Definition of
F
,
G
,
H
Suppose that
F
=
U

στ
. Then when we take the differential, we need to remember to use the
product rule. We obtain:
dF
=
dU

σ
dτ

τ
dσ
Now, we can substitute in the Thermodynamic Identity to obtain:
dF
= 
σ
dτ

p
dV
+
μ
dN
Notice that F is a function now of
τ
,
V
, and
N
. By adding the term 
στ
, we were able to swap
two of the variables,
σ
and
τ
. We call F the Helmholtz Free Energy, and we will soon see why it
is useful.
The quick mind will realize that we could define 6 such energies in total, by successively
swapping all of the variables. It turns out that we'll only be interested in two more. The Enthalpy,
H
, swaps
p
and
V
. We write
H
=
U
+
pV
and obtain
dH
=
τ
dσ
+
V
dp
+
μ
dN
. We also define
the Gibbs Free Energy by utilizing both of these swaps. Letting
G
=
U
+
pV

τσ
, we obtain
dG
=

σ
dτ
+
V
dp
+
μ
dN
.
We say that the energy of any of these types is a function of the variables that appear as
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 Fall '10
 DavidJudd
 Physics, Thermodynamics, Entropy, Helmholtz free energy, thermodynamic identity

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