Note that in the above example, all of the processes maybe spontaneous.
However, only the last two mustbe spontaneous by the laws of thermody-
namics as we know them. The difference between “may” and “must” is impor-
tant for science. Science recognizes that anything mightoccur. It focuses, how-
ever, on what willoccur. These spontaneity conditions help us determine what
will occur.4.3 The Gibbs Free Energy and the Helmholtz Energy
We now define two more energies. The definition of the Helmholtz energy, A,is
AUTS (4.5)
The infinitesimal dAis therefore equal to
dAdUT dSS dT
which becomes, for a reversible process,
dAS dTp dV
where we have used the definition ofdUand the entropy for a reversible
process as substitutions. Parallel to the above conclusions regarding dUand
dH, their natural variables, and spontaneity, we state that the natural variables
ofAare Tand V, and that for an isothermal, isochoric process,
(dA)T,V 0 (4.6)
is sufficient to ensure the spontaneity of a process. Again, the “equal to” part
of the sign applies to processes that occur reversibly. This definition has some
application, since some chemical and physical processes do occur under con-
ditions of constant volume (for example, bomb calorimetry).
We also define the Gibbs energy,or the Gibbs free energy, G,as
GHTS (4.7)
The infinitesimal dGis
dGdHT dSS dT
Substituting for the definition ofdHand again assuming a reversible change,
we get
dGS dT V dp
This equation implies certain natural variables, namely Tand p, such that the
following spontaneity condition is
(dG)T,p 0 (4.8)
This is the spontaneity condition we have been looking for! We therefore make
the following, perhaps premature, statements. Under conditions of constant
pressure and temperature:
IfG 0: the process is spontaneous
IfG0: the process is not spontaneous (4.9)
IfG0: the system is at equilibrium
Since G(and A) are state functions, these statements reflect the fact that
dGG, not G.92 CHAPTER 4 Free Energy and Chemical Potential