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Question #42579

Difficulty: 8/10 | 🟢 Free

The Question

A vertical cylindrical tank of height $H$ contains an ideal gas of molar mass $M$ in hydrostatic equilibrium under a uniform gravitational field $g$ . The temperature of the gas decreases linearly with height. The RMS speed of the gas molecules at the bottom ( $y=0$ ) is $v_0$ , and at the top ( $y=H$ ) is $v_H$ . The pressure at the top of the tank is $P_H$ . A small orifice is opened at the bottom of the tank, allowing the gas to escape into an external environment maintained at pressure $P_H$ . Assuming the efflux velocity is given by $v_e = \sqrt{\frac{2 \Delta P}{\rho_0}}$ , where $\Delta P$ is the pressure difference across the orifice and $\rho_0$ is the density of the gas at the bottom, which of the following expressions correctly represents $v_e$ ?

Options

A.
$v_0 \sqrt{\frac{2}{3} \left[ 1 - \left( \frac{v_H}{v_0} \right)^{\frac{6 g H}{v_0^2 - v_H^2}} \right]}$
B.
$\sqrt{2 g H}$
C.
$v_0 \sqrt{\frac{2}{3} \left[ 1 - \exp\left(-\frac{3 g H}{v_0^2}\right) \right]}$
D.
$v_0 \sqrt{1 - \left( \frac{v_H}{v_0} \right)^{\frac{4 g H}{v_0^2 - v_H^2}}}$

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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

RMS vs Average vs Most Probable Speed|Torricelli’s Efflux Velocity|Hydrostatic Paradox

Why This Question Is Tricky

Common Mistakes

Students often incorrectly apply Torricelli's law ( $v = \sqrt{2gH}$ ) to gases, ignoring compressibility and the variable density of the atmospheric column.

Furthermore, when attempting to find the pressure difference, they may falsely assume an isothermal profile ( $P = P_0 e^{-\rho g h/P}$ ) or fail to link the macroscopic temperature gradient to the microscopic RMS speeds ( $T \propto v_{rms}^2$ ).

The trap lies in assuming pressure depends only on head height ( $P = h\rho g$ ) rather than the integral of the distribution function over a linearly varying temperature profile, leading to incorrect pressure ratios.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	RMS vs Average vs Most Probable Speed
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free