Given below are two statements: (A): The density of the copper (⁶⁴Cu₂₉) nucleus is greater than that of the carbon (¹²C₆) nucleus. (R): The nucleus of mass number A has a radius proportional to A¹/³.

❓ Question

Given below are two statements:

• Assertion (A): The density of the copper ($^{64}\text{Cu}_{29}$) nucleus is greater than that of the carbon ($^{12}\text{C}_6$) nucleus.

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

---

🖼️ Question Image

---

✍️ Short Solution

We need to check (1) whether A is true, (2) whether R is true, and (3) whether R explains A.

🔹 Step 1 — Is the Reason (R) true?

Empirically and from the liquid-drop model, the nuclear radius is given by

$$R=r_0{A}^{1\mathrm{/}3},$$

where $r_0\approx 1.2\text{ fm}$
So R is true.

---

🔹 Step 2 — Use R to examine the Assertion (A)

Density of a nucleus $\rho$ ≈ mass / volume.

• Mass $\propto A$ (number of nucleons).

• Volume $V\propto R^3\propto (A^{1\mathrm{/}3}{)}^3\propto A$

Thus

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

Therefore, to first approximation nuclear density is independent of $A$ — nuclei have roughly the same average density (≈ $2.3\times10^{17}\,\text{kg/m}^3$ or about $0.16$ nucleons/fm³). So the density of $^{64}\text{Cu}$ is not significantly greater than that of $^{12}\text{C}$; they are approximately the same. Hence Assertion (A) is false.

---

🔹 Step 3 — Does R explain A?

Although R is true and is used above to show that density scales out (so densities are roughly constant), it actually contradicts the idea in A that copper’s nuclear density is greater. So R does not support A. R is true but does not explain A (instead R shows why A is false).

---

🧮 Image Solution

---

✅ Conclusion & Video Solution

• Reason (R): True — nuclear radius $R=r_0A^{1/3}$
  

• Assertion (A): False — average nuclear density is approximately constant across nuclei, so copper’s nucleus is not appreciably denser than carbon’s.

• R does not explain A.

$$\boxed{{\displaystyle \text{R is true, A is false; R does not explain A.}\text{<br>}\text{<br>}\text{<br>}}}$$