Answer :
The possible wavelengths that appear In the emission spectrum of the system are [tex]$210 \mathrm{~nm}, 250 \mathrm{~nm}, 410 \mathrm{~nm}, 620 \mathrm{~nm}, 1200 \mathrm{~nm}$[/tex].
Given:
The quantum energy levels allowed In the system are:
[tex]$\begin{aligned}& E_1=1.0 \mathrm{eV} \\& E_2=2.0 \mathrm{eV} \\& E_3=4.0 \mathrm{eV} \\& E_4=7.0 \mathrm{eV}\end{aligned}$[/tex]
What is wavelength of the emission?
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:
[tex]$E_f-E_i=\frac{h c}{\lambda}$[/tex]
Here, [tex]$E_f$[/tex] is the final energy level, [tex]$E_i$[/tex] is the Initlal energy level and [tex]$\lambda$[/tex] is the wavelength of emlsslon.
(1). Transition of electron from [tex]$E_1$[/tex] to [tex]$E_2$[/tex] energy level:
[tex]$\begin{aligned}E_2-E_1 & =\frac{h c}{\lambda} \\\lambda & =\frac{\left(6.63 \times 10^{-34}\right) \times\left(3 \times 10^8\right)}{(2-1)\left(1.6 \times 10^{-19}\right)} \mathrm{m} \\& =1.244 \times 10^{-6} \mathrm{~m} \\& \approx 1200 \mathrm{~nm}\end{aligned}$[/tex]
(2). Transition of electron from [tex]$E_1$[/tex] to [tex]$E_3$[/tex] energy level:
[tex]$\begin{aligned}E_3-E_1 & =\frac{h c}{\lambda} \\\lambda & =\frac{\left(6.63 \times 10^{-34}\right) \times\left(3 \times 10^8\right)}{(4-1)\left(1.6 \times 10^{-19}\right)} \mathrm{m} \\& =0.414 \times 10^{-6} \mathrm{~m} \\& \approx 410 \mathrm{~nm}\end{aligned}$[/tex]
(3). Transition of electron from [tex]$E_1$[/tex] to [tex]$E_4$[/tex] energy level:
[tex]$\begin{aligned}E_4-E_1 & =\frac{h c}{\lambda} \\\lambda & =\frac{\left(6.63 \times 10^{-31}\right) \times\left(3 \times 10^8\right)}{(7-1)\left(1.6 \times 10^{-19}\right)} \mathrm{m} \\& =0.207 \times 10^{-6} \mathrm{~m} \\& \approx 210 \mathrm{~nm}\end{aligned}$[/tex]
(4). Transition of electron from [tex]$E_2$[/tex] to [tex]$E_3$[/tex] energy level:
[tex]$\begin{aligned}E_3-E_2 & =\frac{h c}{\lambda} \\\lambda & =\frac{\left(6.63 \times 10^{-34}\right) \times\left(3 \times 10^8\right)}{(4-2)\left(1.6 \times 10^{-19}\right)} \mathrm{m} \\& =0.621 \times 10^{-6} \mathrm{~m} \\& \approx 620 \mathrm{~nm}\end{aligned}$[/tex]
(5). Transition of electron from[tex]$E_2$[/tex] to [tex]$E_4$[/tex] energy level:
[tex]$\begin{aligned}E_1-E_2 & =\frac{h c}{\lambda} \\\lambda & =\frac{\left(6.63 \times 10^{-34}\right) \times\left(3 \times 10^8\right)}{(7-2)\left(1.6 \times 10^{-19}\right)} \mathrm{m} \\& =0.248 \times 10^{-6} \mathrm{~m} \\& \approx 250 \mathrm{~nm}\end{aligned}$[/tex]
(6). Transition of electron from [tex]$E_3$[/tex] to [tex]$E_4$[/tex] energy level:
[tex]$\begin{aligned}E_4-E_3 & =\frac{h c}{\lambda} \\\lambda & =\frac{\left(6.63 \times 10^{-34}\right) \times\left(3 \times 10^8\right)}{(7-4)\left(1.6 \times 10^{-19}\right)} \mathrm{m} \\& =0.414 \times 10^{-6} \mathrm{~m} \\& \approx 410 \mathrm{~nm}\end{aligned}$[/tex]
Thus, The posslble wavelengths that appear in the emisslon spectrum of the system are[tex]$210 \mathrm{~nm}, 250 \mathrm{~nm}, 410 \mathrm{~nm}, 620 \mathrm{~nm}, 1200 \mathrm{~nm}$[/tex].
Complete question: 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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