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Question #43754

Difficulty: 8/10 | 🟢 Free

The Question

A thermally insulated cylinder with a massless, frictionless piston contains an ideal diatomic gas ( $f=5$ ). Inside the cylinder, a photoelectric cathode of work function $\phi = 2.0 \text{ eV}$ is illuminated by monochromatic light of wavelength $\lambda = 310 \text{ nm}$ . A small collection voltage maintains a saturation photoelectric current $I_s = 14.0 \text{ mA}$ but does negligible work on the electrons. Assume all emitted photoelectrons possess the maximum possible kinetic energy and transfer this energy entirely to the gas molecules via inelastic collisions before reaching the anode. The gas expands isobarically against a constant external pressure $P_0 = 100 \text{ kPa}$ . Calculate the rate of change of the gas volume, $\frac{dV}{dt}$ . (Take $hc = 1240 \text{ eV}\cdot\text{nm}$ )

Options

A.
80 \text{ mm}^3/\text{s}
B.
280 \text{ mm}^3/\text{s}
C.
112 \text{ mm}^3/\text{s}
D.
160 \text{ mm}^3/\text{s}

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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

Photoelectric Saturation|Molecular Degrees of Freedom|Isothermal Process Work

Why This Question Is Tricky

Common Mistakes

This question tests the coupling between quantum optical energy transfer and macroscopic thermodynamics.

The 'Lethal Collision' occurs when students misapply the First Law of Thermodynamics by treating the slow expansion as a purely isothermal process (assuming $\Delta U = 0$ ), incorrectly equating the photoelectric power input directly to the rate of work done ( $P_{\text{in}} = P_0 \frac{dV}{dt}$ ).

This ignores the energy absorbed by the molecular degrees of freedom and yields the trap answer of 280 mm³/s.

Furthermore, it tests the precise definition of the energy transferred: students must use the kinetic energy of the photoelectrons ( $K_{\text{max}}$ ), not the total incident photon energy (which leads to 160 mm³/s), and correctly apply the degrees of freedom ( $f=5$ ) for a diatomic gas in an isobaric process.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Photoelectric Saturation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free