Question

What is the minimum wavelength in the Lyman series of the hydrogen spectrum?

• A. \(91.2\,\text{nm}\)
• B. \(121.6\,\text{nm}\)
• C. \(656.3\,\text{nm}\)
• D. \(364.6\,\text{nm}\)

Hint:

Remember the limiting wavelengths of the important hydrogen series: \[ \begin{aligned} \text{Lyman limit} &\approx 91.2\,nm,\\ \text{Balmer limit} &\approx 364.6\,nm. \end{aligned} \] The minimum wavelength of any spectral series is obtained by taking \[ \boxed{n_{2}\rightarrow\infty.} \]

Answer 1

The Correct Option is A

Concept: The hydrogen spectrum consists of several spectral series that arise due to the transition of an electron from a higher energy level to a lower energy level. The wavelength of the emitted radiation is given by the

Rydberg Formula, \[ \frac{1}{\lambda} = R\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right), \] where \[ R=1.097\times10^{7}\;m^{-1} \] is the Rydberg constant, \[ n_{1}=\text{lower energy level}, \] \[ n_{2}=\text{higher energy level}, \qquad n_{2}\gt n_{1}. \] For the

Lyman series, \[ n_{1}=1, \] and the electron falls from higher energy levels to the ground state. The

minimum wavelength (also called the series limit) is obtained when \[ n_{2}\rightarrow\infty. \]

Step 1: Write the Rydberg formula for the Lyman series.
For the Lyman series, \[ n_{1}=1. \] Hence, \[ \frac{1}{\lambda} = R \left( 1-\frac{1}{n_{2}^{2}} \right). \]

Step 2: Find the limiting wavelength.
The minimum wavelength corresponds to the highest possible transition, \[ n_{2}\rightarrow\infty. \] Therefore, \[ \frac{1}{n_{2}^{2}} = 0. \] Substituting, \[ \frac{1}{\lambda_{\min}} = R. \] Thus, \[ \lambda_{\min} = \frac{1}{R}. \]

Step 3: Substitute the value of the Rydberg constant.
Using, \[ R = 1.097\times10^{7}\;m^{-1}, \] we get, \[ \lambda_{\min} = \frac{1}{1.097\times10^{7}}. \] Therefore, \[ \lambda_{\min} = 9.12\times10^{-8}\;m. \] Converting into nanometres, \[ 1\,nm=10^{-9}\,m, \] hence, \[ \lambda_{\min} = 91.2\,nm. \] Thus, \[ \boxed{\lambda_{\min}=91.2\,nm.} \] Therefore, the correct answer is \[ \boxed{\textbf{Option (A)}}. \]

Additional Information: The important hydrogen spectral series are listed below: \[ \begin{array}{|c|c|c|} \hline \textbf{Series} & \mathbf{n_{1}} & \textbf{Region} \\ \hline \text{Lyman} & 1 & \text{Ultraviolet} \\ \hline \text{Balmer} & 2 & \text{Visible} \\ \hline \text{Paschen} & 3 & \text{Infrared} \\ \hline \text{Brackett} & 4 & \text{Infrared} \\ \hline \text{Pfund} & 5 & \text{Infrared} \\ \hline \end{array} \] The first line of the Lyman series has wavelength \[ 121.6\,nm, \] while the limiting (minimum) wavelength is \[ 91.2\,nm. \]