Question 427: Atoms & Nuclei

Question

Which of the following transitions in hydrogen atom emit photons of highest frequency? ( $\mathrm{n}=$ principle quantum number)

Accepted Answer

The frequency of the emitted photon is directly proportional to the energy difference between the two levels:
$ E = h
u $
For the hydrogen atom, the energy of the $n^{	ext{th}}$ level is:
$ E_n = -13.6 \frac{1}{n^2} \, 	ext{eV} $
The energy difference during a transition from $n_i$ to $n_f$ ($n_i > n_f$) is:
$ \Delta E = E_{n_f} - E_{n_i} = -13.6 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} ight) 	ext{eV} $
Here, a larger value of $|\Delta E|$ means a higher frequency photon is emitted.
Now, let us calculate $|\Delta E|$ for each option:
Option A:
$\mathrm{n}=1$ to $\mathrm{n}=3$ (Actually, this is absorption, not emission, since $n$ increases.)
Option B:
$\mathrm{n}=2$ to $\mathrm{n}=4$ (Also absorption.)
Option C:
$\mathrm{n}=5$ to $\mathrm{n}=3$ (Emission, $n$ decreases.)
Option D:
$\mathrm{n}=2$ to $\mathrm{n}=1$ (Emission.)
Let us consider only emission (from higher $n$ to lower $n$): Options C and D.
Option C: $n_i = 5$ to $n_f = 3$
$ \Delta E = -13.6 \left( \frac{1}{3^2} - \frac{1}{5^2}ight) $
$ = -13.6 \left( \frac{1}{9} - \frac{1}{25}ight) $
$ = -13.6 \left( \frac{25-9}{225} ight) $
$ = -13.6 	imes \frac{16}{225} $
$ = -0.967 	ext{ eV} $
Since emission, take absolute value: $0.967$ eV.
Option D: $n_i = 2$ to $n_f = 1$
$ \Delta E = -13.6 \left( \frac{1}{1^2} - \frac{1}{2^2}ight) $
$ = -13.6 \left(1 - \frac{1}{4}ight) $
$ = -13.6 	imes \frac{3}{4} $
$ = -10.2 	ext{ eV} $
Absolute value: $10.2$ eV.
$10.2$ eV (Option D) is much greater than $0.967$ eV (Option C).
Answer:
Option D ($n=2$ to $n=1$) emits photons of the highest frequency.

Answer

$\mathrm{n}=1$ to $\mathrm{n}=3$

Answer

$\mathrm{n}=2$ to $\mathrm{n}=4$

Answer

$\mathrm{n}=5$ to $\mathrm{n}=3$

Answer

$\mathrm{n}=2$ to $\mathrm{n}=1$

Atoms & Nuclei

Which of the following transitions in hydrogen atom emit photons of highest frequency? ( \mathrm{n}= principle quantum number)

Choose the correct answer: