are free carriers of conductivity because there are plenty of unoccupied 
states in the vicinity of their energy levels. Because of their removal, 
there appears in the 5-band an equal number (z+ = 3__) of unoccupied 
levels or "holes." It was pointed out by Peierls l that the formulas 
of the quantum theory can be given the following interpretation. 
The "holes" behave in every way as if they were free carriers similar to 
the electrons but having the opposite (positive) charge. In this way 
we arrive at the following model of an intrinsic semi-conductor: it 
contains two clouds of conductive carriers, the cloud of free negative 
electrons (G_) and the cloud of "free (positive) holes" (G+). The 
"bound electrons" (G B ) of the 5-band can be left out of consideration 
as far as conduction phenomena are concerned. 

A free electron of the p-band may, of course, drop into a vacancy 
("hole") of the 5-band and so become a bound electron. This process 
can go also in the opposite direction and can be described in the sym- 

1 R. Peierls, Zs. Physik S3, p. 255, 1929. Compare also: Bronstein, Phys. Zs. 
Sovjetunion 2, p. 28, 1932. 

(a) (b) 

FIG. 64. Position of 5- and />-bands 
in (a) intrinsic, (b) impurity semi- 


bols of a chemical reaction as GL + G+ GB = 0. The equation of 
equilibrium is, therefore, according to (17.09), 

^- + (p+ <pB = 0. 

The concentrations of free electrons and of free holes are very small 
so that both clouds can be regarded as non-degenerate in the sense of 
Chapter XVI. We shall assume, as a first approximation, that they 
obey the equations of classical perfect gases. On the other hand, the 
bound electrons are completely degenerate, and their thermodynamic 
potential is reduced to a constant (section 111), namely the mean 
energy level from which the transitions take place. If we measure 
all energies from this level we can ignore <f> B altogether and have the 
same conditions as in the case of equilibrium of positive and negative 
electrons. Therefore, we can apply eqs. (19.28) and (19.29) of section 
128 with the only difference that here the heat of reaction is Ae l 

*_= 4.9 X 10 15 r % exp (-11 600Ae/2r) (20.59) 

(if Ae is expressed in electron-volts), a reasonable result as we shall 
see below. 2 Since the electrons and "holes" are treated as classical 
perfect gases, the electric conductivity can be computed by means of 
the Drude-Lorentz formula which was derived precisely under this 
assumption : X = %e 2 (2irM e kT) " ^lz. The mean free path / is known to 
be inversely proportional to the temperature, / = feo-293/r (feo being 
the value at 20 C). Substituting the numerical values of M e and jfe, 
we can write, therefore, 

X= 1.01 X10 6 r-*/2os. (20.60) 

Using the expression (20.59), and seeing that there are two kinds 
of carriers, we obtain 

X - 1.0 X 10 21 /2oexp(-5800Ae/r). (20.61) 

As /2o is of the order 10 ~ 6 cm, the expression is of a reasonable 
order of magnitude when Ae is of the order 0.1 e-v. 

(B) Impurity semi-conductors. There exist substances which are 
insulators in their pure states but semi-conductors when impure. 
Often as small an admixture of foreign atoms as 1 in 10 6 , or even less, 

1 Already J. Koenigsberger (Jahrb. Radioaktiv. 4, p. 158, 1907) suggested an 
exponential dependence of 2 on 1/3T. 

2 A. H. Wilson (loc. cit.) does not treat the electrons as a perfect gas. His 
formula for * is a little more general, differing from ours by a factor k/ft, where ft 
measures the tightness of binding of the electrons and is a characteristic constant of 
the material. 


is sufficient to produce an appreciable conductivity. Cuprous oxide 
(Cu2O), the only semi-conductor investigated with modern accuracy, 
belongs to this class. The impurity on which the conduction of Cu2O 
depends is free oxygen: it has been shown 1 that the conductivity of 
this material is lowered by glowing in vacuo and increased by heating 
in an oxygen atmosphere. The interpretation in terms of our model 
is as follows: conduction electrons in the p-band are supplied by the 
impurity atoms. The distance between the p- and $-bands is so large 
that in the pure substance the p-band would be empty. On the other 
hand, each impurity atom possesses an electron which can be more 
easily separated because its energy level is only little (Ae) below the 
lowest levels of the p-band (Fig. 646). We denote by 0- the number 
of conduction electrons in the p-band, by z+ the number of ionized, 
and by z of neutral, impurity atoms. In analogy with the preceding 
case, the equation of reaction can be written GL + G<+ d = 0, 
and the equation of equilibrium ^?- + <?>+ ^< = 0. We regard 
again the conduction electrons as a classical perfect gas and use for it 
the expression (19.04). As to (pi = u Ts + pv of the impurity 
atoms, they must be considered as stuck in the crystal lattice and 
having a negligible thermokinetic motion: 2 therefore, u = o, po^ 0, 
while their only entropy is the entropy of mixing given by eqs. (15.03) 
of section 98 or (1 1 .39) of section 80, so that s t -+ $< = -R log (z+i/zt). 
Substituting the numerical values of the chemical constant and of Jfe, 

logio(*-*<+/*<) - 15.69 + f logio r - 5040Ae/r, 

*- * 7.00 X lOVSr* exp(- 5800Ae/jT). (20.62) 

Vogt 3 and Engelhardt measured the coefficient of the exponential 
(i.e. 7.00 X 10 7 \/ZiT^ in our theory) and obtained values between 
1.5 X 10 17 and 9.4 X 10 19 in different samples. This would give for 
Zi numbers from 1 X 10 16 to 4 X 10 20 , or from 4 X 10~ 7 to 2 X 10~ 2 
impurity atoms per molecule of Cu2O. The measured Ae was about 
0.65 e-v. The conductivity, as obtained from (20.60), is 

X - 7.0 X 10 12 Zi"r-*l2Q exp (- SSOOAe/T). (20.63) 

(C) Light-sensitive semi-conductors. There exist substances whose 
conductivity increases many times when they are exposed to light, 

1 R. Engelhardt, he. cit. 

* A more rigorous statement is that the impurity atoms have the same thermo- 
kinetic motion whether they are ionized or not, so that the differences in u and pv 
drop put of the equation. 

W. Vogt., Ann. Physik 7, p. 183, 1930. 


the best-known example being selenium. This case also fits readily 
into the model of Fig. 63a. Here the p-band lies so high that the 
number of its electrons is negligible in the dark. However, when 
light of sufficiently high frequency is absorbed by the crystal, it raises 
electrons into the -band by photoelectric action. The return into 
the s-band requires a radiative transition and may be a very slow 
process, thus assuring an appreciable density of conduction electrons. 
One of the most important aspects of the theory outlined above is 
that it provides a new type of positive carriers of electric conduction : 
the ''free holes" of the 5-band. We have considered under (B) impuri- 
ties which supply free electrons. There may exist other impurities 
which have an affinity for electrons and are able to bind them, thus 
producing "holes". In this way some conductors may have an excess 
of negative, others of positive, carriers. More important still, in the 
more accurate theory the electrons and "holes" are not regarded as 
entirely free. The binding of the two kinds of carriers is different, 
and, for this reason, there is an asymmetry in their actions, even if 
they are present in equal numbers. It was always emphasized by 
E. H. Hall that the peculiarities of the effect bearing his name could 
not be explained without an assumption of this sort. The same 
hypothesis would be helpful in accounting for the great variations in 
the sign of the thermoelectric e.m.f. and of the Thomson coefficient. 


141. General remarks. General rules predicting the direction in 
which a process is influenced by outer forces are very desirable from a 
practical point of view. They may offer a quick orientation in the 
workings of new experimental arrangements and so facilitate their 
theoretical and practical understanding. This applies to thermo- 
dynamics even more than to other parts of physical science because 
the thermodynamical treatment is more abstract and formal and less 
accessible to visualization. The attempts to establish such rules, 
with respect to thermodynamical processes, were all influenced, 
directly or indirectly, by the famous principle of electrodynamics enun- 
ciated in 1833 by Lenz l : "When a force acting on a primary electric 
current induces a secondary current, the direction of this secondary 
current is such that its electrodynamical action opposes the acting 

An analogous principle of thermodynamics was formulated by 
LeCha teller 2 in 1884 and in an extended form by Braun 3 in 1887. 
Their procedure was essentially inductive (although Braun included a 
few formulas in his paper) : they reviewed a large number of examples 
which they tried to state in a form roughly analogous to Lenz's princi- 
ple. They claimed that all these examples could be regarded as special 
cases of a general rule which they proceeded to formulate. However, 
this formulation was so vague that it was impossible to apply the rule 
without ambiguity. This fact was first pointed out by Raveau 4 and 
Ehrenfest 5 and has been generally accepted since. It will serve no 
purpose to mention here the form they gave to their principle, but it is 
well to reproduce two of the examples from which it was derived. 

