Statistical Thermodynamics and Rate Theories/Vibrational partition function of a diatomic molecule

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The general form of the molecular partition function is an infinite sum which is open form, making it difficult to calculate, this is why the sum is approximated as a closed form which leads to an algebraic equation. The derivation of the closed form on the equation is as follows:

The open form of the vibrational partition function:

solving for

(where j=0,1,2...)

and by substituting (i.e., singly degenerate) into the summation for the resulting equation is:

The j is taken outside the brackets by the common exponent rule:

Note that is the equation that represents the energy levels of a harmonic oscillator which is used to approximate the vibrational molecular degree of freedom. The vibrational zero point energy is not negligible and must be defined at n=0.

Next, in order for the open system to be converted into the closed system, the equation must take the form of a geometric series identity, like in calculus.

if x<1,

This is done by first letting

Then,

this converges when x<1, giving and by replacing x with the original expression, you have:

where is the vibrational frequency of the molecule, which can by found by the following equation: =

where k is the spring constant of the molecule and is the reduced mass

Example[edit]

Calculate the population of the ground vibrational state of at 298.15 K. ()

Next the Probability of the ground state can be calculated:

This means that 99.9998% of all molecules are in the ground vibrational state at 298.15 K.