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Statement II: In a constant volume process, no work is produced and all the heat withdrawn goes into the chaotic motion and is reflected by a temperature increase of the ideal gas.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:
We must evaluate basic thermodynamic laws regarding heat capacities (\(C_p\) and \(C_v\)) and the First Law of Thermodynamics for isochoric (constant volume) processes.
Step 2: Key Formula or Approach:
1. Mayer's Relation: \(C_p - C_v = R\) for an ideal gas.
2. First Law of Thermodynamics: \(\Delta U = q + W\). For constant volume, \(W = -P\Delta V = 0\), thus \(\Delta U = q_v\).
Step 3: Detailed Explanation:
Analyzing Statement I:
For an ideal gas, heating at constant pressure requires extra energy to perform expansion work against the surroundings in addition to raising the internal energy. Heating at constant volume only raises internal energy. Therefore, the heat capacity at constant pressure (\(C_p\)) is ALWAYS greater than the heat capacity at constant volume (\(C_v\)).
The statement claims \(C_v>C_p\), which is incorrect. So, Statement I is False.
Analyzing Statement II:
In a constant volume process, \(\Delta V = 0\), so \(W = 0\) (no expansion work is produced).
According to the First Law, \(q_v = \Delta U\). All the heat exchanged directly alters the internal energy.
For an ideal gas, internal energy is purely kinetic (chaotic motion of molecules), which manifests macroscopically as temperature. Heat added ("withdrawn from the surroundings into the system") increases this chaotic motion and raises the temperature.
So, Statement II is True.
Step 4: Final Answer:
Statement I is false but Statement II is true.