1 H. F. Lenz, St. Petersburg Acad. of Sci. 29, XI, 1933; reprinted: Ann. Phys. u. 
Chem. 31, p, 483, 1934. 

2 H. LeChatelier, Comptes Rendus 99, p. 788, 1884. 

1 F. Bra'un, Zs. phys. Chem. 1, p. 269, 1887; Ann. Physik 33, p. 337, 1888. 
* M. C. Raveau, J. Phys. 8, p. 572, 1909. 

' P. Ehrenfest, J. Russ. Phys. Soc. 41, p. 347, 1909; Zs. phys. Chemie, 77, 
p. 227, 1911. 



(A) Influence of pressure on solubility. Substances whose solubility 
increases with pressure dissolve (at constant temperature) with volume 
contraction; those which have the opposite pressure dependence dis- 
solve with volume dilatation. This fact (although not new) was 
rediscovered by Braun a little earlier and was the starting point of his 

(B) Influence of temperature on compressibility. Suppose that we 
compress a gas, increasing the pressure under which it stands by the 
small amount A. We can do this in two different ways: (1) keeping 
the temperature constant (T = const) with the help of a heat bath; 
(2) "leaving the gas to itself" (i.e. adiabatically, 5 = const). The 
compressibility is smaller in the second case 




We selected the two examples so as to show the heterogeneity of 
the cases considered by LeChatelier and Braun. The first involves 
only three parameters of the system: pressure, volume, and concentra- 
tion of the saturated solution. In the second we have an interplay 
of four parameters: pressure, volume, temperature and entropy. 
The closer analysis will show that the two cases are also quite different 
in their mathematical characteristics. In fact, Braun himself dis- 
cussed the two types represented by them separately but claimed, in 
the end, that they are reducible to the same principle. Without 
maintaining that all possible examples belong to the types (A) or (B), 
we assert that the principle of LeChatelier-Braun must be resolved into 
at least two different and unconnected rules which we shall analyze in the 
next two sections. 

142. Displacement of equilibrium involving transformation of 
matter. It will be noticed that example (A) of the preceding section 
is only a special case of the first of the two laws stated at the end of 
section 59. These laws, in their turn, are consequences of eqs. (6.51) 
and (6.53) derived from the fundamental conditions of equilibrium, 
as stated in section 40, namely: 

dU = 0, (21.02) 

with the subsidiary conditions 

55 =0, 5F = 0. (21.03) 

The gist of the argument which led in Chapter VI to the explicit 
equilibrium conditions is as follows. The variables S and V possess 


the additive property. In a heterogeneous system the total entropy 
and the total volume are sums of the contributions of the several phases 

S = S ( , V = 2] F<. (21 .04) 

i J 

As was shown in section 40, this property together with the condi- 
tions (21.02) and (21.03) immediately leads to the uniformity of the 
parameters T and />, conjugate to S and V 


The generalization for systems depending on other variables than 5 
and V is obvious. For instance, in Chapter XX we discussed mag- 
netization and electrostatic polarization. We considered systems 
placed in homogeneous magnetic or electric fields of the strength H 
or E (i.e. being homogeneous and having this strength when the sys- 
tem is not there). The element of work was denoted by HdM 9 
EdP, where M and P are the magnetic and electric moments of the 
system in the direction of the fields. When the system consists of 
several phases, the equations analogous to (21.05) 

> = E, 

are a matter of course since all the phases are placed in the same 
homogeneous field. Moreover, the moments have the additive prop- 

M = Z) Jf ( >\ P =><'>. 

i J 

Finally, it is known from the electromagnetic theory that in equi- 

5M = 0, 6P 0. 

The set of equations governing the properties of these variables is 
formally identical with that applying to the parameters 5, V. 

Passing to the general case we assume a heterogeneous system 
which has the following properties: 

(1) Each phase (j) is described by the variables 

which have the additive property 

EXi 01 - JT* (21.06) 


In particular let Xo (;) stand for the entropy 5 (j) . 

(2) The differential of the internal energy has the form 

y{fldX{*>. (21.07) 

We call the coefficient yi (j) the generalized force conjugate to the 
additive variable Xi ( >\ The temperature T (;) = y$ (j) is classed as a 
generalized force j 1 the pressure p must be counted as a negative force 
(p = - yi (i)). 

(3) The equilibrium of the system is determined by 

8U = 0, 
with the subsidiary conditions 

Mi** 0, (/ = 0, 1,. .. w). (21.08) 

(4) The chemical composition and mass of each phase are deter- 
mined by the mol numbers Nk { ' } of its components (k = 1, 2, . . . 0). 
The variations dN k (i) are given by the reactions possible in the system 
(section 42): dN k <i} oc ^ (y) , 

S^C^G* ^ 0. (21.09) 

The explicit equilibrium conditions can be obtained exactly in the 
same way as in section 40 where a less general case was treated. The 
procedure need not be repeated here but only indicated in its bare out- 
line. The problem is broken up in two partial ones: first, changes of 
composition and mass are excluded (6JV* (y) = 0). The variation of the 
internal energy has then the form 

C 'W>=0, (21.10) 

j i 

with the subsidiary conditions (21.08). This gives immediately (as 
in section 40) 

yz (y) =yz, (/ = 0,1, . . .n). (21.11) 

The generalized forces are uniform throughout the system. To treat 
the second partial problem, we introduce the generalized thermo- 
dynamic potential 

* (21-12) 

1 As compared with (2.02) the coefficients y are defined with the opposite sign. 


whose differential is, according to (21.07), 

d& = - *,< W>, (21.13) 


so that it must be regarded as a function of the variables 

We denote the partial thermodynamic potentials, as before, 

When the generalized forces yi are kept constant, the conditions 
(21.02) and (21.08) can be replaced by 

a$ = o. 

for the same reasons which were adduced in section 36. 

We consider only variations in which yi = const; the condition 
6$ = is then valid and can be written in the form 

(; V ;) =0. (21.15) 

j t 

The partials of A< with respect to the generalized forces are, accord- 
ing to (21.13), 




is the change of the variable Xi (for the whole system) in the trans- 
formation or reaction (21.09) 

Apart from the generalized definition of 3> (/) the condition (21.15) 
is precisely the same as (6.50) of section 42. The consequences derived 
in Chapters VII to XI are based on this condition and remain true. 
In particular, applying the generalized form of the thermodynamic 
potential to the expressions (9.13) or (11.15) of the equilibrium con- 
stant, we find from (21.16) 

9 log* **' 

a* " RT (2L17) 

The increase of the generalized force yi produces a displacement of 
equilibrium attended by a process (transformation or chemical reaction) 
in which the change of the conjugate variable AXi is positive. 


In addition to example (A) of Braun's (preceding section), we shall 
illustrate this principle by considering the influence of a magnetic field 
on the equilibrium: yi = H, X t = M t 

The increase of the magnetic strength of field produces a displace- 
ment of equilibrium in which the magnetic moment of the system (in 
the direction of the field) is increased. The more paramagnetic phases 
and components increase their mass at the expense of the less para- 
magnetic. Since supraconductors are classed as diamagnetics, this 
includes also the equilibrium between metals in the normal and the 
supraconductive states: a magnetic field makes the normal phase 
increase at the expense of the supraconductive. These phenomena 
are manifestations of the same causes which make an inhomogeneous 
magnetic field suck in paramagnetic and push out diamagnetic bodies. 
In a similar way, an electric field favors stronger dielectrics at the 
expense of weaker ones. 

143. The restricted LeChatelier-Braun principle. (Influence of 
secondary forces.) Example (B) of section 141 is a true analogue of 
Lenz's principle: when the temperature is allowed to change, it opposes 
the acting force of compression. This is in keeping with the expres- 
sion (21.07) in which the temperature is treated as a generalized force. 
Another example of the same sort is the longitudinal compression of 
of a solid elastic bar with uniform cross-section. It can be compressed 
by a longitudinal force pi in two ways: (1) keeping the forces (of 
pressure) on the lateral surfaces constant (p2 = const) ; (2) keeping the 
lateral dimensions constant (/2 = const) and preventing a bulging out 
of the cross-section. If the length of the bar is denoted by /i, it is 
known from the theory of elasticity that 






Here, too, the secondary (lateral) force, if permitted to change, 
opposes the acting (longitudinal) force. As it is sometimes said, the 
secondary forces help the system to resist the acting force. 

The analytical connection of the examples of this type with the 
foundations of thermodynamics was cleared up by Ehrenfest. 1 We 
shall call the rules pertaining to class (B) the restricted LeChatelier- 

1 See footnote on p. 375. 


Braun principle. Ehrenfest himself refrained from formulating the 
principle, 1 but the enunciation given below is based on his analysis. 

