Question

The mean square speed $(\langle v^2 \rangle)$ of the molecules of a gas at absolute temperature $T$ is proportional to

• A. $\frac{1}{T}$
• B. $\sqrt{T}$
• C. $T$
• D. $T^2$

Hint:

In kinetic theory: - Mean square speed $\propto T$, - RMS speed $\propto \sqrt{T}$.

Answer 1

The Correct Option is C

Concept: From the kinetic theory of gases, the mean square speed of gas molecules is given by: \[ \langle v^2 \rangle = \frac{3kT}{m} \] where:
• $k$ = Boltzmann constant
• $T$ = absolute temperature
• $m$ = mass of a molecule

Step 1: Write the formula for mean square speed.
\[ \langle v^2 \rangle = \frac{3kT}{m} \]

Step 2: Identify proportionality.
Since $k$ and $m$ are constants: \[ \langle v^2 \rangle \propto T \]

Step 3: Final conclusion.
Thus, the mean square speed is directly proportional to temperature: \[ T \]