Page 107 - Modern physical chemistry

5.11 Key Thermodynamic Relations 97

The energy added to such a system can be classified as heat absorbed q and as work
done w. During an infinitesimal step, the increase in internal energy E is

dE=dq+dw. [5.72]
The heat absorbed in a reversible process goes to increase the entropy S by the formula
dB = dqrev [5.73]
T '
where T is the absolute temperature. The first law tells us that de is exact; the second
law, that dB is exact.
A given system may be taken from one state to another along a reversible path with
all work done work of compression. With P the pressure, V the volume, and S the entropy,
we have
dw = dwrev = -P dV [5.74]

and
dq = dqrev = T dB. [5.75]
Substituting into equation (5.72) yields
dE=TdB-PdV. [5.76]

Relation (5.76) can be applied even though the actual process is irreversible and involves
other kinds of work because an exact differential depends only on the initial and final
points infinitesimally far apart.
By definition, the enthalpy H, Helmholtz free energy A, and Gibbs free energy G are
given by
H=E+PV, [5.77]

A=E-TS, [5.78]

G=H-TS. [5.79]
Differentiating each of these equations and combining the result with condition (5.76) yields
dH=TdB+VdP, [5.80]

dA=-PdV -SdT, [5.81 ]

dG=VdP-SdT. [5.82]
Since differential dA is exact, equation (5.81) fits the form of (4.75) and

( aA) =_P ( aA) =-S. [5.83]
av
T ' aT v
Differentiating the first of equations (5.83) with respect to P, the second with respect to V,
-:~:T = ( ~~ l ( ~~ 1· [5.84]
and changing signs leads to

=

Since differential dG is exact, equation (5.82) similarly yields

( aG) =V , ( aG) =-S. [5.85]
ap
T aT p