DERIVATION OF EQUILfBBIUM CONDITIONS
699
THE DERIVATION OF T H E EQUILIBRIUM CONDITIONS I N PHYSICAL CHEMISTRY ON T H E BASIS OF T H E WORK PRINCIPLE J. N. BRPNSTED University Institute of Physical Chemistry, Copenhagen, Denmark Received November IS, 1989 I. IpTRODUCTION.
THE ENERGETIC BASIS
In some previous publications (3) an attempt has been made to revise the fundamental concepts of energetics and to formulate the fundamental laws of thermodynamics on the basis of new points of view. It is the concept of heat evolution and the relation between heat and work as advanced by classical thermodynamics which has particularly 'called for criticism and necessitated a revision of the accepted theories. As an introduction to the present paper it will be necessary to delineate briefly certain of the elementary considerations which have contributed particularly to this altered orientation. It should be observed first that traditional thermodynamics does not seem to recognize the fundamental difference between the problems of defining, on the one hand, amounts of heat communicated, Le., added to or removed from a system through thermal conduction or radiation, and, on the other hand, heat produced or consumed within the boundary of the system itself. In traditional thermodynamics heat communicated is defined and determined by means of calorimetric methods which are a t any rate selfconsistent. As regards heat produced or consumed, no general definition exists to separate it from heat communicated. The practical determination of heat evolution imitates the methods employed in the determination of heat communicated, the temperature change at the process, irrespective of its reversibility, being introduced as an indicator and a measure of the heat involved. This procedure, however, completely lacks a theoretical foundation, and leads in a great measure to contradiction with the fundamental principle, recognized and made use of also in classical thermodynamics, that heat is a form of energy the production and consumption of which demands an energetic equivalent. A simple example will illustrate this assertion. If by means of a weight falling through a certain distance an ideal gas of pressure p is compressed from the volume u to u - dv, it is claimed by classical thermodynamics that, owing to the rise in temperature, an evolution of heat takes place, which in the case considered has the value pdu. This well-known interpretation of the compression process, however, entirely overlooks the fact,
'io0
J. N. BR$NSTED
+
that on compression the gaseous system, i.e., the gas the surrounding vacuum, has received an amount of work pdu, exactly equal to the loss of work accompanying the transport of the weight. Accordingly, if the energy principle has to be maintained, no room will be left for the additional production of energy such as the pretended evolution of heat would imply. On the other hand, when an ideal gas expands irreversibly, the classical theory postulates the evolution of heat to be zero, because of the rise in temperature being zero. This interpretation cannot be maintained either, since the irreversible expansion, exactly as the irreversible fall of a weight from a higher to a lower position or the irreversible flow of electric charges from higher to lower potential, is accompanied by a loss of work and therefore, if the energy principle is to be fulfilled, must evolve heat. With this lack of clarity in the traditional conception of heat evolution, obviously a corresponding lack of clarity as to the concept of work is closely connected. In the first place, on account of the straight phenomenologic analogy between the various basic processes, it is not permissible to confine the concept of work to processes of a mechanical nature. This is well recognized in the classical treatment, as shown by its introduction of an electric work term as an energetic concept. Exclusion of chemical, thermal, and other kinds of work, however, does not only mean incompleteness and awkwardness in the treatment, but unavoidably leads to contradiction. In the second place, as clearly illustrated by the above example, one finds in classical thermodynamics a fatal uncertainty as regards the purely elementary enumeration of the individual members of work taking part in a process. It is under the impression of these and similar decisive weaknesses in the classical conception, regarding which further information is to be found in the papers quoted above, that a rationalized exposition of the fundamental problems of thermodynamics has been attempted. As an essential part of such an exposition it has proved necessary first to analyze more closely the nature of the fundamental energetic entities and of the elementary components in which the changes occurring in nature can be schematically divided. For this purpose the concepts of quantity and intensity or potential already used by earlier authors have proved suitable, the sense of these concepts appearing particularly clearly through introducing the idea of the basic energetic process. By this term is meant transport or displacement of energetic quantities, in the first place entropy, volume, chemical substance, and electricity, between states which may be characterized by their energetic potentials, i.e., tsmperature, pressure, chemical and electrical potential. To such a process, as a logical extension of the existing mechanical concept of work, there must quite generally be coordinated a loss of work, given as the
