12 Lecture 12: Stability of equilibrium states


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1 12. LECTURE 12: STABILITY OF EQUILIBRIUM STATES Lecture 12: Stability of equilibrium states Summary We can derive the universal stability (and evolution) criterion δ 2 S < 0 (> 0) that is independent of the environmental constraints (say, isothermal, constant volume or not, etc.) (X, Y )/ (x, y) > 0. In equilibrium changes occur in the direction to discourage further changes (to avoid runaway processes) (Le ChatelierBraun principle). Gibbs free energy is a convenient thermodynamic potential to study the systems under T, P constant. G µ i N i, and the direction of a chemical reaction can be studied by G due to reaction. Key words universal stability criterion, universal evolution criterion, positive definite quadratic form, Le Chatelier principle, Le ChatelierBraun principle, GibbsDuhem relation. What you should be able to do To derive the universal stability criterion. To mention some of the crucial conclusions due to the stability criterion (say, C X > 0). To derive the GibbsDuhem relation. Let us perform a few review exercises. What is the sign of ( ) S F L (12.1) for a rubber band? Intuition tells us that it must be positive (To increase F while keeping L, we must invigorate the motion of chains, so we must raise the temperature, resulting in the increase of entropy). Let us check this, starting with our empirical result ( ) T > 0. (12.2) L What you should do first is to rewrite the partial derivative in terms of Jaco S
2 110 bians: ( ) S F L (12.2) is ( ) T L S so we should keep (L, S) and insert (T, S): ( S F ) L (S, L) (F, L) (S, L) (T, S) (T, S) (F, L) (S, L) (F, L). (12.3) (T, S) (L, S), (12.4) (S, L) (T, S) (L, S) (T, S) > 0. (12.5) We have used a Maxwell s relation. One more for a gas: How about the sign of ( ) S? (12.6) V To increase P under constant V, we have to raise the temperature, resulting in the increase of entropy, so the sign must be positive. The cleverest approach may be ( ) S V (S, V ) (P, V ) (S, V ) (T, P ) (T, P ) (P, V ) (S, V ) / ( ) V. (12.7) (T, P ) T P Therefore, (12.6) and ( V/ T ) P, which is positive for any gas, 84 have the same sign, but to understand this we need the stability result (S, V ) (T, P ) > 0. (12.8) Today s main dish is the universal stability criterion. Clausius told us that if a spontaneous change occurs in an isolated system, S 0. (12.9) 84 This inequality is not a thermodynamic inequality. That is, it is not due to the principles of thermodynamics and materialsfree general conditions. In contrast, the inequality (12.8) is thermodynamic (universally true). It is important to distinguish these different kinds of inequalities in thermodynamics.
3 12. LECTURE 12: STABILITY OF EQUILIBRIUM STATES 111 We use the standard trick to study a nonisolated system S as a small part of a huge isolated system (Fig. 12.1) whose intensive variables are kept constant, but their conjugate extensive variables may be exchanged between S and its surrounding reservoir freely. T Pe x e e S Figure 12.1: S is a part of a huge isolated system whose intensive parameters, P e, x e, etc., are kept constant. Their conjugate extensive quantities S (or heat), V, X, etc., can be freely exchanged between the system S and the rest. If something spontaneous can happen, the total entropy must increase. In the system something irreversible might have happened, so we cannot compute S directly with the aid of imported quantities E, V, N, etc. However, for the reservoir, since we assume ti is always in equilibrium, we can write its entropy change as S res 1 E P e V + x e X + µ e N. (12.10) Here, E, etc., are the quantities seen from the system S (+ for importing), so E, V, etc., are the imported quantities to the reservoir. That is why the signs in (12.10) are different from the usual Gibbs relation. Thus, the total entropy change is S + S res : S 1 E P e V + x e X + µ e N 0. (12.11) This is essentially Clausius inequality. If the equilibrium state is stable, then S 1 E P e V + x e X + µ e N < 0. (12.12) Now, let us look at S in more detail. If the change is very small, we can Taylor expand S into a power series of δe E, δv V, etc. (here is replaced by δ to make it clear that all changes are small): S δs + δ 2 S +. (12.13)
