# Statistical Thermodynamics and Rate Theories/Data

Molecule ${\displaystyle \Theta _{r}}$(K) ${\displaystyle \Theta _{v}}$(K)
H2[1] 87.547 6332.52
N2Invalid parameter in <ref> tag 2.0518 3393.54
O2Invalid parameter in <ref> tag 2.0793 2273.60
F2Invalid parameter in <ref> tag 1.2808 1318.87
HFInvalid parameter in <ref> tag 41.345 5954.27
HClInvalid parameter in <ref> tag 15.240 4303.41
NOInvalid parameter in <ref> tag 2.4524 2739.79
C2H2[2] 1.7012

## 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

1. a b http://webbook.nist.gov/cgi/cbook.cgi?ID=1333-74-0
2. E. Plyler, E. Tidwell, and T. Wiggins, (1963). Rotation-Vibration Constants of Acetylene. Journal of Optical Society America. Table 4, Data section in appendix