Question

A gas is compressed isothermally to half its initial volume. The same gas is compressed separately through an adiabatic process until its volume is again reduced to half. Then :

• A. Compressing the gas through adiabatic process will require more work to be done.
• B. Compressing the gas isothermally or adiabatically will require the same amount of work.
• C. Which of the case (whether compression through isothermal or through adiabatic process) requires more work will depend upon the atomicity of the gas.
• D. Compressing the gas isothermally will require more work to be done.

Answer 1

The Correct Option is A

Concept: 
When a gas is compressed, the work done depends on the nature of the thermodynamic process.

Isothermal Process:
In an isothermal process, the temperature remains constant. The work done in compressing a gas from volume $V_i$ to $V_f$ is:
\(W_{\text{iso}} = nRT \ln\left(\frac{V_i}{V_f}\right)\)

Adiabatic Process:
In an adiabatic process, there is no heat exchange. The work done is:
\(W_{\text{adi}} = \frac{P_i V_i - P_f V_f}{\gamma - 1}\)
or derived using:
\(W_{\text{adi}} = \frac{nR(T_i - T_f)}{\gamma - 1}\)
Key Point: In adiabatic compression, temperature rises, so internal energy increases → more work is required.

Since both compressions reduce the volume to half, we compare them:

• Isothermal: Temperature constant → some heat leaves → less work needed.
• Adiabatic: No heat exchange → temperature increases → internal energy increases → more work required.

Correct Answer:
\(\boxed{\text{Option 1: Compressing the gas through adiabatic process will require more work to be done.}}\)