0.014 kg of nitrogen is enclosed in a vessel at a temperature of 27 °C. Find the amount of heat energy to be supplied to the gas to double the rms speed of its molecules.

The root mean square speed is related to absolute temperature T as

${\mathrm{c}}_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{kT}}{\mathrm{m}}}$

For a given gas, m is fixed. Therefore, ${\mathrm{c}}_{\mathrm{rms}}\propto \sqrt{\mathrm{T}}$. Hence in order to double the root mean square speed, the absolute temperature must be increased to four times the initial value. Initial temperature ${\mathrm{T}}_1=273+27=300\mathrm{K}$. Therefore, final temperature ${\mathrm{T}}_2=4{\mathrm{T}}_1=1200\mathrm{K}$.

Since the volume of the vessel is fixed, $∆$V = 0. Hence the heat energy supplied to the gas does no work on the gas; it only increases the internal energy of the molecules.

$$\begin{gathered} \Delta U=n{C}_{v}dT \\ =\frac{m}{M}\times \frac{5R}{2}\times \left(T_2-T_1\right) \\ =\frac{0.014\times {10}^3}{28}\times \frac{5\times 8.314}{2}\times \left(1200-300\right) \\ =9353J \end{gathered}$$