An electron in a hydrogen atom emits wavelengths during the transitions given below. Arrange these wavelengths in ascending order:
(A) from n = 3 to n = 2
(B) from n = 2 to n = 1
(C) from n = ∞ to n = 1
(D) from n = ∞ to n = 3
(E) from n = 4 to n = 1
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (C), (E), (B), (A), (D)
By the Rydberg formula, the reciprocal of the emitted wavelength is proportional to the energy difference of the two levels. A larger energy gap gives a shorter wavelength.
\( \displaystyle \frac{1}{\lambda} = R\left(\frac{1}{n_1^{2}} - \frac{1}{n_2^{2}}\right) \)
The bracket for each transition:
\( (A)\ 3 \rightarrow 2: \ 1/4 - 1/9 = 5/36 \approx 0.139 \)
\( (B)\ 2 \rightarrow 1: \ 1 - 1/4 = 0.75 \)
\( (C)\ \infty \rightarrow 1: \ 1 - 0 = 1 \)
\( (D)\ \infty \rightarrow 3: \ 1/9 - 0 \approx 0.111 \)
\( (E)\ 4 \rightarrow 1: \ 1 - 1/16 = 0.9375 \)
Ascending wavelength means descending value of the bracket:
\( 1\ (C) \gt 0.9375\ (E) \gt 0.75\ (B) \gt 0.139\ (A) \gt 0.111\ (D) \)
The order is C, E, B, A, D.