**Problem : **

Give four different definitions of the chemical potential
*μ*
, as
derivatives of the different energies we have defined.

*μ* =
=
=
=

**Problem : **

Give two definitions of the entropy
*σ*
in terms of derivatives of
the different energies we have defined.

*σ* = -
= -

**Problem : **

Using the definition of temperature that uses the enthalpy, give an
expression for the temperature in terms of
*U*
,
*σ*
,
*p*
, and
*V*
,
following the method used to derive an expression for the pressure
above.

We know that
*τ* =
, and
that
*H* = *U* + *pV*
. We can differentiate the second equation with respect to
*σ*
, holding
*p*
and
*N*
constant, and then set equal to
*τ*
to
obtain:

**Problem : **

Derive the Maxwell relation that relates a derivative of
*μ*
with a
derivative of
*σ*
.

We use
*G*
because
*μ*
and
*σ*
are free in its differential
identity. We can write
= *μ*
and
= - *σ*
.
Taking the partial derivative of the first with respect to
*τ*
, holding
*N*
constant, and taking the partial derivative of the second with respect to
*N*
, holding
*τ*
constant, and setting the two equal, we obtain:

= -

**Problem : **

Derive the Maxwell Relation that relates a derivative of
*τ*
with a
derivative of
*V*
.

We need
*V*
and
*τ*
to be free in the energy, so let us choose the
enthalpy
*H*
. Then we can write
*τ* =
and
*V* =
.
Taking the partial derivative of the first with respect to
*p*
, holding
*σ*
constant, and taking the partial derivative of the second with
respect to
*σ*
, holding
*p*
constant, and setting them equal,
yields:

=