Question

How many spectral lines are seen for hydrogen atoms when electrons jump from $n_2 = 5\;$ to $n_1 = 1\;$ in a visible region?

Answer 1

Hint : In order to solve this question we must know few basic information like what is meant by spectral lines. Spectral series are said to be the set of wavelengths arranged in a sequential fashion which characterizes light or any electromagnetic radiation emitted by the energised atoms.

Complete Step By Step Answer:
Hydrogen atom is said to be the simplest atomic system found in nature, thus it produces the simplest of the spectral series.
When electrons de-excite from higher energy level ( $n_2$ ) to lower energy level ( $n_1$ ) in atomic sample, then number of spectral line observed in the spectrum is given by the formula;
$Total{\text{ }}no{\text{ }}of{\text{ }}the{\text{ }}spectral{\text{ }}lines\; = \dfrac{{\left( {n_2 - n_1} \right)\left( {n_2 - n_1 + 1} \right)}}{2}$
Since it is given in the question that in hydrogen atom electron jumps from $n_2 = 5\;$ to $n_1 = 1\;$
Therefore the total no of the spectral line is found to be;
$Total{\text{ }}no{\text{ }}of{\text{ }}the{\text{ }}spectral{\text{ }}lines\; = \dfrac{{\left( {5 - 1} \right)\left( {5 - 1 + 1} \right)}}{2}$
$Total{\text{ }}no{\text{ }}of{\text{ }}the{\text{ }}spectral{\text{ }}lines\; = 10$
So, the ten lines are;
$5 \to 4,5 \to 3,5 \to 2,5 \to 1,4 \to 3,4 \to 2,4 \to 1,3 \to 2,3 \to 1,2 \to 1\;$ are possible in this case.
Balmer series is displayed when electron transition takes place from higher energy states ( ${{\mathbf{n}}_{\mathbf{h}}} = {\mathbf{3}},{\mathbf{4}},{\mathbf{5}},{\mathbf{6}},{\mathbf{7}}, \ldots$ ) to ${{\mathbf{n}}_{\mathbf{l}}} = {\mathbf{2}}$ energy state.
For Balmer series, ${n_l} = 2$ and ${n_h} = 3,4,5$
Thus,
$5 \to 2,4 \to 2,3 \to 2\;$ it shows three lines lie in the visible region.
Hence the final answer is three spectral lines lie in the visible region for the hydrogen atom when electrons jump from $n_2 = 5\;$ to $n_1 = 1\;$ in the visible region.

Note :
All the wavelength of the Balmer series falls in the visible part of the electromagnetic spectrum( $400nm{\text{ }}to{\text{ }}740nm$ ). Other series are lyman series, paschen series, bracket series, pfund series, Humphreys series. The series is observed at a higher wavelength. The spectral lines are extremely faint and widely spread out. They correspond to highly rare atomic events.