Question

Root mean square velocity of a gas is tripled when the temperature is

• A. Reduced to one third
• B. Reduced to one ninth
• C. Increased by three times
• D. Increased by nine times

Hint:

For gaseous molecules: \[ v_{\text{rms}} \propto \sqrt{T} \] If velocity changes by a factor \(n\), temperature changes by a factor \(n^2\).

Answer 1

The Correct Option is D

Step 1: Root mean square (rms) velocity of a gas is given by: \[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
Step 2: From the formula: \[ v_{\text{rms}} \propto \sqrt{T} \]
Step 3: If rms velocity is tripled: \[ \frac{v_2}{v_1} = 3 \] \[ \sqrt{\frac{T_2}{T_1}} = 3 \]
Step 4: Squaring both sides: \[ \frac{T_2}{T_1} = 9 \] \[ T_2 = 9T_1 \]
Step 5: Hence, temperature must be increased by nine times.