Page 105 - Modern physical chemistry

5.10 Gibbs Free Energy 95

When the temperature is kept constant, operator d commutes with T and

dwrev =d(E-TS)T =dAT· [5.57]
Integration of this yields
[5.58]

In the last steps, the label
A=E-TS [5.59]
has been introduced.
Function A is called the Helmholtz free energy. The energy is free in the sense that
any increase in it may reappear as work when the pertinent process is reversed at the
given temperature.
During a spontaneous process in a system, some dissipation occurs, some of the work
replaces some of the heat, and
dw>dwrev · [5.60]
Combining inequality (5.60) with equality (5.57) gives us
dw:?:dAT· [5.61]
When volume V of the system is also constant, all work is net work. But a sponta-
neous process does not require any net work to be done on the system. So we have

dAT,v ~ 0 [5.62]
in any such process at constant T and V. At constant temperature and volume, the
Helmholtz free energy A of a system decreases spontaneously until it can decrease no
more and equilibrium is reached.

5. 10 Gibbs Free Energy

At constant temperature and pressure the external energy PV as well as the Helmholtz
free energy A are sources for work energy. Thus, work is available for dissipation as long
as A + PV can decrease.
We again consider a specific system. The net work equals the total work minus the
work of compression. For an infinitesimal change, we have
dWnet = dw + P dV. [5.63]

When this is reversible, the relation becomes
dWnet,rev = de - T dS + P dV. [5.64]

When both temperature and pressure are kept constant, operator d commutes with T and
Pand
dWnet,rev = d(E - TS + PV) = d(A + PV) = d(H -TS) = dGT,p· [5.65]
T,P T,P T,P
Integration of this yields
Wnet,rev = A( H -TS )T,P = AGT,p· [5.66]

In the last steps, the label
G=H-TS [5.67]
has been introduced.