In a hydrogen atom an electron makes a transition from \( n_1 \rightarrow n_2 \), where \( n_1 \) and \( n_2 \) are the principal quantum numbers of these two states. Assuming Bohr's model to be valid, if the time period of electron in the initial state is eight times of that in the final state, the possible values of \( n_1 \) and \( n_2 \) are:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( n_1 = 8,\ n_2 = 4 \)
In Bohr's model the radius of the \( n \)th orbit grows as \( n^{2} \) and the speed of the electron falls as \( 1/n \). The time period is the time for one revolution:
\( \displaystyle T = \frac{2\pi r_n}{v_n} \propto \frac{n^{2}}{1/n} = n^{3} \)
\( \displaystyle \frac{T_1}{T_2} = \left(\frac{n_1}{n_2}\right)^{3} = 8 \;\Rightarrow\; \frac{n_1}{n_2} = 2 \)
Only \( n_1 = 8,\ n_2 = 4 \) has \( n_1/n_2 = 2 \).