Question

Model an atom as an electron in a rigid box of length $0.100\mathrm{nm}$, roughly twice the Bohr radius.

a. What are the four lowest energy levels of the electron?

b. Calculate all the wavelengths that would be seen in the emission spectrum of this atom due to quantum jumps between these four energy levels. Give each wavelength a label ${\mathrm{\lambda }}_{\mathrm{n}\to \mathrm{m}}$to indicate the transition.

c. Are these wavelengths in the infrared, visible, or ultraviolet portion of the spectrum?

d. The stationary states of the Bohr hydrogen atom have negative energies. The stationary states of this model of the atom have positive energies. Is this a physically significant difference? Explain.

e. Compare this model of an atom to the Bohr hydrogen atom. In what ways are the two models similar? Other than the signs of the energy levels, in what ways are they different?

Short answer

a.

$$\begin{gathered} {\mathrm{E}}_1=37.61\mathrm{eV} \\ {\mathrm{E}}_2=37.61\times 4=150.44\mathrm{eV} \\ {\mathrm{E}}_3=37.61\times 9=338.5\mathrm{eV} \\ {\mathrm{E}}_4=37.61\times 16=601.7\mathrm{eV} \end{gathered}$$

c.

localid="1650205122542" $$\begin{gathered} {\lambda }_{\mathit{1}}=1.1{\mathrm{A}}^0 \\ {\lambda }_2=0.66{\mathrm{A}}^0 \\ {\lambda }_3=0.4125{\mathrm{A}}^0 \\ {\lambda }_{\mathit{4}}\mathit{=}\mathit{0}\mathit{.}\mathit{275}\mathit{}{A}^{\mathit{0}} \end{gathered}$$

Step-by-step solution

Part (a) Step 1: Given information

We have given,

The box length =$0.1\mathrm{nm}$

We have to calculate the four energy levels of the electron.

Simplify

The energy of the one dimensional box is given by,

$\mathrm{E}=\frac{{\mathrm{n}}^2{\mathrm{h}}^2}{8{\mathrm{mL}}^2}$

Where

$$\begin{gathered} \mathrm{h}=6.632\times {10}^{-34}\mathrm{J}.\mathrm{s} \\ \mathrm{L}={10}^{-10}\mathrm{m} \end{gathered}$$

Then, energy level, n

$$\begin{gathered} \mathrm{E}=\frac{{\mathrm{n}}^2{(6.632\times {10}^{-34}\mathrm{J}.\mathrm{s})}^2}{8\times 9.1\times {10}^{-31}\mathrm{kg}\times {10}^{-20}{\mathrm{m}}^2} \\ \mathrm{E}=6.025\times {10}^{-18}{\mathrm{n}}^2\mathrm{J} \end{gathered}$$

Foe ground sate,

$$\begin{gathered} {\mathrm{E}}_1=6025\times {10}^{-187}\mathrm{J}=37.61\mathrm{eV} \\ {\mathrm{E}}_2=37.61\times 4=150.44\mathrm{eV} \\ {\mathrm{E}}_3=37.61\times 9=338.5\mathrm{eV} \\ {\mathrm{E}}_4=37.61\times 16=601.7\mathrm{eV} \end{gathered}$$

Part (b) Step 1: Given information

We have to plot the transition levels diagram.

Simplify

Part (c) Step 1: Given information

We have to find there wavelengths.

Simplify

Since we know that,

$$\begin{gathered} E=\frac{hc}{\lambda } \\ \lambda =\frac{hc}{E} \end{gathered}$$

Since,

$\mathrm{hc}=1240\mathrm{eV}.\mathrm{nm}$

then,

localid="1650204854787" $$\begin{gathered} {\lambda }_{\mathit{1}}=\frac{hc}{E_{\mathit{1}}} \\ {\lambda }_{\mathit{1}}=\frac{1240}{37.16} \\ {\lambda }_{\mathit{1}}=1.1{\mathrm{A}}^0 \\ {\lambda }_2=0.66{\mathrm{A}}^0 \\ {\lambda }_3=0.4125{\mathrm{A}}^0 \\ {\lambda }_{\mathit{4}}\mathit{=}\mathit{0}\mathit{.}\mathit{275}\mathit{}{A}^{\mathit{0}} \end{gathered}$$

yes, there are in the range of ultraviolet.

Part (d) Step 1: Given information

We have to find the physical significance of the positive energy.

Simplify

There is not have any physical significance.

Because model is correct as found energy is also in same pattern as in the Bohr's hydrogen atom.

Part (e) Step 1: Given information 

We have to find the difference between the one dimensional box and the Bohr hydrogen atom.

Simplify

There is just one different due to electron motion.

The motion of electron in hydrogen atom is circular about the nucleus.

But in the one dimensional box it is moving too and fro translational motion.