Question

Statement I: For an ideal gas, heat capacity at constant volume is always greater than the heat capacity at constant pressure.
Statement II: In a constant volume process, no work is produced and all the heat withdrawn goes into the chaotic motion and is reflected by a temperature increase of the ideal gas.
In the light of the above statements, choose the correct answer:

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Heat capacity at constant pressure ($C_p$) and constant volume ($C_v$) are related by Mayer's formula for an ideal gas: $C_p - C_v = R$. Additionally, the First Law of Thermodynamics ($\Delta U = q + w$) governs the energy distribution during these processes.

Step 2: Key Formula or Approach:
1. Mayer's Relation: $C_p = C_v + R$. Since $R$ is a positive constant, $C_p>C_v$. 2. Constant Volume Process: $\Delta V = 0 \implies W = -P\Delta V = 0$.

Step 3: Detailed Explanation:
1. Statement I analysis: According to Mayer's relation, $C_p$ is always greater than $C_v$ because at constant pressure, heat is used for both increasing internal energy and doing expansion work. Thus, Statement I is false. 2. Statement II analysis: In an isochoric (constant volume) process, work done is zero. Therefore, $q_v = \Delta U$. All heat added increases the internal kinetic energy (chaotic motion) of the molecules, leading to a temperature rise. Thus, Statement II is true.

Step 4: Final Answer:
Statement I is false, but Statement II is true.