Assertion (A): There is no change in internal energy for ideal gas at constant temperature.
Reason (R): Internal energy of an ideal gas is a function of temperature only.
For an ideal gas, internal energy \(U\) depends solely on temperature \(T\). Therefore, if \(T\) is constant, \(Delta U = 0\). Reason (R) correctly explains Assertion (A).
Assertion (A): The atoms of a monoatomic gas have less degrees of freedom as compared to molecules of the diatomic gas.
Reason (R): The ratio of \(\frac{C_p}{C_v}\) for an ideal diatomic gas is more than that for an ideal monoatomic gas (where \(C_p\) and \(C_v\) have usual meaning).
Monoatomic gas has 3 degrees of freedom, diatomic has 5 (or 7). So (A) is true. The ratio \(\gamma = 1 + \frac{2}{f}\) is \(5/3\) for monoatomic and \(7/5\) for diatomic. Since \(5/3 > 7/5\), (R) is false.
Assertion (A): A gas is kept in an insulated cylinder with a movable piston, in compressed state. As the piston is suddenly released, temperature of the gas decreases.
Reason (R): According to the kinetic theory of gas, a molecule colliding with the piston must rebound with less speed than it had before the collision. Hence average speed of the molecules is reduced.
In an adiabatic expansion, the gas does work on the receding piston. Molecules lose kinetic energy upon collision, reducing their average speed and thus the gas temperature. Reason (R) correctly explains Assertion (A).
Assertion (A): At \(0K\), pressure of an ideal gas becomes zero.
Reason (R): At \(0K\), according to ideal gas equation \(PV = 0\), volume cannot be zero hence pressure should be zero to satisfy this equation.
From the ideal gas equation \(PV = nRT\), if \(T = 0K\), then \(PV = 0\). Since volume \(V\) cannot be zero, pressure \(P\) must be zero. Reason (R) directly explains Assertion (A).
Assertion (A): Molar heat capacity of an ideal monoatomic gas at constant volume is a constant at all temperatures.
Reason (R): As the temperature of an monoatomic ideal gas is increased, number of degrees of freedom of gas molecules remains constant.
For an ideal monoatomic gas, \(C_v = \frac{3}{2}R\) as it only has 3 translational degrees of freedom. This number \(f=3\) remains constant with temperature. Thus, \(C_v\) is constant. Reason (R) correctly explains Assertion (A).
Assertion (A): According to kinetic theory of gases the internal energy of a given sample of an ideal gas is only kinetic.
Reason (R): The ideal gas molecules exert force on each other only when they collide.
For an ideal gas, internal energy is purely kinetic due to random motion, so (A) is true. Ideal gas molecules have no intermolecular forces, so (R) is false. Thus, (A) is true and (R) is false.
Assertion (A): Internal energy of an ideal gas \( U = nC_V T \) is due to random motion of gas molecules.
Reason (R): A container is moving with speed \( v \). It is suddenly stopped by a force, temperature of gas increases.
Assertion (A) is true as internal energy comes from random molecular motion. Reason (R) is also true, as kinetic energy converts to internal energy upon sudden stopping. However, (R) does not explain (A).
Assertion (A): Experimental results indicate that the molar specific heat of hydrogen gas at constant volume below \( 50 \text{ K} \) is equal to \( 5/2 R \), where \( R \) is the universal gas constant.
Reason (R): A diatomic hydrogen molecule possesses three translational and two rotational degrees of freedom at all temperatures.
Assertion (A) is false. Below \( 50 \text{ K} \), hydrogen's rotational modes freeze out, so \( C_V \) approaches \( 3/2 R \), not \( 5/2 R \).
Reason (R) is false because degrees of freedom depend on temperature; vibrational modes activate at high T, and rotational modes freeze out at low T.
Assertion (A): When an ideal gas is heated in a rigid non conducting container then pressure becomes double if the temperature is doubled.
Reason (R): Both the frequency of collisions and momentum transferred per collision becomes \( \sqrt{2} \) times.
Assertion (A) is true by ideal gas law \( P \propto T \) at constant volume. Reason (R) is also true, as \( v_{rms} \propto \sqrt{T} \), affecting both collision frequency and momentum transfer per collision by a factor of \( \sqrt{2} \) when T is doubled. (R) correctly explains (A).
Assertion (A): The total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times the product of its pressure and its volume.
Reason (R): The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.
Assertion (A) is true because \( E_k = \frac{3}{2} nRT \) and \( PV = nRT \), so \( E_k = \frac{3}{2} PV \). Reason (R) is also true, as molecules of an ideal gas undergo elastic collisions with each other, changing their individual velocities. However, (R) does not explain (A).