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