356
9 Bioenergetics
Our expression for the chemical potential of an ideal gas, Eq. (9.79), will be
derived in Chap. 10, resulting in Eq. (10.158).
9.11.2 Gibbs–Duhem Relations
As we saw, the equations of state determine the state of the system uniquely because
of the homogeneity of the internal energy E in the dependence on the extensive
variables S, V , and the N k , as shown in Eq. (9.73). But this same homogeneity
implies that not all the equations of state are independent.
We can see this if we differentiate E in Eq. (9.76) and subtract dE from
Eq. (9.102), to find
0 = SdT − V dp +
k
N k dμ k .
(9.82)
This is called a ‘Gibbs–Duhem’ relation. Because of this connection, T , p, and
the μ k cannot be varied arbitrarily. With r different components in the system (i.e.
1 ≤ k ≤ r), there will be r + 1 independent equations of state among the r + 2
variables T , p, μ 1 , μ 2 , · · · , μ r .
To see how the relationship works, let us again consider an ideal gas made from
a mixture of different gases. Since the first law of thermodynamics can be written
as dE = T dS − pdV +
μ k dN k , the equations of state should determine T , p,
and the μ k as a function of S, V , and N . Kinetic theory applied to an ideal gas
tells us that E = Nc v T and p = nRT /V = NkT /V , where c v is the gas specific
heat at constant volume. For a single component gas, we do not need to give a
third equation of state for μ(T , p, N) as a function of three other thermodynamic
variables because we can use the Gibbs–Duhem relation. If we take, for the moment,
a constant-temperature process to do the integration, then Eq. (9.82) gives
μ =
V
n
dp = RT
dp
p
.
For any temperature and pressure,
μ(T , p, {N i }) = RT ln
p
p o
+ μ
o (T , {N i }) ,
(9.83)
where μ o (T {N i }) is a function of temperature but does not depend on pressure. 39
39 The chemical potential is often given in units of joules per mole. If we want the chemical
potential in units of energy per molecule, we must replace R by k in the Eq. (9.83).
9 Bioenergetics
Our expression for the chemical potential of an ideal gas, Eq. (9.79), will be
derived in Chap. 10, resulting in Eq. (10.158).
9.11.2 Gibbs–Duhem Relations
As we saw, the equations of state determine the state of the system uniquely because
of the homogeneity of the internal energy E in the dependence on the extensive
variables S, V , and the N k , as shown in Eq. (9.73). But this same homogeneity
implies that not all the equations of state are independent.
We can see this if we differentiate E in Eq. (9.76) and subtract dE from
Eq. (9.102), to find
0 = SdT − V dp +
k
N k dμ k .
(9.82)
This is called a ‘Gibbs–Duhem’ relation. Because of this connection, T , p, and
the μ k cannot be varied arbitrarily. With r different components in the system (i.e.
1 ≤ k ≤ r), there will be r + 1 independent equations of state among the r + 2
variables T , p, μ 1 , μ 2 , · · · , μ r .
To see how the relationship works, let us again consider an ideal gas made from
a mixture of different gases. Since the first law of thermodynamics can be written
as dE = T dS − pdV +
μ k dN k , the equations of state should determine T , p,
and the μ k as a function of S, V , and N . Kinetic theory applied to an ideal gas
tells us that E = Nc v T and p = nRT /V = NkT /V , where c v is the gas specific
heat at constant volume. For a single component gas, we do not need to give a
third equation of state for μ(T , p, N) as a function of three other thermodynamic
variables because we can use the Gibbs–Duhem relation. If we take, for the moment,
a constant-temperature process to do the integration, then Eq. (9.82) gives
μ =
V
n
dp = RT
dp
p
.
For any temperature and pressure,
μ(T , p, {N i }) = RT ln
p
p o
+ μ
o (T , {N i }) ,
(9.83)
where μ o (T {N i }) is a function of temperature but does not depend on pressure. 39
39 The chemical potential is often given in units of joules per mole. If we want the chemical
potential in units of energy per molecule, we must replace R by k in the Eq. (9.83).