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Equations

In the previous section, we derived the basic equations on which the Bohr theory is based, for the hydrogen atom.  These are:

where the symbols in these equations are as follows:   is the mass of the electron, and r, v, e and E are its orbital radius, orbital velocity, charge and energy, respectively.  In addition, h is Planck’s constant,  is the permeability of free space, and n is an integer.

The task, then, is to rearrange these equation to find r, and E in terms of the other quantities.  This is not difficult, and we find that:

where we have defined a new quantity

Results for the Hydrogen Atom

We can see from these results that:

The allowed orbits of the electron vary as

The energies of those orbits vary as

Now let’s substitute in the values for the constants that appear in the expressions for radius and energy.  When we do this, and noting that 1 eV (electron-volt) is equivalent to 1.6 10^-19 J of energy, we find that

             metres

             eV

Thus, when n=1, the radius of the first allowed electron orbit is 5.3 10^-11 metres and the energy of the electron in that orbit is -13.6 eV.

We can now draw a picture to show the first few electron orbits around the nucleus, and the corresponding energy level diagram, obtained by varying n in the expressions above.  Note that this diagram is not drawn to scale!

Hydrogen Spectrum

We are now in a position to derive the electronic spectrum for hydrogen, and to do this we need to make a final assumption.  We will assume that when an electron moves from a higher orbit to a lower orbit, it emits electromagnetic radiation with a frequency given by:

where the left-hand side is the difference between the initial and final energy states, and f is the frequency of the emitted radiation.  It can be seen that since the electron can only sit in discrete energy levels, there will only be a certain number of allowed values of f.  Consider transitions from an energy level n (> 1) to the first orbit (n=1).  The energy of the radiation emitted is then equal to

We can convert this to a radiation wavelength by noting that the energy of the radiation is given by

Consider the transition from n=2 to n=1.  The energy of the radiation emitted during this transition is equal to 10.2 eV = 1.63 10^-19 J.  From the expression for wavelength above, the wavelength of the emitted radiation is 1.21 10^-7 m, or in more convenient units, 121 nm.  Within the electronic spectrum of hydrogen, this radiation appears as a line at a position in the spectrum corresponding to a wavelength of 121 nm (see the figure below).  This radiation lies in the ultra-violet region of the electromagnetic spectrum.

Transitions to the n=1 orbit produce what is known as the Lyman Series of spectral lines.  Transitions can also occur to any of the other orbits (from higher orbits of course).  For example, transitions to the n=2 orbit give rise to the Balmer Series of lines, and transitions to the n=3 orbit give rise to the Paschen Series of lines.

The following diagram shows the electronic spectrum for hydrogen:

An electronic spectrum such as this is produced by taking low pressure hydrogen gas and passing an electrical discharge through it.  The supplied energy excites the electrons in the hydrogen atoms to high orbits, and subsequently the return to lower orbits, with the emission of radiation as discussed above.  The emitted radiation can then be analysed to identify the various components of the radiation spectrum, the result being a spectrum as shown above.  The transition we considered (n=2 to n=1) giving rise to 121 nm radiation is marked as “Ly-α” in the figure above.

On to Corrections to the Bohr Theory