Step 1: Understanding the Concept:
The time period (\(T\)) of an electron's revolution is defined as the time taken to complete one full orbit.
In Bohr's theory, the physical parameters of the electron change as it transitions between different energy levels (shells).
We know that the time period is given by the ratio of the distance traveled to the speed:
\[ T = \frac{2\pi r}{v} \]
This means we need to know how the radius \(r\) and velocity \(v\) scale with the quantum number \(n\).
Step 2: Key Formula or Approach:
From the Bohr model postulates:
1. Radius \(r_n \propto n^2\).
2. Velocity \(v_n \propto 1/n\).
Substituting these into the time period formula:
\[ T_n \propto \frac{n^2}{1/n} \implies T_n \propto n^3 \]
Step 3: Detailed Explanation:
The problem provides a relationship between the initial time period (\(T_1\)) and the final time period (\(T_2\)):
\[ T_1 = 8 T_2 \]
Using the derived proportionality (\(T \propto n^3\)):
\[ \frac{T_1}{T_2} = \left( \frac{n_1}{n_2} \right)^3 \]
Substitute the given value:
\[ 8 = \left( \frac{n_1}{n_2} \right)^3 \]
Taking the cube root of both sides:
\[ \sqrt[3]{8} = \frac{n_1}{n_2} \]
\[ 2 = \frac{n_1}{n_2} \implies n_1 = 2 \cdot n_2 \]
This implies that the initial quantum number must be exactly double the final quantum number.
Let's evaluate the given options:
(A) \(n_1 = 8, n_2 = 4 \implies 8 = 2 \cdot 4\) (Correct)
(B) \(n_1 = 4, n_2 = 8 \implies 4 \neq 2 \cdot 8\) (Incorrect)
(C) \(n_1 = 8, n_2 = 2 \implies 8 \neq 2 \cdot 2\) (Incorrect)
(D) \(n_1 = 2, n_2 = 8 \implies 2 \neq 2 \cdot 8\) (Incorrect)
Thus, option A is the only choice that fits the calculated ratio.
Step 4: Final Answer:
The possible values for the transition are \(n_1 = 8\) and \(n_2 = 4\).