Bohr's Model for Hydrogen Atom

Why Bohr Reworked the Atom

Rutherford's nuclear model placed the electron in orbit around a tiny dense nucleus, but classical electromagnetism predicted that an orbiting (accelerating) charge must continuously radiate energy and spiral into the nucleus within a fraction of a second. That model could not explain why atoms are stable, nor why excited hydrogen emits light only at certain discrete frequencies rather than a continuous band.

When an electric discharge is passed through gaseous hydrogen, the $\ce{H2}$ molecules dissociate and the energetically excited $\ce{H}$ atoms emit electromagnetic radiation of discrete frequencies. Passed through a prism, this light separates into a set of sharp lines — the hydrogen line spectrum — rather than a rainbow. Bohr's central achievement was to make a model that produces exactly such a discrete spectrum.

The Four Postulates

Bohr's model for the hydrogen atom rests on four postulates. The first two introduce stationary states; the third connects energy gaps to radiation; the fourth quantises angular momentum, which is the engine that produces all the quantitative results.

The angular momentum used in postulate 4 follows from treating the electron of mass $m_e$ on a circular path of radius $r$ as a rotating body. With moment of inertia $I = m_e r^2$ and angular velocity $\omega = v/r$,

$$\text{angular momentum} = I \times \omega = m_e r^2 \times \frac{v}{r} = m_e v r.$$

Quantising this quantity — allowing it to take only the values $n h / 2\pi$ — is the single radical assumption. Radiation is emitted or absorbed only when the electron jumps from one quantised value of angular momentum to another, so Maxwell's classical electromagnetic theory does not apply and only certain fixed orbits are allowed. This idea grew directly out of Planck's quantisation of energy; for the experimental clues that pushed physics toward it, see developments leading to Bohr's model.

Radius of the Stationary States

Bohr's theory gives the radii of the allowed orbits of hydrogen as

$$r_n = n^2 a_0, \qquad a_0 = 52.9\ \text{pm}.$$

The first stationary state ($n = 1$), called the Bohr orbit, has radius 52.9 pm, and the hydrogen electron is normally found here. As $n$ increases the radius increases as $n^2$, so the electron sits progressively farther from the nucleus. The stationary states are numbered $n = 1, 2, 3, \dots$; these integers are the principal quantum numbers.

Energy of the Stationary States

The most important property of a stationary state is its energy. For hydrogen,

$$E_n = -R_H\left(\frac{1}{n^2}\right), \qquad R_H = 2.18 \times 10^{-18}\ \text{J}.$$

Here $R_H$ is the Rydberg constant expressed as an energy. The ground state ($n = 1$) is the lowest and most stable: $E_1 = -2.18 \times 10^{-18}\,(1/1^2) = -2.18 \times 10^{-18}$ J. For $n = 2$, $E_2 = -2.18 \times 10^{-18}\,(1/2^2) = -0.545 \times 10^{-18}$ J. In electron-volts the same expression reads $E_n = -13.6/n^2$ eV, so $E_1 = -13.6$ eV.

Hydrogen-like Ions

Bohr's theory works for any species that has just one electron — the so-called hydrogen-like ions such as $\ce{He+}$, $\ce{Li^2+}$ and $\ce{Be^3+}$. The energy and radius then depend on the atomic number $Z$:

$$E_n = -2.18 \times 10^{-18}\left(\frac{Z^2}{n^2}\right)\ \text{J}, \qquad r_n = 52.9\left(\frac{n^2}{Z}\right)\ \text{pm}.$$

As $Z$ increases, the energy becomes more negative and the radius becomes smaller, so the electron is bound more tightly to the more highly charged nucleus. The velocity of the orbiting electron, by contrast, increases with $Z$ and decreases with $n$.

Explaining the Line Spectrum

The triumph of the model is that it explains the hydrogen line spectrum quantitatively. Energy is absorbed when the electron jumps from a lower orbit to a higher one, and emitted when it falls from a higher orbit to a lower one. The gap between two orbits is

$$\Delta E = E_f - E_i = R_H\left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right) = 2.18 \times 10^{-18}\left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right)\ \text{J}.$$

By the Bohr frequency rule the photon's frequency is $\nu = \Delta E / h$, which works out as

$$\nu = 3.29 \times 10^{15}\left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right)\ \text{Hz}.$$

Dividing by the speed of light gives the wavenumber form, which is exactly the empirical formula Johannes Rydberg had fitted to the data years earlier:

$$\bar{\nu} = 1.09677 \times 10^{7}\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\ \text{m}^{-1}.$$

The value $109677\ \text{cm}^{-1}$ is the Rydberg constant for hydrogen. Because the right-hand side can take only specific values (the $n$'s are integers), only specific photon energies are possible — and those are exactly the sharp lines that are observed. For absorption $n_f > n_i$ and the bracket is positive; for emission $n_i > n_f$ and $\Delta E$ is negative, meaning energy is released.

The Spectral Series

The lines group themselves into series, each defined by the orbit $n_f$ at which the electron lands. Of all the elements, hydrogen has the simplest line spectrum; the Balmer series (landing on $n = 2$) is the only one in the visible region.

Worked Examples

Limitations of the Model

Bohr's model was a major advance over Rutherford's, accounting for the stability and the line spectrum of hydrogen and hydrogen-like ions. Yet it is too simple to survive closer scrutiny.

These failures stem from a deeper problem: the model treats the electron as a particle on a definite circular path, which violates the wave nature of matter and the Heisenberg uncertainty principle. Recognising this drove physicists toward a fully quantum mechanical description of the atom.