Vibrational temperature

The vibrational temperature is commonly used in thermodynamics, to simplify certain equations. It has units of temperature and is defined as

${\displaystyle \theta _{\text{vib}}={\frac {h{\tilde {\nu }}c}{k_{\text{B}}}}={\frac {h\nu }{k_{\text{B}}}}}$

where ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant, ${\displaystyle c}$ is the speed of light, ${\displaystyle {\tilde {\nu }}}$ is the wavenumber, and ${\displaystyle \nu }$ (Greek letter nu) is the characteristic frequency of the oscillator.

The vibrational temperature is used commonly when finding the vibrational partition function.

Molecule	${\displaystyle {\tilde {\nu }}}$	${\displaystyle \theta _{vib}}$
N2	2446 cm−1	3521 K
O2	1568 cm−1	2256 K
F2	917 cm−1	1320 K
HF	4138 cm−1	5957 K
HCl	2991 cm−1	4303 K