Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). 
Assertion (A): The density of the copper ($^{64}Cu$) nucleus is greater than that of the carbon ($^{12}C$) nucleus.
Reason (R): The nucleus of mass number A has a radius proportional to $A^{1/3}$. 
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. (A) is correct but (R) is not correct
• B. (A) is not correct but (R) is correct
• C. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• D. Both (A) and (R) are correct but (R) is not the correct explanation of (A)

Hint:

Nuclear density remains approximately constant for all nuclei. The formula \( R \propto A^{1/3} \) describes the relationship between the radius and the mass number, but does not affect the overall density of the nucleus.

Answer 1

The Correct Option is B

To determine the correctness of the Assertion (A) and the Reason (R), let's analyze both statements individually and in conjunction:

1. Assertion (A): The density of the copper ($^{64}Cu$) nucleus is greater than that of the carbon ($^{12}C$) nucleus.

The assertion is about the density comparison of copper and carbon nuclei. Nuclear density is considered to be approximately constant for all nuclei, regardless of their size or the element. This is because the volume of a nucleus is proportional to the number of nucleons \( A \) since radius \( r \) is proportional to \( A^{1/3} \). Hence, the density \( \rho \), which is mass per unit volume, turns out to be relatively constant. Therefore, the assertion that copper nucleus density is greater than carbon nucleus density is incorrect.

2. Reason (R): The nucleus of mass number \( A \) has a radius proportional to \( A^{1/3} \).

This statement is a well-established scientific fact in nuclear physics. The radius \( R \) of a nucleus can be defined as:

R = R_0 \cdot A^{1/3}

where \( R_0 \) is a constant. This relation indicates that the radius grows with the cube root of the mass number. The reason given here is indeed correct.

In summarizing:

• Assertion (A) is incorrect because nuclear density doesn't vary significantly between different nuclei and remains approximately constant across different elements.
• Reason (R) is correct because it accurately describes the relationship between the radius of a nucleus and its mass number.

Thus, the correct answer is:

(A) is not correct but (R) is correct.

Answer 2

Step 1: Understanding Assertion (A)
- The assertion states that the density of the copper nucleus is greater than that of the carbon nucleus. However, this is incorrect. The density of atomic nuclei is approximately constant across different elements, regardless of the specific element. This is due to the fact that nuclear density depends mainly on the nuclear force and not on the element. Hence, assertion (A) is not correct. 

Step 2: Understanding Reason (R)
- The radius of a nucleus is proportional to \(A^{1/3}\), where \(A\) is the mass number (total number of nucleons). This is a well-established empirical relation known as the "nuclear radius formula." This relation holds for all nuclei, including those of copper and carbon. 
Therefore, reason (R) is correct. 

Step 3: Connecting Assertion and Reason
- Although reason (R) is correct, it does not explain assertion (A) because the density of a nucleus does not depend on \(A^{1/3}\) in the way the assertion implies. The radius \(A^{1/3}\) only affects the volume, not the density in the way described in assertion (A). 
Therefore, reason (R) is correct, but it is not the explanation for assertion (A).