Question

An atom \({}_3^8 X\) is bombarded with electrons, neutrons and protons and in 10 sec, 10 electrons, 10 protons and 9 neutrons are absorbed. If final surface area is x% of initial area, find x : -

• A. 250%
• B. 350%
• C. 225%
• D. 900%

Hint:

Always ignore electrons in nuclear physics problems unless they participate in Beta decay. Only nucleons (protons and neutrons) change the nuclear radius and mass number.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The radius of a nucleus depends on its mass number \(A\) according to the relation \(R = R_0 A^{1/3}\).
The surface area of the nucleus (assuming it to be spherical) is \(S = 4\pi R^2\), thus \(S \propto A^{2/3}\).
Step 2: Key Formula or Approach:
Ratio of surface areas: \(\frac{S_f}{S_i} = \left( \frac{A_f}{A_i} \right)^{2/3}\).
Step 3: Detailed Explanation:
Initial atom is \({}_3^8 X\). Mass number \(A_i = 8\).
The atom absorbs 10 protons and 9 neutrons. Electrons do not contribute to the mass of the nucleus.
Final mass number \(A_f = A_i + (\text{added protons}) + (\text{added neutrons})\).
\[ A_f = 8 + 10 + 9 = 27 \]
Now, calculate the ratio of the surface areas:
\[ \frac{S_f}{S_i} = \left( \frac{27}{8} \right)^{2/3} = \left[ \left( \frac{27}{8} \right)^{1/3} \right]^2 = \left( \frac{3}{2} \right)^2 = \frac{9}{4} \]
Converting this ratio to a percentage:
\[ x = \frac{S_f}{S_i} \times 100 = 2.25 \times 100 = 225% \]
Step 4: Final Answer:
The final surface area is 225% of the initial area.