498
9 Chemical Equilibrium
approximation, in which the equilibrium separation r e , the molecular force constant
κ, and chemical dissociation energy D e are the same for all isotopologues, so that
we may therefore deduce the values of the characteristic rotational and vibrational
temperatures for the heteronuclear and 37 Cl 2 homonuclear isotopologues of Cl 2
from the experimental values for the 35 Cl 2 isotopologue. We could, in principle, also
deduce values for the spectroscopic dissociation energy, D 0 , for these isotopologues.
However, it is relatively difficult to obtain very accurate values for D 0 without
carrying out an elaborate fit of the potential energy function to an extensive set of
accurate spectroscopic data. As even a small error in the value of D 0 ( 35 Cl 2 ) could
lead to a significant error in the values for K P (T ), we shall attempt to avoid doing
this here.
To this end, we recall that the characteristic rotational temperature rot for a
linear molecule A is defined as
rot (A) ≡
¯
h 2
2m r,A r 2
e (A)
,
with m r,A denoting the reduced mass of A. Because all three isotopologues have
the same value of r e , the product m r rot = ¯
h 2 /(2r e ) is also fixed for the other
isotopologues, consistent with the Born–Oppenheimer approximation. Thus, from a
knowledge of the reduced masses of the various isotopologues and the value of rot
for one of them, the others may be determined from
rot (B) =
m r,A
m r,B
rot (A) .
In a similar manner, given that the characteristic vibrational temperature for
isotopologue A is given by vib (A) ≡ (2π) −1
κ/m r,A , we have a means of
determining vib for the other isotopologues from that for isotopologue A, namely,
vib (B) =
m r,A
m r,B
vib (A) .
Finally, as the spectroscopic dissociation energy D 0 and the chemical dissociation
energy D e (i.e., the depth of the potential well) are related to vib by −D 0 = −D e +
1
2 k B vib , we also have a relation connecting the spectroscopic dissociation energies
of two isotopologues and their respective characteristic vibrational temperatures,
namely,
D 0 (B) = D 0 (A) +
1
2 k B [ vib (A) − vib (B)] .
However, rather than employing the values of D 0 evaluated in this fashion, then
substituting them into the expression for U
◦
0 , we see that we may directly obtain
a Born–Oppenheimer expression for U
◦
0 for our isotopologues reaction as
9 Chemical Equilibrium
approximation, in which the equilibrium separation r e , the molecular force constant
κ, and chemical dissociation energy D e are the same for all isotopologues, so that
we may therefore deduce the values of the characteristic rotational and vibrational
temperatures for the heteronuclear and 37 Cl 2 homonuclear isotopologues of Cl 2
from the experimental values for the 35 Cl 2 isotopologue. We could, in principle, also
deduce values for the spectroscopic dissociation energy, D 0 , for these isotopologues.
However, it is relatively difficult to obtain very accurate values for D 0 without
carrying out an elaborate fit of the potential energy function to an extensive set of
accurate spectroscopic data. As even a small error in the value of D 0 ( 35 Cl 2 ) could
lead to a significant error in the values for K P (T ), we shall attempt to avoid doing
this here.
To this end, we recall that the characteristic rotational temperature rot for a
linear molecule A is defined as
rot (A) ≡
¯
h 2
2m r,A r 2
e (A)
,
with m r,A denoting the reduced mass of A. Because all three isotopologues have
the same value of r e , the product m r rot = ¯
h 2 /(2r e ) is also fixed for the other
isotopologues, consistent with the Born–Oppenheimer approximation. Thus, from a
knowledge of the reduced masses of the various isotopologues and the value of rot
for one of them, the others may be determined from
rot (B) =
m r,A
m r,B
rot (A) .
In a similar manner, given that the characteristic vibrational temperature for
isotopologue A is given by vib (A) ≡ (2π) −1
κ/m r,A , we have a means of
determining vib for the other isotopologues from that for isotopologue A, namely,
vib (B) =
m r,A
m r,B
vib (A) .
Finally, as the spectroscopic dissociation energy D 0 and the chemical dissociation
energy D e (i.e., the depth of the potential well) are related to vib by −D 0 = −D e +
1
2 k B vib , we also have a relation connecting the spectroscopic dissociation energies
of two isotopologues and their respective characteristic vibrational temperatures,
namely,
D 0 (B) = D 0 (A) +
1
2 k B [ vib (A) − vib (B)] .
However, rather than employing the values of D 0 evaluated in this fashion, then
substituting them into the expression for U
◦
0 , we see that we may directly obtain
a Born–Oppenheimer expression for U
◦
0 for our isotopologues reaction as