Thermal capacity is \(C = mc\), where \(m\) is mass and \(c\) is specific heat. If \(m < 1\) (e.g., in kg), then \(c\) will be numerically greater than \(C\). Thus, (A) is true. (R) states mass can be less than unity, which is true and correctly explains (A).

During melting at constant temperature, the average kinetic energy of molecules (related to temperature) remains constant.

Latent heat increases potential energy. So (A) is true. (R) correctly defines latent heat.

However, (R) does not explain why kinetic energy remains unchanged.

For an ideal gas, internal energy \(U\) depends only on temperature \(T\). If \(U\) is constant, then \(T\) is constant. For a constant temperature process (isothermal), pressure \(P\) and volume \(V\) can change while \(T\) remains constant (as \(PV = nRT\)). Thus, both assertion and reason are true, and the reason correctly explains the assertion.

Assertion (A) is true; average kinetic energy of molecules is directly proportional to absolute temperature. Reason (R) is false because some substances, like water between \(0^{circ}text{C}\) and \(4^{circ}text{C}\,) contract upon heating. Thus, (A) is true and (R) is false.

Assertion (A) is true. For the same volume expansion, work done \(W = int P dV\). In isothermal expansion, \(P\) drops slower than in adiabatic expansion (due to heat supply), so the area under the \(P-V\) curve is greater for isothermal. Reason (R) is also true and explains why \(P\) behaves differently, leading to different work done.

When ice melts to water, its volume decreases \(\Delta V < 0\). Work done *by* the atmosphere *on* the system is \(-P\Delta V\), which is positive. So (A) is true. During melting, latent heat is absorbed, increasing internal energy \(\Delta U = Q - W\). Since \(Q\) is positive and \(W\) (work by system) is negative, \(\Delta U\) is positive. So (R) is true. However, the increase in internal energy is not the reason for the work done by the atmosphere; it's the volume change. So (R) does not explain (A).

During free expansion of an ideal gas, no work is done \(W=0\) and no heat is exchanged \(Q=0\). Therefore, change in internal energy \(Delta U = Q - W = 0\). For an ideal gas, \(Delta U = 0\) implies \(Delta T = 0\). Internal energy itself is not zero (it just doesn't change). Free expansion is an irreversible process, so entropy *increases* \(Delta S > 0\), it's not zero. Thus, both (A) and (R) are false.

Assertion (A) is true. For an ideal monatomic gas, molecules only have translational degrees of freedom, and there are no intermolecular forces, so internal energy consists solely of translational kinetic energy. Reason (R) is false; it describes contributions to internal energy from rotational, vibrational, and potential energies which are absent in an *ideal monatomic gas*.

Concept: Internal energy of an ideal gas depends only on temperature.
Formula: For ideal gas, \( U = f(T) \). For monoatomic, \( U = \frac{3}{2} nRT \). Isothermal process implies \( T \) is constant.
Solution: If \( U \) is constant, then \( T \) is constant. An isothermal process allows simultaneous change in \( P \) and \( V \) while \( T \) (and thus \( U \)) remains constant. Reason correctly explains this.

Concept: Temperature is a measure of average kinetic energy. Thermal expansion.
Formula: Average Kinetic Energy \( \propto T \).
Solution: Increasing temperature increases molecular kinetic energy. However, not all substances expand on heating (e.2.g., water between \( 0^{\circ}\text{C} \) and \( 4^{\circ}\text{C} \)). So R is false.