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

 To distinguish which process requires more work during compression, we compare isothermal and adiabatic processes for a gas. Let's explore this step by step:

1. In an isothermal process, the temperature remains constant. The work done in compressing a gas isothermaly can be represented as: \(W_{\text{iso}} = nRT \ln{\left(\frac{V_i}{V_f}\right)}\), where:
   • \(n\) is the number of moles.
   • \(R\) is the universal gas constant.
   • \(T\) is the absolute temperature.
   • \(V_i\) is the initial volume.
   • \(V_f\) is the final volume.
2. In an adiabatic process, no heat exchange occurs. The work done can be expressed as: \(W_{\text{adi}} = \frac{C_v (T_i - T_f)}{R} \times \ln{\left(\frac{V_i}{V_f}\right)}\), where the relationship between initial and final temperatures is determined by: \(T_i V_i^{\gamma - 1} = T_f V_f^{\gamma - 1}\), and
   • \(\gamma\) is the heat capacity ratio (\(\gamma = \frac{C_p}{C_v}\)).
   • \(C_v\) is the molar heat capacity at constant volume.
3. Since we have \(\gamma > 1\) for any gas: \(\Rightarrow W_{\text{adi}} > W_{\text{iso}}\).
4. Thus, for the same change in volume, an adiabatic process involves a larger amount of work because it additionally requires an alteration in temperature, while isothermal compression occurs without changing temperature.

Therefore, the correct answer is: Compressing the gas through adiabatic process will require more work to be done.