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Vibrational Motions
A summation can be used to obtain the vibrational partition function and the vibrational contribution to the thermal energy.
With the classical picture, gas molecules fly around, tumble end over end, and vibrate. The atoms of a diatomic molecule are described as vibrating against one another, as do two balls joined by a spring. But the vibrational energy is quantized, and the expression for the allowed vibrational energies was developed. Now we see how the molecules of a macroscopic sample are distributed throughout these allowed vibrational states.
Vibrational partition function: each way in which a molecule can vibrate leads to a set of vibrational states with energies given by

For each vibrational mode we can thus write

Now, in contrast to translational and rotational energy spacings, the vibrational spacings are appreciable compared with kT, and therefore only a few of the terms in the series will contribute appreciably to the partition function. The partition function sum cannot be replaced by an integral, but we can obtain an expression by developing the summation
qvib mode = 1 + e-hv vib/(kT) + e -2hv vib/(kT) + ……
With the introduction of the convenient symbol

This becomes
qvib mode = 1 + e-x + e-2x + …. = 1 + (e-x)1 + (e-x)2 + ……
This series can be recognized as the binomial expansion of (1 – e-x)-1, and thus we have

Example: studies of the absorption of infrared radiation, which is primarily due to vibrational-energy changes, shows that radiation with v⁻ = 1000 cm-1 and a frequency 3 × 1013 s-1 is typical of that required to promote molecules from one vibrational energy to the next higher energy. Calculate the energy, in joules, for this representative vibrational-energy separation. What is the vibrational partition function for such a vibrational mode at (a) 25˚C and (b) 1000˚C?
Solution: the energy of the photons of this infrared radiation is calculated from ε = hv as ε = (6.626 × 10-34 J s) (3.0 × 1013 s-1) = 20 × 10-21 J. now we can proceed to the calculation of x = hv vib/(kT) and then to the first expression
At 25˚C = 298 K we have

And

The vibrational energy spacing hvvib is so much greater than kT at this temperature that little more than the lowest-energy vibrational state is available to the molecules.
At 1000˚C = 1273 K the results are
X = 1.13 and qvib = 1.48
At this temperature the excited vibrational states are appreciably available to the molecules.
Vibrational energy: the thermal energy for any degree of freedom is obtained from the general expression. The vibrational partition function gives

The thermal energy contributed by a vibrational mode is then obtained as

Alternatively, with x replaced by hv vib/(kT) and R by Nk, this can be written

Plots of (U – U0) vib mode, for various values of T as a function of the vibrational-energy-level spacing are suitable in this amplitude.
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With the classical picture, gas molecules fly around, tumble end over end, and vibrate. The atoms of a diatomic molecule are described as vibrating against one another, as do two balls joined by a spring. But the vibrational energy is quantized, and the expression for the allowed vibrational energies was developed. Now we see how the molecules of a macroscopic sample are distributed throughout these allowed vibrational states.
Vibrational partition function: each way in which a molecule can vibrate leads to a set of vibrational states with energies given by
For each vibrational mode we can thus write
Now, in contrast to translational and rotational energy spacings, the vibrational spacings are appreciable compared with kT, and therefore only a few of the terms in the series will contribute appreciably to the partition function. The partition function sum cannot be replaced by an integral, but we can obtain an expression by developing the summation
qvib mode = 1 + e-hv vib/(kT) + e -2hv vib/(kT) + ……
With the introduction of the convenient symbol
This becomes
qvib mode = 1 + e-x + e-2x + …. = 1 + (e-x)1 + (e-x)2 + ……
This series can be recognized as the binomial expansion of (1 – e-x)-1, and thus we have
Example: studies of the absorption of infrared radiation, which is primarily due to vibrational-energy changes, shows that radiation with v⁻ = 1000 cm-1 and a frequency 3 × 1013 s-1 is typical of that required to promote molecules from one vibrational energy to the next higher energy. Calculate the energy, in joules, for this representative vibrational-energy separation. What is the vibrational partition function for such a vibrational mode at (a) 25˚C and (b) 1000˚C?
Solution: the energy of the photons of this infrared radiation is calculated from ε = hv as ε = (6.626 × 10-34 J s) (3.0 × 1013 s-1) = 20 × 10-21 J. now we can proceed to the calculation of x = hv vib/(kT) and then to the first expression
At 25˚C = 298 K we have
And
The vibrational energy spacing hvvib is so much greater than kT at this temperature that little more than the lowest-energy vibrational state is available to the molecules.
At 1000˚C = 1273 K the results are
X = 1.13 and qvib = 1.48
At this temperature the excited vibrational states are appreciably available to the molecules.
Vibrational energy: the thermal energy for any degree of freedom is obtained from the general expression. The vibrational partition function gives
The thermal energy contributed by a vibrational mode is then obtained as
Alternatively, with x replaced by hv vib/(kT) and R by Nk, this can be written
Plots of (U – U0) vib mode, for various values of T as a function of the vibrational-energy-level spacing are suitable in this amplitude.
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