DEHlVATlON OF EQUILIBRIUM CONDlTIONS
70 1
product of the amount of quantity transported and the difference between the corresponding potentials in the two states : This loss of work is said to take place within the syster.1 which contains the quantities transported. In particular we may write for the thermal loss of work: 6A = (TI - TZ)6S (2)
T being the temperature and S the entropy.’ The basic processes described here are necessary constituents of all natural changes. The changes consisting of basic processes alone are reversible changes. The fundamental law governing such changes is the work principle, which postulates that the total loss of work involved in the reversible process is zero : Z6A = 0
(3)
This equation, which emphasizes the perfect symmetry with which all basic processes enter in the process, offers a simple and exhaustive description of the reversible phenomena in nature and does not in any way lead to contradiction. In the irreversible changes, in addition to the basic processes, an evolution of heat will take place. Now the work principle is no longer valid, since a positive loss of work will occur in the system, which, irrespective of the nature of the basic processes, is equal to the heat produced:
ZSA
=
SQ
(4)
This heat, which we shall call the energetic heat evolution, and which obviously satisfies an extended equivalence principle dealing with work in a general sense, is defined as the heat communication of the system on reversible reproduction of the irreversible process considered. I t is related to the entropy formed simultaneously through the general equation: 6Q = T6S
(5)
The theory is thus in conformity with the classical assertion that entropy increases in all irreversible processes in an adiabatically isolated system. -4s the summary of the above considerations it may thus be maintained that all natural processes are characterized by the fact that either the loss 1 As emphasized by Callendar (4) and also in the first of the papers mentioned in reference 3, Carnot’s conception of the “motive power of heat” ( 5 ) is in agreement with this expression if the term “calorique” with which he operates is identified with entropy. The expression is also t o a certain extent in conformity with Zeuner’s (7) conception of the function of the heat engine.
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J. N. B R ~ N S T E D
of work is zero (equation 3), in which case the process is reversible, or else the loss of work is positive and equal to the evolution of heat (equation 4), in which case the process is irreversible. Processes in which loss of work and evolution of heat are negative are not possible according to this theory. It is the object of the present paper to show how the various equilibrium equations in thermodynamics and physical chemistry can be derived on the basis given here. Since every transformation occurring through a state of equilibrium is a reversible process, it is the work principle upon which these calculations should be based. This method is not only theoretically unassailable, but, on account of the perspicuity of the work principle, also possesses a considerable didactic significance. For the application of the work principle in a given case it is necessary that the reversible character of the change in question should be ascertained. We shall begin our exposition by proving the reversibility of certain groups of processes of partiwlar significance for our purpose. 11. THE WORK THEOREM FOR INFINITESIMALLY DIVERGING SYSTEMS
I n this and the following sections we shall suppose the systems in question to be constituted exclusively by the independently variable quantities, entropy, volume, and chemical components. We shall consider two systems, I and 11, homogeneous with respect to quantity and potential and so constituted that a certain fraction of I contains the quantities : S, V , n 1 , n z . . . . . . . . .
(6)
and a certain fraction of I1 the quantities:
S
+ dS, V + dV, + dni , n~ + dnz , . . . . 121
(7)
The quantities indicated, which define the systems completely, are measured in reservoirs each communicating only a single quantity. The potentials, corresponding to the quantities given by expressions 6 and 7, are T , p , P I , pz * ’ * . ‘ ’ ’ * ’ * (8) and
+
+
+
+
T dT, P dp, PI dPi, ccz dpz, * . (9) respectively. The two systems are supposed completely miscible. We now consider a process completely cut off from the surroundings and consisting in the transference from I to I1 of the fraction of I given by expression 6. The transference is supposed to be differential, which means that the amount transferred is so small in comparison to the total amount of the two systems that the changes in potential resulting from the transference are negligible compared to the potential differences already exkting.
