# Statistical Thermodynamics and Rate Theories/Data

Molecule $\Theta _{r}$ (K) $\Theta _{v}$ (K)
H2 87.547 6332.52
N2Invalid <ref> tag; invalid names, e.g. too many 2.0518 3393.54
O2Invalid <ref> tag; invalid names, e.g. too many 2.0793 2273.60
F2Invalid <ref> tag; invalid names, e.g. too many 1.2808 1318.87
HFInvalid <ref> tag; invalid names, e.g. too many 41.345 5954.27
HClInvalid <ref> tag; invalid names, e.g. too many 15.240 4303.41
NOInvalid <ref> tag; invalid names, e.g. too many 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

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

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.

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

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