Lesson 2: The atomic emission spectrum and Bohr's atomic model

2.1 Atomic emission (line) spectrum

• When the atoms of a pure element in the gaseous or vapour state are heated to a high temperature, or are subjected to an electric discharge at low pressure in a discharge tube, they emit radiation.
• When this light is passed through a slit and a prism in a spectrometer, it does not give a continuous rainbow.
• Instead it appears as a limited number of coloured lines, each of definite wavelength and frequency, separated by dark regions.
• It is therefore called the line (emission) spectrum.

• Experimentally, the line spectrum of every element is a characteristic property: no two elements have the same line spectrum, like a fingerprint.
• For this reason spectra are used to identify elements (for example in stars and in flame tests).

The visible spectrum of hydrogen (discharge tube containing H₂ at low pressure) shows four lines: violet (410 nm), blue (434 nm), green (486 nm) and red (656 nm).

2.2 Bohr's atomic model (1913)

• The Danish scientist Niels Bohr explained the hydrogen spectrum in 1913 (Nobel Prize in Physics, 1922).
• He kept three ideas of Rutherford: (1) a positive nucleus at the centre; (2) the number of electrons equals the number of positive charges in the nucleus; (3) a centrifugal force balances the attraction of the nucleus.
• Then he added the following postulates:

4. Electrons move rapidly around the nucleus without losing or gaining energy while they stay in a given orbit.
5. Electrons move in a number of definite energy levels; the spaces between them are forbidden regions for electron motion.
6. An electron has a definite energy depending on the distance of its level from the nucleus; the energy increases as the radius of the level increases. Each level is labelled by an integer, the principal quantum number n (1, 2, 3, …).
7. Normally the electron stays in the lowest possible energy level (ground state, stable). If it gains a quantum of energy (by heating or electric discharge) the atom becomes excited and the electron jumps to a higher level corresponding to the quantum absorbed. This state is unstable; the electron soon returns, losing the same quantum as light of a characteristic wavelength and frequency — producing a spectral line.
8. Many atoms absorb different quanta at the same time and many emit others, so a set of lines appears, indicating the energy levels between which the electrons jump. This explains the hydrogen lines.

A quantum is the amount of energy gained or emitted when an electron moves from one energy level to another.

2.3 The hydrogen spectrum explained

• When the excited electron falls to the second level (n = 2) from n = 3, 4, 5, 6, the energy released is visible light: n = 3→2 gives the red line (656 nm); 4→2 gives the green (486 nm); 5→2 blue (434 nm); 6→2 violet (410 nm). These make up the visible line spectrum of hydrogen.
• When the electron returns to the first level (n = 1), the energy released is greater and the lines are in the ultraviolet region, invisible to the eye.
• Transitions to levels n = 3 and higher give lines in the infrared region, also invisible.

Important observations of Bohr's calculations:

1. The energy differences between neighbouring levels are not equal; they decrease as we move away from the nucleus. Hence the quantum needed to move an electron between different levels is not constant, and each transition gives a different line.
2. The electron never stays between levels; it moves in definite jumps from one level to another.
3. The larger the energy gap, the shorter the wavelength (higher frequency) of the emitted light, so 6→2 (violet) has a larger gap than 3→2 (red).

Worked example 2.1. Which transition of hydrogen gives the line of the greatest energy among the four visible lines? The violet line, 410 nm (the shortest wavelength), arising from the electron falling from n = 6 to n = 2. The red line (3→2, 656 nm) has the smallest energy.

2.4 Success of Bohr's model

• (a) It explained correctly the line spectrum of the hydrogen atom.
• (b) It was the first theory to introduce the quantum concept into the determination of electron energy in the different levels.

2.5 Shortcomings of Bohr's model

Many quantitative results did not agree with experiment. The most important shortcomings:

1. It dealt mainly with hydrogen, the simplest atom. It explained only its spectrum, and failed to explain the spectrum of any other element, even helium with only two electrons.
2. The electron was treated purely as a negative material particle; its wave properties were ignored.
3. It assumed that the position and the speed of the electron can be determined precisely at the same time, which is practically impossible (Heisenberg).
4. Bohr's equations treat the electron as moving in a circular plane, so the hydrogen atom would be flat (planar); atoms are known to be three-dimensional.

These shortcomings led scientists to continue the search that produced the modern atomic theory.