REVIEW PROBLEMS
4.3 THE GIBBS AND HELMHOLTZ ENERGIES
We now use (4.8) to deduce conditions for material equilibrium in terms of state func-tions of the system. We first examine material equilibrium in a system held at constant T and V. Here dV 0 and dT 0 throughout the irreversible approach to equilibrium.
The inequality (4.8) involves dS and dV, since dw P dV for P-V work only. To introduce dT into (4.8), we add and subtract S dT on the right. Note that S dT has the dimensions of entropy times temperature, the same dimensions as the term T dS that appears in (4.8), so we are allowed to add and subtract S dT. We have
(4.9) The differential relation d(uy) u dy y du [Eq. (1.28)] gives d(TS) T dS S dT, and Eq. (4.9) becomes
(4.10) The relation d(u y) du dy [Eq. (1.28)] gives dU d(TS) d(U TS), and (4.10) becomes
(4.11) If the system can do only P-V work, then dw P dV (we use dwrevsince we are as-suming mechanical equilibrium). We have
(4.12) At constant T and V, we have dT 0 dV and (4.12) becomes
(4.13) where the equality sign holds at material equilibrium.
Therefore, for a closed system held at constant T and V, the state function U TS continually decreases during the spontaneous, irreversible processes of chemical reaction and matter transport between phases until material equilibrium is reached. At material equilibrium, d(U TS) equals 0, and U TS has reached a minimum. Any spontaneous change at constant T and V away from equilibrium (in either direction) would mean an increase in U TS, which, working back through the preceding equa-tions from (4.13) to (4.3), would mean a decrease in Suniv Ssyst Ssurr. This decrease would violate the second law. The approach to and achievement of material equilib-rium is a consequence of the second law.
The condition for material equilibrium in a closed system capable of doing only P-V work and held at constant T and V is minimization of the system’s state function U TS. This state function is called the Helmholtz free energy, the Helmholtz energy, the Helmholtz function, or the work function and is symbolized by A:
(4.14)*
Now consider material equilibrium for constant T and P conditions, dP 0, dT 0. To introduce dP and dT into (4.8) with dw P dV, we add and subtract S dT and V dP:
(4.15) d1H TS2 S dT V dP
d1U PV TS2 S dT V dP
dU d1TS2 S dT d1PV2 V dP
dU T dS S dT S dT P dV V dP V dP A⬅ U TS
const. T and V, closed syst. in
d1U TS2 0 therm. and mech. equilib., P-V work only d1U TS2 S dT P dV
d1U TS2 S dT dw dU d1TS2 S dT dw dU T dS S dT S dT dw
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Section 4.3 The Gibbs and Helmholtz Energies 113
Const. T, P
Time Equilibrium
reached G
Figure 4.2
For a closed system with P-V work only, the Gibbs energy is
minimized if equilibrium is reached under conditions of constant T and P.
Therefore, for a material change at constant T and P in a closed system in mechanical and thermal equilibrium and capable of doing only P-V work, we have
(4.16) where the equality sign holds at material equilibrium.
Thus, the state function H TS continually decreases during material changes at constant T and P until equilibrium is reached. The condition for material equilibrium at constant T and P in a closed system doing P-V work only is minimization of the sys-tem’s state function H TS. This state function is called the Gibbs function, the Gibbs energy, or the Gibbs free energy and is symbolized by G:
(4.17)*
G decreases during the approach to equilibrium at constant T and P, reaching a mini-mum at equilibrium (Fig. 4.2). As G of the system decreases at constant T and P, Suniv increases [see Eq. (4.21)]. Since U, V, and S are extensive, G is extensive.
Both A and G have units of energy (J or cal). However, they are not energies in the sense of being conserved. Gsyst Gsurrneed not be constant in a process, nor need Asyst Asurrremain constant. Note that A and G are defined for any system to which meaning-ful values of U, T, S, P, V can be assigned, not just for systems held at constant T and V or constant T and P.
Summarizing, we have shown that:
In a closed system capable of doing only P-V work, the constant-T-and-V material-equilibrium condition is the minimization of the Helmholtz energy A, and the constant-T-and-P material-equilibrium condition is the minimization of the Gibbs energy G:
(4.18)*
(4.19)*
where dG is the infinitesimal change in G due to an infinitesimal amount of chemical reaction or phase change at constant T and P.
EXAMPLE 4.1
G and A for a phase changeCalculate G and A for the vaporization of 1.00 mol of H2O at 1.00 atm and 100°C. Use data from Prob. 2.49.
We have G⬅ H TS. For this process, T is constant and G G2 G1 H2 TS2 (H1 TS1) H T S:
(4.20) The process is reversible and isothermal, so dS dq/T and S q/T [Eq. (3.24)].
Since P is constant and only P-V work is done, we have H qP q. Therefore (4.20) gives G q T(q/T ) 0. The result G 0 makes sense because a reversible (equilibrium) process in a system at constant T and P has dG 0 [Eq. (4.19)].
