Question

Monochromatic radiation of wavelength $\lambda$ is incident on a hydrogen sample containing ground state. Hydrogen atoms absorb the light and subsequently emit radiations of ten different wavelengths. The value $\lambda$ of is:
A. $95nm$
B. $103nm$
C. $73nm$
D. $88nm$

Answer 1

The Rydberg formula is a mathematical expression used to determine the wavelength of light emitted by an electron moving in between the energy levels of an atom. Each element has a unique line emission spectrum. The hydrogen spectrum consists of several series of lines

Formula used:
$\dfrac{{n\left( {n - 1} \right)}}{2}$
$\dfrac{1}{\lambda } = R\left( {\dfrac{1}{{{n_f}^2}} - \dfrac{1}{{{n_i}^2}}} \right)$

Complete step-by-step answer:
Spectral lines are produced by transitions of electrons within the atoms or ions. When an electron jumps from ${n^{th}}$ orbit to ground state, total $\dfrac{{n\left( {n - 1} \right)}}{2}$ number of spectral lines are formed. Each element has its own unique line emission of spectrum.
Since, it is given in the question that hydrogen atoms absorb the light and subsequently emit radiations of ten different wavelengths.
So, number of spectral lines emitted are,
$\Rightarrow \dfrac{{n\left( {n - 1} \right)}}{2} = 10$
$\Rightarrow n = 5$
Now, using Rydberg’s formula,
$\dfrac{1}{\lambda } = R\left( {\dfrac{1}{{{n_f}^2}} - \dfrac{1}{{{n_i}^2}}} \right)$ ; where ${n_1}$ and ${n_2}$ are integers and ${n_2}$ is always greater than ${n_1}$. $R$ is a constant called Rydberg constant.
Substituting the value of $n = 5$ in Rydberg formula we get,
$\dfrac{1}{\lambda } = R\left[ {\dfrac{1}{{{1^2}}} - \dfrac{1}{{{5^2}}}} \right]$
$\Rightarrow \dfrac{1}{\lambda } = R\left[ {\dfrac{{25 - 1}}{{25}}} \right]$
$\Rightarrow \dfrac{1}{\lambda } = \dfrac{{25}}{{24 \times 109677}}cm$
$\Rightarrow \lambda = 95nm$

So, the correct answer is “Option A”.

Additional Information: Splitting of spectral lines are of two types:
Stark effect: the splitting of spectral lines in the presence of electric fields is known as Stark effect.
Zeeman effect: the splitting of spectral lines in the presence of magnetic fields is known as Zeeman effect.

Note: When an electron jumps from an orbital with high energy to a lower energy state, a photon of light is generated. A photon of light gets absorbed by the atom when the electron shifts from lower energy to a higher energy state.