Statistical Thermodynamics and Rate Theories/Data

Molecule	${\mathrm{\Theta }}_r$(K)	${\mathrm{\Theta }}_v$(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 (${\mathrm{\Theta }}_r$) and characteristic vibrational (${\mathrm{\Theta }}_v$) temperatures for molecular hydrogen, H₂.

${\mathrm{\Theta }}_r=\frac{{\hslash }^2}{2k_B\mu {r_e}^2}$

Where $\overline{h}=h/2\pi$ 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.

${\mathrm{\Theta }}_r=\frac{{1.0546\times 10^{-34}\text{J s}}^2}{21.3806\times 10^{-23}\frac{J}{K}\times 8.35942\times 10^{-28}\text{kg}\times {7.4144\times 10^{-11}\text{m}}^2}$

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

${\mathrm{\Theta }}_r=87.54_7\text{K}$

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

${\mathrm{\Theta }}_v=\frac{h\nu }{k_B}$

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.

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

${\mathrm{\Theta }}_v=6332.5_2\text{K}$