From A ⬅ U TS, we get A U T S at constant T. Use of U q w and S q/T gives A q w q w. The work is reversible P-V work at constant pressure, so w 兰21P dV P V. From the 100°C density in Prob. 2.49, the molar volume of H2O(l) at 100°C is 18.8 cm3/mol. We can
¢G ¢H T ¢S const. T dG 0 at equilib., const. T, P dA 0 at equilib., const. T, V
G⬅ H TS ⬅ U PV TS d1H TS2 0 const. T, P
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Chapter 4 Material Equilibrium
114
accurately estimate Vmof the gas from the ideal-gas law: Vm RT/P 30.6 103cm3/mol. Therefore V 30.6 103cm3and
Exercise
Find G and A for the freezing of 1.00 mol of H2O at 0°C and 1 atm. Use data from Prob. 2.49. (Answer: 0, 0.165J.)
What is the relation between the minimization-of-G equilibrium condition at con-stant T and P and the maximization-of-Sunivequilibrium condition? Consider a system in mechanical and thermal equilibrium undergoing an irreversible chemical reaction or phase change at constant T and P. Since the surroundings undergo a reversible isothermal process, Ssurr qsurr/T qsyst/T. Since P is constant, qsyst Hsystand
Ssurr Hsyst/T. We have Suniv Ssurr Ssystand
(4.21) where (4.20) was used. The decrease in Gsystas the system proceeds to equilibrium at constant T and P corresponds to a proportional increase in Suniv. The occurrence of a reaction is favored by having Ssyst positive and by having Ssurr positive. Having
Hsystnegative (an exothermic reaction) favors the reaction’s occurrence because the heat transferred to the surroundings increases the entropy of the surroundings (Ssurr
Hsyst/T ).
The names “work function” and “Gibbs free energy” arise as follows. Let us drop the restriction that only P-V work be performed. From (4.11) we have for a closed system in thermal and mechanical equilibrium that dA S dT dw. For a constant-temperature process in such a system, dA dw. For a finite isothermal process, A w.
Our convention is that w is the work done on the system. The work wbydone by the sys-tem on its surroundings is wby w, and A wby for an isothermal process.
Multiplication of an inequality by 1 reverses the direction of the inequality; therefore (4.22) The term “work function” (Arbeitsfunktion) for A arises from (4.22). The work done by the system in an isothermal process is less than or equal to the negative of the change in the state function A. The equality sign in (4.22) holds for a reversible process. Moreover, A is a fixed quantity for a given change of state. Hence the max-imum work output by a closed system for an isothermal process between two given states is obtained when the process is carried out reversibly.
Note that the work wbydone by a system can be greater than or less than U, the internal energy decrease of the system. For any process in a closed system, wby
U q. The heat q that flows into the system is the source of energy that allows wby
to differ from U. Recall the Carnot cycle, where U 0 and wby 0.
Now consider G. From G A PV, we have dG dA P dV V dP, and use of (4.11) for dA gives dG S dT dw P dV V dP for a closed system in ther-mal and mechanical equilibrium. For a process at constant T and P in such a system (4.23) Let us divide the work into P-V work and non-P-V work wnon-P-V. (The most common kind of wnon-P-Vis electrical work.) If the P-V work is done in a mechanically reversible
dG dw P dV const. T and P, closed syst.
wby ¢A const. T, closed syst.
¢Suniv ¢Gsyst>T closed syst., const. T and P, P-V work only
¢Suniv ¢Hsyst>T ¢Ssyst 1¢Hsyst T ¢Ssyst2>T ¢Gsyst>T w 130.6 103 cm3 atm2 18.314 J2>182.06 cm3 atm2 3.10 kJ ¢A
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manner, then dw P dV dwnon-P-V; Eq. (4.23) becomes dG dwnon-P-Vor G wnon-P-V wby,non-P-V. Therefore
(4.24) For a reversible change, the equality sign holds and wby,non-P-V G. In many cases (for example, a battery, a living organism), the P-V expansion work is not useful work, but wby,non-P-Vis the useful work output. The quantity G equals the maximum possi-ble nonexpansion work output wby,non-P-V done by a system in a constant-T-and-P process. Hence the term “free energy.” (Of course, for a system with P-V work only, dwby,non-P-V 0 and dG 0 for a reversible, isothermal, isobaric process.) Examples of nonexpansion work in biological systems are the work of contracting muscles and of transmitting nerve impulses (Sec. 13.15).
Summary
The maximization of Suniv leads to the following equilibrium conditions. When a closed system capable of only P-V work is held at constant T and V, the condition for material equilibrium (meaning phase equilibrium and reaction equilibrium) is that the Helmholtz function A (defined by A⬅ U TS) is minimized. When such a system is held at constant T and P, the material-equilibrium condition is the minimization of the Gibbs function G⬅ H TS.