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703
It is easy to see, then, that such a transference i s a reversible process. As a matter of fact, on account of the complete miscibility which, as mentioned before, is presupposed, the amount presented by expression 6 after being isolated from I and then brought into contact with I1 will dissolve spontaneously in this system, while reversely the amount presented by expression 7 after being isolated from I1 and brought into contact with I will spontaneously dissolve in this phase. Since these two spontaneous processes are opposite with respect to direction and only infinitesimally different from each other, they deviate only infinitesimally from truly reversible processes. This important result now permits the application of the work principle to the process of transfer defined above. According to the work principle the total work lost in the process is zero. For each kind of work the loss is obtained by multiplying the quantity appearing in expression 6 by the corresponding ditrerence in potential appearing from expressions 8 and 9. Thus the equation & = I-SdT+
Vdp
- nldpl - n z d p ~- ..... = 0
(10)
is obtained as an expression of the work theorem for infinitesimally diverging system. The total loss of work being zero means in the present ease that i t is small compared to the individual work terms entering in the equation, or a t any rate to some of them. The contents of equation 10 may therefore be expressed in a quite unambiguous way by:
ZKiPi
21&pij
(.
(11)
where the denominator indicates the sum of the numerical values of all work terms involved. Actually this equation can be taken as a characterization of the reversible process. Since the natural processes are never completely reversible, but are always accompanied by a certain net loss of work, the question of the definition of reversibility must be that of the definition of the deviation from reversibility, Le., of the irreversibility. It would be possible to introduce the absolute loss of work as a measure of the irreversibility, but it would no doubt be more rational to use the loss of work in proportion to the total work involved. It is this conception which is the basis of the reversibility principle established above and mathematically formulated in equation 11. It should be carefully noted that the theorem concerning the zero value of the transference work applies only to fractional amounts having the composition of the systems with respect to all quantities. In special cases one obtains simplified expressions. If, for instance, the temperature and
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J. N . BRGNSTED
pressure differences between systems I and 11 are zero, cquation 10 is reduced to the following form:
6A = - n l dpl - n2dp2 -
. .. = 0
(12)
an expression the validity of which for two components has been demonstrated by the author in an earlier publication (2; cf. 1). From the expression given there for the differential transport of a mixture of mole fraction a from its own solution to a mixture of mole fraction z, one obtains in the case of ideal miscibility conditions the expression:
which shows that when a - z is small, A changes proportionally to the square of the difference in concentration. This expresses the validity of equation 12 in the special case under consideration. It will be of interest finally to compare the work theorem for infinitesimally diverging systems with the fundamental Gibbs equation for the potential variations of a homogeneous system (6). Like equation 10 the Gibbs equation expresses that the sum ZKidPi is zero. But the treatment of Gibbs does not interpret this sum as representing the transference work considered here or any other work whatsoever. In Gibbs’ treatment the equation expresses the correlation between the potential changes generated in the system through the addition of arbitrary infinitesimal amounts of quantity or through intrinsic irreversible changes. It is possible, of course, as an alternative procedure, to base our considerations upon the Gibbs equation and establish its character as a work expression under the assumption of reversibility. Through this method, however, one will to a certain extent conceal the fundamental idea on which our exposition is based, and which it is the special purpose of this paper to emphasize and elucidate. 111. THE HETEROGENEOUS EQUILIBRIUM
The work theorem, which in the previous section has been shown to hold for infinitesimally diverging systems, forms the immediate basis for the derivation of all formulas in physical chemistry for heterogeneous and homogeneous equilibria. We shall first consider the simple two-phase equilibrium in a onecomponent system. We imagine the two phases in equilibrium with each other in a system I, where the potentials are T , p , and p , and in another dT, p dp, and p dp. The system 11, where the potentials are T quantities in a certain section of phase A in system I are
+
+
+
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DERIVATION OF EQUILIBRIUM CONDI’FIOAS
and in a certain section of phase B in the same system SB
9
VB
(14)
nB
In system I1 the corresponding quantities are
SA-k dSA, VA -k dVA, n~ -k dnA and
(15)
+
SB dSB, VB -k dVB n~ -k dnB (16) respectively. We now pass a fraction of system I A, given by expression 13, differentially from I A to 11 A, and a fraction of I1 B, given by expression 16, differentially from I1 B to I B. Since systems I and I1 are differing only infinitesimally from each other, the loss of work involved in each of these processes will be zero according to the work theorem. Disregarding members which are small of second order we can thus write for the total loss of work associated with the transport:
