JEE Physics 2023 Question with Solution

Given: An assertion about the order of nuclear densities of different nuclides, and the relation $R = R_0 A^{1/3}$.

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

For a nucleus, volume is proportional to $R^3$. Using

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

we get

$$V \propto R^3 \propto A$$

Since nuclear mass is also proportional to $A$, the nuclear density is

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

So, nuclear density is approximately independent of mass number $A$.

Hence, the assertion giving a specific order of nuclear densities is false.

The reason statement

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

is true.

Therefore, the correct option is B: A is false but R is true.

Given: $R = R_0 A^{1/3}$.

Find: Whether this supports the assertion about different nuclear densities.

Using the spherical nucleus model,

$$V = \frac{4}{3}\pi R^3$$

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

$$V = \frac{4}{3}\pi \left(R_0 A^{1/3}\right)^3$$ $$V = \frac{4}{3}\pi R_0^3 A$$

Thus volume is directly proportional to $A$.

Also, nuclear mass is proportional to the number of nucleons, so

$$M \propto A$$

Therefore,

$$\rho = \frac{M}{V} \propto \frac{A}{A} = \text{constant}$$

So nuclei such as $^{10}_{5}B$, $^{6}_{3}Li$, $^{56}_{26}Fe$, $^{20}_{10}Ne$, and $^{209}_{83}Bi$ have nearly the same nuclear density. Hence the asserted ordering is incorrect.

The solution concludes that the correct option is B.