Question

Find the ratio of minimum to maximum wavelength of radiations emitted when an electron jumps from higher energy state into ground state of hydrogen atom.

Hint:

To find the minimum and maximum wavelengths, use the Rydberg formula for transitions between energy levels. The maximum wavelength occurs for the transition \( n_2 = 2 \) to \( n_1 = 1 \), and the minimum wavelength occurs for a transition from \( n_2 = \infty \) to \( n_1 = 1 \).

Answer 1

The Rydberg formula describes the wavelengths of radiation emitted during electron transitions between energy levels in a hydrogen atom:\[\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\]where:- \( \lambda \) denotes the emitted radiation's wavelength,- \( R_H \) is the Rydberg constant for hydrogen (\( R_H = 1.097 \times 10^7 \ \text{m}^{-1} \)),- \( n_1 \) and \( n_2 \) represent the principal quantum numbers of the initial and final energy levels, respectively. Maximum Wavelength:This occurs during the transition from the first excited state (\( n_2 = 2 \)) to the ground state (\( n_1 = 1 \)):\[\frac{1}{\lambda_{\text{max}}} = R_H \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R_H \left( 1 - \frac{1}{4} \right) = \frac{3}{4} R_H\]Therefore,\[\lambda_{\text{max}} = \frac{4}{3 R_H}\] Minimum Wavelength:This corresponds to the transition from the highest possible energy state (\( n_2 \to \infty \)) to the ground state (\( n_1 = 1 \)):\[\frac{1}{\lambda_{\text{min}}} = R_H \left( \frac{1}{1^2} - \frac{1}{\infty^2} \right) = R_H\]Thus,\[\lambda_{\text{min}} = \frac{1}{R_H}\] Ratio of Minimum to Maximum Wavelength:The ratio is calculated as:\[\frac{\lambda_{\text{min}}}{\lambda_{\text{max}}} = \frac{\frac{1}{R_H}}{\frac{4}{3 R_H}} = \frac{3}{4}\] Final Answer: The ratio of the minimum to maximum wavelength is \( \boxed{\frac{3}{4}} \).