Given below are two statements:
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Statement II: In a constant volume process, no work is produced and all the heat withdrawn goes into the chaotic motion and is reflected by a temperature increase of the ideal gas.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 statement, choose the correct answer from the options given below.  

For one mole of an ideal gas, the molar heat capacities at constant volume and at constant pressure are denoted by $$C_V$$ and $$C_P$$, respectively.

From the first-law relation $$dU = \delta q - \delta w$$ and the ideal-gas equation $$PV = RT$$, it can be shown that

$$C_P - C_V = R \; \; -(1)$$

Because the universal gas constant $$R$$ is positive, equation $$(1)$$ gives

$$C_P \gt C_V$$

Therefore, the heat capacity at constant pressure is always greater than that at constant volume for an ideal gas, not the other way around. Hence Statement I (which says $$C_V \gt C_P$$) is false.

In a constant-volume process, the volume is fixed, so the work term $$\delta w = P\,dV$$ equals zero. The first law reduces to $$dU = \delta q$$. For an ideal gas, internal energy $$U$$ depends only on temperature. Thus any heat added or removed at constant volume changes only the internal (chaotic) kinetic energy of the molecules and appears entirely as a change in temperature. Statement II is therefore true.

Combining the results: Statement I is false, Statement II is true. The correct option is

Option D which is: Statement I is false but Statement II is true