Example

Question:

A flask contains argon and chlorine in the ratio of 2:1 by mass. The temperature of the mixture is 27°C. Obtain the ratio of (i) average kinetic energy per molecule, and (ii) root mean square speed \(v_\mathrm{rms}\) of the molecules of the two gases. Atomic mass of argon = 39.9 u; Molecular mass of chlorine = 70.9 u.

Solution:

(i) The average kinetic energy per molecule of an ideal gas is \(\frac{3}{2}k_B T\), same for all gases at the same temperature. So, the ratio is \(1:1\).

(ii) For root mean square speed: \[ \frac{1}{2} m v_\mathrm{rms}^2 = \frac{3}{2} k_B T \implies v_\mathrm{rms}^2 = \frac{3 k_B T}{m} \] Therefore, \[ \frac{v^2_{\mathrm{rms,Ar}}}{v^2_{\mathrm{rms,Cl}}} = \frac{m_{\mathrm{Cl}}}{m_\mathrm{Ar}} = \frac{M_{\mathrm{Cl}}}{M_\mathrm{Ar}} = \frac{70.9}{39.9} = 1.77 \] Taking square root, \[ \frac{v_{\mathrm{rms,Ar}}}{v_{\mathrm{rms,Cl}}} = 1.33 \] The speed above for argon is 1.33 times that for chlorine.