5.2 Why Is the Chemical Potential Negative?
223
G = −Nk B T ln
1
nn 3
,
(5.2.3)
from which we may obtain the chemical potential (Gibbs energy per particle) μ as
μ = k B T ln(nn
3 ) .
(5.2.4)
From expression (5.2.4), it is clear that μ will be positive, zero, or negative
depending upon whether nn 3 is greater than, equal to, or less than 1, respectively.
For a fixed value n 0 of the number density, we can obtain a value T 0 for the
temperature at which n 0 3 will have the value unity: this temperature then serves
as a characteristic temperature and provides a characteristic thermal energy k B T 0 ,
which we may use to define a dimensionless chemical potential μ/(k B T 0 ) and a
dimensionless temperature T /T 0 .
To proceed further, we need to specify a number density. It will be convenient
to choose n 0 ≡ 5 × 10 25 m −3 , which is a value close to the number density for an
ideal gas at STP (i.e., T = 298.15 K, P = 1 bar), and then examine the behaviour
of μ(T ) for this fixed value of n 0 . From the defining relation (3.2.22) for (T ), we
may write 3 (T ) as
3 (T ) 5.3215 × 10
−27 (mT )
−
3
2 m
3 ,
(5.2.5)
with mass m expressed in atomic mass units (amu) and temperature T expressed in
K. For n = n 0 , we can write the chemical potential as
μ(T )
k B T 0
=
T
T 0
ln
5.3215 × 10 −27 n 0
(mT )
3
2
.
(5.2.6)
To proceed further, it will also be necessary to choose a specific mass.
Let us choose Ar, with mass m = 39.95 amu, as a substance with a typical mass.
For Ar, we find that n 0 3 = 1 occurs for temperature T 0 0.0104 K, a very low
temperature indeed, and roughly three orders of magnitude smaller than the freezing
temperature (T f = 83.8 K) for Ar. The condition for μ Ar (T ) to be a maximum is
given by (dμ/dT ) T =T max = 0 or, equivalently, ln[n 0 3 (T = T max )] =
3
2 . We thus
find that T max 3.8 mK, so that T max /T 0 0.3654. The characteristic behaviour
of the chemical potential for a classical gas is shown in Fig. 5.1.
Even for a gas made up of H atoms, the corresponding values for T 0 and T max
are T 0 0.414 K and T max 0.152 K for n 0 = 5 × 10 25 m −3 . It is clear from
our discussion above that nn 3 will be less than unity for all physically realizable
situations for a classical gas, and hence μ(T ) is, for all practical purposes, always
negative.
223
G = −Nk B T ln
1
nn 3
,
(5.2.3)
from which we may obtain the chemical potential (Gibbs energy per particle) μ as
μ = k B T ln(nn
3 ) .
(5.2.4)
From expression (5.2.4), it is clear that μ will be positive, zero, or negative
depending upon whether nn 3 is greater than, equal to, or less than 1, respectively.
For a fixed value n 0 of the number density, we can obtain a value T 0 for the
temperature at which n 0 3 will have the value unity: this temperature then serves
as a characteristic temperature and provides a characteristic thermal energy k B T 0 ,
which we may use to define a dimensionless chemical potential μ/(k B T 0 ) and a
dimensionless temperature T /T 0 .
To proceed further, we need to specify a number density. It will be convenient
to choose n 0 ≡ 5 × 10 25 m −3 , which is a value close to the number density for an
ideal gas at STP (i.e., T = 298.15 K, P = 1 bar), and then examine the behaviour
of μ(T ) for this fixed value of n 0 . From the defining relation (3.2.22) for (T ), we
may write 3 (T ) as
3 (T ) 5.3215 × 10
−27 (mT )
−
3
2 m
3 ,
(5.2.5)
with mass m expressed in atomic mass units (amu) and temperature T expressed in
K. For n = n 0 , we can write the chemical potential as
μ(T )
k B T 0
=
T
T 0
ln
5.3215 × 10 −27 n 0
(mT )
3
2
.
(5.2.6)
To proceed further, it will also be necessary to choose a specific mass.
Let us choose Ar, with mass m = 39.95 amu, as a substance with a typical mass.
For Ar, we find that n 0 3 = 1 occurs for temperature T 0 0.0104 K, a very low
temperature indeed, and roughly three orders of magnitude smaller than the freezing
temperature (T f = 83.8 K) for Ar. The condition for μ Ar (T ) to be a maximum is
given by (dμ/dT ) T =T max = 0 or, equivalently, ln[n 0 3 (T = T max )] =
3
2 . We thus
find that T max 3.8 mK, so that T max /T 0 0.3654. The characteristic behaviour
of the chemical potential for a classical gas is shown in Fig. 5.1.
Even for a gas made up of H atoms, the corresponding values for T 0 and T max
are T 0 0.414 K and T max 0.152 K for n 0 = 5 × 10 25 m −3 . It is clear from
our discussion above that nn 3 will be less than unity for all physically realizable
situations for a classical gas, and hence μ(T ) is, for all practical purposes, always
negative.