6.2 Diatomic Molecules
307
In addition we have the ideal gas equation of state,
P V = Nk B T ,
and we can express the chemical potential μ(T , P ) as
μ(T , P ) =
G(T , P )
N
= μ
◦ (T ) + k B T ln P ,
(6.2.133)
with the standard chemical potential μ ◦ (T ) given by
μ
◦ (T ) = H
◦
f 0 − k B T
ln
2π(m 1 + m 2 )k B T
h 2
3
2
k B T
+ ln
T
σ σ rot
− ln(1 − e
− vib /T ) + ln ω e1
.
(6.2.134)
6.2.6 Direct Evaluation of Internal State Contributions
Not only is it possible to determine the potential energy function for a typical
linear molecule, especially a diatomic molecule, extremely accurately from direct
fits of an appropriate functional form to the large set of spectroscopic transition
frequencies that can be measured with a high degree of precision, but there also exist
very accurate computational procedures capable of providing all bound (and quasibound) energy levels for the molecule. These energy level values can be utilized
directly to compute both the relevant partition function for the molecule and the
thermodynamic state functions pertaining to this species. We shall briefly examine
the expressions required for the determination of the temperature dependence of the
internal state contributions to the thermodynamic state functions in this subsection.
We begin with the fundamental relation (4.1.3b) between the thermodynamic
internal energy U and the canonical partition function Z(T , V ; N), together with
expression (6.1.3) resulting from the rigorous decoupling between translational
and internal state motions. We have also seen in Sect. 6.1 that, because the
thermodynamic state functions depend in general only upon ln Z and/or its partial
derivatives, the translational and internal state contributions to these functions will
be additive. The internal energy U(T , V ; N) may then be written in the form
U(V ; N) = U trans (T , V ; N) + U int (T ; N) ,
(6.2.135)
307
In addition we have the ideal gas equation of state,
P V = Nk B T ,
and we can express the chemical potential μ(T , P ) as
μ(T , P ) =
G(T , P )
N
= μ
◦ (T ) + k B T ln P ,
(6.2.133)
with the standard chemical potential μ ◦ (T ) given by
μ
◦ (T ) = H
◦
f 0 − k B T
ln
2π(m 1 + m 2 )k B T
h 2
3
2
k B T
+ ln
T
σ σ rot
− ln(1 − e
− vib /T ) + ln ω e1
.
(6.2.134)
6.2.6 Direct Evaluation of Internal State Contributions
Not only is it possible to determine the potential energy function for a typical
linear molecule, especially a diatomic molecule, extremely accurately from direct
fits of an appropriate functional form to the large set of spectroscopic transition
frequencies that can be measured with a high degree of precision, but there also exist
very accurate computational procedures capable of providing all bound (and quasibound) energy levels for the molecule. These energy level values can be utilized
directly to compute both the relevant partition function for the molecule and the
thermodynamic state functions pertaining to this species. We shall briefly examine
the expressions required for the determination of the temperature dependence of the
internal state contributions to the thermodynamic state functions in this subsection.
We begin with the fundamental relation (4.1.3b) between the thermodynamic
internal energy U and the canonical partition function Z(T , V ; N), together with
expression (6.1.3) resulting from the rigorous decoupling between translational
and internal state motions. We have also seen in Sect. 6.1 that, because the
thermodynamic state functions depend in general only upon ln Z and/or its partial
derivatives, the translational and internal state contributions to these functions will
be additive. The internal energy U(T , V ; N) may then be written in the form
U(V ; N) = U trans (T , V ; N) + U int (T ; N) ,
(6.2.135)