Statistical Thermodynamics and Rate Theories/Data

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.

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

$$\Theta_v = 6332.5_2 \text{K} $$