Page 144 - Physical Chemistry

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curves can be calculated using Eqs. (4.60) and (4.63), and u and s of vaporization Section 4.6
of water. Chemical Potentials
and Material Equilibrium
Calculation of G and A
From G H TS and the equation that follows (2.48), we have G G G
2
1
H (TS) H T S S T S T. However, thermodynamics does not
1
1
define entropies but only gives entropy changes. Thus S is undefined in the expression
1
for G. Therefore G is undefined unless T 0. For an isothermal process, the de-
finition G H TS gives [Eq. (4.20)]
¢G ¢H T ¢S const. T (4.64)
Thus G is defined for an isothermal process. To calculate G for an isothermal pro-
cess, we first calculate H and S (Secs. 2.9, 3.4, and 4.5) and then use (4.64).
Alternatively, G for an isothermal process that does not involve an irreversible com-
position change can be found from (
G/
P) V [Eq. (4.51)] as
T
P 2
¢G V dP const. T (4.65)
P 1
A special case is G for a reversible process at constant T and P in a system with
P-V work only. Here, H q and S q/T. Equation (4.64) gives
¢G 0 rev. proc. at const. T and P; P-V work only (4.66)
An important example is a reversible phase change. For example, G 0 for melting
ice or freezing water at 0°C and 1 atm (but G 0 for the freezing of supercooled
water at 10°C and 1 atm). Equation (4.66) is no surprise, since the equilibrium con-
dition for a closed system (P-V work only) held at constant T and P is the minimiza-
tion of G (dG 0).
As with G, we are interested in A only for processes with T 0, since A is
2
undefined if T changes. We use A U T S or A PdV to find A for
1
an isothermal process.

4.6 CHEMICAL POTENTIALS AND MATERIAL EQUILIBRIUM
Figure 4.5
The basic equation dU TdS PdV and the related equations (4.34) to (4.36) for
dH, dA, and dG do not apply when the composition is changing due to interchange of Specific entropy of H O(g) versus
2
matter with the surroundings or to irreversible chemical reaction or irreversible inter- T and versus P.
phase transport of matter within the system. We now develop equations that hold dur-
ing such processes.

The Gibbs Equations for Nonequilibrium Systems
Consider a one-phase system that is in thermal and mechanical equilibrium but not
necessarily in material equilibrium. Since thermal and mechanical equilibrium exist, T
and P have well-defined values and the system’s thermodynamic state is defined by the
values of T, P, n , n , . . . , n , where the n ’s (i 1, 2, . . . , k) are the mole numbers of
1 2 k i
the k components of the one-phase system. The state functions U, H, A, and G can each
be expressed as functions of T, P, and the n ’s.
i
At any instant during a chemical process in the system, the Gibbs energy is
G G1T, P, n , . . . , n 2 (4.67)
k
1
Let T, P, and the n ’s change by the infinitesimal amounts dT, dP, dn , . . . , dn as the
i 1 k
result of an irreversible chemical reaction or irreversible transport of matter into the
system. We want dG for this infinitesimal process. Since G is a state function, we shall