30.3 Bohr’s Theory of the Hydrogen Atom

A hypothetical one-electron atom in its highest excited state can only emit photons of energy 2E, 3E, and 5E before reaching the ground state. Which of the following represents the complete set of energy levels for this atom?

1. 0, 3E, 5E
2. 0, 2E, 3E
3. 0, 2E, 3E, 5E
4. 0, 5E, 8E, 10E

The ground state of a certain type of atom has energy –E₀. What is the wavelength of a photon with enough energy to ionize an atom in the ground state and give the ejected electron a kinetic energy of 2E₀?

1. $\frac{h c}{3 E_{0}}$
2. $\frac{h c}{2 E_{0}}$
3. $\frac{h c}{E_{0}}$
4. $\frac{2 h c}{E_{0}}$

By calculating its wavelength, show that the first line in the Lyman series is UV radiation.

Look up the values of the quantities in $a_{\text{B}} = \frac{h^{2}}{\left(4 \pi\right)^{2} m_{e} \text{kq}_{e}^{2}}$, and verify that the Bohr radius $a_{\text{B}}$ is $\text{0}.\text{529} \times \text{10}^{- \text{10}} \text{ m}$.

If a hydrogen atom has its electron in the $n = 4$ state, how much energy in eV is needed to ionize it?

Find the radius of a hydrogen atom in the $n = 2$ state according to Bohr’s theory.

What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?

Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.

A wavelength of $4 . \text{653 }\mu\text{m}$ is observed in a hydrogen spectrum for a transition that ends in the $n_{\text{f}} = 5$ level. What was $n_{\text{i}}$ for the initial level of the electron?

A beryllium ion with a single electron (denoted $\text{Be}^{3 + }$) is in an excited state with radius the same as that of the ground state of hydrogen.

(a) What is $n$ for the $\text{Be}^{3 + }$ ion?

(b) How much energy in eV is needed to ionize the ion from this excited state?

Verify Equations $r_{n} = \frac{n^{2}}{Z} a_{\text{B}}$ and $a_{B} = \frac{h^{2}}{\left(4 \pi\right)^{2} m_{e} \text{kq}_{e}^{2}} = \text{0}.\text{529} \times \text{10}^{- \text{10}} \text{ m}$ using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization.