# Statistical Thermodynamics and Rate Theories/Molecular partition functions of polyatomic gases

## Rotational partition functions

 Rotational partition function of a linear rotor ${\displaystyle q_{rot}={\frac {8\pi Ik_{B}T}{h^{2}}}}$

Partition function of a spherical rotor

 Rotational partition function of a spherical rotor ${\displaystyle q_{rot}={\frac {\pi ^{1/2}}{\sigma }}\left({\frac {8\pi ^{2}Ik_{B}T}{h^{2}}}\right)^{3/2}}$

Partition function of a symmetric top

 Rotational partition function of a symmetric rotor ${\displaystyle q_{rot}={\frac {\pi ^{1/2}}{\sigma }}\left({\frac {8\pi ^{2}I_{A}k_{B}T}{h^{2}}}\right)\left({\frac {8\pi ^{2}I_{B}k_{B}T}{h^{2}}}\right)^{1/2}}$
 Rotational partition function of an asymmetric rotor ${\displaystyle q_{rot}={\frac {\pi ^{1/2}}{\sigma }}\left({\frac {8\pi ^{2}I_{A}k_{B}T}{h^{2}}}\right)^{1/2}\left({\frac {8\pi ^{2}I_{B}k_{B}T}{h^{2}}}\right)^{1/2}\left({\frac {8\pi ^{2}I_{C}k_{B}T}{h^{2}}}\right)^{1/2}}$

## Vibrational partition functions

 Vibrational partition function of a polyatomic molecule ${\displaystyle q_{vib}=\prod _{i}{\frac {1}{1-{\textrm {exp}}\left({\frac {-hv}{k_{B}T}}\right)}}}$