Page 136 - Physical Chemistry

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Collecting the expressions for dU, dH, dA, and dG, we have Section 4.4
Thermodynamic Relations for a
dU T dS P dV System in Equilibrium
(4.33)*
dH T dS V dP closed syst., rev. proc.,
u (4.34)
dA S dT P dV P-V work only
(4.35)
dG S dT V dP
(4.36)*
These are the Gibbs equations. The first can be written down from the first law dU
dq dw and knowledge of the expressions for dw rev and dq . The other three can be
rev
quickly derived from the first by use of the definitions of H, A, and G. Thus they need
not be memorized. The expression for dG is used so often, however, that it saves time
to memorize it.
The Gibbs equation dU TdS PdV implies that U is being considered a func-
tion of the variables S and V. From U U(S, V), we have [Eq. (1.30)]
0U 0U
dU a b dS a b dV
0S V 0V S
Since dS and dV are arbitrary and independent of each other, comparison of this equa-
tion with dU TdS PdV gives

0U 0U
a b T, a b P (4.37)
0S V 0V S
A quick way to get these two equations is to first put dV 0 in dU TdS PdV to
give (
U/
S) T and then put dS 0 in dU TdS PdV to give (
U/
V) P.
V
S
[Note from the first equation in (4.37) that an increase in internal energy at constant vol-
ume will always increase the entropy.] The other three Gibbs equations (4.34) to (4.36)
give in a similar manner (
H/
S) T,(
H/
P) V,(
A/
T) S,(
A/
V) P,
P
S
V
T
and
0G 0G
a b S, a b V (4.38)
0T P 0P T
Our aim is to be able to express any thermodynamic property of an equilibrium
system in terms of easily measured quantities. The power of thermodynamics is that it
enables properties that are difficult to measure to be expressed in terms of easily mea-
sured properties. The easily measured properties most commonly used for this purpose
are [Eqs. (1.43) and (1.44)]

1 0V 1 0V
C 1T, P2, a1T, P2 a b , k1T, P2 a b (4.39)*
P
V 0T P V 0P T
Since these are state functions, they are functions of T, P, and composition. We are
considering mainly constant-composition systems, so we omit the composition depen-
dence. Note that a and k can be found from the equation of state V V(T, P) if this is
known.

The Euler Reciprocity Relation
To relate a desired property to C , a, and k, we use the basic equations (4.25) to (4.31)
P
and mathematical partial-derivative identities. Before proceeding, there is another
partial-derivative identity we shall need. If z is a function of x and y, then [Eq. (1.30)]
0z 0z
dz a b dx a b dy M dx N dy (4.40)
0x y 0y x