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Question #42231

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The Question

A closed vertical cylinder of height $H$ is filled with a mixture of two ideal gases, A and B, having molar masses $M$ and $2M$ respectively. The system is in hydrostatic equilibrium in a uniform downward gravitational field $g$ . The temperature of the gas mixture varies with height $z$ from the base as $T(z) = T_0(1 + \alpha z)$ , where $T_0$ and $\alpha$ are positive constants. If the partial pressures of gases A and B at the base ( $z=0$ ) are both equal to $P_0$ , which of the following expressions gives the total pressure of the gas mixture at height $z=H$ ? (Here $R$ is the universal gas constant).

Options

A.
$P_0 \left[ (1+\alpha H)^{-\frac{Mg}{\alpha R T_0}} + (1+\alpha H)^{-\frac{2Mg}{\alpha R T_0}} \right]$
B.
$2P_0 (1+\alpha H)^{-\frac{3Mg}{2\alpha R T_0}}$
C.
$P_0 \left[ (1+\alpha H)^{-\frac{Mg}{R T_0}} + (1+\alpha H)^{-\frac{2Mg}{R T_0}} \right]$
D.
$2P_0 \exp\left(-\frac{3MgH}{2 R T_0 (1+\alpha H)}\right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Pascal’s Law Limitations|Dalton's Law of Partial Pressures|Approximation (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the application of hydrostatic equilibrium and Dalton's Law in a non-isothermal, multi-component gas mixture.

The primary trap (Option B) anticipates students treating the gas mixture as a single fluid with a constant average molar mass ( $1.5M$ ) calculated at the base.

However, because the gases have different molar masses, they settle independently according to their own density gradients, causing the mixture's composition and average molar mass to vary continuously with height.

Students must realize that the hydrostatic equation $dP/dz = -\rho g$ must be integrated for each partial pressure separately, coupling it with the local temperature $T(z)$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Pascal’s Law Limitations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free