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

To solve the problem, we need to determine the percentage change in the surface area of the atom when it absorbs additional electrons, neutrons, and protons. Here's the step-by-step explanation:

1. Initially, the atom \({}_3^8X\) has the atomic number \( Z = 3 \) and mass number \( A = 8 \). This indicates it contains 3 protons and 5 neutrons (\(8 - 3 = 5\)) and, normally, 3 electrons.
2. Upon absorption:
   • The atom absorbs 10 electrons, resulting in \(3 + 10 = 13\) electrons.
   • The atom absorbs 10 protons, leading to \(3 + 10 = 13\) protons. Thus, the new atomic number becomes \(13\).
   • The atom absorbs 9 neutrons, yielding \(5 + 9 = 14\) neutrons. Therefore, the new mass number is \(13 + 14 = 27\).
3. After these absorptions, the new element becomes \({}_{13}^{27}X'\).
4. Assuming the atom is a sphere, the surface area \(S\) is proportional to the square of the atomic radius \(r\):

   \(S \propto r^2\)

5. The radius of a nucleus approximately depends on \(A^{1/3}\), where \(A\) is the atomic mass number. The ratio of the new radius to the old radius is:

   \(\frac{r_{\text{new}}}{r_{\text{old}}} = \left(\frac{A_{\text{new}}}{A_{\text{old}}}\right)^{1/3} = \left(\frac{27}{8}\right)^{1/3}\)

6. Calculating the cube root:

   \(\left(\frac{27}{8}\right)^{1/3} = \frac{3}{2}\)

7. Thus, the new surface area compared to the old one is:

   \(\left(\frac{r_{\text{new}}}{r_{\text{old}}}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}\)

8. The percentage increase in surface area is calculated as:

   \(x = \frac{9}{4} \times 100\% = 225\%\)

9. Therefore, the final surface area is 225% of the initial area. Hence, the correct answer is 225%.