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Question #42604

Difficulty: 8/10 | 🟢 Free

The Question

A thermally insulated rigid vessel contains an ideal diatomic gas (with active rotational but frozen vibrational degrees of freedom) at temperature $T$ . A microscopic hole is opened, allowing the gas to effuse into a vacuum. Let $N$ be the number of gas molecules remaining in the vessel. Which of the following expressions correctly relates the instantaneous fractional rate of change of temperature $\left(\frac{1}{T}\frac{dT}{dt}\right)$ to the fractional rate of change of the number of molecules $\left(\frac{1}{N}\frac{dN}{dt}\right)$ ?

Options

A.
$\frac{1}{T}\frac{dT}{dt} = \frac{1}{5}\frac{1}{N}\frac{dN}{dt}$
B.
$\frac{1}{T}\frac{dT}{dt} = 0$
C.
$\frac{1}{T}\frac{dT}{dt} = \frac{2}{5}\frac{1}{N}\frac{dN}{dt}$
D.
$\frac{1}{T}\frac{dT}{dt} = \frac{1}{7}\frac{1}{N}\frac{dN}{dt}$

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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|Mean vs. RMS Values|Energy Partitioning

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of energy partitioning and velocity distributions during effusion.

A common trap is assuming that effusing molecules carry away the same average energy as the molecules inside the container ( $\frac{5}{2}kT$ ), which incorrectly leads to the conclusion that the temperature remains constant ( $dT=0$ ).

However, because the effusion rate is proportional to molecular speed ( $v$ ), the velocity distribution of escaping molecules is skewed.

This shifts the average translational kinetic energy of an effusing molecule from $\frac{3}{2}kT$ to $2kT$ .

Combined with $1kT$ of rotational energy, each molecule removes $3kT$ , causing the remaining gas to cool.

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