Unlike our treatment in section 142, we may restrict our con- 
siderations to homogeneous systems described by the parameters 
Xo, Xi, . . . X n so that the differential of the internal energy has the 

dU = y Q dX + yidXi + . . . + y n dX n , (21.20) 

defining the generalized forces yo 9 yi, . . . y n - As dU is an exact dif- 
ferential, the following reciprocity relations hold: 

In addition to them, we shall assume the validity of the inequalities 

~ ^ 0, (21.22) 

U**- I 

which are often called conditions of stability. If the variables X are 
those of section 142, the first of these conditions is QT/QS ^ 0, which 
expresses the fact that the temperature rises when heat is imparted to 
the system. As to the condition relating to the volume (X\ = F), it 
must be borne in mind that y\ = />, so that dp/QV 3* 0. In fact 
the sign generally given to the hydrostatic pressure is an anomaly 
among the generalized forces. In hydrodynamics and elasticity the 
tension, and not the pressure, is taken as positive. The inequality 
dp/dV ^ was discussed at length in section 5, where it was shown 
to be a necessary condition of stability. The same applies in all cases 
where the generalized force yi can be envisaged as different in the 
system and in the environment. If the condition (21.22) were not 
fulfilled, the difference would tend to increase and a stable equilibrium 
between system and environment would be impossible. However, this 
argument does not apply to cases like magnetization and polarization 
(sections 134 and 138) where the generalized forces are defined as H 
and E 9 as they would be in the absence of the system. Here the 
validity of the conditions (21.22) is simply a coincidence, as far as 
thermodynamics is concerned. As a matter of fact, diamagnetic bodies 
form an exception to the rule: for them QH/QM < 0, if M is the mag- 
netization in the direction of the field. 2 Fortunately, the principle we 

1 The reason for this was that he did not segregate class (A) into a separate 
principle and was looking for a single rule embracing all cases. Planck (Ann. Physik 
19, p. 759, 1934) also failed to draw the distinction. 

* It would not help to define yi * M because this would reverse the sign of 
i in eq. (21.20). 


are going to formulate need not be amended because of diamagnetics 
(see below). Sometimes it is said that the inequalities (21.22) hold 
because the internal energy U has its minimum in equilibrium. But 
such a conclusion may not be drawn when the minimum is contin- 
gent upon subsidiary conditions of the type (21.03). In this case, 
the equations of stability (21.22) are not, in general, invariant with 
respect to transformations of coordinates. 

j All we shall need for the proof of the principle are the relations 
(21.20), (21.21), and (21.22). It is immaterial whether the parameters 
Xi have the additive property or not; therefore, we drop the requirement 
of additivity. We are now ready for the enunciation. 

(I) Suppose that the system is described by the variables Xo, . . . X n 
and the generalized forces yo, . . . y n for which eqs. (21.20) and (21.21) 
and the inequalities (21.22) are satisfied. 

(II) We keep all the variables X constant except two, Xi and Xi, and 
we give a primary increment dyi to the force yi which acts directly on its 
conjugate variable Xi. We let this increment change Xi in two different 

(a) keeping the (secondary) force y^ constant (yi = const) ; 
(6) keeping the variable Xi constant (Xi = const) and allowing 
the force yi to change. 

(III) We assert that the primary increment 8yi is less effective in 
case (b) 

(21 . 23) 

In other words t the secondary force yi, if permitted to act, opposes the 
primary force yi. 

The proof is simple: the generalized forces yi and y^ as all the 
coefficients of the expression (21.20), are functions of the variables X. 
But since only Xi and Xi are allowed to change, we need to bring in 
evidence only these two parameters 

y, - yi(X lt X<), yi - y(jr, f Xi). (21.24) 

We can resolve the second equation with respect to X < 

Xi-Xi(Xi, yi ). (21.25) 

Now dyi/dXi is the reciprocal of QXi/dVit so that the condition 
(21.23) can be also written 

According to the rules of partial differentiation 


i.e. yi in (21.24) must be differentiated in so far as it contains 
explicitly and in so far as it contains it through the medium of 
From the second eq. (21.24) we find, as in section 3, 

We substitute this, taking into account (21.21) 

. (2127) 

V ' ' 

This finishes the proof of the inequalities (21.26) and (21.23) since 
all three partials (dyi/dXi) v ., (dyi/dXi) Xi , (dXi/dyi) Xt are positive 
because of (21.22). 

It will be seen that diamagnetic bodies do not contradict the 
principle although they do not satisfy the conditions (21.22). In 
fact, for them Xi = M, yi = II and (dM/QXi) = 0, since the mag- 
netic moment depends only on //, as was pointed out in section 134. 
For diamagnetics the second term on the right side of (21.27) vanishes 
and they fulfill the limiting case of (21.23) corresponding to the sign 
of equality. It is conceivable, of course, that applications may arise 
in which the stability conditions (21.23) are not satisfied any more 
than in diamagnetics but which are not so innocuous and break 
through the law (21.23). It is obvious how the principle can be 
extended to take care of such contingencies: one must write the 
inequality (21.23) for the absolute values of the partials and must 
make the sense (< or >) depend on whether (dyi/QXi) and (Qyt/QXi) 
have the same or opposite signs. 

144. Intensive and extensive quantities. The principle of 
LeChatelier-Braun is sometimes brought in connection with the 
division of physical variables into " Quantitaeten " and " Inten- 
sitaeten " proposed by the defunct school of energeticists. 1 This 
nomenclature is still used by German and Dutch writers and has 
some relationship to the English terms: extensive and intensive quan- 
tities. As the reader is likely to come across both designations in his 

l Main proponents: Mach, Ostwald, Helm. Compare G. Helm, Energetik. 
Leipzig 1898. 


study of foreign and domestic literature, it will be well to explain here, 
briefly, their similarities and dissimilarities. 

As defined by Tolman, 1 extensive quantities are those which have 
the additive property, for instance, the variables Xo, . . . X n of section 
142. Every quantity which is not additive is called by him intensive. 
The division is intended to be exhaustive so that every variable must 
fall into one of these two classes. It is possible, and often convenient, 
to use as parameters describing the system not the variables X but 
simple functions of them (like the reciprocals 1/X or the squares X 2 ) 
which are not additive. In this case, the system would be described 
only by intensive variables. It must be said, however, that Tolman's 
definitions are not universally accepted and that some authors use the 
term intensive in the same sense in which we have used the words 
specific quantity in section 39 (i.e. a homogeneous function of the 
degree zero in the mol numbers). 

On the other hand, the classification into Quantitaeten and Inten- 
sitaeten is not meant to be exhaustive. They are exceptional or 
preferred variables, and a parameter chosen at random need not 
belong to either of the two groups. Ostwald and Helm give no sharp 
definition, saying only that the element of work (section 7) can be 
represented as a sum of terms, each the product of a " Quantitaet " 
and an "Intensitaet". It is safe to say that the additive variables X 
of section 142 (i.e. the extensive quantities) belong to the " Quanti- 
taeten " and the generalized forces y, conjugate to them, to the 
" Intensitaeten". But it is vague what other parameters (if any) 
may be classed in these groups. The only practicable suggestion seems 
to be that of Mesdames Ehrenfest-Afanassiewa and De Haas-Lorentz, 2 
who propose to restrict these terms to such variables and conjugate 
generalized forces which leave invariant eqs. (21.20) and (21.22) of 
the preceding section. 

1 R. C. Tolman, Phys. Rev. 9, p. 234, 1917. 

1 T. Ehrenfest-Afanassiewa and G. L. DeHaas-Lorentz, Physica 2, p. 743, 1935. 


145. The statistical point of view. The statistical interpretation 
of the entropy concept was treated in section 30. It was stated there 
that the entropy of a thermodynamical system can be brought in con- 
nection with its probability P by means of Boltzmann's principle (4.67) 

5 = klogP. (22.01) 

While we emphasized in section 30 the parallelism between the 
thermodynamical and the statistical points of view, we shall dwell now 
on the discrepancies between them which have produced in the last 
decades a profound change in the outlook of science upon the theory of 
heat and, especially, upon the second law. 

The identification of entropy and probability involves the following 
difficulty. According to the second law, the entropy of an adiabatically 
isolated system can never decrease: AS ^ 0. If we leave such a 
system to itself, its entropy will either increase monotonically or 
remain stationary. On the other hand, the probability P shows a 
different behavior. Owing to the interplay of atomic and molecular 
movements and forces, the system undergoes, continuously, small but 
erratic changes. In general they occur in the direction of more probable 
states. However, this is only an average effect: it lies in the nature of 
the concept of probability that the less probable states are not totally 
excluded but only less frequent in their occurrence. Therefore, the 
probability of a system will, occasionally, decrease. In fact, it was 
rigorously proved by Poincar 2 that a finite system, subject to the 
Hamiltonian equations of dynamics, returns again and again to any 
state through which it once has passed. 

