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): 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.
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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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.
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 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): 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).
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.
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 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).
Assertion (A): Molar heat capacity at constant pressure can be less than molar heat capacity at constant volume.
Reason (R): \( C_p – C_V = R \) is valid only for ideal monoatomic gas.
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 false; \( C_p \) is always greater than \( C_V \) because work is done at constant pressure.
Reason (R) is false; Mayer's relation, \( C_p - C_V = R \), is valid for all ideal gases, regardless of atomicity.
Assertion (A): An ideal gas is enclosed within a container fitted with a piston when volume of this enclosed gas is increased at constant temperature. The pressure exerted by the gas on the piston decreases.
Reason (R): In the above situation the rate of molecules striking the piston decreases. Therefore pressure exerted by gas on piston decreases.
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 Boyle's Law (\( PV = \text{constant} \) at constant \( T \)). Reason (R) explains (A) microscopically: increasing volume at constant temperature reduces the density of molecules and thus the frequency of collisions with the piston, leading to decreased pressure.
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): 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.
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 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): The specific heat of a monatomic gas may have value between \(0\) and \(\infty\).
Reason (R): \(C_p = \frac{5}{2} R\) and \(C_v = \frac{3}{2} R\) for a monatomic gas.
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; specific heat depends on the process and can range from \(0\) (adiabatic) to \(\infty\) (isothermal). Reason (R) is true; the specific heats for a monatomic gas are correctly given.
However, R provides specific values and does not explain the general range of specific heat values mentioned in A. Therefore, R is not the correct explanation of A.