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JEE Advanced · Physics · Kinetic Theory of Gases

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1
The left and right compartments of a thermally isolated container of length LL are separated by a thermally conducting, movable piston of area AA. The left and right compartments are filled with 32\dfrac32 and 11 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant kk and natural length 2L5\dfrac{2L}{5}. In thermodynamic equilibrium, the piston is at a distance L2\dfrac L2 from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is P=kLAαP=\dfrac{kL}{A}\alpha, then the value of α\alpha is ____
Not attempted
Your answer: —
Correct answer: 0.2
2
An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is nn. The internal energy of one mole of the gas is UnU_n and the speed of sound in the gas is vnv_n. At a fixed temperature and pressure, which of the following is the correct option?
Not attempted
A
v3<v6v_3 < v_6 and U3>U6U_3 > U_6
B
v5>v3v_5 > v_3 and U3>U5U_3 > U_5
C
v5>v7v_5 > v_7 and U5<U7U_5 < U_7
D
v6<v7v_6 < v_7 and U6<U7U_6 < U_7
3
A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas (γ=5/3\gamma=5/3) and one mole of an ideal diatomic gas (γ=7/5\gamma=7/5). Here, γ\gamma is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 6666 Joule when heated at constant pressure. The change in its internal energy is ________ Joule.
Not attempted
Your answer: —
Correct answer: 121
4
As shown schematically in the figure, two vessels contain water solutions (at temperature TT) of potassium permanganate (KMnO4_4) of different concentrations n1n_1 and n2n_2 (n1>n2n_1>n_2) molecules per unit volume, with Δn=(n1−n2)≪n1\Delta n=(n_1-n_2)\ll n_1. When they are connected by a tube of small length ll and cross-sectional area SS, KMnO4_4 starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases, and the difference in their partial pressure in the two vessels causing the diffusion. The speed vv of the molecules is limited by the viscous force −βv-\beta v on each molecule, where β\beta is a constant. Neglecting all terms of the order (Δn)2(\Delta n)^2, which of the following is/are correct? (kBk_B is the Boltzmann constant)
Not attempted
A
the force causing the molecules to move across the tube is Δn kBTS\Delta n\,k_BTS
B
force balance implies n1βvl=Δn kBTn_1\beta v l = \Delta n\,k_BT
C
total number of molecules going across the tube per sec is (Δnl)(kBTβ)S\left(\dfrac{\Delta n}{l}\right)\left(\dfrac{k_BT}{\beta}\right)S
D
rate of molecules getting transferred through the tube does not change with time
5
A flat plate is moving normal to its plane through a gas under the action of a constant force FF. The gas is kept at a very low pressure. The speed of the plate vv is much less than the average speed uu of the gas molecules. Which of the following options is/are true?
Not attempted
A
The pressure difference between the leading and trailing faces of the plate is proportional to uvuv
B
The resistive force experienced by the plate is proportional to vv
C
The plate will continue to move with constant non-zero acceleration, at all times
D
At a later time the external force FF balances the resistive force
6
A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature TT. Assuming the gases are ideal, the correct statement(s) is(are)
Not attempted
A
The average energy per mole of the gas mixture is 2RT2RT.
B
The ratio of speed of sound in the gas mixture to that in helium gas is 6/5\sqrt{6/5}.
C
The ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/21/2.
D
The ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/21/\sqrt2.
7
Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2:32:3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4:34:3. The ratio of their densities is
Not attempted
A
1:41:4
B
1:21:2
C
6:96:9
D
8:98:9
ρ=PMRT\rho=\dfrac{PM}{RT} at fixed volume and temperature.
Deconstruction\textbf{Deconstruction}: Two ideal gases share the same vessel (same volume VV) and temperature TT, with atomic mass ratio M1:M2=2:3M_1:M_2=2:3 and partial pressure ratio P1:P2=4:3P_1:P_2=4:3. Find the density ratio ρ1:ρ2\rho_1:\rho_2. Strategy\textbf{Strategy}: From PV=nRT=mMRTPV=nRT=\dfrac{m}{M}RT, derive ρ=mV=PMRT\rho=\dfrac{m}{V}=\dfrac{PM}{RT}, then take the ratio at common V,TV,T. Execution\textbf{Execution}: ρ1ρ2=P1M1P2M2=43×23=89\dfrac{\rho_1}{\rho_2}=\dfrac{P_1M_1}{P_2M_2}=\dfrac{4}{3}\times\dfrac{2}{3}=\dfrac{8}{9}. Verification\textbf{Verification}: Both gases occupy the same volume at the same temperature, so density is directly proportional to PMPM by the ideal gas law; substituting the given ratios directly gives 8:98:9 with no hidden assumptions beyond ideal-gas behaviour. Final Answer\textbf{Final Answer}: (D) 8:98:9.