Explaining hydrogen's emission spectrum

The Balmer and Rydberg Equations By an amazing bit of mathematical insight, in 1885 Balmer came up with a simple formula for predicting the wavelength of any of the lines in what we now know as the Balmer series. Three years later, Rydberg generalised this so that it was possible to work out the wavelengths of any of the lines in the hydrogen emission spectrum. What Rydberg came up with was: RH is a constant known as the Rydberg constant. n1 and n2 are integers (whole numbers). n2 has to be greater than n1. In other words, if n1 is, say, 2 then n2 can be any whole number between 3 and infinity. The various combinations of numbers that you can slot into this formula let you calculate the wavelength of any of the lines in the hydrogen emission spectrum - and there is close agreement between the wavelengths that you get using this formula and those found by analysing a real spectrum.

You can also use a modified version of the Rydberg equation to calculate the frequency of each of the lines. You can work out this version from the previous equation and the formula relating wavelength and frequency further up the page.

The origin of the hydrogen emission spectrum The lines in the hydrogen emission spectrum form regular patterns and can be represented by a (relatively) simple equation. Each line can be calculated from a combination of simple whole numbers. Why does hydrogen emit light when it is excited by being exposed to a high voltage and what is the significance of those whole numbers? When nothing is exciting it, hydrogen's electron is in the first energy level - the level closest to the nucleus. But if you supply energy to the atom, the electron gets excited into a higher energy level - or even removed from the atom altogether. The high voltage in a discharge tube provides that energy. Hydrogen molecules are first broken up into hydrogen atoms (hence the atomic hydrogen emission spectrum) and electrons are then promoted into higher energy levels. Suppose a particular electron was excited into the third energy level. This would tend to lose energy again by falling back down to a lower level. It could do this in two different ways. It could fall all the way back down to the first level again, or it could fall back to the second level - and then, in a second jump, down to the first level. Tying particular electron jumps to individual lines in the spectrum If an electron falls from the 3-level to the 2-level, it has to lose an amount of energy exactly the same as the energy gap between those two levels. That energy which the electron loses comes out as light (where "light" includes UV and IR as well as visible). Each frequency of light is associated with a particular energy by the equation: The higher the frequency, the higher the energy of the light. If an electron falls from the 3-level to the 2-level, red light is seen. This is the origin of the red line in the hydrogen spectrum. By measuring the frequency of the red light, you can work out its energy. That energy must be exactly the same as the energy gap between the 3-level and the 2-level in the hydrogen atom. The last equation can therefore be re-written as a measure of the energy gap between two electron levels. The greatest possible fall in energy will therefore produce the highest frequency line in the spectrum. The greatest fall will be from the infinity level to the 1-level. (The significance of the infinity level will be made clear later.) The next few diagrams are in two parts - with the energy levels at the top and the spectrum at the bottom. If an electron fell from the 6-level, the fall is a little bit less, and so the frequency will be a little bit lower. (Because of the scale of the diagram, it is impossible to draw in all the jumps involving all the levels between 7 and infinity!) . . . and as you work your way through the other possible jumps to the 1-level, you have accounted for the whole of the Lyman series. The spacings between the lines in the spectrum reflect the way the spacings between the energy levels change. If you do the same thing for jumps down to the 2-level, you end up with the lines in the Balmer series. These energy gaps are all much smaller than in the Lyman series, and so the frequencies produced are also much lower. The Paschen series would be produced by jumps down to the 3-level, but the diagram is going to get very messy if I include those as well - not to mention all the other series with jumps down to the 4-level, the 5-level and so on. The significance of the numbers in the Rydberg equation n1 and n2 in the Rydberg equation are simply the energy levels at either end of the jump producing a particular line in the spectrum. For example, in the Lyman series, n1 is always 1. Electrons are falling to the 1-level to produce lines in the Lyman series. For the Balmer series, n1 is always 2, because electrons are falling to the 2-level. n2 is the level being jumped from. We have already mentioned that the red line is produced by electrons falling from the 3-level to the 2-level. In this case, then, n2 is equal to 3. The significance of the infinity level The infinity level represents the highest possible energy an electron can have as a part of a hydrogen atom. So what happens if the electron exceeds that energy by even the tiniest bit? The electron is no longer a part of the atom. The infinity level represents the point at which ionisation of the atom occurs to form a positively charged ion. Using the spectrum to find hydrogen's ionisation energy When there is no additional energy supplied to it, hydrogen's electron is found at the 1-level. This is known as its ground state. If you supply enough energy to move the electron up to the infinity level, you have ionised the hydrogen. The ionisation energy per electron is therefore a measure of the distance between the 1-level and the infinity level. If you look back at the last few diagrams, you will find that that particular energy jump produces the series limit of the Lyman series.