DERIVATION of the CRITERIA for IRREVERSIBLE CHANGES, EQUILIBRIUM, and STABILITY BASED o n the UNCOMPENSATED HEAT PIERRE VAN RYSSELBERGHE Stanford University, Stanford University, California
For many of the systems dealt with in ordinary physical chemistry dWd is simply done The Two L~~~ of ~ h ~ and the ~ un.~ ~ ~ the work ~ ~ against ~ the exi ternal pressure and we have compensated Heat.-The purpose of the present paper is to call to the attention of teachers of physical chemd W = PdV (2) istry and of chemical thermodynamics *at a simple, in d is the increase of the of the system convincing, and entirely general way of establisF!?i! the various criteria for irreversible changes, equ~hb- corresponding to the infinitesimal change of state, The second law of is written asfollows, rium, and stability can be based upon the concept of T d S = dQ + dQ' with dQ' 2 0 uncmfiensated heat, which originated with Clausius (3) and was systematically applied to chemical th-0in which T is the absolute temperature, dS the increase the particular of the entropy of the system, dQ the heat received by dynamics by De Dander. In fact, criteria reduce to a single general one which states that the system, ~ Q the I uncompensated h a t corresponding the uncompensated heat can never be negative in a to the infinitesimal change of state. This uncompennatural process. A complete exposition of the theory of chemical thermodynamics based upon the use of uncompensated heat, degree of advancement of the reaction, and affinity was recently published by De Donder and the present author (1). In this paper we shall consider closed systems, i. e., systems whose total masses stay constant in the course of time. No quantities of matter are added from, or removed to the surroundmgs. Changes of composition occur only through chemical reactions (homogeneous or heterogeneous, changes of state). Moreover, thermal and mechanical equilibrium prevail throughout the system. I In other words, temperature and pressure are uniform. We shall assume that all thermodynamic funcI tions, including entropy and free energy, are defined 5 for all instantaneous states of the system, regardless F~oune 1 of the degree of irreversibility of the transformation undergone by the system. These precautions being taken, the two laws of thermodynamics can be forsated heat dQ' is positive for all real irreversible changes. mulated as follows. The first law of thermodynamics is written as fol- It vanishes when the change is reversible. A justificalows, for an infinitesimal change in the state of the tion for the name uncompensated heat given to dQ' is the system, following. Consider a finite irreversible change AB. We have, d E = dQ - d W from (3), in which dE represents the increase of the energy of the f n s ~ =d Qns ~ Q'na with Q'rs > 0 (4) system during the lapse of time dt; dQ is the amount of heat (thermal energy) received by the system; dW If the system can be made to come back from B to A is the work done by the system on its surroundings. through the same succession of states we have I.
INTRODUCTION
'
~
I)
+
476
f
s n ~ =d QBA ~
+ Q'sa with Q ' m
>0
(5)
Adding (4) and (5) we get
f
+ Q'ABAwith Q'rar > 0
A ~ A T ~= SQABA
W = Q = J T d S - Q' (6)
but the integral f miTds is since the areas f ,,TdS and , f ~ a T d S(see Figure 1) are equaljn absolute value but have opposite signs. From (6) we xet Q'nsr
=
-Qmr
>0
thermal cycle one easily finds from the foregoing equations
(7)
is the total heat evolved along ABA. but -Q,,, This heat evolved is thus seen to be equal to the total uncompensated heat. On account of the irreversibility
(12)
For a reversible cycle consisting of the same succes,ion of states as the irreversible cycle under considera. we have W , = Q. = f T d S (13) since Q' is now zero. Subtracting (12) from (13) we get U,-W=Q,-Q=Q'>o
(14)
The work done by a reversible cycle is larger than that done by an irreversible cycle consisting of the same succession of states, the difference being equal to the total uncompensated heat of the irreversible cycle.
TI
XI.