-(SA - SB)d T
+ ( V A - VB) dp - ( n -~
?ZB) d p
=0
(17)
This is the general expression, valid for the two-phase equilibrium] Le., for the changes in potential caused by arbitrary changes of the state of equilibrium. The usual expression, which contains only the potentials T and p, is obtained by putting n~ = ?zB = 1. Equation 17 is then reduced to: -(SA - SB)dT (VA - VB)dp = 0 (1%)
+
the quantities involved now referring to unit amount of substance. Equation 18 may also be written:
index “Eq” referring to the equilibrium condition. Equation 18 is identical with the vapor pressure formula of Clapeyron and the William Thomson equation for the dependence of the melting point on pressure. It is brought into formal agreement with these equations by introducing T(SA - SB) = g, Le., the heat of evaporation or fusion. In a quite similar manner one derives from equation 17
(SA-
SB)
dT
+
- TLB) dp
(I~A
= 0
(20)
where the quantities are those contained in the unit of volume, and (%A - nB) dp - (VA - VB) dp = 0 (21) where the quantities are those associated with unit amount of entropy.
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J. N. BR6NSTED
The general two-phase equilibrium expression for systems containing two chemical components is deduced analogously to equation 17. The amounts SA,VA nlA , n2A are transferred from system I to system 11, and the amounts SB VB n 1 ~, %B are transferred in the opposite direction, the net process being a transport of: SA- SB V.4 - VB nlA - n l B n2A - 7 ~ from 2 ~ I to 11. On the basis of the work principle for infinitesimally diverging systems we put the transference work equal to zero. Hence,
(SB- SA)d T - (VB - VA) dp
+ ( n i ~-
niA)
dpi
+
(?ZZB
- nzA) dpz = 0 (22)
an equation which is symmetrical with respect to all quantities and POtentials involved. I n the special case of nZB= nzA= 1 the quantities involved will be those associated with unit amount of the second chemical component. The last term in equation 22 then disappears. Further simplification into the equilibrium expressions usually employed in physical chemistry, containing only two variable potentials, is obtained by putting one of the remaining potential differences equal to zero, Thus, if one wishes to determine the equilibrium a t constant temperature,ane puts d T = 0, obtaining
- (VB - VA) dp + (niB - 7 2 1 ~ )dpi = 0
(23)
or
while for constant pressure
or
From the general equations 22 to 26, thus deduced on the basis of the work principle, all the equations and qualitative laws concerning physicochemical equilibria are derivable by simple transformations with or without introduction of special equations of state for vapors and solutions. For example, the laws on the osmotic pressure, on vapor pressure lowering and freezing point depression, the influence of temperature and pressure on the solubility of solid substances, Konowalow's rule of the composition of liquids and saturated vapors, the laws of electromotive force, etc.
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707
The law on the freezing point depression, for instance, is obtained by introducing in equation 26 the following equations:
and the gas law expression
where s1 and sz are the differential entropy in the liquid mixture for 1 g. of the first and second chemical component, respectively, and q the melting heat appearing in the freezing point equation:
i.e., the differential heat of dissolution for 1 g. of the second component in the solid state a t the existing equilibrium. IV. THE HOMOGENEOUS EQUILIBRIUM
The law of “chemical mass action” in its usual formulation deals with concepts and unities which are of molecular-theoretical origin. Moreover it is valid only for systems of particular simplicity. The main law for homogeneous equilibria is therefore essentially different in character from the purely energetic laws with which we have been concerned in the previous sections. As a basis of the traditional mass action expression, valid a t constant temperature and pressure, there exists, however, an expression of a purely energetic character, which may be termed the “energetic mass action law.” In order to derive this general law we shall consider a reversible basic process, consisting in the transport of quantity between two homogeneous systems I and 11. While in the case of the processes considered in the previous sections the fraction transported had the same composition as the system from which it was taken, we are here dealing with the transport of an assemblage of quantities, the composition of which differs from that of the communicating systems, and which in general, therefore, can only be
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J. N. BRdNSTED
conducted reversibly if coupled to another working system, for instance, a work rservoir. It is essential to note that this transport a t the same time w being defined by the initial and final states of the quantities transported generally may be effected in a number of different ways, owing to the fact that the chemical systems between which the transport takes place are themselves describable in terms of a number of freely selected components. Let us suppose that a homogeneous system, in addition to entropy, volume, and some material substances which need not be specified, is constituted by a chemical system A, containing the components AI , A2 + . in the amounts V A ,~ V A ~. . . , and let us further suppose that the same homogeneous system may just as well be made up of the quantities mentioned above, if a B system, containing the components BI , Bz . . . . in , y~~ . . . . be substituted for the A system. Between the the amounts two systems A and B the stoichiometric relation
-
+
VA,AI
VAJZ
+ * V B ~ B+I vg,Bz +
* .
