4.1 Canonical Ensemble: Closed Systems
171
Example 4.1 Internal energy of an ideal gas in a gravitational field [1].
The single-particle canonical partition function for an ideal gas in a (weak)
gravitational field can be shown [see Example 7.1, Chap. 7] to be
z(T , V ) =
V
3 (T )
1 − e −η
η
,
in which (T ) is the thermal de Broglie wavelength of Eq. (3.2.22), while η ≡
βmgH , with m the particle mass, β = (k B T ) −1 , g the acceleration due to gravity,
H the height of the gas column, parameterizes the gravitational field. The singleparticle internal energy u(T ) for this ideal gas is given formally as
u(T ) = −
∂ ln z
∂β
V
,
from which the internal energy is obtained as
u(T ) =
3
2β
−
∂
∂β
ln(1 − e
−η ) +
∂
∂β
ln η,
or, upon simplification,
u(T ) =
5
2
−
η
e η − 1
k B T .
This expression for the internal energy of a structureless particle subjected to a weak
gravitational field may be rewritten as
u(T ) =
3
2 k B T
1 +
2
3
1 −
η
e η − 1
,
so that the second term within the square brackets explicitly represents the contribution due to the presence of the gravitational field.
For a single particle of mass 4 amu and H = 1 km, temperature T = 298 K, g =
9.80665 m s −2 (exactly), the gravitational parameter η has a value of approximately
1.6 × 10 −2 , which, in turn, leads to a value of approximately 5.1×10 −4 for the
correction factor
2
3 [1 − η/(e η − 1)] in our expression for the internal energy. As
U(T ) for an ideal gas is given by U(T ) = Nu(T ), the same correction factor also
applies to U(T ). This example illustrates why ordinary thermodynamics does not
need to take the Earth’s gravitational field into account.
Now, let us consider the variance, ((E) 2 , and standard deviation, σ E , of E,
namely
((E)
2
= =E
2
− −E
2
≡ σ
2
E .