Assertion (A): When a hot liquid is mixed with a cold liquid, the temperature of the mixer is undefined for some time and then becomes nearly constant.
Reason (R): If two bodies at different temperature are mixed in a calorimeter, the total energy of the two bodies remains conserved.
When liquids of different temperatures are mixed, it takes time to reach thermal equilibrium, so temperature is not uniform initially. So (A) is true. In an ideal calorimeter, total energy is conserved. So (R) is true. However, (R) does not explain the time-dependent nature of temperature equilibration in (A).
Assertion (A): A bottle is filled with water at \(40^{\circ}\text{C}\text{ on opening it at moon, water will boil}\).
Reason (R): Atmospheric pressure on the surface of moon is zero and boiling point is proportional to pressure.
The moon has a negligible atmosphere, leading to near-zero surface pressure. At such low pressures, water's boiling point decreases significantly. Consequently, water at \(40^{\circ}\text{C}\text{ would readily boil}\). Both (A) and (R) are true, and (R) provides the correct explanation for (A).
Assertion (A): The expanded length l of a rod of original length l_0 is not correctly given by assuming \(\alpha\) to be constant with T \( \l = \l_0 (1 + \alpha \Delta T)\), if \(\alpha \Delta T\) is large.
Reason (R): It is given by \(l = \l_0 \text{e}^{\alpha \Delta T}\), which cannot be treated as being approximately equal to \(l_0 (1 + \alpha \Delta T)\text{ for large value of }\alpha \Delta T\).
The standard linear expansion formula \(ell = \ell_0 (1 + \alpha \Delta T)\text{ is an approximation valid for small }\alpha \Delta T\), derived from the exponential form \(ell = \ell_0 \text{e}^{\alpha \Delta T}\). If \(alpha \Delta T\) is large, this approximation fails. Both (A) and (R) are true, and (R) correctly explains (A).
Assertion (A): Energy of molecules increase on increasing the temperature.
Reason (R): All substances expand on increasing the temperature.
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): Work done by a gas in isothermal expansion is more than the work done by the gas in the same expansion adiabatically.
Reason (R): Temperature remains constant in isothermal expansion but not in adiabatic expansion.
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.
Assertion (A): During the melting of a slab of ice at \(273\text{ K}\) at \(1\text{ atm}\) positive work is done on the ice-water system by the atmosphere.
Reason (R): In above process, the internal energy of ice-water system increases.
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).
Assertion (A): During free expansion of an Ideal gas, entropy is zero.
Reason (R): Internal energy of an ideal gas is zero during free expansion.
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): In an ideal monoatomic gas, The Internal energy of gas is equal to translational Kinetic energy of all its molecules
Reason (R): The Internal energy may get contributes from Translational, Rotatory, vibrationally as well as from the Potential energy corresponding to the molecular force.
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*.
Assertion (A): It is possible for both the pressure and volume of a monoatomic ideal gas of a given amount to change simultaneously without causing the internal energy of the gas to change.
Reason (R): The internal energy of an ideal gas of a given amount remains constant if temperature does not change. It is possible to have a process in which pressure and volume are changed such that temperature remains constant.
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): It is possible for both the pressure and volume of a monoatomic ideal gas of a given amount to change simultaneously without causing the internal energy of the gas to change.
Reason (R): The internal energy of an ideal gas of a given amount remains constant if temperature does not change. It is possible to have a process in which pressure and volume are changed such that temperature remains constant.
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.