CRITERIA FOR IRREVERSIBLE CHANGES
Returning to (3) we have
ao
~ Q =I T ~ S d~
This formula can be regarded as the fundamental criterion of irreversible change and also, when the equality sign holds, that of reversible change and equilibrium. The opposite inequality
I
T ~S d~
P
0
(17)
Since T is positive, we also have
An example of an irreversible transformation to which dS > o (18) formulas (4) to (7) apply is the expansion of a gas followed by its compression in a cylinder provided with a c hi^ is clausiusPcelebrated criterion, which he applied piston exerting friction on the walls of the cylinder. to the Universe as a whole, ( u ~ ~i ~ ~ der wdt ~ If the piston is perfectly frictionless, the transformation ist ~i~ ~~~~~~i~ der wdt strebt einem can be regarded as reversible and the equalities (8) zu,~) apply. The Pressure is, of course, assumed to be 2. ~f the system is made to keep its entropy conuniform throughout the volume of the gas. stant we havedS = 0 and Another important property of Q' is its connection with the work done by irreversible cycles. Let dQ' = -dQ > 0 (19) us first consider an isothermal cycle. We have (see The criterion of irreversible change is that heat be Figure 2) liberated. 3. At the absolute zero we would always have ~f m r d = ~ QABA Q'ABA = 0 (') TdS = 0 and hence But, according to (1)
+
dQ' = -dQ
Q = ~ ~ E + w = o + w = w
(lo)
The integral of dE, like that of dS, is equal to zero, since both E and S are functions of state. From (9) and (10) we deduce
w=Q'0
as in (19). 4. Combining (1) and (3) we get
- d W - dQ'
(20)
+ TdS - d W > 0
(21)
dE = TdS
or also dQ' = -dE
For S constant anddWequal to zero we have dQ' = - d ~>
o
(22)
g
i
The criterion of irreversible change is that the energy E should decrease. If dW = PdV, (22) holds when S and V are constant. For E constant and dW = 0 (for instance, V constant when dW = Pd V) we have dQ'
-
TdS 2 0 or dS 2 0
(23)
in the function A is the reversible work, which is precisely equal to -dA. The irreversible work is smaller than -dA by an amount equal to the uncompensated heat. These properties are a justification for the name maximum work given to the function A . For A and T constant we get from (36)
as in (17) and (18). For E and S constant, dQ'= - d W 2 O
dQ1= - d W 2 0 (24)
In particular, for dW = Pd V,
In particular, for dW = Pd V,
dQ'= - P d V > O a r d V < O
o
~ Q I= - P ~ V 2
(40)
(25)
The pressure P being always positive, we also have
(41)
For A constant and dW = 0 (for instance V constant when dW = Pd V)we have dQ'= - S d T > O o r d T < O
5. Introducing the heat content' H=E+PV
(27)
since entropy can always be assumed to be positive. For T constant and dW = 0 we have dQ' = -dA 2 0
we deduce from (20) dH = TdS
-dW+d(Pv
-dQ'
(28)
We shall only consider here the case dW = Pd V , We get from (28)
+ TdS + VdP 2 0
dQ' = -dH
F=E-TS+PV
we get from (20) dF = -SdT - dW
dQ' = -dF
For H a n d P constant
+ d(PV) - dQ'
(45)
- SdT
- (dW - PdV) f VdP
>0
(46)
For T and P constant
dQ' = TdS 2 OordS 2 0
dQ' = -dF - (dW - PdV) S 0
(31)
dW,
dQ' = VdP 2 0
(32)
dP
dW
>0
(33)
we get from (20)
- dW
- dQ'
- PdV
=
-dF
If the change is irreversihle dQ'
-dF
+
(36)
(49)
(37)
o
(50)
Instead of (47), for T and P constant,
2O
(51)
Instead of (48), for reversihle changes, dQ' = -dF = 0
If the change is reversible dQ' = 0 and dW, = -dA
> 0 and
- dQ'
0
(86)
or also in terms of the reaction velocity v =
dt dt (87)
The general criterion of equilibrium is then A =0
and that of stability, a t T, P constant for instance,
SUMMARY
1. The two laws of thermodynamics are recalled, the second one being expressed in terms of the uncompensated heat. The significance of this quantity and its connection with the work of irreversible cycles is discussed. 2. All the particular criteria for irreversible changes are derived from the fundamental one which states that in no natural process can the uncompensated heat be negative. 3. The corres~ondinecriteria of eauilibrium and stability are established.- Special attention is paid to Gibbs' criteria. 4. The chemical variable is defined and used in establishing new and convenient f o m s for the criteria of equilibrium and stability. The general definition of affinity is recalled.
LITERATURE CITED E. A., cf. DONNAN, F. G. AND A. HAAS,editors, (1) DE DONDER. TH.,AND P. VANR Y S S E L B E R E ~ E , , " T ~ ~(4) ~ OMILNE, ~Yn a m ~theory of affinity: A book of pnnclples, Stan"A commentary on the scientific writings of J. W. Gibbs," ford University Press, Stanford University, California. Yale University Press, New Haven, Connecticut, 1936. E. A,, "Modern thermodynamics by the 1936. (5) GUGGENHEIM, methods of J. W. Gibbs," Methuen and Co.. Ltd., (2) LEWIS, G. N. AND M. RANDALL, "Thymadynamics and the free energy of chemlcal substances. McGraw-Hill Book London, 1933. (6) VANRYSSELBERGHE, P., Chem. Reviews, 16, 29, 37 (1935). Co., New Pork City, 1923. (3) GIBBS,J. W., "The collected works of J. Willard Gibbs. (7) ZEMANSKY. M. W., "Heat and thermodynamics," McGrawHill Book Co., New York City, 1937. Volume I, Thermodynamics," Longmans, Green and Co., New York City. 1928.
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