Assertion (A): Absolute zero temperature is not the zero energy temperature.
Reason (R): At absolute zero temperature the gas may possess potential energy.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Concept: Absolute zero and molecular energy. At absolute zero 0 K, the average kinetic energy of molecules is at its minimum (or zero for an ideal gas classically). However, molecules can still possess potential energy due to intermolecular forces or external fields. Hence, total energy is not necessarily zero. Both (A) and (R) are true, and (R) correctly explains (A).
Assertion (A): For gas molecules absolute zero temperature is not the temperature of zero energy.
Reason (R): Only the kinetic energy of the molecules is represented by temperature.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Concept: Temperature and molecular energy. Temperature is a direct measure of the average translational kinetic energy of molecules. While kinetic energy is minimal at 0 K, gas molecules can still have potential energy from intermolecular interactions. Thus,0 K is not zero total energy. Both (A) and (R) are true, and (R) explains (A).
Assertion (A): Molar heat capacity of a gas in any process can have any value ( -infty ) to ( +infty ).
Reason (R): Molar heat capacity of a gas in an isothermal process is ( infty ).
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Concept: Molar heat capacity \( C = \frac{dQ}{dT} \). For an isothermal process, ( dT = 0 ), so ( C = infty ). For an adiabatic process, ( dQ = 0 ), so ( C = 0 ). Thus, molar heat capacity can range from ( -infty ) to ( +infty ). Both (A) and (R) are true, and (R) explains (A).
Assertion (A): The internal energy of a real gas is function of both, temperature and volume.
Reason (R): For any gas internal kinetic energy depends on temperature and internal potential energy depends on volume.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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For an ideal gas, internal energy depends only on temperature. For a real gas, intermolecular forces exist, giving rise to internal potential energy in addition to kinetic energy. The internal kinetic energy depends on temperature, while the internal potential energy depends on the average distance between molecules, which is related to the volume. Thus, Assertion (A) is true, and Reason (R) is true. Reason (R) provides the accurate explanation for why the internal energy of a real gas is a function of both temperature and volume.
Assertion (A): Vibrational energy of molecule at temperature \(T\) is \(kT\).
Reason (R): For every molecule, vibrational degree of freedom is 2.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true as each vibrational mode contributes \(kT\) to the internal energy. Reason (R) is false; vibrational degrees of freedom vary by molecular structure (e.g., diatomic molecules have 1). Thus, (A) is true and (R) is false.
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).
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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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.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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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): 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.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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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.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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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): 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.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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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).