Question

Air is mostly N₂, diatomic nitrogen, with an effective spring constant of 2.3 x 10³N/m, and an effective oscillating mass of half the atomic mass. For roughly what temperatures should vibration contribute to its heat capacity?

Short answer

Vibration will contribute to heat capacity at temperatures of the order of 2200 K.

Step-by-step solution

Given data

Nitrogen has 14 protons in its nucleus. Atomic mass of Nitrogen is

$\begin{array}{rcl}m& =& 14\times 1.67\times {10}^{-27}\text{kg}\\ & =& 23.38\times {10}^{-27}\text{kg}\end{array}$

Spring constant is

$k=2.3\times {10}^3\text{N/m}$

Difference in energy of quantum harmonic oscillator states and energy of a diatomic gas

Difference in energy of quantum states of a quantum harmonic oscillator is

$\Delta E=\hslash \sqrt{\frac{k}{m/2}}$ ,,,,,(I)

Here $\hslash$is the reduced Planck's constant of value

$\hslash =1.05\times {10}^{-34}J·s$

Energy of a diatomic gas is

$E=\frac{3}{2}{k}_{B}T$ .....(II)

Here $k_B$is the Boltzmann constant of value

$k_B=1.38\times {10}^{-23}J/K$

Determining the temperature for which vibrational energy contributes to heat capacity

Equate equations (I) and (II) to get

$\begin{array}{rcl}\frac{3}{2}{k}_{B}T& =& \hslash \sqrt{\frac{k}{m/2}}\\ T& =& \frac{2\hslash }{3k_B}\sqrt{\frac{k}{m/2}}\\ & & \end{array}$

Substitute the values to get

$\begin{array}{rcl}T& =& \frac{2\times 1.05\times {10}^{-34}\text{J}·\text{s}}{3\times 1.38\times {10}^{-23}\text{J/K}}\sqrt{\frac{2\times 2.3\times {10}^3\text{N/m}}{23.38\times {10}^{-27}\text{kg}}}\\ & =& 0.5\times {10}^{-11}\text{s/K}\times \sqrt{0.2\times {10}^{30}·\left(1\text{N}\times \frac{1\text{kg}·{\text{m/s}}^2}{1\text{N}}\right)·\left(\frac{1}{1\text{m}}\right)·\left(\frac{1}{1\text{kg}}\right)}\\ & =& 0.22\times {10}^4\text{K}\\ & =& 2200\text{K}\end{array}$

The required temperature is of the order 2200 K.