.
(27)
must then hold true, where no condition is imposed upon the figures v beyond that implied by this stoichiometric relation. It should be noticed in particular that the components chosen need not be “chemical individuals. ” If systems I and I1 are characterized by such an alternative possibility of constitution, the transference of system A from I to I1 will be chemically equivalent to the corresponding transference of system B. If entropy and volume do not participate in the transport, the total loss of work will only be of chemical nature and consequently, if the two systems differ only infinitesimally from each other, the potential in I, being dp higher than the potential in 11, will be expressible as: dpA1 -
~ A= A
-VA~
8Ag =
- V B ~ dpcg,
dpA2 -
V A ~
* * * *
= - 2 V A I dPA1
(28)
and
-
V B dlsg, ~
- .... =
- Z V B ~dpB1
(29)
for the transport of system A and system B, respectively. Now according to the work principle the loss of work associated with reversible processes depends only upon the initial and final states of the system involved. Hence: - ~ V A ]
d k 1 d-
2Vgl
dpB1 = 0
(30)
By integration or by direct application of the work principle to systems with final potential differences, one obtains
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709
I being a constant the value of which depends on the zero states chosen for the chemical potentials. By adequate choice of these the constant will be zero, and we may write:
+
- Z Y A ~ P A ~ Z Y B ~ P= B ~0
Putting Z V A ~ P= A ~PA
and Z V B , P B= ~ Pn
the last equation simplifies into -PA
+
PB =
0
(35)
Equations 30, 31, 32, and 35 are equivalent expressions for the energetic mass action law. It should be emphasized in particular that this law, as appears from the deduction, is not confined to changes a t constant temperature and pressure, but is of a quite general character. In order to derive the usual mass action law from equations 30 to 35 one should first introduce molecular unities, thus changing the figures Y into integers and scheme 27 into the ordinary chemical reaction scheme. Further-as a condition for the validity of the usual mass action law-the gas laws must be introduced, for instance in the form where PA, and zA,are corresponding values for the molar chemical potential and mole fraction of component AI in a system maintained a t constant temperature T and constant pressure. As appears from equation 36, the magnitude p ~ formally ~ ~ equals ~ the , potential of AI for X A ~= 1 a t the same temperature and pressure. The equation indicates how the potential of a molecular component depends upon its concentration under ideal conditions a t constant temperature and pressure. By summation over all the components involved equation 36, on application of equations 33 and 34, gives where
Introducing the energetic equilibrium condition 35 in this expression we finally obtain PA^
- PB,
=
RT In (K=)E~
(39)
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J. N. B R ~ ~ N S T E D
(K.)Q being the value of K , in the state of equilibrium. As appears from equation 39, ( K = ) Eis~constant under the given temperature and pressure conditions, which is an expression for the traditional mass action law. By this law evidently a structure definition is implied for the homogeneous mixture which has not been required in the case of the energetic expressions established previously. The chemical “composition” as given by the figures v in these expressions was defined from a purely energetic standpoint as the amounts of given components necessary for making up the system in question, and this definition was implicitly supposed not to lead to any kind of ambiguity. The possibility of the same system being constructed from different components A and B led to the energetic mam action law. The traditional mass action law, on the other hand, is based upon the possibility of a logical discrimination between these component systems within the mixture itself, and upon a detailed structure determination based upon this possibility. In such a case the two reciprocal systems A and B cannot under all conditions yield identical mixtures. Starting