When Boltzmann's principle as expressed in eq. (22.01) was first 
advanced, scientific thought divided itself into two schools. One school 

1 With the kind permission of Yale University Press parts of this chapter were 
patterned after the exposition by P. S. Epstein, Commentary on the Scientific Writ- 
ings of J. W. Gibbs, Article 0, Section 16. 

* H. Poincare, Acta Mathematica 13, p. 1, 1890. 



regarded the lack of conformity between thermodynamics and the 
kinetic theory as a serious objection against the statistical interpreta- 
tion. The other contended that the principle of the increase of entropy 
had itself only a statistical validity and was true only in the time 
average. As Gibbs put it as early as 1876: "The impossibility 
of an uncompensated decrease of the entropy seems reduced to an 
improbability." The views of the two schools were brought to a focus 
in a very interesting polemic between Zermelo 1 and Boltzmann. 2 
The wide attention which this discussion received stimulated new work 
on the subject, and, before long, the controversy was definitely settled 
in favor of the statistical point of view through the investigations of 
Von Smoluchowski 3 and of Einstein. 4 These authors showed that 
deviations from the entropy principle do, actually, occur and can be 
observed provided the system is sufficiently small. They accomplished 
this by turning eq. (22.01) into a heuristic method for the treatment of 
problems lying outside the scope of classical thermodynamics. Let / 
be one of the parameters describing the state of the system, and let /o 
be its value in the normal state of maximum entropy 5. We ask the 
following question: what is the probability of this parameter assum- 
ing a value between /o + A/ and /o + A/ + dl? Let the entropy, 
corresponding to /o + A/, be So + AS, where AS is, necessarily, nega- 
tive, So being the possible maximum. Inverting the relation (22.01) 
between entropy and probability, we find that P must be proportional 
to exp [(So + AS)/]. On the other hand, the probability must be 
also proportional to the interval dl. Therefore, we can write for it 

Pidl = Ce* s/ *dl. (22.02) 

The factor C may depend on the normal values of all the parame- 
ters of the system. 6 It is determined by the condition 

Pidl - 1, (22.03) 

which expresses the fact that the value of / will lie, certainly, in one 
of the intervals dl. 

E. Zermelo, Ann. Physik 57, p. 485, 1896; 59, p. 743, 1897. 

1 L. Boltzmann, Ann. Physik 57, p. 773, 1896; 60, p. 392, 1897. 

1 M. Von Smoluchowski, Boltzmann-Festschrift, p. 626. Leipzig 1904; Ann. 
Physik 21, p. 756, 1906; 25, p. 205, 1908. 

4 A. Einstein, Ann. Physik 17, p. 549, 1905; 19, p. 373, 1906; 33, p. 1275, 1910. 

* Strictly speaking, the parameter / must be selected in a definite way to make C 
depend only on the normal values /o and not on the / themselves. We cannot enter 
here into the rules of selection as they represent a problem of statistical mechanics. 


146. Brownian movements. Classical thermodynamics cannot 
account for the fact that small particles dispersed in a liquid or gas 
form a permanent suspension, that is to say, float in the medium and 
are in a state of irregular agitation called Brownian movement. From 
the point of view of that theory the particle is merely a part of the 
boundary of the liquid, and its position is determined not by thermo- 
dynamical but by dynamical laws. These laws tell us that the position 
of equilibrium for the particle is its motionless state at the bottom of 
the vessel. Let us suppose the vessel to be of heat-insulating material 
so that our thermodynamical system (liquid or gas) is adiabatically 
isolated. In this case, the dynamical equilibrium of the particle 
corresponds also to the maximum of entropy of the liquid (or gas). In 
fact, it would require the energy expenditure e = M'gx to raise the 
particle from the bottom to the height x, g being the acceleration of the 
gravity field and M' the effective mass of the particle (its true mass 
minus the mass of the displaced medium). This energy can be supplied 
only by the liquid (or gas) which undergoes a corresponding loss of 
internal energy A 7 = e. In general the entropy is expressed by 
AS = (AZ7 + &W)/T\ in our particular case, the volume of the 
liquid remains unchanged when the particle is raised and no work is 
done by the system (AW = 0). Therefore, AS = e/T: this entropy 
change is negative, and the strict point of view on the second law 
would deny the possibility of the particle being permanently afloat. 

On the other hand, the statistical point of view admits the existence 
of small entropy fluctuations. The probability of the floating particle's 
being found at a height between x and x + dx is given directly by the 
formula (22.02) 

-. _M*g* 

P x dx = Ce trdx = Ce *** dx. (22.04) 

It is equally easy to find the probability for the particle being in 
motion. Suppose the components of its velocity c lie between the 
limits c z and c x + dc x , c v and c y + dc v , c g and c g + dc f while its kinetic 
energy is Mc 2 /2. We conclude, as in the preceding case, that this 
energy is supplied by the liquid which experiences the changes 
AJ7 = - Mc 2 /2, AS = - Mc*/2T. The formula (22.02) gives for the 
probability of this state of the particle 

Pdctdc^c, - C f exp (-Mc 2 /2kT)dc x dc v dc t . (22.05) 

If the suspension consists of a large number of identical particles, 
each having the mass Af, the formulas (22.04) and (22.05) determine 
their numbers z at the height x and in the velocity interval 


The first equation can be written 

z = zo exp (-M'gx/kT), (22.06) 

where z is the number of particles (referred to unit volume) at the 
height x, and zo at the bottom (x = 0). It is identical with the baro- 
metric law of Laplace, which was deduced in section 107 in a generalized 
form starting from the law of perfect gases. We see now that the 
particles of a suspension also obey the barometric law, and this implies 
that their osmotic pressure is that of perfect gases: pv = RT or 
p =5 zkT (section 66). The simplest way of showing this is to invert 
the usual form of the derivation of Laplace's formula. At the height x 
the pressure is higher than at x + dx by the amount dp because of the 
weight of the particles in the layer dx. This weight is dp = zM'gdx 
per unit area. On the other hand, we find from (20.06) x = kT 
X log (Z/ZQ)/ M'g and dx = kTdz/zM'g, whence p = zkT. 

The formula (22.05) represents Maxwell's velocity distribution and 
is equivalent to the caloric equation of state u = %RT + #o. This 
follows immediately from calculating the total kinetic energy of all 
particles by integrating \Mf f J 9 c 2 Pdc ai dCj / dc t and keeping in mind 
thatffrpdcyidcydcg = 1. Of course, the particles have also rotational 
kinetic energy which is not included in our calculation of u. It is 
easy to show that its amount is RT per mol, but the above examples 
will suffice. 

Extensive experimental work on suspensions was done by Perrin l 
and Svedberg. 2 These authors tested the barometric formula (20.06) 
directly, and Maxwell's distribution law (20.05) indirectly, and found 
both in excellent agreement with experiments. 

Brownian movements affect not only particles in a suspension 
but indeed every movable object. For instance, a delicate tor- 
sion balance continuously undergoes small deflections from its zero 
position. If the directing force is a and the angle of deflection tf, 
the potential energy of the balance becomes e tf = ^ad 2 . As in the 
case just treated, this energy is taken from the surrounding medium 
(air) whose internal energy and entropy experience the changes 
A 7 = - atf 2 , AS = - Ja# 2 /T. Again we find from (20.02) as the 
probability of a deflection between tf and & + d& 

Pjto = Cexp (-%atf*/kr) d&. (20.07) 

1 J. Perrin, Brownian Movement and Molecular Reality. London 1910. 
* The Svedberg, Die Existenz der Molekflle. Leipzig 1912. 


The mean potential energy of the balance is, therefore, 


Determining the coefficient from the condition (22.03) we find 
2e - ^ kT, (20.08) 

corresponding to equipartition of energy (section 30). Observation 
on a torsion balance were carried out by Kappler. 1 He found that 
the distribution of the deflections is accurately represented by the 
Gaussian curve (20.07). Substituting into (20.08) the measured mean 
at? 2 , he obtained an experimental value for k and derived from it 
(because of n A = R/k) the result n A = (6.06 0.06) X 10 ~ 23 mol" 1 
for the Avogadro number. 

