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): 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): 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).
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): 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.