0.4 Quantum electron energy levels in an atom  (Page 3/6)

There doesn’t appear to be a pattern of any sort in the data in [link] , just a collection of numbers. The first important conclusion to remember, though, is that only these frequencies are observed. Hydrogen atoms will not emit radiation with a frequency between these numbers, for example. Given the infinite possibilities for what the frequency of light can be, this is a very small set of numbers observed in the spectroscopy experiment.

As it turns out, there is a pattern in the data, but it is quite hard to see. Each frequency in the spectrum can be predicted by a very simple formula, in which each line corresponds to a specific choice of two simple positive integers, n and m . If we pick any two small integers (1, 2, 3, …) for n and m , we can calculate a frequency ν , from the equation:

$\nu =R\times \left(\frac{1}{n^2}\mathrm{-}\frac{1}{m^2}\right)$

This is called the Rydberg equation, and the constant R is called the Rydberg constant, which has value 3.29 ≥ 10 ¹⁵ sec ^-1 . This is a truly remarkable equation, both because it is so simple and because it is complete. Every choice of n and m will produce a value of ν which is observed experimentally. And every ν that is observed in the experiment corresponds to a choice of n and m . There are no exceptions, no simple choices of n and m which don’t produce a frequency observed in the spectrum, and no frequency in the spectrum for which we can’t find simple choices of n and m .

A very interesting question, then, is what n and m mean. They must have some physical significance to the hydrogen atom, since they predict a physical property of the hydrogen atom. The spectrum alone does not provide any further insights, so we’ll need additional observations.

What about the spectrum of atoms of other elements? Experimentally, we find that the atomic spectrum of every element consists of a limited set of frequencies which are observed. Each element has a unique atomic spectrum, which a set of frequencies which uniquely identify that element. However, the Rydberg equation only predicts the frequencies for the hydrogen atom spectrum. There does not exist a similarly simple formula for any other atom.

Observation 2: the photoelectric effect

To understand the spectrum of hydrogen and other elements, we need to have a better understanding of the energy associated with electromagnetic radiation. To begin, we should be clear that electromagnetic radiation is a form of energy. There are some simple ways to see this in everyday life, including the fact that water left in sunlight will become hotter just as if it had been heated over a flame. It is also possible to generate electrical power from light, and it is even possible to “push” an object using energy absorbed from light. The question we now ask is, “How much energy is contained in light?” Even more specifically, “How is the energy contained in the light related to the frequency and intensity of the light source?”

We need a way to measure the energy. One way is via the “photoelectric effect.” When light is directed at a metal surface under the correct conditions, experimental observations show that electrons are ejected from the metal. These electrons can be collected to produce a usable electric current. There are a number of simple applications of the photoelectric effect, including in remote control devices or “electric eye” door openers.