Chapter 7, Problem 7.117QP

Interpretation Introduction

Interpretation:

An equation that relates ${\text{λ}}_{\text{1}}{\text{ to λ}}_{\text{2}}{\text{ and λ}}_{\text{3}}$ in which an electron in an excited state in a hydrogen atom can return to the ground state either by direct transition or by an intermediate excited state should be derived.

Concept Introduction:

A wave is a disturbance or variation that travels through a medium transporting energy without transporting matter.  The wavelength is the distance between identical points on successive waves.  The frequency is the number of waves that pass through any particular point in 1 second.

Figure 1

The speed, wavelength and frequency of a wave are related by the equation: $\text{c }=\text{ λν}$ where $\lambda \text{ and }\nu$ are expressed in meters ($\text{m}$) and reciprocal seconds (${\text{s}}^{-1}$) respectively.  Hence, rearranging the equation for getting frequency is

$\text{ν }=\frac{\text{c}}{\text{λ}}$

Planck’s quantum theory

1. 1. Different atoms and molecules can emit or absorb energy in discreet quantities only.  The smallest amount of energy that can be emitted or absorbed in the form of electromagnetic radiation is known as quantum.
2. 2. The energy of the radiation absorbed or emitted is directly proportional to the frequency of the radiation.  The energy of radiation is expressed in terms of frequency as,

$\text{E = hν}$

Where,

$\text{E}$ = energy of the radiation

$\text{h}$ = Planck’s constant ($6.626\times {10}^{–34}\text{ Js}$)

$\text{ν}$ = Frequency of radiation

Substituting the frequency formula in this equation,

$\text{E }=\frac{\text{hc}}{\text{λ}}$