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
![{\displaystyle \epsilon _{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d007888875e2294ff7ef04b13349b27ec46e5a)
![{\displaystyle \epsilon _{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d007888875e2294ff7ef04b13349b27ec46e5a)
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743)
![{\displaystyle (j+{\frac {1}{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d58b96de965c99f52462e7bf77dc8d7ec39ff0ae)
![{\displaystyle -h\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee3e189d574f7d172b9b2f2a11a57aa56b33771)
![{\displaystyle \epsilon _{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d007888875e2294ff7ef04b13349b27ec46e5a)
![{\displaystyle =h\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7f75479d26d04b4d337e0dcc7d5dcaa13a1897c)
![{\displaystyle j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0)
![{\displaystyle h\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc23768f43b26085f80e1882a94b31d24abd653)
![{\displaystyle -}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36)
![{\displaystyle {\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55)
![{\displaystyle h\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc23768f43b26085f80e1882a94b31d24abd653)
![{\displaystyle =h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a83597a2a0bdb8caca0c17ae6cdcc3c850e5fd8)
![{\displaystyle \nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468)
(where j=0,1,2...)
and by substituting ![{\displaystyle g_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc0c1f707e4786c1121db1ba0a608fe85a94e2d)
(i.e., singly degenerate) into the summation for
the resulting equation is:
![{\displaystyle q_{vib}=\sum _{j=0}^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b44945dba8b74d676eaba3925729cfcb33f765f4)
The j is taken outside the brackets by the common exponent rule:
Note that ![{\displaystyle E_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6b82f2a00af6c9efd4c16d4e99329605645c0c)
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743)
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, ![{\displaystyle \sum _{i=0}^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b261337f4466ff77a582b269344fa14e784e9d4)
![{\displaystyle x^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be8cd88951c7c0e3b181f956531b0a878bbed203)
This is done by first letting
Then,
this converges when x<1, giving
and by replacing x with the original expression, you have:
![{\displaystyle q_{vib}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43daf87bd8e477a7047b1cd13e4692deb0094220)
where
is the vibrational frequency of the molecule, which can by found by the following equation:
=![{\displaystyle {\frac {1}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/129204d50704b07e6a4223870954242b21170354)
where k is the spring constant of the molecule and
is the reduced mass
Calculate the population of the ground vibrational state of
at 298.15 K. (
)
![{\displaystyle q_{vib}={\frac {1}{1-\exp \left({\frac {-h\nu j}{k_{B}T}}\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6785fe233cb631aa0fbd7fa5002de3bd0bd7e32a)
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.