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Question #42593

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The Question

In a Young's Double Slit Experiment, a transparent tube of length $L$ is placed in front of the upper slit. The tube is initially divided into two compartments of lengths $L_1$ and $L_2$ ( $L_1 + L_2 = L$ ) by a thin, transparent partition. Compartment 1 contains Gas A (molar mass $M_1$ ) at pressure $P_1$ , with a refractive index $n_1$ . Compartment 2 contains Gas B (molar mass $M_2$ ) at pressure $P_2$ , with a refractive index $n_2$ . Both gases are ideal, non-reacting, and maintained at a constant temperature $T$ . The refractive index $n$ of each gas satisfies the Gladstone-Dale relation $(n-1) \propto \rho$ , where $\rho$ is the density of the gas. The partition is suddenly removed, allowing the gases to mix completely. Let $\mu = \frac{M_1 M_2}{M_1 + M_2}$ be the reduced mass of the gas pair. If $\lambda$ is the wavelength of the light used, what is the magnitude of the shift in the central interference maximum (in number of fringes) after the mixing is complete?

Options

A.
0
B.
$\frac{|(n_1 - 1)L_2 - (n_2 - 1)L_1|}{\lambda}$
C.
$\frac{(n_1 - 1)L_1 + (n_2 - 1)L_2}{\lambda} \left( \frac{\mu}{M_1 + M_2} \right)$
D.
$\frac{|n_1 - n_2| L_1 L_2}{\lambda (L_1 + L_2)}$

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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

Dalton's Law of Partial Pressures|Reduced Mass Effect|Phase and Path Difference

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the optical path length in gaseous mixtures and the application of Dalton's Law of partial pressures alongside the Gladstone-Dale relation.

A common trap is assuming that the non-linear mixing of molar masses (often incorrectly invoking reduced mass or average molar mass for density calculations) alters the total optical path.

Students frequently fail to realize that the optical path length $(n-1)L$ is directly proportional to the column density (total number of molecules per unit area in the light path), which remains strictly conserved when the partition is removed.

The inclusion of the reduced mass $\mu$ serves as a red herring to bait students into complex, erroneous algebraic manipulations instead of relying on the superposition principle of partial pressures and polarizabilities.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Dalton's Law of Partial Pressures
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free