The Atom: Rutherford, Bohr and Spectra: Quick Revision

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

Rutherford's atom

• Experiment: most alpha particles go straight through gold, a few turn a lot, and about 1 in 8000 turns back.
• Meaning: all the positive charge and nearly all the mass sit in a tiny nucleus.
• Closest approach (head-on): $r_0 = \dfrac{2Ze^2}{4\pi\varepsilon_0 K} = \dfrac{2Z \times 1.44}{K}$ fm, with $K$ in MeV.
• Angle: $\tan\dfrac{\theta}{2} = \dfrac{r_0}{2b}$. At $b = \dfrac{r_0}{2}$, $\theta = 90^\circ$.
• Problem: an orbiting electron accelerates and should radiate and spiral in.

Bohr's model

• Rules: stationary orbits, $mvr = n\hbar$, and photons from jumps, $h\nu = E_i - E_f$.
• Radius: $r_n = \dfrac{0.0529\ \text{nm}\,n^2}{Z}$.
• Speed: $v_n = \dfrac{2.19 \times 10^6\ \text{m/s}\,Z}{n}$.
• Energy: $E_n = -\dfrac{13.6\ \text{eV}\,Z^2}{n^2}$, with $K = -E$ and $U = 2E$.
• de Broglie: $2\pi r = n\lambda$ gives $mvr = \dfrac{nh}{2\pi}$.
• Limits: one-electron atoms only, and no line intensities.

Spectra of hydrogen

• Formula: $\dfrac{1}{\lambda} = RZ^2\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)$, with $R = 1.097 \times 10^7\ \text{m}^{-1}$.
• Series: Lyman 1 (ultraviolet), Balmer 2 (visible), Paschen 3, Brackett 4, Pfund 5 (infrared).
• Limit: $\lambda = \dfrac{n_f^2}{R}$, the shortest wavelength of a series.
• Lines from the level $n$: $\dfrac{n(n-1)}{2}$.

A worked example

The ion $\text{Li}^{2+}$ has $Z = 3$. Find the photon from the jump $n = 5$ to $n = 2$.

1. Energy. $E = 13.6 \times 9 \times \left(\dfrac{1}{4} - \dfrac{1}{25}\right)\ \text{eV} = 25.7\ \text{eV}$.
2. Wavelength. $\lambda = \dfrac{1240\ \text{eV nm}}{25.7\ \text{eV}} = 48.2\ \text{nm}$, in the ultraviolet.

Practice with new numbers

A program to try

The program works through the example above.

Mistakes to avoid

• Leaving out the charge $2e$ of the alpha particle in $r_0$.
• Forgetting $Z^2$ in the energies of an ion.
• Using $\dfrac{1}{n}$ in place of $\dfrac{1}{n^2}$ in the formula for the lines.
• Taking the first line of a series as the shortest. The limit is the shortest.

Check yourself

1. What is the closest approach when the energy of the alpha particles doubles?

2. What are the potential energy and the total energy in the first orbit of hydrogen, if the kinetic energy is $13.6\ \text{eV}$?

3. Which series of hydrogen has lines in the visible region?

4. How many different lines can atoms in the level $n = 6$ give out, in all?