147, Theory of fluctuations. According to classical thermody- 
namics, any substance is in equilibrium quite uniform and not subject 
to spontaneous changes. The statistical view on the second law, on 
the contrary, permits us to put the question as to local deviations 
from uniformity. Let us focus our attention on a small part of the 
system, a group of adjacent molecules, and let us calculate the proba- 
bility that the parameter / has in it an abnormal value. We denote 
by MI, M2 the masses of the small part and of the remaining large 
part of the system (M 2 MI), by so, k the normal specific values 
of the entropy and of the parameter /. We consider the case, however, 
when these quantities have in the two parts the slightly abnormal 
values so + Asi, h + A/i, and SQ + As 2 , / + A/ 2 . The total devia- 
tion of entropy is, therefore, AS = MiAsi + M 2 As 2 or expanding 
into a Taylor series with respect to A/i, A/2 (which we assume to be 
very small quantities) 

AS - ^ 

Since the entropy has its maximum in the normal state, the first 
term must vanish: AfiA/i + Af 2 A/ 2 =0 or A/2 = (Mi /M 2) A/I. 
In the second term M 2 (A/ 2 ) 2 = Mi(A/i) 2 Afi/Af 2 becomes then 
negligible, whence 


The entropy deviation in the small part of the system is expressed 
1 E. Kappler, Ann. Physik 11, p. 233, 1931; 15, p. 545, 1932. 


in terms of its own properties. We may, therefore, drop the subscript 
(1) treating it as if it were alone. Equation (22.02) takes the form 

Pidl = Cexp 

The mean quadratic deviation of the parameter / (in the small 
part of the system) is by definition 


(S/) 5 = C I 


(A/) 2 P,<tf, 

while C is to be determined from (22.03). It must be remembered 
that Qs 2 /Ql 2 is necessarily negative: S being a maximum, the deviation 
can have only the negative sign. Therefore, the integration is easily 
carried out giving 

(22 - 10) 

148. Density fluctuations and light scattering. We shall apply 
this formula to compute the magnitude of the spontaneous density 
fluctuations in a gas. Let us take as the parameter the molal volume 
in the small selected portion of the gas: / = v. Considering the 
fluctuations of one parameter only implies that the others drop out 
in the averaging process, being on the average constant. The ques- 
tion arises, therefore, which other parameter we have to regard as 
constant in taking the partial of 5 with respect to v. The answer is 
simple in this particular case because we know from section 31 that a 
subsidiary condition of equilibrium is dU = 0. Unless we want to 
calculate the fluctuations of the internal energy itself, we must always 
consider changes at u = const. We have, therefore, for the relative 

/A-,\2 / A A2 I. 


where p oc i/v is the density, whence Ap oc Av/t^. From (4.15), 
(9s/9i/)t = P/T, the result of the second differentiation at u = const 
is little different from that at T = 0. We may write with a good 
approximation (d 2 s/dv*)u = (dp/dv) T /T, or denoting the compressi- 
bility _^^ 

^' (22.12) 


In the case of the ideal gas, /3 =!//>, V = kZT/p, Z being the 
total number of molecules in the considered small portion of the gas. 
We obtain, therefore, _ 

/A.\2 1 


as found already by Bernoulli for the fluctuations of independent units 
(molecules). This formula shows us that the mean relative deviation 
of the density from its normal value is very small for a large mass of gas 
but becomes appreciable when the mass decreases. Measuring the 
density of 1 cm of air (Z = 2.7 X 10 19 ) we are not likely to find any 
deviation from the norm, as it is in the mean only 2 X 10~ 10 . How- 
ever, the deviation is measurable in an air-filled cube whose edge is 
of the order of a wave length of light (5 X 10~ 5 cm) because here 
Z = 3.6 X 10 6 whence 1/Z W = 0.53 X 10~ 3 . There exist, therefore, 
in the gas numerous condensed and rarefied regions of this size and 
smaller. This inhomogeneity must produce a scattering action on 
light passing through the gas, and the amount of scattering to be 
expected on the basis of the formula (22.12) was computed by Von 
Smoluchowski l and by Einstein. 2 

A simple way of deriving this formula is as follows. 3 The theory 
of optics gives the following expression for the intensity of light 
scattered by a small dielectric particle under the angle # from the 
direction of the incident plane wave (whose intensity is /) and at the 
distance r from the particle 

AJ, ^F2(AD)2 1 + cos* tf 

~T ~ ~~ (22 ' 14) 

(Rayleigh scattering). AZ> is the difference between the dielectric 
constants of the particle and of the surrounding medium. It is sup- 
posed that AD is small ; under this assumption the scattered intensity 
depends only on the volume V of the particle and not on its shape. 
Ao is the wave length of the incident light, as it would be in vacua. The 
formula applies only to particles whose dimensions are small compared 
with the wave length. We can regard the spontaneous inhomogeneities 
of a gas as particles of this sort. Its whole bulk consists of such 
inhomogeneities lying side by side, but if we focus our attention on one 
of them, we can say that the average dielectric constant of its environ- 

1 See footnote on p. 386. 

2 See last paper in footnote on p. 386. 

8 P. S. Epstein, EnzyklopSdie der math. Wiss. V 3, p. 519, Leipzig, 1915. 


merit has the normal value D while its own dielectric constant deviates 
from it by AD. This deviation is connected with the density fluctua- 
tion (22.12) by the well-known formula (D - !)/(!> + 2)p const, 

^ = A, P- 1)^ + 2). (22 15) 

P 3 

We now put the question as to the scattering of a mass of gas of 
the volume V M . The inhomogeneities are distributed in it quite irregu- 
larly, and there are no phase relations between the amounts of light 
scattered by them. The total scattering is, therefore, simply the sum 
of these amounts. According to our previous analysis the inhomo- 
geneities of small volume greatly dominate, so that there is no objec- 
tion to applying the formula (22.14): The summation of the expres- 
sions for all the homogeneities in V M will lead to F 2 (AZ>) 2 being 
replaced by 

Substituting from (22.15) and (22.12) 
Using the abbreviation 

8-JT 3 

a = ^ kT ^ D ~ " 1 ) S ( Z? + 2 > 2 ' < 22 ' 16 > 

the scattering by a unit volume of gas becomes 

1 - L c2 '\ (22.17) 


"" 16* r 2 

or integrated over all space directions 

- a. 

a has, therefore, the meaning of the coefficient of extinction through 

In the particular case of permanent gases D is very little different 
from 1, ft is practically 1/p, and kTft = 1/z, where z is the number of 
molecules per unit volume: 

Sir 3 

- (D - I) 2 . (22.19) 


The last three formulas were first given by Lord Rayleigh 1 in his 
theory of atmospheric scattering. He explained the blue color of the 
sky as a consequence of the factor 1/Ao 4 : light of short wave length is 
scattered more strongly so that the sunlight which indirectly reaches 
our eyes, after being scattered in the upper layers of the atmosphere, 
has its intensity maximum in the blue. Numerous measurements 
confirmed the dependence on the wave length and on the angle # 
given by these formulas. 2 The direct observations of the coefficient of 
extinction give Sit*(D - l) 2 /3z == 1.00 X 10 ~ 24 cm- 1 , reduced to 1 atm 
and C, the calculated value being 1.04 X 10 ~ 24 cm- 1 . The more 
general formula (22. 16) shows that light scattering must be particularly 
strong when the gas is in the vicinity of the critical state because the 
compressibility $ is then very large. This explains the phenomenon of 
the so-called opalescence in the critical state. It has been investigated 
quantitatively by Keesom 3 and found in agreement with the theory. 

In conclusion, we repeat that the entropy principle cannot account 
for Brownian movements or density fluctuations. The existence of 
these phenomena is conclusive proof that an uncompensated decrease 
of the entropy is not impossible but only highly improbable. This 
fact does not impair the value of thermodynamics as a method of 
analyzing physical reality: in systems of appreciable size the change 
of a deviation from the second law is so extremely small as to be 
entirely negligible. 

*Lord Rayleigh, Phil. Mag. 41, pp. 107, 274, 447, 1871; 12, p. 81, 1881; 44, 
p. 28, 1897; 47, p. 375, 1899. 

2 H. Dember, Ber. Saxon Acad. 64, p. 289, 1912; F. E. Fowle, Astrophys. J. Obs. 
Smithsonian Inst. Ill, Washington 1914; E. Kron, Ann. Physik 45, p. 377, 1914. 

* W. H. Keesom, Ann. Physik 35, p. 591, 1911. 


There are in use four major and numerous minor systems of thermodynamical 
notations. The major systems are as follows: (1) the Gibbs notations; (2) the 
system developed by the physicists Clausius, Helmholtz, and Planck (C.-H.-P.); 
(3) the notations adopted by the larger part of the American chemists, especially 
Lewis and Randall (L.-R.); (4) the system used in technical thermodynamics. 

