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): 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): 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): A real gas behaves as an ideal gas at high temperature and low pressure.
Reason (R): At low pressure and high temperature intermolecular forces vanish away and volume of gas molecules is negligible.
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. Real gases approximate ideal gas behavior under conditions of high temperature (high kinetic energy overcomes intermolecular forces) and low pressure (molecules are far apart, making their own volume negligible).
Reason (R) accurately states these conditions as the underlying cause for ideal gas behavior. Thus, R is the correct explanation for A.
Assertion (A): On a V-T graph, the slope of an isobar increases with pressure.
Reason (R): At constant temperature, for an ideal gas its volume is directly proportional to its pressure.
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 isobar, \( V = (\frac{nR}{P})T \). The slope on a V-T graph is \( \frac{nR}{P} \). As P increases, slope decreases, so (A) is false. Boyle's law states that at constant T, \( V \propto \frac{1}{P} \), i.e., V is inversely proportional to P, so (R) is false.
Assertion (A): For an ideal gas, at constant temperature, the product of the pressure and volume is constant.
Reason (R): The mean square velocity of gas molecules is inversely proportional to mass of molecule.
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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Boyle's Law states that for an ideal gas at constant T, \( PV = \text{constant} \). So (A) is true. The mean square velocity \( = \frac{3kT}{m} \), so it is inversely proportional to molecular mass m.
So (R) is true. However, (R) does not explain Boyle's law (A).