Before You Begin

Question #42159

Difficulty: 8/10 | 🟢 Free

The Question

A thermally isolated radioactive sample of initial mass $m_0$ and constant specific heat capacity $c$ is placed in a vacuum at absolute zero. The sample generates heat at a rate $P_{\text{in}}(t) = P_0 e^{-\lambda t}$ due to radioactive decay, and its mass decreases as $m(t) = m_0 e^{-\lambda t}$ . The sample loses energy via Stefan-Boltzmann radiation from its constant surface area $A$ , such that $P_{\text{out}}(t) = \sigma A T^4$ . The temperature $T$ of the sample is governed by the energy balance $P_{\text{in}}(t) = P_{\text{out}}(t) + m(t) c \frac{dT}{dt}$ . If the initial temperature of the sample is $0\text{ K}$ , what is the value of $\frac{d^2T}{dt^2}$ at $t=0$ ?

Options

A.
$0$
B.
$- \frac{\lambda P_0}{m_0 c}$
C.
$\frac{\lambda^2}{16} \left( \frac{P_0}{\sigma A} \right)^{1/4}$
D.
$\frac{\lambda P_0}{m_0 c}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Stefan-Boltzmann Power Law|Radioactive Decay Law Dynamics|Differential Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly set up and differentiate the energy balance equation for a system with time-varying mass and temperature.

The primary trap is treating the mass $m_0$ as a constant in the thermal capacity term $mc(dT/dt)$ .

If a student forgets to apply the product rule to $m(t) = m_0 e^{-\lambda t}$ , they will miss the $-\lambda m_0 c (dT/dt)$ term and incorrectly arrive at $-\lambda P_0 / (m_0 c)$ .

Another trap is assuming a quasi-steady state ( $P_{\text{in}} = P_{\text{out}}$ ), which ignores thermal inertia entirely.

By asking for the second derivative at $t=0$ , the problem elegantly bypasses an unsolvable non-linear integration while strictly enforcing the mathematical consequences of the decaying mass.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Stefan-Boltzmann Power Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free