Bohr's Model

Niels Bohr ^[1]

In 1913, the physicist Niels Bohr introduced a model of the atom that contributed a greater understanding to its structure and quantum mechanics. Atoms are the basic units of chemical elements and were once believed to be the smallest indivisible structures of matter.

The concept and terminology of the atom date as far back as ancient Greece, and different models were proposed and refined over time. The most famous are attributed to John Dalton, J.J.Thompson and Ernest Rutherford.

Each atomic model has contributed to a deeper understanding of the behavior of atoms and subatomic particles. The Bohr model was the first to propose quantum energy levels, where electrons orbit the nucleus at predefined distances and must overcome an energy barrier to move into a new orbital. Bohr was awarded a Nobel prize in 1922 for his investigations into atomic structure.

The key difference between Bohr's atomic model and earlier atomic models is that the electron can only move around the nucleus in orbits of specific, allowed radii. Another way to phrase this is to say that the electron can only occupy certain regions of space.

Bohr postulated the following regarding atomic structure:

1. The electrons revolve around the nucleus in special orbits called discrete orbits to overcome the loss of energy. When an electron revolves around the nucleus in this orbit, it does not radiate energy. This proved that the electrons need not lose energy and fall into the nucleus.

2. Each orbit is called a shell or energy level, and each level contains a specific amount of energy. The Rutherford–Bohr model of a hydrogen atom, where the negative electron is confined to an atomic shell, encircles a positively charged nucleus and where an electron jump between orbits (from $n=3$ to $n=2$) emits or absorbs an amount of electromagnetic energy ($hf$).[1] The 3 → 2 jump here is the first line of the Balmer series, and for hydrogen (Z = 1) it emits a photon of wavelength 656 nm (red light). ^[2]

3. An electron will absorb energy when moving from a lower energy level to a higher energy level. This is called an excited state.

4. An electron will radiate energy when moving from a higher energy level to a lower energy level.

5. When electrons move from one orbit to another, they emit photons, producing light in characteristic absorption and emission spectra. Since each element has its own signature, the spectra can be used to determine the composition of a material. This principle has been harnessed in many types of spectroscopy. Emission spectra are also responsible for the colors seen in neon signs and fireworks.

6. Orbits closer to the nucleus (those that have lower energy levels) are more stable. (An electron in its orbit with the lowest possible energy is said to be in its ground state.)

7. Out of the infinite number of possible circular orbitals around the nucleus, the electron can revolve only in those orbits whose angular momentum is an integral multiple of $\frac h{2\pi}$, i.e. angular momentum is quantized and $mvr = \frac{nh}{2\pi},$ where $m$ = mass of an electron, $v$ = velocity of an electron, $r$ = radius of the orbit, and $n$ = number of the orbit.

1. Bohr's model only explains the spectra of species that have a single electron, such as the hydrogen atom $(\ce{H})$, $\ce{He+, Li^2+, Be^3+,}$ etc.

2. Bohr's theory predicts the origin of only one spectral line from an electron between any two given energy states. Under a spectroscope of strong resolution, a single line is found to split into a number of very closely related lines. Bohr's theory could not explain this multiple or fine structure of spectral lines. The appearance of the several lines implies that there are several sub energy levels of nearly similar energy for each principal quantum number, n. This necessitates the existence of new quantum numbers.

3. It does not explain the splitting of spectral lines under the influence of a magnetic field (the Zeeman effect) or under the influence of an electric field (the Stark effect).

4. The pictorial concept of electrons jumping from one orbit to another orbit is not justified because of the uncertainty in their positions and velocities.

The force of attraction between the electron and proton for an atom with atomic number $Z$ is

$$\begin{aligned} F_A=\text K\dfrac{q_1q_2}{r^2}=\text K\dfrac{(Ze)(-e)}{r^2}=-\text K\dfrac{Ze^2}{r^2}.\end{aligned}$$

And the centrifugal force is given by

$$F_C =-\dfrac{mv^2}{r}.$$

But the force of attraction is equal to the centrifugal force, so

$$\begin{aligned} \text- K\dfrac{Ze^2}{r^2}&=-\dfrac{mv^2}{r}\\\\ v^2&=\text K\dfrac{Ze^2}{mr}. \end{aligned}$$

But from Bohr's theory

$$\begin{aligned} mvr =\dfrac{nh}{2\pi}\implies v&=\dfrac{nh}{2\pi m r}\\\\ v^2&=\dfrac{n^2h^2}{4\pi^2 m^2 r^2}. \end{aligned}$$

