Kinetic Theory
6 Board Physics previous year questions on Kinetic Theory — options free on every question; 1 include the answer & explanation free, the rest unlock with PYQ Pass.
For a diatomic gas:
- Let \({\gamma }_{1}=\frac{{C}_{p}}{{C}_{v}}\) for a rigid molecule (without vibrational modes).
- Let \({\gamma }_{2}=\frac{{C}_{p}}{{C}_{v}}\) for a diatomic molecule with vibrational modes included.
Determine the relationship between \({\gamma }_{1}\) and \({\gamma }_{2}\).
(Shift II Memory Based)
\({\gamma }_{2}<{\gamma }_{1}\)
-
Degrees of Freedom:
- A rigid diatomic molecule has 5 degrees of freedom (3 translational + 2 rotational).
- A diatomic molecule with vibrational modes has additional vibrational degrees of freedom.
-
Specific Heats (\({C}_{v}\) and \({C}_{p}\)):
- The internal energy is proportional to the degrees of freedom. For a rigid diatomic gas: \({C}_{v}=\frac{5R}{2}\)
- When vibrational modes are included, the total degrees of freedom increase. Each vibrational mode adds \(2R\), making: \({C}_{v}=\frac{5R}{2}+2R=\frac{9R}{2}\)
- \({C}_{p}\) is related to \({C}_{v}\) by: \({C}_{p}={C}_{v}+R\)
-
Adiabatic Index (\(\gamma\)γ):
- The adiabatic index \(\gamma\)γ is defined as: \(\gamma =\frac{{C}_{p}}{{C}_{v}}=1+\frac{R}{{C}_{v}}\)
- For the rigid diatomic gas: \({\gamma }_{1}=\frac{{C}_{p}}{{C}_{v}}=\frac{\frac{7R}{2}}{\frac{5R}{2}}=\frac{7}{5}=1.4\)
- For the diatomic gas with vibrational modes: \({\gamma }_{2}=\frac{{C}_{p}}{{C}_{v}}=\frac{\frac{11R}{2}}{\frac{9R}{2}}=\frac{11}{9}\approx 1.22\)
Clearly, \({\gamma }_{2}<{\gamma }_{1}\) because the inclusion of vibrational modes increases \({C}_{v}\) more significantly than \({C}_{p}\), leading to a smaller \(\gamma\).
Final Answer:(A) \({\gamma }_{2}<{\gamma }_{1}\)
For a diatomic gas, if \({\gamma }_{1}=\left(\frac{Cp}{Cv}\right)\) for rigid molecules and \({\gamma }_{2}=\left(\frac{Cp}{Cv}\right)\) for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct ? (Cp and Cv are specific heats of the gas at constant pressure and volume)
[JEE Main 2025, 22 Jan (Shift 2)]
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The translational kinetic energy of molecules of \(50 \mathrm{~g} \quad\mathrm{of}\quad \mathrm{CO}_2\) gas at \(17^{\circ} \mathrm{C}\) is:
(Shift - II Memory Based)
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A flask contains Hydrogen and Argon in the ratio 2 : 1 by mass. The temperature of the mixture is 30°C. The ratio of average kinetic energy per molecule of the two gases
(Kargon / Khydrogen) is: (Given: Atomic Weight of Ar = 39.9)
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A container has two chambers of volumes \({\mathrm{V}}_{1}=2\) litres and \({\mathrm{V}}_{2}=3\) litres separated by a partition made of a thermal insulator. The chambers contains \({\mathrm{n}}_{1}=5\) and \({\mathrm{n}}_{2}=4\) moles of ideal gas at pressures \({\mathrm{p}}_{1}=1\mathrm{atm}\) and \({\mathrm{p}}_{2}=2\mathrm{atm}\), respectively. When the partition is removed, the mixture attains an equilibrium presšure of :
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A cylinder contains hydrogen gas at pressure of \( 249 \mathrm{kPa} \) and temperature \( 27^{\circ} \mathrm{C} \). Its density is \( :\left(\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right) \)
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