Question

Model a hydrogen atom as an electron in a cubical box with side length L. Set the value of L so that the volume of the box equals the volume of a sphere of radius, he Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.

Short answer

The energy separation between the ground and the first excited levels for the cubical box, $∆\mathrm{E}=2.49\times {10}^{-17}\mathrm{J}=155\mathrm{eV}$and the Bohr model is$∆\mathrm{E}=10.2\mathrm{eV}$ .

The energy gap between the ground level and the first excited state in the cubical box is much larger than the Bohr model predicts.

Step-by-step solution

Energy levels of the hydrogen atom

In a cubical box with side length L, the allowed energies for a particle of mass m are given by;

In the Bohr model, the energy levels of the hydrogen atom are given by;

The energy separation between the ground and first excited levels

Given,

Bohr radius is $a_0=5.29\times {10}^{-11}m$

The mass of an electron is $m_e=9.11\times {10}^{-31}kg$

For a cube with side length L and volume equal to a sphere with radius r,

$$\begin{gathered} V_{cube}=V_{sphere} \\ {L}^3=\frac{4}{3}\pi r^3 \\ {L}^3={\left(\frac{4\pi }{3}\right)}^{1/3}r \end{gathered}$$

As an alternative to $\mathrm{r}={\mathrm{a}}_0$, consider the side length of a cubical box with the same volume as a sphere of radius ${\mathrm{a}}_0$

The combination $\left(n_x,n_y,n_z\right)=\left(1,1,1\right)$corresponds to the ground level.

Therefore,

Any of the combinations corresponds to the first excited level.

Therefore,

The energy separation between the ground level and the first excited state can be calculated using the equation:

Now, putting the numbers together;

In the Bohr model, the energy separation between the ground level n=1 and the first excited state n=2 is given by the equation;

Hence, the energy gap between the ground level and the first excited state in the cubical box is much larger than the Bohr model predicts.