Equating both the results for $v^2$ gives

$$\begin{aligned} \text K\dfrac{Ze^2}{Mr}&=\dfrac{n^2h^2}{4\pi m^2r^2}\\\\ \Rightarrow r&=\dfrac{n^2h^2}{4\pi^2 m\text KZe^2}. \end{aligned}$$

Finally, substituting for the constants produces

$$\begin{aligned} \boxed{(\text{Radius})=r=\dfrac{n^2h^2}{4\pi^2 m \text K Ze^2}}\\ \approx 0.529 \dfrac{n^2}{Z}\si{\angstrom}. \end{aligned}$$

Deriving the energy of the electron in the $n^\text{th}$ orbit is quite easy; the total energy of an electron is the sum of its kinetic and potential energies:

$$\begin{aligned} \textrm{P.E.}&=(\text{Force of Attraction})\times (\text{Radius})\\ &=-\text K\dfrac{Ze^2}{r} \\ \textrm{K.E.}&=\dfrac 12mv^2 \\ &=\dfrac 12m\times \text K\dfrac{Ze^2}{mr}\\ &=\dfrac 12K\dfrac{Ze^2}{r}. \end{aligned}$$

Thus the total energy is given by the sum of the two results:

$$\begin{aligned} (\textrm{Total Energy}) &=-\text K\dfrac{Ze^2}{r}+\dfrac 12K\dfrac{Ze^2}{r}\\ &=-\dfrac 12 \text K\dfrac{Ze^2}{r}. \end{aligned}$$

Replacing the expression for $r$ returns

$$\text E_n= -\dfrac 12 \text K\dfrac{Ze^2}{n^2h^2} \times 4\pi^2m\text KZe^2,$$

which gives

$$\begin{aligned} \boxed{(\text{Energy})=E_n=-\dfrac{2\pi^2 m \text K^2Z^2e^4}{n^2h^2}}&\approx -13.6\dfrac{Z^2}{n^2} \text{eV/atom}\\ &\approx -1312\dfrac{Z^2}{n^2} \text{kJ/mol}\\ &\approx -21.6 \times 10^{-19}\dfrac{Z^2}{n^2} \text{J/atom}\\ &\approx -313\dfrac{Z^2}{n^2} \text{kcal/mol}. \end{aligned}$$

From Bohr's theory $$\begin{aligned} mvr=\dfrac{nh}{2\pi} \implies v&=\dfrac{nh}{2\pi mr}\\ &=\dfrac{nh}{2\pi m}\times \dfrac{4\pi^2 m\text KZe^2}{n^2h^2}\\ &=\boxed{\dfrac{2\pi \text KZe^2}{nh}=(\text{Velocity})}\\ &\approx 2.188\times 10^6 \dfrac Zn m/s. \end{aligned}$$

Rotations per second is the velocity of the electron by its circumference, which is given by

$$\begin{aligned} \textrm{RPS}=\dfrac{(\text{Velocity})}{(\text{Circumference})}&=\dfrac{\hspace{3mm} \dfrac{2\pi \text K Ze^2}{nh}\hspace{3mm} }{2\pi r}\\ &=\dfrac{\text KZe^2}{nh}\times \dfrac{4\pi^2m\text KZe^2}{n^2h^2}\\ &=\boxed{\dfrac{4\pi^2\text K^2mZ^2e^4}{n^3h^3}=\text{RPS}}\\ &\approx 6.58\times 10^{15}\dfrac{Z^2}{n^3}. \end{aligned}$$

Time period and frequency are related as

$$\text{T.P.}=\dfrac 1{\text{RPS}}.$$

Thus the expression for time period is as follows:

$$\begin{aligned} \boxed{\text{T.P.}=\dfrac{n^3h^3}{4\pi^2m\text KZ^2e^4}} \approx 1.52\times 10^{-16}\dfrac{n^3}{Z^2}\text{ sec}. \end{aligned}$$

1. AB Lagrelius AND Westphal, . Niels Bohr, physicist.. Retrieved August 24, 2016, from https://commons.m.wikimedia.org/wiki/File:Niels_Bohr.jpg
2. JabberWok, . Bohr-atom. Retrieved August 24, 2016, from https://en.wikipedia.org/wiki/Bohr_model#/media/File:Bohr-atom-PAR.svg