Chapter 7, Problem 7.25P

Interpretation Introduction

Interpretation:

The wavelength $\left(\text{nm}\right)$ of the least energetic spectral line in the infrared series of the $\text{H}$ atom is to be determined.

Concept introduction:

An atom of hydrogen contains one electron. But the spectrum of hydrogen consists of a large number of lines. This is so because a sample of hydrogen contains a very large number of atoms. When energy is supplied to a sample of gaseous atoms of hydrogen, different atoms absorb different amounts of energy. Therefore, the electrons in different atoms jump to different energy levels. Upon losing the energies gained initially, the electrons jump back to lower energy levels and release radiations of different wavelengths.

The equation used to predict the position and wavelength of any line in a given series is called the Rydberg’s equation.

Rydberg’s equation is as follows:

$\frac{1}{\lambda }=R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$        (1)

Here,

$\lambda$ is the wavelength of the line.

$n_1$ and $n_2$ are positive integers, with ${\text{n}}_2>n_1$.

$R$ is the Rydberg’s constant.

The conversion factor to convert wavelength from $\text{nm}$ to $\text{m}$ is,

$1\text{nm}=1\times {10}^{-9}\text{ m}$