Before You Begin

Question #43810

Difficulty: 8/10 | 🟢 Free

The Question

According to the Liquid Drop Model, a heavy nucleus of mass number $A$ and atomic number $Z$ can be modeled as a spherical drop of incompressible nuclear fluid of radius $R = R_0 A^{1/3}$ . The strong nuclear force creates an effective surface tension $\gamma$ and an internal pressure $P_{\text{int}}$ to maintain mechanical equilibrium. The protons are assumed to be uniformly distributed throughout the nuclear volume, creating an electrostatic Coulomb energy $E_c = \frac{3}{5} \frac{(Ze)^2}{4\pi \epsilon_0 R}$ . Which of the following expressions gives the correct internal pressure $P_{\text{int}}$ required to maintain mechanical equilibrium of the nucleus?

Options

A.
$\frac{2\gamma}{R_0 A^{1/3}} - \frac{3 Z^2 e^2}{80 \pi^2 \epsilon_0 R_0^4 A^{4/3}}$
B.
$\frac{2\gamma}{R_0 A^{1/3}} + \frac{3 Z^2 e^2}{80 \pi^2 \epsilon_0 R_0^4 A^{4/3}}$
C.
$\frac{2\gamma}{R_0 A^{1/3}} - \frac{9 Z^2 e^2}{80 \pi^2 \epsilon_0 R_0^4 A^{4/3}}$
D.
$\frac{2\gamma}{R_0 A^{1/3}}$

Check Your Answer

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

Nuclear Radius Scaling|Excess Pressure (Laplace Law)|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to adapt macroscopic thermodynamic and fluid mechanics concepts (Laplace pressure) to a quantum/nuclear system (Liquid Drop Model).

The primary trap is blindly applying the standard Laplace excess pressure formula ( $\Delta P = 2\gamma/R$ ) without accounting for the outward 'negative pressure' exerted by the Coulomb repulsion of the protons.

A secondary trap involves incorrectly calculating the Coulomb pressure by simply dividing the Coulomb energy by the volume ( $E_c/V$ ) rather than taking the proper thermodynamic volume derivative ( $-\partial E_c/\partial V$ ).

This highlights the misconception between energy density and thermodynamic pressure in systems with non-linear volume dependencies.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Radius Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free