Statistical Thermodynamics and Rate Theories/Data

Molecule	$\Theta _{r}$(K)	$\Theta _{v}$(K)
H2	87.547	6332.52
N2	2.0518	3393.54
O2	2.0793	2273.60
F2	1.2808	1318.87
HF	41.345	5954.27
HCl	15.240	4303.41
NO	2.4524	2739.79
C2H2	1.7012

Example

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

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

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

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

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

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

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

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

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

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

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