Example

Calculate the ground state characteristic rotational (${\displaystyle \Theta _{r}}$) and characteristic vibrational (${\displaystyle \Theta _{v}}$) temperatures for molecular hydrogen, H₂.

${\displaystyle \Theta _{r}={\frac {\hbar ^{2}}{2k_{B}\mu {r_{e}}^{2}}}}$

Where ${\displaystyle {\bar {h}}=h/2\pi }$ is the reduced Planck constant, ${\displaystyle r_{e}}$ is the internuclear distance for ground state hydrogen^[1], ${\displaystyle k_{B}}$ is the Boltzmann constant, and ${\displaystyle \mu }$ is the reduced mass.

${\displaystyle \Theta _{r}={\frac {{1.0546\times 10^{-34}{\text{J s}}}^{2}}{2{1.3806\times 10^{-23}{J \over K}}\times {8.35942\times 10^{-28}{\text{kg}}}\times {{7.4144\times 10^{-11}{\text{m}}}^{2}}}}}$

${\displaystyle \Theta _{r}={\frac {1.11212\times 10^{-68}{\text{J}}^{2}{\text{s}}^{2}}{1.2703\times 10^{-70}{\text{J}}^{2}{\text{K}}^{-1}}}}$

${\displaystyle \Theta _{r}=87.54_{7}{\text{K}}}$

The characteristic vibrational temperature (${\displaystyle \Theta _{v}}$) is calculated using the following equation

${\displaystyle \Theta _{v}={\frac {h\nu }{k_{B}}}}$

Where ${\displaystyle h}$ is Planck's constant, ${\displaystyle k_{B}}$ is the Boltzmann constant, and ${\displaystyle \nu }$ is the vibrational frequency of the molecule. To retain units of K the vibrational frequency must be changed to units of s^-1.

${\displaystyle \Theta _{v}={\frac {{6.6261\times 10^{-34}{\text{J s}}}\times {4401.21{\text{cm}}^{-1}}\times {2.998\times 10^{10}{cm \over s}}}{1.3806\times 10^{-23}{J \over K}}}}$

${\displaystyle \Theta _{v}=6332.5_{2}{\text{K}}}$

References