10.6 Partition Function for a Gas
445
the secular determinant into block-diagonal form. In any case, numerical methods
can be successfully used. 25
10.6.8 Molecular Rotation-Partition Functions for ‘High’
Temperatures
If the temperatures are ‘high’ (i.e. T >> θ rot,B , which is satisfied by room and body
temperatures), then the summation expressions for the molecular partition functions
can be approximated by an integral (since then many large j values contribute).
In fact, because room temperatures are sufficient to cause many molecules in a
gas to rotate with j significantly bigger than 1, room temperature is ‘high’. These
characteristic temperatures θ rot,B are shown in Table 10.3 for various common
gases, together with their measured rotational coefficients A, B, and C, as well as
their three moments of inertia.
By approximating the rotational partition function sum by an integral, the
molecular partition function for rotation for linear molecules is given by
z
2D
rot ≈
1
σ
8π 2 I B k B T
h 2
,
(10.55)
while for spherical and asymmetric molecules,
z
3D
rot ≈
1
σ
√
π
8π 2 I A k B T
h 2
1/2
8π 2 I B k B T
h 2
1/2
8π 2 I C k B T
h 2
1/2
.
(10.56)
Since system energies can be found by the negative derivative of the log of the
partition function with respect to β = 1/(k B T ), we see that at high temperatures,
the thermodynamic energy will have contributions of Nk B T /2 from each rotational
degree of freedom, a reflection of the equipartition of energy theorem.
10.6.9 The Vibrational Part
The vibrational excitation energies of molecules generally are at least an order
greater in magnitude than rotational ones. Most of the molecules in a gas at room
temperature will not be in vibrational excitation. In quantum theory, the vibrational
energy for low amplitude motion is harmonic, of the form (1/2)μ ˙
x 2 +(1/2k(x−x o ) 2
25 Values for r (κ, J, τ) up to J = 10 were first given by G.W. King, R.M. Hainer, and P.C.
Cross,The asymmetric rotor I. Calculation and symmetry classification of energy levels, J Chem
Phys 11, 27 (1943).
445
the secular determinant into block-diagonal form. In any case, numerical methods
can be successfully used. 25
10.6.8 Molecular Rotation-Partition Functions for ‘High’
Temperatures
If the temperatures are ‘high’ (i.e. T >> θ rot,B , which is satisfied by room and body
temperatures), then the summation expressions for the molecular partition functions
can be approximated by an integral (since then many large j values contribute).
In fact, because room temperatures are sufficient to cause many molecules in a
gas to rotate with j significantly bigger than 1, room temperature is ‘high’. These
characteristic temperatures θ rot,B are shown in Table 10.3 for various common
gases, together with their measured rotational coefficients A, B, and C, as well as
their three moments of inertia.
By approximating the rotational partition function sum by an integral, the
molecular partition function for rotation for linear molecules is given by
z
2D
rot ≈
1
σ
8π 2 I B k B T
h 2
,
(10.55)
while for spherical and asymmetric molecules,
z
3D
rot ≈
1
σ
√
π
8π 2 I A k B T
h 2
1/2
8π 2 I B k B T
h 2
1/2
8π 2 I C k B T
h 2
1/2
.
(10.56)
Since system energies can be found by the negative derivative of the log of the
partition function with respect to β = 1/(k B T ), we see that at high temperatures,
the thermodynamic energy will have contributions of Nk B T /2 from each rotational
degree of freedom, a reflection of the equipartition of energy theorem.
10.6.9 The Vibrational Part
The vibrational excitation energies of molecules generally are at least an order
greater in magnitude than rotational ones. Most of the molecules in a gas at room
temperature will not be in vibrational excitation. In quantum theory, the vibrational
energy for low amplitude motion is harmonic, of the form (1/2)μ ˙
x 2 +(1/2k(x−x o ) 2
25 Values for r (κ, J, τ) up to J = 10 were first given by G.W. King, R.M. Hainer, and P.C.
Cross,The asymmetric rotor I. Calculation and symmetry classification of energy levels, J Chem
Phys 11, 27 (1943).