Question:
If the radius of a nucleus having \(13\) protons and \(14\) neutrons is \(3.6\,\text{fm}\), then the ratio of the volume to surface area of a nucleus having \(53\) protons and \(72\) neutrons is
If the radius of a nucleus having \(13\) protons and \(14\) neutrons is \(3.6\,\text{fm}\), then the ratio of the volume to surface area of a nucleus having \(53\) protons and \(72\) neutrons is
Show Hint
For a spherical nucleus,
\[
R=R_0A^{1/3}.
\]
Also,
\[
\frac{\text{Volume}}{\text{Surface Area}}
=
\frac{R}{3}.
\]
So once the nuclear radius is known, the required ratio can be obtained directly.
Updated On: Jul 9, 2026
- \(12\,\text{fm}\)
- \(3\,\text{fm}\)
- \(6\,\text{fm}\)
- \(2\,\text{fm}\) \bigskip
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The Correct Option is D
Solution and Explanation
Concept:
The nuclear radius is given by
\[
R=R_0A^{1/3},
\]
where \(A\) is the mass number.
Also,
\[
\text{Volume}=\frac{4}{3}\pi R^3,
\]
\[
\text{Surface Area}=4\pi R^2.
\]
Hence,
\[
\frac{\text{Volume}}{\text{Surface Area}}
=
\frac{\frac{4}{3}\pi R^3}{4\pi R^2}
=
\frac{R}{3}.
\]
Step 1: Find the radius constant \(R_0\). For the first nucleus, \[ A_1=13+14=27. \] Given, \[ R_1=3.6\,\text{fm}. \] Using \[ R_1=R_0A_1^{1/3}, \] \[ 3.6=R_0(27)^{1/3}. \] \[ 3.6=3R_0. \] \[ R_0=1.2\,\text{fm}. \]
Step 2: Find the radius of the second nucleus. For the second nucleus, \[ A_2=53+72=125. \] Therefore, \[ R_2 = R_0A_2^{1/3}. \] \[ R_2 = 1.2(125)^{1/3}. \] \[ R_2 = 1.2\times5. \] \[ R_2=6\,\text{fm}. \]
Step 3: Calculate the ratio of volume to surface area. \[ \frac{V}{S} = \frac{R_2}{3}. \] \[ \frac{V}{S} = \frac{6}{3}. \] \[ \frac{V}{S} = 2\,\text{fm}. \]
Step 4: Write the final answer. \[ \boxed{\frac{V}{S}=2\,\text{fm}} \] \[ \boxed{\text{Answer = (D)}} \]
Step 1: Find the radius constant \(R_0\). For the first nucleus, \[ A_1=13+14=27. \] Given, \[ R_1=3.6\,\text{fm}. \] Using \[ R_1=R_0A_1^{1/3}, \] \[ 3.6=R_0(27)^{1/3}. \] \[ 3.6=3R_0. \] \[ R_0=1.2\,\text{fm}. \]
Step 2: Find the radius of the second nucleus. For the second nucleus, \[ A_2=53+72=125. \] Therefore, \[ R_2 = R_0A_2^{1/3}. \] \[ R_2 = 1.2(125)^{1/3}. \] \[ R_2 = 1.2\times5. \] \[ R_2=6\,\text{fm}. \]
Step 3: Calculate the ratio of volume to surface area. \[ \frac{V}{S} = \frac{R_2}{3}. \] \[ \frac{V}{S} = \frac{6}{3}. \] \[ \frac{V}{S} = 2\,\text{fm}. \]
Step 4: Write the final answer. \[ \boxed{\frac{V}{S}=2\,\text{fm}} \] \[ \boxed{\text{Answer = (D)}} \]
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