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Question #42601

Difficulty: 8/10 | 🟢 Free

The Question

An ideal gas of molecular mass $m$ is maintained at a constant pressure $P$ and a measured temperature $T$ inside a large container. The gas effuses through a small hole of area $A$ into a vacuum. The temperature measurement has a small uncertainty $\Delta T$ . Assuming the gas follows the Maxwell-Boltzmann velocity distribution, which of the following expressions correctly represents the absolute uncertainty $\Delta \Phi$ in the effusion rate $\Phi$ (defined as the total number of molecules escaping per unit time)? (Here $k$ is the Boltzmann constant)

Options

A.
$\frac{P A \Delta T}{4 T} \sqrt{\frac{2}{\pi m k T}}$
B.
$\frac{P A \Delta T}{8 T} \sqrt{\frac{3}{m k T}}$
C.
$\frac{P A \Delta T}{2 T} \sqrt{\frac{2}{\pi m k T}}$
D.
$\frac{P A \Delta T}{4 T} \sqrt{\frac{3}{m k T}}$

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Correct Answer

Step-by-Step Solution

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

Effusion and Diffusion|Maxwell-Boltzmann Distribution|Relative Error Propagation in Powers

Why This Question Is Tricky

Common Mistakes

This question tests the precise formulation of the effusion rate from the Maxwell-Boltzmann distribution and the subsequent error propagation.

A common trap is to incorrectly use the root-mean-square speed ( $v_{rms} = \sqrt{3kT/m}$ ) instead of the mean speed ( $\langle v \rangle = \sqrt{8kT/\pi m}$ ) for effusion, leading to an incorrect constant factor.

Additionally, students may forget the $1/2$ factor from differentiating the $T^{-1/2}$ dependence or fail to account for the temperature dependence of the number density $n = P/kT$ at constant pressure.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Effusion and Diffusion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free