Thermodynamics and Chemistry

CHAPTER 5 THERMODYNAMIC POTENTIALS

5.4 CLOSEDSYSTEMS 139

dADdUTdSSdT (5.3.5)

dGDdUTdSSdTCpdVCVdp (5.3.6)

These expressions for dH, dA, and dGare general expressions for any system or phase
with uniformT andp. They arenottotal differentials ofH,A, andG, as the variables in
the differentials in each expression are not independent.
A useful property of the enthalpy in a closed system can be found by replacing dUin
Eq.5.3.4by the first law expression∂qpdVC∂w^0 , to obtain dHD∂qCVdpC∂w^0.
Thus, in a process at constant pressure (dpD 0 ) with expansion work only (∂w^0 D 0 ), we
have

dHD∂q (5.3.7)

(closed system, constantp,

∂w^0 D 0 )

The enthalpy change under these conditions is equal to the heat. The integrated form of this
relation is

R

dHD

R

∂q, or

ÅHDq (5.3.8)

(closed system, constantp,

w^0 D 0 )

Equation5.3.7is analogous to the following relation involving the internal energy, ob-
tained from the first law:

dUD∂q (5.3.9)

(closed system, constantV,

∂w^0 D 0 )

That is, in a process at constant volume with expansion work only, the internal energy
change is equal to the heat.

5.4 Closed Systems

In order to find expressions for the total differentials ofH,A, andGin a closed system
with one component in one phase, we must replace dUin Eqs.5.3.4–5.3.6with

dUDTdSpdV (5.4.1)

to obtain

dHDTdSCVdp (5.4.2)

dADSdTpdV (5.4.3)

dGDSdTCVdp (5.4.4)

Equations5.4.1–5.4.4are sometimes called theGibbs equations. They are expressions
for the total differentials of the thermodynamic potentialsU,H,A, andGin closed sys-
tems of one component in one phase with expansion work only. Each equation shows how