Question

Show that of hydrogen’s spectral series—Lyman, Balmer, Paschen, and so on—only the four Balmer lines of Section 3 are in visible range $\left(400-700nm\right)$

Short answer

It is shown that only the first four lines of the Balmer series are in the visible range, that is between $\left(400-700\text{nm}\right)$.

Step-by-step solution

Given data

The frequency range of the visible spectrum is $\left(400-700\text{nm}\right)$.

Wavelength of Hydrogen lines

The wavelength of emission when electron jumps from orbit $n_1$ to $n_2$orbit is

role="math" localid="1659966880476" $\mathit{\lambda }\mathbf{=}{\mathit{R}}^{\mathbf{-}\mathbf{1}}{\mathbf{(}\frac{\mathbf{1}}{{\mathbf{n}}_{\mathbf{1}}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{1}}{{\mathbf{n}}_{\mathbf{2}}^{\mathbf{2}}}\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}.....\left(I\right)$

Here R is the Rydberg constant of value

$\mathit{R}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{097}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}\mathbf{}{\mathit{m}}^{\mathbf{-}\mathbf{1}}$

Determining the wavelength of Hydrogen lines in the visible range

For Lyman series $n_1=1$. The maximum wavelength obtained from Lyman series is when $n_2=2$the value of which from equation (I) is

$$\begin{gathered} \lambda ={\left(1.097\times {10}^7{\text{m}}^{\text{- 1}}\right)}^{-1}{\left(\frac{1}{1^2}-\frac{1}{2^2}\right)}^{-1} \\ =121.5\text{nm} \end{gathered}$$

This is less than the visible range. So Lyman series is not in the visible range.

For Paschen series $n_1=3$. The minimum wavelength obtained from Paschen series is when $n_2=\infty$the value of which from equation (I) is

$$\begin{gathered} \lambda ={\left(1.097\times {10}^7{\text{m}}^{\text{- 1}}\right)}^{-1}{\left(\frac{1}{3^2}-\frac{1}{\infty }\right)}^{-1} \\ =820.4\text{nm} \end{gathered}$$

This is greater than the visible range. So Paschen series is not in the visible range. Thus only the Balmer series can be in the visible range.

For Balmer series $n_1=2$. The wavelength of the first line obtained from the Balmer series is when $n_2=3$the value of which from equation (I) is

$$\begin{gathered} \lambda ={\left(1.097\times {10}^7{\text{m}}^{\text{- 1}}\right)}^{-1}{\left(\frac{1}{2^2}-\frac{1}{3^2}\right)}^{-1} \\ =656.3\text{nm} \end{gathered}$$

The wavelength of the second line obtained from the Balmer series is when $n_2=4$the value of which from equation (I) is

$$\begin{gathered} \lambda ={\left(1.097\times {10}^7{\text{m}}^{\text{- 1}}\right)}^{-1}{\left(\frac{1}{2^2}-\frac{1}{4^2}\right)}^{-1} \\ =486.2\text{nm} \end{gathered}$$

The wavelength of the third line obtained from the Balmer series is when $n_2=5$ the value of which from equation (I) is

$$\begin{gathered} \lambda ={\left(1.097\times {10}^7{\text{m}}^{\text{- 1}}\right)}^{-1}{\left(\frac{1}{2^2}-\frac{1}{5^2}\right)}^{-1} \\ =434.1\text{nm} \end{gathered}$$

The wavelength of the fourth line obtained from the Balmer series is when $n_2=6$the value of which from equation (I) is

$$\begin{gathered} \lambda ={\left(1.097\times {10}^7{\text{m}}^{\text{- 1}}\right)}^{-1}{\left(\frac{1}{2^2}-\frac{1}{6^2}\right)}^{-1} \\ =410.2\text{nm} \end{gathered}$$

The wavelength of the fifth line obtained from the Balmer series is when $n_2=7$the value of which from equation (I) is

$$\begin{gathered} \lambda ={\left(1.097\times {10}^7{\text{m}}^{\text{- 1}}\right)}^{-1}{\left(\frac{1}{2^2}-\frac{1}{7^2}\right)}^{-1} \\ =397\text{nm} \end{gathered}$$

Thus only the first four lines are in the visible range