Question

Which of the following relations is correct for root means square speed of a gas at temperature T?

(m = mass of molecule, K = Boltzmann constant)

Options

• \[C_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{KT}}{\mathrm{m}}}\]

• \[C_{\mathrm{rms}}=\sqrt{\frac{8\mathrm{KT}}{\mathrm{m}}}\]

• \[C_{\mathrm{rms}}=\sqrt{\frac{2\mathrm{KT}}{\mathrm{m}}}\]

• \[C_{\mathrm{rms}}=\sqrt{\frac{\mathrm{m}}{3\mathrm{KT}}}\]

Answer 1

\[C_{\mathrm{rms}}=\sqrt{\frac{3\mathrm{KT}}{\mathrm{m}}}\]

Explanation:

From the kinetic theory of gases, the average kinetic energy of a molecule is:

\[\frac{1}{2}mv^2=\frac{3}{2}KT\]

Solving for \[v_{rms}{:}\]

\[v_{rms}^2=\frac{3KT}{m}\]

\[C_{rms}=\sqrt{\frac{3KT}{m}}\]

Where:

• K = Boltzmann constant
• TT = absolute temperature
• mm = mass of one molecule

The factor 3 comes from the 3 degrees of freedom (x, y, z directions) of a gas molecule.