304
6 Molecular Systems
By employing these redefined factors, the electronic contributions to the internal
energy U(T , V ) and the Helmholtz energy A(T , V ), for example, are expressed
directly as
U el (T ) = − ND 0 ,
A el (T ) = ND 0 + Nk B T ln ω e1
(6.2.127)
in terms of experimentally accessible quantities.
The dissociation energy D 0 , which represents the amount of energy that has to
be added to an individual diatomic molecule to cause it to be dissociated into its
component noninteracting atoms (equivalently, at infinite separation) can also be
interpreted as the negative of the (internal) energy, ◦
0 = U
◦
0 /N 0 , per molecule at
T = 0 K (at which the translational component vanishes). This latter quantity can
also be looked upon as the heat of formation of the molecule from its constituent
atoms at 0 K, i.e., we may write
U
◦
0 = U
◦
f 0 = H
◦
f 0 ,
with the latter equality coming from the ideal gas law P V = Nk B T . This result
allows us to express the molar internal energy in the form
U − H
◦
f 0 =
5
2 N 0 k B T +
N 0 k B vib
e vib /T − 1
.
(6.2.128)
We note that this form is particularly convenient for comparisons between our
calculated contributions from the various internal degrees of freedom, as the righthand side of this expression no longer contains D 0 , which is the least accessible
molecular parameter. Determination of the chemical dissociation energy is not as
great a problem for diatomic molecules as it is for polyatomic molecules, as we
shall see below. Nonetheless, the determination of accurate values for heats of
formation has been an item of concern to chemists for more than a century, and
they have devised a number of means for determining them for a very large number
of substances. Similar expressions to Eq. (2.7.4) for U may be obtained for A, H, G
and μ ◦ . These are the quantities whose values are listed in the NIST-JANAF 1
thermochemical tables [21].
Some caution must be applied here, however, as our statistical mechanical
expressions are based upon the SSA zero of energy, so that H
◦
f 0 for molecules
like N 2 will be nonzero, while the heats of formation of atoms like N will be zero
by definition. This must be compared with the more practical convention chosen
in thermodynamics, which sets H
◦
f to zero for the elements in their naturally
occurring forms (at earth surface temperatures!). Thus H
◦
f 0 (thermo) for N 2 is zero,
while H
◦
f 0 (thermo) for N is nonzero. Let us illustrate the calculation of the heat of
1 JANAF is an acronym standing for Joint Army Navy Air Force: the funding for the creation of
these tabulations of thermodynamic data was provided by these three agencies.
6 Molecular Systems
By employing these redefined factors, the electronic contributions to the internal
energy U(T , V ) and the Helmholtz energy A(T , V ), for example, are expressed
directly as
U el (T ) = − ND 0 ,
A el (T ) = ND 0 + Nk B T ln ω e1
(6.2.127)
in terms of experimentally accessible quantities.
The dissociation energy D 0 , which represents the amount of energy that has to
be added to an individual diatomic molecule to cause it to be dissociated into its
component noninteracting atoms (equivalently, at infinite separation) can also be
interpreted as the negative of the (internal) energy, ◦
0 = U
◦
0 /N 0 , per molecule at
T = 0 K (at which the translational component vanishes). This latter quantity can
also be looked upon as the heat of formation of the molecule from its constituent
atoms at 0 K, i.e., we may write
U
◦
0 = U
◦
f 0 = H
◦
f 0 ,
with the latter equality coming from the ideal gas law P V = Nk B T . This result
allows us to express the molar internal energy in the form
U − H
◦
f 0 =
5
2 N 0 k B T +
N 0 k B vib
e vib /T − 1
.
(6.2.128)
We note that this form is particularly convenient for comparisons between our
calculated contributions from the various internal degrees of freedom, as the righthand side of this expression no longer contains D 0 , which is the least accessible
molecular parameter. Determination of the chemical dissociation energy is not as
great a problem for diatomic molecules as it is for polyatomic molecules, as we
shall see below. Nonetheless, the determination of accurate values for heats of
formation has been an item of concern to chemists for more than a century, and
they have devised a number of means for determining them for a very large number
of substances. Similar expressions to Eq. (2.7.4) for U may be obtained for A, H, G
and μ ◦ . These are the quantities whose values are listed in the NIST-JANAF 1
thermochemical tables [21].
Some caution must be applied here, however, as our statistical mechanical
expressions are based upon the SSA zero of energy, so that H
◦
f 0 for molecules
like N 2 will be nonzero, while the heats of formation of atoms like N will be zero
by definition. This must be compared with the more practical convention chosen
in thermodynamics, which sets H
◦
f to zero for the elements in their naturally
occurring forms (at earth surface temperatures!). Thus H
◦
f 0 (thermo) for N 2 is zero,
while H
◦
f 0 (thermo) for N is nonzero. Let us illustrate the calculation of the heat of
1 JANAF is an acronym standing for Joint Army Navy Air Force: the funding for the creation of
these tabulations of thermodynamic data was provided by these three agencies.