4 112 The first order term reads δs 1 δe + P e δv x e δx µ e δn, (12.14) because the derivatives are computed around the equilibrium state. Combining this expression, (12.12) and (12.13), we conclude that the stability condition of the equilibrium state is δ 2 S < 0 (12.15) irrespective of the constraints imposed on the system S (that is, independent of whether some extensive quantities are allowed to be exchanged or not). Thus, this is the universal stability condition for the equilibrium state. (12.12) may be rearranged as E > S P e V + x e X + µ e N. (12.16) Now, restricting the variations to be small ones, we can Taylor expand E just as we did for S. You should immediately realize that a very similar logic as above can give us another, but equivalent universal stability criterion δ 2 E > 0. (12.17) Notice that (12.15) was concluded for isolated systems before as the max entropy principle, but here this does not imply max entropy; irrespective of S max or not, δ 2 S < 0 is the stability condition. Let us study the consequences of the stability criterion (12.17): a general expression is 2 E δx i δx j > 0. (12.18) X i X j i,j This is a positive definite quadratic form, so we can express it as ( T ) ( T ) ( T ) S V,N V S,N N S,V (ds, dv, dn) ( ) ( ) ( ) ds S) V,N V) S,N N) S,V dv > 0. (12.19) dn ( µ S V,N ( µ V S,N Let us assume N is constant. The sign of (12.19) must always be positive irrespective of the choice of ds and dv. Therefore, all the diagonal terms must be positive: ( ) T 0 < T, (12.20) S C V V ( µ N S,V
5 12. LECTURE 12: STABILITY OF EQUILIBRIUM STATES 113 and in terms of the adiabatic compressibility κ S ( V/ ) S /V ( ) 0 < 1/V κ S. (12.21) V You must imagine what happens if these signs are flipped. The diagonal inequalities are called LeChatelier s principle. 85 We can verbally state the consequence as follows: In equilibrium changes occur in the direction to discourage further changes (to avoid runaway processes). For example, if S is injected (heat is injected) into the system, its temperature goes up, which usually discourages further injection of heat. In the case of compressibility, decrease of the volume of the system increase pressure resisting further squishing. Thus, no runaway phenomenon is realized in the world we live in. A necessary and sufficient condition for (12.19) is the positivity of all the principal minors 86 of the matrix in (12.19). Therefore, in particular, S (T, P ) (S, V ) > 0. (12.22) Generally, (X, Y ) > 0, (12.23) (x, y) where (x, X) and (y, Y ) are conjugate pairs. This is perhaps the last formula you should remember when you use the Jacobian technique. The positivity of the diagonal terms may be used to derive the following inequality. Since dx ( ) x dx + ( ) x dy + X Y Y X ( ) ( ) ( ) ( ) x x x Y +, (12.24) X y X Y Y X X y ( ) ( ) x x (Y, y) (X, x) (X, Y ) + (12.25) X Y Y X (X, x) (X, Y ) (X, y) ( ) ( ) 2 ( ) x x Y (12.26) X Y y Y 85 Henry Louis Le Chatelier ( ): Compt. rend., 99, 786 (1884). 86 You sample the same row and column numbers (say, 1, 3, 7 and 8th columns and 1, 3, 7 and 8th rows from the original matrix and make a determinant det(a ij ), where i, j {1, 3, 7, 8}). Such determinants are called principal minors. X X
6 114 Here, a Maxwell s relation has been used. That is, ( ) ( ) x x < X X y Y. (12.27) This implies that the indirect change occurs in the direction to reduce the effect of the perturbation (Le ChatelierBraun s principle). 87 A typical example is C V C P : larger specific heat implies that it is harder to warm up, that is, the system becomes more stable against heat injection. If we wish to study a closed system under constant T and P, we should further change the independent variables from T, V, N, to T, P, N,. The necessary Legendre transformation is (notice that the conjugate quantity of V is P, not P ) A + P V E T S + P V G, (12.28) which is called the Gibbs free energy. We have dg SdT + V dp + µdn. (12.29) Stability condition (12.16) can read under constant temperature as A > P e V + δ 2 A > 0, (12.30) and under constant T and P as G > + δ 2 G > 0. (12.31) Let us pursue a consequence of the fourth law of thermodynamics ( thermodynamic observables are either extensive or intensive). If we increase the amount of all materials in the system from N i to (1 + λ)n i, then all the extensive quantities are 87 Karl Ferdinand Braun ( ): Z. physik. Chen., 1, 269 (1887), Ann. Physik, 33, 337 (1888) [the inventor of the cathoderay tube, the discoverer of principle of semiconductor diode, shared the Nobel prize with Marconi for wireless technology]. The history of this principle can be found in J. de Heer, The principle of le Chatelier and Braun, J. Chem. Educ., 34, 375 (1957). The form stated here is due to Ehrenfest.
