Question

The RMS velocity of a gas at temperature \( T \) is \( v \). If temperature becomes \( 4T \), new RMS velocity is:

• A. \( v \)
• B. \( 2v \)
• C. \( 4v \)
• D. \( v/2 \)

Hint:

When temperature increases by a factor of \( n \), the RMS velocity increases by a factor of \( \sqrt{n} \). Since the temperature quadrupled (\( 4 \times \)), the velocity doubled (\( \sqrt{4} = 2 \)).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The Root Mean Square (RMS) velocity is a statistical representation of the speed of gas molecules within a system.
According to the kinetic theory of gases, the absolute temperature of a gas is a direct measure of the average translational kinetic energy of its molecules.
Because kinetic energy is proportional to the square of the velocity (\(E_k = 1/2 mv^2\)), it follows that the temperature is proportional to the square of the RMS velocity.
Conversely, the RMS velocity is proportional to the square root of the absolute temperature.
This explains why molecules move significantly faster in hotter gases, leading to higher pressure and more rapid diffusion.
Step 2: Key Formula or Approach:
1. Formula: \(v_{rms} = \sqrt{\frac{3RT}{M}}\).
2. Proportionality: \(v_{rms} \propto \sqrt{T}\) for a constant gas mass \(M\).
Step 3: Detailed Explanation:
Let the initial state be defined by temperature \(T_1 = T\) and RMS velocity \(v_1 = v\).
The final state is defined by temperature \(T_2 = 4T\) and new RMS velocity \(v_2\).
From the proportionality relation:
\[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \]
Substituting the values provided in the question:
\[ \frac{v_2}{v} = \sqrt{\frac{4T}{T}} = \sqrt{4} = 2 \]
Therefore, \(v_2 = 2v\).
This demonstrates that quadrupling the absolute temperature only doubles the speed of the molecules, reflecting the non-linear relationship between temperature and molecular speed.
Step 4: Final Answer:
The new RMS velocity is \(2v\).