This question has Statement-1 and Statement-2. Of the four choices given after the Statements, choose the one that best describes the two Statements.
Statement 1: The internal energy of a perfect gas is entirely kinetic and depends only on absolute temperature of the gas and not on its pressure or volume.
Statement 2: A perfect gas is heated keeping pressure constant and later at constant volume. For the same amount of heat the temperature of the gas at constant pressure is lower than that at constant volume.

Let us analyze Statement-1 and Statement-2 step by step.

First, consider Statement-1: "The internal energy of a perfect gas is entirely kinetic and depends only on absolute temperature of the gas and not on its pressure or volume." A perfect gas, also known as an ideal gas, follows the ideal gas law and has no intermolecular forces. Therefore, the internal energy of an ideal gas is solely due to the kinetic energy of its molecules and does not include any potential energy. According to the kinetic theory of gases, this internal energy is a function of temperature alone and is independent of pressure or volume. Hence, Statement-1 is true.

Now, consider Statement-2: "A perfect gas is heated keeping pressure constant and later at constant volume. For the same amount of heat the temperature of the gas at constant pressure is lower than that at constant volume." We need to compare the temperature change when the same amount of heat is supplied to the gas under constant pressure and constant volume conditions.

Let the heat supplied be $$ Q $$. For a constant volume process, the heat supplied is given by $$ Q = n C_v \Delta T_v $$, where $$ n $$ is the number of moles, $$ C_v $$ is the molar specific heat at constant volume, and $$ \Delta T_v $$ is the temperature change at constant volume.

For a constant pressure process, the heat supplied is $$ Q = n C_p \Delta T_p $$, where $$ C_p $$ is the molar specific heat at constant pressure, and $$ \Delta T_p $$ is the temperature change at constant pressure.

Since the same heat $$ Q $$ is supplied in both cases:

$$ n C_v \Delta T_v = n C_p \Delta T_p $$

Canceling $$ n $$ (assuming $$ n \neq 0 $$):

$$ C_v \Delta T_v = C_p \Delta T_p $$

For an ideal gas, $$ C_p $$ and $$ C_v $$ are related by $$ C_p = C_v + R $$, where $$ R $$ is the gas constant, and $$ C_p > C_v $$ because at constant pressure, some heat is used to do work during expansion, whereas at constant volume, all heat goes into increasing internal energy.

Substituting $$ C_p = C_v + R $$:

$$ C_v \Delta T_v = (C_v + R) \Delta T_p $$

Rearranging for $$ \Delta T_v $$:

$$ \Delta T_v = \frac{C_v + R}{C_v} \Delta T_p $$

$$ \Delta T_v = \left(1 + \frac{R}{C_v}\right) \Delta T_p $$

Since $$ \frac{R}{C_v} > 0 $$, the term $$ 1 + \frac{R}{C_v} > 1 $$, so:

$$ \Delta T_v > \Delta T_p $$

This means the temperature change at constant volume ($$ \Delta T_v $$) is greater than the temperature change at constant pressure ($$ \Delta T_p $$) for the same heat input. Therefore, for the same amount of heat, the temperature of the gas at constant pressure is lower than at constant volume (assuming the same initial temperature). Hence, Statement-2 is true.

Now, we must determine if Statement-2 is the correct explanation for Statement-1. Statement-1 states that internal energy depends only on temperature, which is a fundamental property of ideal gases derived from kinetic theory. Statement-2 describes the difference in temperature changes under constant pressure and constant volume heating, which arises because at constant pressure, part of the heat is used for work done in expansion, leaving less heat to increase temperature and internal energy. While Statement-2 relies on the fact that internal energy is a function of temperature only (so that at constant volume, all heat increases internal energy), it does not explain why internal energy is entirely kinetic and temperature-dependent in the first place. Thus, Statement-2 is true and consistent with Statement-1 but does not provide the explanation for it.

Therefore, both statements are true, but Statement-2 is not the correct explanation of Statement-1.

Hence, the correct answer is Option C.