JEE Physics 2025 Question with Solution

Given: Assertion (A): The density of the copper nucleus is greater than that of the carbon nucleus. Reason (R): The nucleus of mass number $A$ has radius proportional to $A^{1/3}$.

Find: Which option correctly describes the truth of Assertion (A) and Reason (R).

Nuclear radius is given by

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

So nuclear volume is proportional to

$$V \propto R^3 \propto \left(A^{1/3}\right)^3 \propto A$$

The mass of the nucleus is also proportional to $A$. Therefore nuclear density is

$$\rho = \frac{\text{mass}}{\text{volume}} \propto \frac{A}{A} = \text{constant}$$

Hence, nuclear density remains approximately the same for different nuclei.

So, the assertion that the copper nucleus has greater density than the carbon nucleus is incorrect.

The reason is correct because the radius of a nucleus does vary as $A^{1/3}$.

Therefore, (A) is not correct but (R) is correct. The correct option is B.

Given: Comparison of nuclear densities of $^{64}\text{Cu}$ and $^{12}\text{C}$, and the relation between nuclear radius and mass number.

Find: Whether Assertion (A) and Reason (R) are correct, and whether the reason explains the assertion.

The assertion claims that the density of the copper nucleus is greater than that of the carbon nucleus. This is incorrect because nuclear density is approximately constant for all nuclei.

The reason states that nuclear radius is proportional to $A^{1/3}$, which is a standard empirical relation:

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

Using this, the volume becomes proportional to $A$, and since nuclear mass is also proportional to $A$, density remains nearly constant.

Thus:

• Assertion (A) is false.
• Reason (R) is true.

Therefore, the correct option is B.