7 12. LECTURE 12: STABILITY OF EQUILIBRIUM STATES 115 multiplied by 1 + λ, and all the intensive quantities remain unaltered. Therefore, (8.23) now reads d[(1 + λ)e] T d[(1 + λ)s] P d[(1 + λ)v ] + i µ i d[(1 + λ)n i ] +, (12.32) or Edλ T Sdλ P V dλ + i µ i N i dλ +. (12.33) That is, we have E T S P V + i µ i N i +. (12.34) Combining the total differential of this formula and the Gibbs relation (8.23), we arrive at SdT V dp + N i dµ i + 0. (12.35) This important relation is called the GibbsDuhem relation. By definition we have A P V + i µ i N i +, (12.36) G i µ i N i +, (12.37) but we can derive these relations directly using the fourth law, mimicking the argument for E starting with (12.32). The last formula in the exercise implies that for a simple pure fluid µ G/N. (12.38) Perhaps the most interesting application of stability is in chemical reactions. Here, an elementary exposition of equilibrium chemical reactions is given. Without chemical reactions no atomism was possible (see Chapter 1). Furthermore, the idea of detailed balance originated from chemical reactions. Also to understand chemical reactions is becoming increasingly important even for physicists because we living organisms are chemical machines. Here, we do not discuss details of chemical reaction thermodynamics (see Section
8 of Invitation for a standard introduction). What we must recognize is that in every chemical reaction, the reactant system consisting of several chemicals and the product system again consisting of some chemicals are in equilibrium usually under constant T and P in a closed system. Hence, the equilibrium condition is that the Gibbs free energy reaches its minimum. Let G G P G R, where P denotes the product system and R the reaction system. Thanks to (12.37) if we know chemical potentials, we can write G R µ R i N R i, G P µ P i N P i. (12.39) The chemical potential can be measured in principle by the method illustrated in Fig semipermeable membrane Figure 12.2: We prepare a injector with a semipermeable membrane selectively allowing only the ith chemical to path through. We apply some force to inject dn i of molecules, and measure the required work µ i dn i. If G < 0, then the reaction proceeds. Notice that δ G SδT + V δp, (12.40) where S Q R /T with Q R being the heat of reaction, and V is the change of volume due to the reaction. Suppose the reaction increases the total volume V > 0. Then, ( ) G V > 0. (12.41) T That is, increasing P increases G, the reaction proceed backward. Since the reaction promotes increase of the total volume, the change due to increase of P occurs to reduce the effect (an example of Le Chatelier s principle). Similarly, if the reaction is exothermic, Q R < 0 ( exothermic implies the system loses energy; pay attention to the sign convention), so ( ) G Q R /T < 0; (12.42) T P increasing T reduces G. That is, the reaction is encouraged to proceed forward to reduce the temperature decrease (another example of Le Chatelier s principle).
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