We believe that this lack of unity is neither strange nor particularly regrettable. 
Thermodynamics is primarily a method for deriving relations between measured 
quantities. Various branches of science make use of it, each applies it to its own 
distinctive material, and in each the problem of harmonizing the notations and 
avoiding duplications and ambiguities is entirely different. It would be extremely 
difficult, if not impossible, to devise a system satisfying all these complicated require- 
ments. On the other hand, the inconvenience to the reader caused by the lack of 
unity is slight, as the relation of the different systems of notations can be set down 
in a simple comparative table. 

As to the intrinsic merits of the different systems there is not much to choose 
between them. A systematic principle was first introduced in the C.-H.-P. system 
in the form of a rule with respect to additive or extensive quantities (sections 39 
and 144): the total quantities (i.e. those referred to the whole system or, at least, to 
a phase) are denoted by capital, the molal and specific quantities by lower-case 
letters. L.-R. modify this by using for molal quantities a special size of type, an 
expedient which seems less convenient in classroom teaching. 

We seriously considered adopting the L.-R. system, which has a large following 
among the chemists of this country. But their choice of the notations E and H 
for the internal energy and heat function (enthalpy) is too repugnant to the physicist 
who has to treat electric and magnetic applications of thermodynamics and is 
accustomed to denote by E and H the electromagnetic field strengths. Worse still, 
according to the above rule the molal energy and enthalpy would have to be denoted 
by e and h, letters which are already badly overtaxed. We have, therefore, chosen 
to use the C.-H.-P. system with the following slight corrections. (1) The letter F 
for the work function ( U- TS) seemed undesirable as it is used by L.-R. for the func- 
tion U TS -f- pV; therefore, we substituted for it *, the choice of Gibbs. (2) The 
use of UK for the mol number (i.e. number of mols of the component h in a system) 
and of No for Avogadro's number is inconsistent with the above rule since the first 
quantity is total and the second molal. We write, therefore Nh and n A . 

In the following table we list the more important notations, omitting those which 
are identical in all systems. 













Internal energy 




N h 






n A 

















n h 



Heat function (enthalpy) 


Work function (UTS) 

Thermodynamic potential 

Partial molal thermodynamic potential 
Mol number 


Avogadro's number 



Mol fraction 



N A 



1 norm, atmosphere 

1 013 249 X 10 G dyne cm" 2 

6 005 717 

1 (15) g-calorie 

4.1852 joule 

621 720 

t < 

4 1852 X 10 7 erg 

7 621 720 

< i 

426 78 g m 

2 630 199 

1 00 095 (20) g-calorie 

OftOO 4.1? 

4 1 

99 976 (mean) g-calorie 

T 999 896 

1 e-v. (electron-volt) 

2 0017 X 10~ 12 erg 

15 30 140 

n A e-v 

1 2139 X 10 12 erg mol"" 1 

1? OS 4.10 


29 003 cal mol ~ l 

4 46 245 

1 Partially based on the critical compilations by R. T. Birge, Phys. Rev. Supple- 
ment 1, p. 1, 1929; Phys. Rev. 40, p. 207, 1932. 





fe c 


T ^ *- - . 

^-t-(-i QQ c^io 

OOO'' H -**'^OO*-<*-' iH 







r- r- o 

^ ^ * MM 

1 1 1 I. 1 1 5 


















Absolute entropy, see: entropy constant 
Absolute temperature, 8, 58, 74 

realization of scale, 74-76 
Absorption of gases in liquids, 161, 162 

coefficient of , 161 
Activity, 199, 287 

coefficient, 201, 293, 294 

function of electrolytes, 203-205, 
290, 294, 295 

of gas mixtures, 205-208 
Additive property of entropy, 59, 60 
Additive quantities, 103, 377, 384 
Adiabatic, equation of , 48 

in gas mixtures, 320 

in perfect gases, 49, 263 

in simple systems, 320 
Adiabatic process, 48, 49 
Adsorption in capillary layer, 223 
Affinity, chemical , 92, 93, 225 
Air, composition of , 308 

specific heat of , 309 
Annihilation of matter, 336, 343, 344 
Atmospheric scattering, 392, 393 
Auxiliary constraints, 99-102 
Avogadro's number, 6, 7, 398 


Barrett effect, 350 

Beattie and Bridgeman, eq. of state 
of , 16 

Binary gas mixtures, 13, 205-208 

Binary systems, 179-194 

Black radiation, energy and entropy 
of, 331-333 

Boiling point, 116-123 
elevation of due to presence of sol- 
utes, 166, 167, 202 

influence of surface tension on , 
217, 218 

Boiling pressure, 11, 116-123 
lowering of by solutes, 163, 166, 202 
influence of capillary layer on , 


of water, 1 19 

Boiling temperature, see: boiling point 
Boltzmann's constant, 7, 78, 398 
Boltzmann's principle, 78, 79, 260, 296, 

Born and Stern, formula for surface 

tension of solids of , 220 
Brownian movements, 249, 387-389. 

Caloric eq. of state (see: eq. of state), 40 
Caloric properties of matter, 27 
Calorie, 26, 397 
Capillarity, 209-224 
Capillary layer, 210, 213 
Capillary rise, 215 

Carbon dioxide, curve of fusion and 
sublimation of , 127, 128 

isothermals of , 17 
Carnot's cyclic process, 50-52, 54-56 

efficiency of , 52. 
Centigrade scale of temperatures, 5 
Cells, galvanic (see also: galvanic cells), 

94, 95, 287, 288 
Characteristic functions, 38, 46, 62, 85 

construction of , 85 
Charged gases, 266-293 

equilibrium in , 268-270 
Chemical affinity, 92, 93, 225 
Chemical compound, 149 
Chemical constant (see also: entropy 
constant of gases and Sackur- 
Tetrode formula), 236-238 

of degenerate perfect gases, 257 

of diatomic gases, 313-314 

of electron gas, 278 

of metallic ions, 284 

of monatomic gases, 312, 313 

of polyatomic gases, 314 




Chemical equilibrium, 
general conditions of , 112, 113 

in charged gases, 270 

in perfect gases, 137-147, 239 

in processes involving radiation, 
335, 336 

in terms of activities, 200, 201 
influence of electric field on , 270 
influence of gravitational field on , 

271, 344 

of solutes in dilute solutions, 

Chlorine, specific heat of , 308, 309, 3 1 1 
Clapeyron-Clausius equation, 118, 120 
Compressibility, coefficient of , 3, 8 

in critical state, 12, 393 
Concentration cells, 288 
Concentrations of ions, 288 
Condensation of the Einstein-Bose de- 
generate gas, 258, 259 

Condensation, process of , 218 
Conservation of energy, see: first law of 

Constant, of entropy, see: entropy 

of internal energy, see: internal 

Constants, table of fundamental , 398 
Contact potential, 275 

classical theory of , 273-286 

quantum theory of , 283 
Continuity of liquid and gaseous states, 

Conversion factors (see also: efficiency), 

52, 397 
Corresponding states, 12 

extended law of , 68, 69, 123-125, 

law of , 12, 16 
Covolume, 9 

Critical constants, 12, 14, 15 
Critical data, 14, 15 
Critical state, 12, 13, 127, 128 

opalescence in , 393 
Cryohydrates, 188 

Crystals, entropy of mixed , 77, 78, 
246, 248 


Debye's formula for specific heat of 
solids, 324 

Debye-Huckel, theory of strong electro- 
lytes, 290-295 

limiting law of , 294 
Degenerate electron gas, 277, 278, 368 
Degenerate perfect gases, 252-265 

numerical amount of degeneration in 

, 263-265 

Density fluctuations, 390-392 
Deuterium, specific heat of , 306, 307 

equilibrium of and hydrogen, 340 
Diamagnetism, 347, 351 
Dielectrics, 359, 360 
Differential, exact , 20 
Differential expression, linear , 20, 21 
Differential Joule-Thomson effect, 70 
Dilute solutions, 152-177 

conditions of equilibrium in , 155- 

Displacement of equilibrium, 156, 376- 

Dissociation, degree of , 140 

depression of by various agents, 

144-148, 178 
Distillation, 183 
Distribution law, 157, 158 
Doublet heat, see: specific heat 
Duhem's equation, 103 
Dulong and Petit, law of , 45, 232 
Dushman's formula for thermionic cur- 
rents, 279 

Efficiency, of Carnot's process, 52 

of galvanic cells, 95 

of heat engines, 51, 52, 55, 95 
Einstein-Bose gas, see: degenerate per- 
fect gases 

condensation of , 258, 259 
Einstein's formula for specific heat of 

solids, 233 
Einstein's law of equivalence of mass 

and energy, 330 

Electric field, influence of on equilib- 
rium in charged gases, 270 
influence of on equilibrium of drop- 
lets, 219 

