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Question #42598

Difficulty: 8/10 | 🟢 Free

The Question

A small solid sphere of density $\rho_s$ and radius $r$ is placed in a tall closed cylinder filled with an isothermal mixture of two ideal gases, A and B, at a constant temperature $T$ . The molar masses of the gases are $M_A$ and $M_B$ . The sphere is in static equilibrium at $z=0$ , where the partial pressures of the gases are $P_A$ and $P_B$ . The sphere is slightly displaced vertically. Assuming the gas viscosity $\eta$ is uniform and the drag force is given by Stokes' law, what is the critical radius $r_c$ of the sphere for which its perturbed motion is critically damped? (Assume the universal gas constant is $R$ and acceleration due to gravity is $g$ in the $-z$ direction)

Options

A.
$\left[ \frac{81 \eta^2 (RT)^2}{16 \rho_s g^2 (M_A^2 P_A + M_B^2 P_B)} \right]^{1/4}$
B.
$\left[ \frac{9 \eta^2 (RT)^2}{4 \rho_s g^2 (M_A^2 P_A + M_B^2 P_B)} \right]^{1/4}$
C.
$\left[ \frac{81 \eta^2 (RT)^2}{16 \rho_s g^2 (M_A P_A + M_B P_B)^2} \right]^{1/4}$
D.
$\left[ \frac{81 \eta^2 RT}{16 \rho_s g^2 (M_A^2 P_A + M_B^2 P_B)} \right]^{1/4}$

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Correct Answer

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Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Equation of State for Mixtures|Equilibrium Stability|Viscosity and Stokes' Law

Why This Question Is Tricky

Common Mistakes

This question tests the ability to couple the thermodynamic equation of state for a stratified gas mixture with the mechanical stability of a submerged object.

A common trap is assuming the mixture behaves as a single gas with an average molar mass when calculating the density gradient, which incorrectly leads to a dependence on $(M_A P_A + M_B P_B)^2$ instead of the sum of squares $M_A^2 P_A + M_B^2 P_B$ .

Furthermore, students must correctly linearize the buoyancy force to find the effective spring constant and apply the critical damping condition from the damped harmonic oscillator model, integrating Stokes' law as the damping term.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Equation of State for Mixtures
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free