Question

A perfect gas of volume 10 litre is compressed isothermally to a volume of 1 litre. The rms speed of the molecules will

• A. decrease 5 times
• B. remain unchanged
• C. increase 5 times
• D. increase 10 times

Hint:

Always associate "isothermal" directly with "constant temperature" and "constant internal energy". For an ideal gas, if $T$ is constant, average kinetic energy and rms speed must also be constant regardless of pressure or volume changes.

Answer 1

The Correct Option is B

Step 1: Read the process.
A gas is compressed from $10$ litre to $1$ litre, but the key word is "isothermally". We must say what happens to the rms speed of the molecules.

Step 2: Recall the rms speed formula.
The root mean square speed is \[ v_{rms}=\sqrt{\frac{3RT}{M}}, \] where $T$ is the absolute temperature and $M$ is the molar mass.

Step 3: See what the formula depends on.
Notice the formula has $T$ and $M$ only. It does not contain volume or pressure at all. So rms speed cares only about temperature (for a fixed gas).

Step 4: Use the meaning of isothermal.
"Isothermal" means the temperature stays the same all through the process. So $T$ does not change.

Step 5: Check the other quantities.
$R$ is a fixed constant and $M$ is fixed because it is the same gas. So every part of the formula stays the same.

Step 6: Conclude about volume change.
Shrinking the volume from $10$ L to $1$ L changes pressure, but since $T$ is constant, it does not touch $v_{rms}$.

Step 7: State the result.
The rms speed stays the same, which is option (2).
\[ \boxed{\text{rms speed remains unchanged}} \]