Bohr's Atomic Model

From the scattering of alpha-particles by a thin gold foil, Rutherford concluded:

• The atom has a very small, dense, positively charged nucleus at its centre that holds almost all the mass.
• Electrons revolve around the nucleus; most of the atom is empty space.

It failed on two counts, both of which Bohr later resolved:

• Stability — a revolving electron is accelerating, so by classical electromagnetism it must continuously radiate energy, spiral inward and collapse into the nucleus. Atoms are in fact stable.
• Line spectrum — a spiralling electron would emit a continuous spectrum, but hydrogen gives a discrete line spectrum.

Bohr's postulates for a hydrogen-like atom:

• The electron revolves only in certain stationary orbits of fixed energy; while in an orbit it does not radiate energy.
• The angular momentum is quantized: $mvr = \dfrac{nh}{2\pi}$, i.e. an integer multiple of $\dfrac{h}{2\pi}$.
• Energy is absorbed or emitted only when the electron jumps between orbits, with $\Delta E = h\nu$.
• So for the $n$th orbit the angular momentum is simply $L = \dfrac{nh}{2\pi}$ — it grows in equal steps of $\dfrac{h}{2\pi}$ and is **independent of $Z$**.

Radius of the $n$th orbit of a hydrogen-like species:

• $r_n = 0.529\,\dfrac{n^2}{Z}\ \text{\AA} = 52.9\,\dfrac{n^2}{Z}\ \text{pm}$.
• The first orbit of hydrogen ($n=1,\ Z=1$) is the Bohr radius, $a_0 = 0.529\ \text{\AA} = 52.9\ \text{pm}$.
• Radius increases as $n^2$ and decreases as $1/Z$; so a higher charge pulls the electron into a tighter orbit.
• Unit reminder: $1\ \text{\AA} = 100\ \text{pm} = 10^{-10}\ \text{m}$.

Energy of the electron in the $n$th orbit of a hydrogen-like species:

• $E_n = -13.6\,\dfrac{Z^2}{n^2}\ \text{eV} = -2.18 \times 10^{-18}\,\dfrac{Z^2}{n^2}\ \text{J}$.
• Per mole: $E_n = -1312\,\dfrac{Z^2}{n^2}\ \text{kJ mol}^{-1}$.
• The energy is always negative (a bound electron); its magnitude falls as $1/n^2$ and rises as $Z^2$.
• For hydrogen ($Z=1,\ n=1$) the ground state is $-13.6\ \text{eV} = -2.18 \times 10^{-18}\ \text{J}$; the constant $R_H = 2.18 \times 10^{-18}\ \text{J}$ is this same number.

Speed of the electron in the $n$th orbit of a hydrogen-like species:

• $v_n = 2.18 \times 10^{6}\,\dfrac{Z}{n}\ \text{m s}^{-1}$.
• Velocity increases with $Z$ and decreases with $n$ (as $1/n$, not $1/n^2$).
• For hydrogen's first orbit ($Z=1,\ n=1$) this gives $2.18 \times 10^{6}\ \text{m s}^{-1}$.
• It follows from the quantization rule: since $mvr = \dfrac{nh}{2\pi}$ and $r \propto \dfrac{n^2}{Z}$, the $n^2$ cancels one $n$ to leave $v \propto \dfrac{Z}{n}$.

Transitions and ionization for a hydrogen-like species:

• Energy gap between orbits $n_1$ and $n_2$: $\Delta E = E_{n_2} - E_{n_1} = 13.6\,Z^2\left(\dfrac{1}{n_1^2} - \dfrac{1}{n_2^2}\right)\ \text{eV}$ (positive when absorbed).
• Ionization energy = energy to take the electron from its orbit to $n = \infty$ (where $E = 0$): $\text{I.E.} = E_\infty - E_n = 0 - E_n = +13.6\,\dfrac{Z^2}{n^2}\ \text{eV}$.
• From the ground state ($n=1$): $\text{I.E.} = 13.6\,Z^2\ \text{eV}$. For hydrogen this is 13.6 eV ($= 2.18 \times 10^{-18}\ \text{J}$).
• The emitted/absorbed photon carries $\Delta E = h\nu = \dfrac{hc}{\lambda}$.

Hydrogen-like (single-electron) species — the only ones the formulas apply to:

• Must have exactly one electron: $\text{H}$ (Z=1), $\text{He}^+$ (Z=2), $\text{Li}^{2+}$ (Z=3), $\text{Be}^{3+}$ (Z=4), $\text{B}^{4+}$ (Z=5).
• $\text{Li}^{+}$ has two electrons, so it is NOT hydrogen-like.

What Bohr's model could NOT explain (its failures):

• The spectra of multi-electron atoms (only hydrogen works).
• The finer details (splitting) of even the hydrogen spectrum.
• The Zeeman effect (splitting of lines in a magnetic field) and the Stark effect (in an electric field).
• The ability of atoms to form chemical bonds (molecules).

Note the contrast with Rutherford's model, which could not describe the energies or arrangement of electrons at all — that gap is exactly what Bohr filled.