Question

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$
Assertion (A): The nuclear density of nuclides ${ }_5^{10} B ,{ }_3^6 Li ,{ }_{26}^{56} Fe ,{ }_{10}^{20} Ne$ and ${ }_{83}^{209} Bi$ can be arranged as $\rho_{ Bi }^N>\rho_{ Fe }^N>\rho_{ Ne }^N>\rho_{ B }^N>\rho_{ Li }^N$.
Reason R: The radius $R$ of nucleus is related to its mass number $A$ as $R=R_0 A^{1 / 3}$, where $R_0$ is a constant.
In the light of the above statements, choose the correct answer from the options given below

• A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A$
• B. $A$ is false but $R$ is true
• C. Both $A$ and $R$ are true but $R$ is NOT the correct explanation of $A$
• D. A is true but $R$ is false

Answer 1

The Correct Option is B

The question requires us to evaluate the correctness of the assertion and reason given, as well as determine the correct relationship between different nuclides' nuclear densities. 

1. Understanding the Assertion ($A$): The assertion claims that the nuclear densities of the nuclides ${}_5^{10} B$, ${}_3^6 Li$, ${}_{26}^{56} Fe$, ${}_{10}^{20} Ne$, and ${}_{83}^{209} Bi$ can be arranged in the order: \(\rho_{Bi}^N > \rho_{Fe}^N > \rho_{Ne}^N > \rho_{B}^N > \rho_{Li}^N\).
2. Nuclear Density: The nuclear density is given by \(\rho^N = \frac{m}{V}\), where \(m\) is the mass and \(V\) is the volume of the nucleus. For a nucleus, the volume is proportional to \(R^3\), where \(R\) is the radius defined by the formula:
   • \(R = R_0 A^{1/3}\)
     • \(A\) is the mass number.
     • \(R_0\) is a constant.
3. The formula implies that nuclear density \(\rho^N\) is approximately constant for most nuclei, as both \(m\) and \(V\) are functions of the mass number \(A\) which cancel each other out (assuming the strong nuclear force gives rise to a roughly constant nuclear density regardless of the nucleus size).
4. Evaluating the Assertion ($A$): Since the nuclear density is roughly constant across nuclei, the nuclear densities can't be distinctly ranked like normal densities can. Therefore, the assertion $A$ is false.
5. Understanding the Reason ($R$): The radius \(R\) of a nucleus is indeed related to its mass number \(A\) by the formula \(R = R_0 A^{1/3}\). Thus, the reason $R$ is true and is a well-known equation in nuclear physics.
6. Conclusion: Option "$A$ is false but $R$ is true" is the correct choice. Although the reason is accurate, it doesn't explain why the assertion is wrong.