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MCQEasyJEE 2025Atomic Mass & Binding Energy

JEE Physics 2025 Question with Solution

For a nucleus of mass number AA and radius RR, the mass density of the nucleus can be represented as:

  • A

    23A\frac{2}{3} A

  • B

    13A\frac{1}{3} A

  • C

    A3A^3

  • D

    Independent of AA

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: A nucleus has mass number AA and radius RR.

Find: How the mass density ρ\rho depends on AA.

The mass density of a nucleus is defined as:

ρ=mass of nucleusvolume of nucleus\rho = \frac{\text{mass of nucleus}}{\text{volume of nucleus}}

The mass of the nucleus is proportional to the mass number AA. The volume of a spherical nucleus is:

V=43πR3V = \frac{4}{3}\pi R^3

For a nucleus, the radius is related to mass number by:

R=R0A1/3R = R_0 A^{1/3}

where R0R_0 is a constant.

Substituting into the volume expression:

V=43π(R0A1/3)3=43πR03AV = \frac{4}{3}\pi (R_0 A^{1/3})^3 = \frac{4}{3}\pi R_0^3 A

Now substitute in the density formula:

ρ=A43πR03A\rho = \frac{A}{\frac{4}{3}\pi R_0^3 A}

So,

ρ=143πR03\rho = \frac{1}{\frac{4}{3}\pi R_0^3}

The factor AA cancels out, so the density does not depend on AA. Therefore, the correct option is D.

Scaling Argument

Given: Nuclear mass is proportional to AA and nuclear radius follows R∝A1/3R \propto A^{1/3}.

Find: Whether nuclear density changes with AA.

Since volume depends on radius cubed,

V∝R3V \propto R^3

Using R∝A1/3R \propto A^{1/3},

V∝(A1/3)3=AV \propto \left(A^{1/3}\right)^3 = A

Thus both mass and volume are proportional to AA. Hence,

ρ∝AA=1\rho \propto \frac{A}{A} = 1

So the nuclear mass density is constant and is independent of AA. Therefore, the correct option is D.

Common mistakes

  • Using ρ∝A\rho \propto A by considering only the mass and ignoring that the nuclear volume also increases with AA. Density depends on both mass and volume, so you must use ρ=MV\rho = \frac{M}{V}.

  • Forgetting the relation R=R0A1/3R = R_0 A^{1/3} and treating RR as independent of AA. This gives the wrong dependence of density on mass number.

  • Not cubing the radius properly while finding volume. Since V∝R3V \propto R^3, substituting R∝A1/3R \propto A^{1/3} gives V∝AV \propto A, not A1/3A^{1/3}.

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