Physics, 29.07.2019 21:30 HMBELITE

The allowed energies of a quantum system are 1.0 ev, 2.0 ev, 4.0 ev, and 7.0 ev. what wavelengths appear in the system's emission spectrum

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Answer from: kezin

The possible wavelengths that appear in the emission spectrum of the system are $\boxed{210\,{\text{nm,}}\,{\text{250}}\,{\text{nm,}}\,{\text{410}}\,{\text{nm,}}\,{\text{620}}\,{\text{nm,}}\,{\text{1200}}\,{\text{nm}}}$ .

Further Explanation:

Given:

The quantum energy levels allowed in the system are:

$\begin{aligned}{E_1}&=1.0\,{\text{eV}}\hfill\\{E_2}&=2.0\,{\text{eV}}\hfill\\{E_3}&=4.0\,{\text{eV}}\hfill\\{E_4}&=7.0\,{\text{eV}}\hfill\\\end{aligned}$

Concept:

As the transition of electron takes from one energy level to another, there is an emission of particular wavelength from the transition. The relation between the wavelength of the emission and the energy of the energy level is expressed as:

$\boxed{{E_f}-{E_i}=\frac{{hc}}{\lambda }}$

Here, ${E_f}$ is the final energy level, ${E_i}$ is the initial energy level and $\lambda$ is the wavelength of emission.

(1). Transition of electron from ${E_1}$ to ${E_2}$ energy level:

$\begin{aligned}{E_2}-{E_1}&=\frac{{hc}}{\lambda }\\\lambda &=\frac{{\left({6.63\times{{10}^{-34}}}\right)\times\left({3\times{{10}^8}}\right)}}{{\left({2-1}\right)\left({1.6\times{{10}^{-19}}}\right)}}\,{\text{m}}\\&{\text{=1}}{\text{.244}}\times{\text{1}}{{\text{0}}^{-6}}\,{\text{m}}\\&\approx{\text{1200}}\,{\text{nm}}\\\end{aligned}$

(2). Transition of electron from ${E_1}$ to ${E_3}$ energy level:

$\begin{aligned}{E_3} - {E_1}&=\frac{{hc}}{\lambda }\\\lambda &=\frac{{\left({6.63\times{{10}^{-34}}}\right)\times\left({3\times{{10}^8}}\right)}}{{\left({4-1}\right)\left({1.6\times{{10}^{-19}}}\right)}}\,{\text{m}}\\&{\text{=0}}{\text{.414}}\times{\text{1}}{{\text{0}}^{-6}}\,{\text{m}}\\&\approx410\,{\text{nm}}\\\end{aligned}$

(3). Transition of electron from ${E_1}$ to ${E_4}$ energy level:

$\begin{aligned}{E_4} - {E_1}&=\frac{{hc}}{\lambda }\\\lambda &=\frac{{\left({6.63\times{{10}^{-34}}}\right)\times\left({3\times{{10}^8}}\right)}}{{\left({7-1}\right)\left({1.6\times{{10}^{-19}}}\right)}}\,{\text{m}}\\&{\text{=0}}{\text{.207}}\times{\text{1}}{{\text{0}}^{-6}}\,{\text{m}}\\&\approx210\,{\text{nm}}\\\end{aligned}$

(4). Transition of electron from ${E_2}$ to ${E_3}$ energy level:

$\begin{aligned}{E_3} - {E_2}&=\frac{{hc}}{\lambda }\\\lambda &=\frac{{\left({6.63\times{{10}^{-34}}}\right)\times\left({3\times{{10}^8}}\right)}}{{\left({4-2}\right)\left({1.6\times{{10}^{-19}}}\right)}}\,{\text{m}}\\&{\text{=0}}{\text{.621}}\times{\text{1}}{{\text{0}}^{-6}}\,{\text{m}}\\&\approx620\,{\text{nm}}\\\end{aligned}$

(5). Transition of electron from ${E_2}$ to ${E_4}$ energy level:

$\begin{aligned}{E_4} - {E_2}&=\frac{{hc}}{\lambda }\\\lambda &=\frac{{\left({6.63\times{{10}^{-34}}}\right)\times\left({3\times{{10}^8}}\right)}}{{\left({7-2}\right)\left({1.6\times{{10}^{-19}}}\right)}}\,{\text{m}}\\&{\text{=0}}{\text{.248}}\times{\text{1}}{{\text{0}}^{-6}}\,{\text{m}}\\&\approx250\,{\text{nm}}\\\end{aligned}$

(6). Transition of electron from ${E_3}$ to ${E_4}$ energy level:

$\begin{aligned}{E_4} - {E_3}&=\frac{{hc}}{\lambda }\\\lambda &=\frac{{\left({6.63\times{{10}^{-34}}}\right)\times\left({3\times{{10}^8}}\right)}}{{\left({7-4}\right)\left({1.6\times{{10}^{-19}}}\right)}}\,{\text{m}}\\&{\text{=0}}{\text{.414}}\times{\text{1}}{{\text{0}}^{-6}}\,{\text{m}}\\&\approx410\,{\text{nm}}\\\end{aligned}$

Thus, The possible wavelengths that appear in the emission spectrum of the system are $\boxed{210\,{\text{nm,}}\,{\text{250}}\,{\text{nm,}}\,{\text{410}}\,{\text{nm,}}\,{\text{620}}\,{\text{nm,}}\,{\text{1200}}\,{\text{nm}}}$ .

Learn More:

1. Calculate the wavelength of an electron (m = 9.11 × 10-28 g) moving at 3.66 × 106 m/s

2. Microwave ovens emit microwave energy with a wavelength of 12.5 cm. What is the energy of exactly one photon of this microwave radiation

3. What is the frequency of light for which the wavelength is 7.1 × 102 nm

Answer Details:

Grade: College

Subject: Physics

Chapter: Modern Physics

Keywords:

Allowed energies, quantum system, energy levels, wavelengths appear, emission spectrum, 1.0eV, 2.0eV, 4.0eV, 7.0eV, transition of electrons.

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