from either A or B there may be formed mixtures which alter their properties by time, indicating chemical instability. At an arbitrary stage of this transformation definite amounts of components from both systems will be required for the construction of the mixture, the intrinsic composition of which is defined exactly through these amounts. Supposing such a homogeneous change of state to take place, a new energetic variable, which is the degree of reaction, will come into the calculation. It is only when considering this intrinsic composition of the homogeneous system that the influence of temperature and pressure makes itself noticed. The energetic mass action law as represented by equations 30 to 35 obviously does not reflect such influence. The effect of temperature upon the equilibrium constant may be obtained by differentiation of equation 39 with respect to temperature. This gives the well-known equation
H being the heat function, while differentiation with respect to pressure gives VA o
The lack of symmetry displayed by these expressions is due to the unsymmetrical manner in which T and p enter in the gas law equation (equation 36). Equations equivalent with these, but containing entropy and volume of
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DERIVATION OF EQUILIBRIUM CONDITIONS
the reactants in the equilibrium mixture itself, are obtained by introducing the new variable, the degree of reaction. We shall write for p~ and pg , given by equations 33 and 34,
and
whence
+ dpB = (SA- SB)d T
f?) ]da:
- (VA - VB) dp - [("e -) 6a
T.9
a:
(44)
T.p
or by introduction of equation 36
Now supposing the system to be in a state of equilibrium, Le., -dpA dpB = 0, the influence of temperature on this state is calculated from equation 45 by introducing dp = 0, and
+
where the first term on the right-hand side is zero. One then obtains
In a similar manner one obtains a t constant temperature
That equation 47 is identical with equation 40 follows from the gas law expression (expression 361, since this by differentiation with respect to temperature gives -SAI =
-sA , ( ~ )-k
In
$A,
(49)
That equation 48 is identical with equation 41 is seen in a similar manner, equation 36 on differentiation with respect to pressure giving
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J. N. BRE~NBTED
As seen by comparing equations 43 and 44, the quantities are defined by the relations
SA^ and
VA~
and
(2) a,T
VAI
thus being the symbols for entropy and volume in mixtures a t constant a. On the other hand S A ~ ,and E ~v A I , E q given by
and
represent the differential quantities a t equilibrium. V. SUMMARY
1. The assumption of traditional thermodynamics that heat can be produced or consumed in reversible processes 'is caused by an incomplete analysis of the basic phenomena associated with such processes. 2. The fundamental law for all reversible processes is the work principle 26A = 0, indicating that the total work expended or lost in a reversible process is zero. 3. Since the reversible process is a change through states of equilibrium, the work principle, valid for such processes, can be made the basis for the deduction of all expressions of physicochemical equilibria. 4. The ditrerential transport of a fraction of a homogeneous system from a certain state into another deviating infinitesimally from the first is a reversible process. The work associated with it is therefore zero. This is called the work theorem f o r infinitesimally diverging systems. 5. On the basis thus given a detailed deduction of some important equilibrium equations in physical chemistry has been presented. REFERENCES (1) BJERRUM, N.: Z. physik. Chem. 104, 411 (1923). (2) BR$NSTED, J. N.: Aflinity Studies 111, p. 2. Copenhagen (1908). . N.: , Kgl. Danske Videnskab. Selskab Math.-fys. Medd. 16, No. 4 (3) B R ~ K S T EJD (1937); 16, No. 10 (1939); Phil. Mag. (1939). (4) CALLENDAR, H. L.: Proc. Roy. SOC. (London) 13, 153 (1910-11). (5) CARNOT,S.: Reflexions sur la puissance motrice du f e u (1824). ( 6 ) GIBBS, W.: Collected Works, Vol. I, p. 88 (1928). (7) ZEUNER, C. : Grundruge der mechanischen Wdrmetheorie (1877).
