Question

When the r.m.s. velocity of a gas is denoted by '\(v\)' at temperature '\(T\)'. Out of the following relations, the true one is

• A. \(v\cdot T^2 = \text{constant}\)
• B. \(\frac{v}{T^2} = \text{constant}\)
• C. \(v^2T = \text{constant}\)
• D. \(\frac{v^2}{T} = \text{constant}\)

Hint:

v_rms = sqrt(3RT/M), so v squared is proportional to T.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The root mean square speed of gas molecules depends on the temperature and the molar mass. For a given gas the molar mass is fixed.

Step 2: Key Formula or Approach:
\[ v = \sqrt{\frac{3RT}{M}} \]

Step 3: Detailed Explanation:
Square both sides:
\[ v^2 = \frac{3R}{M}\,T \]
For a given gas, \(\dfrac{3R}{M}\) is a constant. So
\[ \frac{v^2}{T} = \text{constant} \]
This is option (D). Options (A), (B) and (C) have the wrong powers: for instance \(v\,T^2\) constant would mean that \(v\) falls as the temperature rises, which is not the case.

Final Answer:
The true relation is \(\dfrac{v^2}{T} = \text{constant}\), option (D). \[ \boxed{\frac{v^2}{T}=\text{constant} \text{ (D)}} \]