Clas Blomberg - Physics of life-Elsevier Science (2007)

120 Part IV. Going further with thermodynamics
The differential relation for this is:
dF SdT work (14.3)
This implies that the entropy is minus the temperature derivative of the free energy. (Thus,
the free energy should always decrease with temperature). The energy is also given by the
temperature dependence of the free energy. Consider the derivative or F/T:
⎛ ( FT / ) ⎞ ⎛
F
F
S
( U TS)
U
1 ⎞
⎝⎜
T ⎠⎟
⎝⎜
T ⎠⎟
T T
2
T T
2
T
2
(14.4)
The energy is given by the temperature derivative of free energy divided by the absolute
temperature. For statistical mechanics calculations, considered at other places, there are ways
to primarily calculate the free energy and then relations as these are very important for calculating
energy and entropy. Here we will show other important applications of the relations.
Besides F, we have the other free energy concept, the free enthalpy (see Section 7D):
G H TS.
We add here that literature can be confusing about these concepts. What we call G here is
often called F (and F may be called A). “Free enthalpy” appears a very suitable term, but one
often calls G simply “free energy” or “Gibbs free energy”.
An important property of G is seen in the differential form. If we now write
“work” PdV, which is a proper relation, primarily for gas processes, then we have:
Energy change: dU TdS PdV (14.5)
As G U PV TS:
dG SdT VdP (14.6)
All these differential expressions, of course, describe changes of the various energy/
entropy quantities in terms of changes of other state variables—energy change in terms of
entropy, free energy change in terms of temperature and volume change and free enthalpy
in terms of temperature and pressure change. What distinguishes free enthalpy from all the
other quantities is that its change is expressed in changes of intensive variables, pressure
and temperature. Thus, the change of G written in this form does not seem to depend on
the extension of the system. Of course, this cannot be quite true, and the statement is due
to the fact that we up to now have neglected one relevant variable, i.e. the number of
molecules in this system. Of course, in many applications, the number of molecules does
not vary.