264
6 Molecular Systems
Notice that due to the structures assumed by z tr , z rot , z vib , and z el , the equation of
state for the diatomic gas, given by
P = k B T
∂ ln Z
∂V
N,T
= Nk B T
∂ ln z tr
∂V
N,T
=
Nk B T
V
,
(6.2.14)
does not depend upon z rot , z vib , or z el .
6.2.2 The Diatomic Vibrational Partition Function
We have seen that a diatomic molecule has a total of six degrees of freedom: five of
them are taken up by the three translational and two rotational degrees of freedom
associated with the molecule as a rigid entity, leaving only a single vibrational
degree of freedom. The only type of vibrational motion that a diatomic molecule
can undergo is the stretching and compression of the chemical bond holding the
diatomic molecule together: the energy associated with this vibrational motion thus
depends upon the displacement between the two nuclei at any given time relative to
their (equilibrium) displacement determined by the position of the minimum in the
potential energy function.
The Simple Harmonic Oscillator Approximation
We shall approximate the potential energy for the vibrational motion of a diatomic
molecule as that of a simple harmonic oscillator (SHO) with a potential energy
minimum of −D e located at an internuclear separation R e (see Fig. 6.1), and given
by
V (R) = −D e +
1
2 k(R − R e )
2 ,
(6.2.15)
with the zero of energy the separated stationary atoms (SSA) zero, and R e the
equilibrium separation of the two nuclei. When describing the SHO vibrational
contribution to the partition function, it will prove more convenient to use the bottom
of the potential well as the zero for vibrational energy than to use the SSA zero.
Accordingly, we shall drop D e from V (R), and use the form ω e1 e βD e for z el , as in
Eq. (6.2.10).
In order to quantify our description of the vibrational motion of a diatomic
molecule, we shall choose a Cartesian axis system, with the z-axis coincident with
the internuclear axis. The nuclei, with masses m 1 and m 2 , are then located on the zaxis with z 1 < z 2 (without loss of generality), as shown in Fig. 6.2. The internuclear
distance is then given as R = z 2 −z 1 , and the SHO potential energy function (6.2.15)
becomes
6 Molecular Systems
Notice that due to the structures assumed by z tr , z rot , z vib , and z el , the equation of
state for the diatomic gas, given by
P = k B T
∂ ln Z
∂V
N,T
= Nk B T
∂ ln z tr
∂V
N,T
=
Nk B T
V
,
(6.2.14)
does not depend upon z rot , z vib , or z el .
6.2.2 The Diatomic Vibrational Partition Function
We have seen that a diatomic molecule has a total of six degrees of freedom: five of
them are taken up by the three translational and two rotational degrees of freedom
associated with the molecule as a rigid entity, leaving only a single vibrational
degree of freedom. The only type of vibrational motion that a diatomic molecule
can undergo is the stretching and compression of the chemical bond holding the
diatomic molecule together: the energy associated with this vibrational motion thus
depends upon the displacement between the two nuclei at any given time relative to
their (equilibrium) displacement determined by the position of the minimum in the
potential energy function.
The Simple Harmonic Oscillator Approximation
We shall approximate the potential energy for the vibrational motion of a diatomic
molecule as that of a simple harmonic oscillator (SHO) with a potential energy
minimum of −D e located at an internuclear separation R e (see Fig. 6.1), and given
by
V (R) = −D e +
1
2 k(R − R e )
2 ,
(6.2.15)
with the zero of energy the separated stationary atoms (SSA) zero, and R e the
equilibrium separation of the two nuclei. When describing the SHO vibrational
contribution to the partition function, it will prove more convenient to use the bottom
of the potential well as the zero for vibrational energy than to use the SSA zero.
Accordingly, we shall drop D e from V (R), and use the form ω e1 e βD e for z el , as in
Eq. (6.2.10).
In order to quantify our description of the vibrational motion of a diatomic
molecule, we shall choose a Cartesian axis system, with the z-axis coincident with
the internuclear axis. The nuclei, with masses m 1 and m 2 , are then located on the zaxis with z 1 < z 2 (without loss of generality), as shown in Fig. 6.2. The internuclear
distance is then given as R = z 2 −z 1 , and the SHO potential energy function (6.2.15)
becomes