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Question #43544

Difficulty: 8/10 | 🟢 Free

The Question

A crystalline solid has an interatomic potential well modeled by $U(x) = \frac{1}{2} k x^2 - \frac{1}{3} \gamma x^3$ , where $x = r - r_0$ is the displacement from the equilibrium interatomic distance $r_0$ at absolute zero. The constants $k$ and $\gamma$ are positive. At high temperatures, classical statistical mechanics applies. The macroscopic breaking stress $\sigma_b$ of the material under tension is determined by the maximum interatomic restoring force per unit cross-sectional area $r_0^2$ . If $\alpha$ is the linear coefficient of thermal expansion of the solid, which of the following expressions correctly represents $\sigma_b$ in terms of Boltzmann's constant $k_B$ , $\alpha$ , and $r_0$ ?

Options

A.
$\frac{k_B}{4 \alpha r_0^3}$
B.
$\frac{k_B}{2 \alpha r_0^3}$
C.
$\frac{3 k_B}{4 \alpha r_0^3}$
D.
$\frac{k_B}{\alpha r_0^3}$

Check Your Answer

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

Thermal Stress in Constrained Rods|Interatomic Force Curve|Breaking Stress Concept

Why This Question Is Tricky

Common Mistakes

This question targets the 'Lethal Collision' between macroscopic linear elasticity and microscopic anharmonicity.

Students typically treat thermal expansion ( $\alpha$ ) and breaking stress ( $\sigma_b$ ) as independent empirical constants, relying on the linear assumption $Y = \text{constant}$ .

By introducing the interatomic force curve, the student must realize that both thermal expansion and fracture stem from the exact same non-linear region of the potential well.

A common trap is attempting to use macroscopic linear formulas (like $\Delta L = \alpha L \Delta T$ or Stress = $Y \epsilon$ ) which fail near the rupture point.

The correct approach requires applying classical statistical mechanics to derive $\alpha$ from the asymmetric potential, and basic calculus to find the maximum restoring force, thereby bridging thermodynamics and solid mechanics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Thermal Stress in Constrained Rods
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free