Electrolytes (see also: weak and strong 
electrolytes, and activity function), 
94, 204 
specific heat of , 329 



Electromotive force, 94 

of galvanic cells, 94, 289, 290 

of thermoelectric couples, 364, 365, 

Electron clouds, 266-285 
Electron gas, 266-285 

degenerate , 277, 278, 368 
Electron pairs, 337, 338 
Electronic specific heat in metals, 278, 


Electrons, positive , see: positrons 
equilibrium of positive and negative 

, 337, 338 
Electrostriction, 360 
Elevation of boiling point by solutes, 

166, 167, 202 
Endothermic process, 27 
Energy, 27 

conservation of , see: first law of 

internal , see: internal energy 

of black radiation, 333 
Enthalpy, see: heat function 
Entropy, 58, 59, 60 

additivity of , 59, 60 
measurement of , 66, 229, 230 

of black radiation, 333 

of degenerate perfect gas, 256, 


of dilute and perfect solutions, 

153, 154 

of electron gas, 278 

of gas mixtures, 136, 207 

of mixed crystals, 77, 78, 246, 


of perfect gases, 63 

of supercooled liquids, 248 

of Van der Waals gases, 65 
statistical interpretation of , 76-79, 

385, 386 
Entropy constant, 59, 230 

of gases (see also: chemical con- 
stant), 235, 236, 337 

Entropy principle, 60, 61, 82 

Eotvos* formula for surface tension, 

Equation of state (caloric), 40, 68, 347 

of degenerate perfect gases, 255 

of perfect gases, 43 

of Van der Waals gases, 65 

Equation of state (magnetic), 347 

of Langevin-Brillouin, 348 
Equation of state (thermal), 3 

of Beattie and Bridgeman, 16 

of Kamerlingh Onnes, 16 

of degenerate perfect gases, 252, 253 

of perfect gases, 6, 7 

of Van der Waals, 12 

of Van der Waals-Lorentz for 
binary gas mixtures, 13, 206 

Equilibrium, 1 

chemical , see: chemical equilibrium 
displacement of , 156, 376-380 
fundamental conditions of , 82, 92, 

93, 376-379 
influence of capillary forces on , 

influence of electric forces on , 

219, 270, 380 
influence of gravitational forces on , 

271, 344, 345 
influence of magnetic forces on , 380 

in processes involving radiation, 

role of internal energy in , 82, 376, 

role of thermodynamic potential in , 


role of work function in , 92 
Equilibrium conditions (explicit) 

general statement of , 112, 113, 200, 


in dilute solutions, 155-157 

in processes involving radiation, 

in terms of activities, 200, 201 
Equilibrium constant, 139, 156, 200, 

235, 236, 239, 336 
Equipartition of energy, 79 
Euler's theorem (about homogeneous 

functions), 102, 104 
Eutectic alloy, 188 
Eutectic point, 188 

ternary , 196 
Exothermic process, 27 
Expansion (thermal), 1 

coefficient of , 3, 5 

in vicinity of T = 0, 228, 258 
Extensive quantities, 384 
Extinction, atmospheric , 392, 393 



Fermi- Dirac gas, see: degenerate perfect 

First law of thermodynamics, 27-51 

formulation of , 37-39 

history of -, 27-34. 
Fluctuations, 389, 390 

of density, 390-392 

Free energy, see: work function and 

thermodynamic potential 
Freezing point (see also: fusion), 116 
^"ugacity, 197 

of perfect gas, 197 

of Van der Waals gas, 198 
Fundamental point (see also: 

triple point), 110, 126-128 
Fundamental state, 110, 126-128 
Fusion, point of , 116, 119 

lowering of due to solutes, 166, 

167, 202, 203 
Fusion, process of , 219 

Gadolinium sulphate, specific heat of , 


magnetization of , 349 
Galvanic cells, 94, 95, 287, 288 

electronomotive force of , 94, 289, 290 
Galvanomagnetic effects, 368 
Gas constant, 6, 7,398 
Gas mixtures, 7, 13 
activities of binary , 205 
chemical equilibrium in , 137-147, 

236-239, 270, 284, 336 
heat capacity in , 315-318 
velocity of sound in , 318-321 
Generic probability definition, 78, 246 
connection of with principle of 
indetermination, 260 
Gibbs-Helmholtz equation, 93, 96, 225 
Gram-molecule, 6 

Gravitational forces, influence of on 
equilibrium, 271, 344 


Heat, 25 

latent , see: latent heat 
of reaction, 27, 67 
molal (see also: specific heat), 26 
specific , see: specific heat 

Heat capacity (see also: specific heat). 

at constant magnetic field, 351, 352 

at constant magnetization, 351, 352 

at constant pressure, 40, 46, 65, 66 

at constant volume, 40, 65, 66 

of electron pairs, 388 

of reacting gas mixtures, 315-318 
Heat content, see: heat function 
Heat function, 46, 85, 86 

connection of with latent heat and 
heat of reaction, 67 

of electron pairs, 388 

of perfect gases, 47 

role of in Joule-Thomson proc- 
ess, 69 
Helium, curve of fusion and sublimation 

in , 128 

equilibrium of and hydrogen, 342 
inversion point of , 72 
supposed second order transformation 

in 129 

Henry's law, 159-162 
Heterogeneous system, 2 
Homogeneous system, 2 
Hydrogen, annihilation of , 343 

equilibrium of and deuterium, 340 
equilibrium of and helium, 342 
solubility of in molten metals, 162 
specific heat of , 44, 303 
Hydrogen bromide, dissociation of , 


Hydrogen iodide, dissociation of , 144, 


Image force, 273, 274 
Independent component, 107-110 

systems with one , 115-133 
Independent variables, 98, 99, 107 
Integrability, 22, 23 
Integrating multiplier, 22, 58, 74 
Intensitaet, 383, 384 
Intensive quantities, 102, 384 
Internal energy, 38, 83 

constant of , 330 

of degenerate perfect gases, 255 

of dilute and perfect solutions, 
151, 152 

of electron gas, 278 



Internal energy, Continued 

of perfect gas, 43 

of Van der Waals gas, 65 
Internal pressure, 9 

Inversion temperature in Joule-Thomson 

effect, 71, 72 
Iodine, dissociation of , 148 

distribution of between water and 

carbon tetrachloride, 158 
Ion clouds, 266-287 

lonization in solar atmosphere, 283-285 
Irreversible process, 35, 36 
Isobaric process, 46, 47 
Isochoric process, 47 
Isosmotic solutions, 170 
Isothermal process, 47, 48 

work in , 87 
Isothermals, of Van der Waals gas, 10 

of CO 2 , 17 
Isotonic, see: isosmotic 

Joule-Thomson effect, 70-73 

differential , 70 

integral , 73 
Joule-Thomson process, 42 

theory of , 70-73 


Kamerlingh-Onnes' empirical reduced 
equation of state, 16 

Kelvin's formula for influence of capil- 
larity on boiling pressure, 214 

Kelvin's scale of temperatures, see: 

absolute temperature 

Lagrangean multipliers, 98 

Langevin's theory of magnetization, 

Langevin-Brillouin, magnetic equation 

of state of , 348 
Latent heat, 26, 46, 67, 118, 121 

temperature dependence of , 121 
LeChatelier-Braun, principle of , 375 

restricted principle of , 380-383 
Legendre transformation, 85 
Lenz's principle of electrodynamics, 375 
Liquidus curve, 185 
Liquidus surface, 195 

Lowering, of boiling pressure, 162-167 

of point of fusion, 166, 167, 202, 203 


Magnetic equation of state, 347 

of Langevin-Brillouin, 348 
Magnetics, 346 

perfect , 348, 350, 353 
Magnetization, 346 

influence of on equilibrium, 380 
Magnetocaloric cooling, 352-354 
Magnetostriction, 350 
Mass fraction, 109, 182, 192, 193, 196 
Mass law, 138-140, 176-178, 270, 271, 

Mixture, 149 

of gases, see: gas mixtures 
Mobile equilibrium, principle of , 156 

fraction, 13, 102, 136 

Molal heat (see also: specific heat), 26 
Molal volume, 6 

of dilute and perfect solutions, 152 
Multiplier, integrating , 22 

Lagrangean , 98 


Nagaoka-Honda effect, 350 
Nernst's distribution law, 157, 158 
Nernst's heat postulate, 225-250 

generality of , 246-250 

independence of from second law, 

Nernst's own formulation of , 226, 
227, 230 

Planck's formulation of , 230, 231 
Neumann's law, 45, 327 
Neutrons, 260, 339, 343 

equilibrium of and protons, 339 
Nitric oxide, specific heat of , 44, 310 
Nitrogen, solubility of in water, 161 

specific heat of , 307, 308 
Nitrogen tetroxide, dissociation of , 
141, 142 

dissociation of in chloroform solu- 
tion, 177 

heat capacity of , 319 
Notations, 395 

comparative table of , 396 



Opalescence in critical state, 393 
Osmotic pressure, 167-178 
equation for , 169 
Van t'Hoff s equation for in dilute 

solutions, 171 

Ostwald-Freundlich, formula for solu- 
bility of small particles of , 218 
Oxygen, specific heat of , 44, 307, 308 

