Heat Capacity and Work in Ideal Gas Processes

To determine the correctness of the given statements, let us analyze them individually using the principles of thermodynamics for an ideal gas:

Analysis of Statement I:

For an ideal gas, the relationship between the molar heat capacity at constant pressure ($C_p$) and the molar heat capacity at constant volume ($C_v$) is given by Mayer's relation: $C_p - C_v = R$ where $R$ is the universal gas constant ($R \approx 8.314 \text{ J mol}^{-1}\text{K}^{-1} > 0$).

Since $R > 0$, it follows directly that: $C_p > C_v$ Thus, the heat capacity at constant pressure ($C_p$) is always greater than the heat capacity at constant volume ($C_v$).

Therefore, Statement I is false.

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Analysis of Statement II:

In a constant volume process (isochoric process), $dV = 0$. The work done by or on the gas is given by: $w = -\int P \, dV = 0$ Hence, no pressure-volume work is done during the process.

According to the First Law of Thermodynamics: $\Delta U = q_v + w$ Since $w = 0$: $\Delta U = q_v$

For an ideal gas, internal energy ($U$) is purely a function of temperature ($T$) and represents the total kinetic energy associated with the random/chaotic motion of the gas molecules. Therefore: $q_v = \Delta U = n C_v \Delta T$ This means that all the heat transferred to the system goes entirely into increasing the internal energy (chaotic molecular motion), which is directly reflected as a rise in the temperature of the gas.

Therefore, Statement II is true.

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Conclusion:

• Statement I is false.
• Statement II is true.

This matches Option D.