Light, Spectra and the Bohr Model: Quick Revision

A brief note on Light, Spectra and the Bohr Model. Read it before a test. Try each question at the end before you open its answer.

Light

• Wave: $c = \nu\lambda$, $\tilde{\nu} = \dfrac{1}{\lambda}$, with $c = 3.00 \times 10^{8}\ \text{m s}^{-1}$.
• Photon: $E = h\nu = \dfrac{1240}{\lambda\ (\text{nm})}$ eV.
• Photoelectric effect: $h\nu = W + KE_{\max}$. Brightness changes the number of electrons; frequency changes their energy.

The hydrogen spectrum

$\tilde{\nu} = R_H\left(\dfrac{1}{n_1^2} - \dfrac{1}{n_2^2}\right)$, with $R_H = 1.097 \times 10^{7}\ \text{m}^{-1}$.

Bohr's model

It works only for one-electron species: H, $\text{He}^+$, $\text{Li}^{2+}$.

A worked example

Find the energy and the wavelength of the photon from $\text{He}^+$ when its electron falls from $n = 5$ to $n = 3$.

1. Energy. $13.6 \times 4 \times \left(\dfrac{1}{9} - \dfrac{1}{25}\right) = 54.4 \times 0.0711 = 3.87\ \text{eV}$.
2. Wavelength. $\dfrac{1240}{3.87} = 320\ \text{nm}$, in the ultraviolet.

Practice with new numbers

A program to try

The program works out the worked example.

Mistakes to avoid

• Forgetting $Z^2$ for an ion such as $\text{He}^+$.
• Thinking brighter light gives faster electrons. Only the frequency does that.
• Mixing up the series. The series is named by the lower level, $n_1$.
• Using Bohr's model for helium atoms, which have two electrons.

Check yourself

1. Which has more energy per photon, red light or blue light?

2. A metal's work function is $2.0\ \text{eV}$. Photons of $3.5\ \text{eV}$ fall on it. What is the greatest kinetic energy of the electrons?

3. Which series of hydrogen lies in the ultraviolet?

4. What is the radius of the first orbit of $\text{He}^+$?