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): 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.
Assertion (A): P-T graph of all gases at low density meet at \(0 K\).
Reason (R): Absolute zero kelvin is less than \(0^{\circ}C\) in Celsius scale.
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. Extrapolation of the P-T (isochoric) graphs for ideal gases shows they converge to zero pressure at \(0 K\). Reason (R) is true; \(0 K\) is equal to \(-273.15^{\circ}C\), which is indeed less than \(0^{\circ}C\). However, R is a statement about temperature scale conversion and does not explain the behavior of the P-T graph.
Assertion (A): An ideal gas has infinitely many molar specific heats.
Reason (R): Specific heat is amount of heat needed to raise the temperature of \(1\) mole of gas by \(1K\).
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. An ideal gas can undergo various thermodynamic processes (isobaric, isochoric, adiabatic, polytropic, etc.), each associated with a unique specific heat capacity.
Reason (R) is true; it is the definition of molar specific heat. However, the definition does not explain *why* there are infinitely many such values; this stems from the different possible thermodynamic paths.
Assertion (A): On increasing the temperature, the height of the peak of the Maxwell’s velocity distribution curve increases.
Reason (R): The height of the peak of the Maxwell’s velocity distribution curve represents most probable speed.
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. As temperature increases, the Maxwell-Boltzmann distribution curve broadens, and its peak height *decreases*, shifting to higher speeds. Reason (R) is false. The *x-coordinate* (speed value) of the peak represents the most probable speed; the *height* of the peak represents the fraction of molecules possessing that speed, not the speed value itself. Since both A and R are false, option (4) is correct.
Assertion (A): All molecular motion ceases at \(-273.15^{\circ}C\).
Reason (R): Temperature \(0K\) cannot be attained.
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. \(-273.15^{circ}C\) is equivalent to \(0 K\), at which point theoretical classical molecular motion ceases. Reason (R) is true, as the Third Law of Thermodynamics states that absolute zero cannot be reached. However, R describes the attainability of \(0 K\), not the phenomenon of molecular motion ceasing at that temperature. Thus, R is not the correct explanation of A.
Assertion (A): In Maxwell’s speed distribution graph, for a given amount of gas, the area under the graph increases as the temperature of the gas increases.
Reason (R): Decrease in temperature broadening the curve.
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. The area under the Maxwell distribution curve represents the total number of gas molecules, which remains constant regardless of temperature changes. Reason (R) is false. A decrease in temperature causes the speed distribution curve to become narrower and taller, not broader. Since both A and R are false, option (4) is correct.
Assertion (A): The pressure exerted by an enclosed ideal gas does not depend on the shape of the container.
Reason (R): The pressure of an ideal gas depends on the number of moles, temperature and volume of the enclosure.
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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Pressure of an ideal gas is given by \( PV = nRT \). For a fixed amount of gas at a given temperature, P depends on V, not shape. So, (A) is true. Also, \( P = \frac{nRT}{V} \), so P depends on n, T, V. So, (R) is true. (R) correctly explains that since the ideal gas law depends only on V (not shape for a given V), A is true.
Assertion (A): The ratio \( \frac{C_P}{C_V} \) is more for helium gas than for hydrogen gas.
Reason (R): Atomic mass of helium is more than that of hydrogen.
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 Helium (monoatomic), \( \gamma = 5/3 \). For Hydrogen (diatomic), \( \gamma = 7/5 \). Since \( 5/3 > 7/5 \), (A) is true. Atomic mass of He is 4 amu, H is 1 amu (H2 is 2 amu), so (R) is true.
However, \( \gamma \) depends on degrees of freedom (monoatomic vs diatomic), not atomic mass. So, (R) is not the correct explanation.
Assertion (A): Internal energy of real gas is always negative at absolute zero temperature.
Reason (R): Potential energy of a bounded system is negative.
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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At absolute zero, kinetic energy is minimal (zero for ideal gas). For a real gas, attractive intermolecular forces mean potential energy is negative (relative to infinite separation). So, total internal energy is negative. Thus, (A) is true. (R) is also true, as attractive forces in a bounded system lead to negative potential energy. (R) explains (A).