Physics Q44 – JEE-MAIN 2025

For a nucleus of mass number

A

and radius

R

, the mass density of nucleus can be represented as

Clue 1 of 3: Nuclear Structure

Recall that the mass of a nucleus is proportional to its mass number $A$ , and its volume is related to its radius $R$ .

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Ninja Strategy•Constant Nuclear Density Principle

A fundamental property of nuclei is that their mass density is approximately constant for all nuclei, regardless of their mass number A.

🧠 Full Solution Path

Step 1: Express Nuclear Mass and Volume in terms of Mass Number A✦ Active

The mass of a nucleus ( $M$ ) is approximately proportional to its mass number ( $A$ ), so $M \approx A \cdot m_{p}$ , where $m_{p}$ is the average mass of a nucleon. The volume of a nucleus ( $V$ ) is given by the formula for a sphere: $V = \frac{4}{3} π R^{3}$ .

Step 2: Substitute the Empirical Nuclear Radius Formula○ Expand

The nuclear radius ( $R$ ) is empirically related to the mass number ( $A$ ) by $R = R_{0} A^{\frac{1}{3}}$ , where $R_{0}$ is a constant. Substitute this into the volume formula:

V = \frac{4}{3} π (R_{0} A^{\frac{1}{3}})^{3} = \frac{4}{3} π R_{0}^{3} A

Step 3: Calculate Nuclear Mass Density○ Expand

Now, calculate the mass density ( $ρ$ ) using the expressions for mass and volume:

ρ = \frac{M}{V} = \frac{A \cdot m_{p}}{\frac{4}{3} π R_{0}^{3} A}

The mass number $A$ cancels out, leaving:

ρ = \frac{m_{p}}{\frac{4}{3} π R_{0}^{3}}

Since $m_{p}$ and $R_{0}$ are constants, the mass density of the nucleus is approximately constant and independent of the mass number $A$ .

💡 Teacher's Secret Hint

Remember that nuclear matter is highly incompressible, leading to a nearly constant density for all nuclei.

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Physics Question 44 – JEE-MAIN 2025

Clue 1 of 3: Nuclear Structure

🧠 Full Solution Path