Question

A cylinder-piston system contains N atoms of an ideal gas. If 𝑡_𝑎𝑣𝑔 is the average time between successive collisions of a given atom with other atoms. If the temperature T of the gas is increased isobarically, then 𝑡_𝑎𝑣𝑔 is proportional to :

• A. \(\sqrt{T}\)
• B. \(\frac{1}{\sqrt{T}}\)
• C. T
• D. \(\frac{1}{T}\)

Answer 1

The Correct Option is A

The average time between successive collisions, \( t_{\text{avg}} \), for an atom in an ideal gas depends on the mean speed of the atoms, which is proportional to the square root of the temperature \( T \).

Step 1: Mean Speed of Atoms

The mean speed, \( v_{\text{mean}} \), is given by:

\[ v_{\text{mean}} \propto \sqrt{T} \]

Step 2: Relation Between \( t_{\text{avg}} \) and Collision Frequency

The average time between collisions, \( t_{\text{avg}} \), is inversely proportional to the collision frequency. Since the collision frequency depends on the mean speed of the atoms, we have:

\[ t_{\text{avg}} \propto \frac{1}{v_{\text{mean}}} \]

Step 3: Proportionality with Temperature

Substituting \( v_{\text{mean}} \propto \sqrt{T} \):

\[ t_{\text{avg}} \propto \sqrt{T} \]

Conclusion:

Thus, \( t_{\text{avg}} \) is directly proportional to \( \sqrt{T} \).