Page 145 - Chemical equilibria Volume 4

Determination of the Values Associated with Reactions – Equilibrium Calculations 121
For simplicity’s sake, we shall base our discussion on a diatomic
molecule. Consider the curve (Figure 4.1) showing the variations of the
vibration energy as a function of the distance between the nuclei. At
equilibrium between the forces of attraction and repulsion, the nuclei are a
distance r 0 apart (which is shown by the minimum of the curve).
If the molecule receives energy, a vibratory motion occurs around the
equilibrium state, so that the distance between the nuclei oscillates between a
maximal value r max and a minimal value r min. The vibration is not entirely
harmonic, because the forces increase more quickly as the nuclei draw near
to one another than they decrease as the nuclei move apart. The vibrational
energy which would yield a parabola in the case of a harmonic vibration, in
reality, exhibits the asymmetrical form shown in Figure 4.1. The vibrational
energy can then be expressed in the form:

E = (v1/ 2 hν+ ) vibr + (v1/ 2+ ) 1/2 h x ν vibr [4.25]
v

This expression includes two terms. The first is due to a harmonic
oscillation, and the second is a correction of the first term, and contains the
coefficient of anharmonicity x given by:

hν
x = vibr [4.26]
4E ∞

The term v, which is an integer that may have a variety of positive values,
gives us quantified energies represented by the horizontal lines in Figure 4.1.

The energy difference between two consecutive vibrational terms (i.e. the
shift from v to v + 1) is therefore:

⎡ hν ⎤
+
)⎤
+
+
−
12x
Δ v1 E = hν vibr ⎣ ⎡ − (v1 = hν vibr ⎢ 12 vibr (v1 ) ⎥ [4.27]
⎦
v
⎣ 4E ∞ ⎦
This energy difference has a corresponding absorption band of frequency
ν abs such that:
E = hν abs [4.28]