Partial molal quantities, 84 
Partial molal thermodynamic potential, 

in dilute solutions, 155, 200 

in gases, 137 

in strong electrolytes, 293 
Peltier coefficient, 362 
Perfect gases, 4-8 

degenerate , 252-265, 277, 278 

mixtures of , 7, 134-147 
Perfect magnetics, 348, 350, 353 
Perfect solutions, 151, 179 
Perpetual motion machine of the second 

kind, 54 
Phase, 2 
Phase equilibrium, 100, 101, 104-114 

displacement of , 156, 376-380 

in binary systems, 179-193 

in dilute solutions, 154-167 

in pure substances, 115-128 
number of phases in , 110, 111 

of higher order, 128-133 
temperature and pressure in , 104 

Phase rule, 110, 111 

Phosphonium chloride, curve of fusion 

and sublimation of , 128 
Photoelectric effect, 275, 280 
Piezoelectricity, 360, 361 
Point of fusion, 116-128 
influence of capillarity on , 219 
lowering of by solutes, 166, 167, 

202, 203 

Point of inversion, see: inversion point 
Point of transition, see: transition point 
Polarization, electric , 359, 360 
Positrons (or positive electrons), 337 
equilibrium of and negative elec- 
trons, 337, 338 

Principle, Boltzmann's , 78, 79, 240, 

260, 296, 385 
entropy , 60-62, 82 
Lenz's of electrodynamics, 375 

of indetermination, 260 

of LeChatelier-Braun, 375, 380-383 

of mobile equilibrium, 156 

of Thomsen-Berthelot, 226 
Probability, 76-79, 246, 247, 260 

generic definition of 78, 246 
relation of generic definition to prin- 
ciple of indetermination, 260 
specific definition of , 78 
Protons, annihilation of , 343, 344 
equilibrium of and neutrons, 339 
Pure substances, phase equilibrium in , 

Pyroelectricity, 361 


Quadruple point, 179 
Quantitaet, 383, 384 


Radiation, black , 331 
energy of , 332, 333 
entropy of , 332, 333 
pressure of , 332 

Ramsay-Shields, formula for surface 
tension of , 221 

Raoult's law, 162, 163, 164 

Rayleigh's formula for atmospheric light 
scattering, 392 

Reciprocity relations, 21, 64, 70, 89, 
350, 360, 381 

Reduced variables, 12, 16, 68, 69, 71, 73, 

Regelation, 120 

Reversible process, 35 

Richardson's formula for thermionic cur- 
rent, 276 

Rigid envelope, 100, 101 

Sackur-Tetrode formula (for absolute 

entropy of gases), 237-241, 257 
Saturated solution, 174, 176 
Scale of temperatures, 1 

absolute ,8,58, 74-76 

centrigrade , 5 




Scattering of light, 391, 392 

Second law of thermodynamics, 53, 54, 
58, 59, 62, 82 

Semi-conductors, 368-374 

thermoelectric effect in , 276, 365 

Semipermeable membrane, 101, 135, 168 

Sieverts' law (solubility of gases in molt- 
en metals), 162 

Simon and Glatzel, formula for pressure 
of fusion of t 122 

Simple system, 2 

Solidus curve, 185 

Solidus surface, 195 

Solubility, mutual (complete or par- 
tial), 149, 150, 184, 186, 189, 190 

of gases, see: Henry's law 

of solids in liquids, 174, 218 
Solute, 151 

Solution, 149-151 
dilute (see also: 

152, 177 

perfect ,151, T' 

Solvent, 151 .ula for electronic 

Sommerf eld's >f metals, 278, 326 

specific heanula for inner potential 
Sommerf eld's f' 

of met ; ' ula for thermoelectric 
Sommerf ela 068 

power, 2&\ also: heat capacity), 
Specific heat (s\ 

26, 296-329 sure, 40, 46, 65, 66 

at constanprefc. 4Q ( 55 ? ^ 

atconstajvolumct^ 327 
doublet , >1, 310, 3 ?2 p 

in vicinr f T Q t 

of elect <ytes, 329 

of elect" 18 i n metals, 278, 326 

of gas*, 44, 298-311 

of liqi*s, 328, 329 

of pei** gases, 44 

of sois, 45, 233, 321-327 

of suites, 329 

rotaticA 1 ~ 29 ^ 300, 304, 305, 307 

transl/onal ,299 

vibraanal , 302, 303, 307-309, 311 
Specifitf> roDa bih'ty definition, 78 
Specifi. quantities and properties, 102, 

Stabi# conditions of dynamical , 11, 


Stability, Continued 
conditions of thermodynamical , 


average, 76 

point of view, 76-79, 246-248, 
385, 386 

principles, 76-79, 259, 260, 296 
Strong electrolytes, 204, 287 

activity function of ^ 204 205, 294 

theory of , 290-29* 361 
Sublimation, 126, lX 
Sugar solution, '*notic pressure of , 


s . states, 297, 300-304 
^Jpercooled liquids, 248 
Superheated state, 11, 117, 126 
Supraconductive transition points, 355 
Supraconductivity, 354-358 
Surface tension, 209-224 
influence of on boiling and melting 

points, 214, 216-219 
temperature dependence of , 224 
Symmetry number, 300, 313, 314 
System, thermodynamical 2 


comparative -1 of notations, 396 
-- of conv ersion factorSf 397 

of f f andamental constants, 398 

ratine, 1 

a)F ,olute ,8,58, 74-76 
centrigrade , 5 
Tension, coefficient of , 3 
Ternary systems, 194-196 
Thermal equation of state (see: equation 

of state), 3 

Thermal equilibrium, see: equilib- 

Thermal expansion (see also: expansion) 
coefficient of , 3 

in vicinity of T 0, 228 
Thermal properties of matter, 27 
Thermionic emission, 274 

in classical theory, 279 

in quantum theory, 279 
Thermionic work function, 274, 276, 279, 

280, 282 



Thermodymanic potential, 90 
connection of with non-mechanical 
work, 90 

of degenerate perfect gas, 257 

of electron gas, 278 

of gas mixtures, 137, 207 

of perfect gas, 91, 331 

of Van der Waals' gas, 91 

role of in equilibrium, 91-93, 106, 

112, 113,115 
Thermodynamics 1 

limitations of , 387-393 

subject of , 2 
Thermoelectric effect, 3oi <^% 

empirical ,364 

in classical theory, 273, 367 

in quantum theory, 282, 368 
Thermoelectric e.m.f., 368 
Thermoelectric power, 364, 366, 368 
Thermomagnetic effects, 368 

Third law of thermodynamics, see: 
Nernst's postulate 

Thomsen-Berthelot, principle of , 226 

Thomson coefficient (in thermoelec- 
tricity), 362, 365, 366 

Transformation pressure and tempera- 
ture, 116, 121, J22 

in binary systems. 183-193 

01A 910 

influence of capillarity 7 on ~~ Z1 *' ' 
influence of solutes on " 162 " 167 
Transition points of supravr onductors ' 
355 __ 

influence of magnetic field J ' 

Trichloroacetic acid, distribution of 

between water and ethyl ether, 158 
Triple point, 125-128 
Trouton and Deprez, rule of , 124, 



Unattainability of absolute zero, 243-245 
Units, conversion factors of , 397 

Van der Waals' 

equation of state, 9, 12 

equation of state for binary gas 

mixtures, 13, 206 

formula for vapor pressure, 121 

formula for surface tension, 221 
Van t'Hoff's equation for osmotic 

pressure, 171 
Vapor pressure, see: boiling pressure 

in binary systems, 180, 182 
'iations, 82 

_ 'tv of sound, 49 

_ j n Jegenerate gases, 264-265 

Virtual chan ixtures ' 318 ~ 321 

Virtual dispfi- 82 t t , 

97-99 "nents, method of , 


Water, activity and f i 

^9g of liquid , 

Water vapor, dissociatit 

Weak electrolytes, 20*i of , 143 

Work, 19 - 287 

element of 18, 
Work function, '**'^ 210 

-of perfect^ 7 ' 

-of stron^ f 8 ' 8 ? \ 

s electrolytes 293 

- - of Van der Waals ga&R9 
role of in equilibriumAl, 92 
thermionic , see: ther^onic work