Electron transit in hydrogen atoms

From the previous post, we've learnt that external power supplied to hydrogen gas could generate a discrete line emission spectrum, like the one shown below. It is due to the movement of electrons, or electron transit in hydrogen atoms after absorbing external energy.

In this post, we'll discuss two observations from the spectrum above, and understand their implications.

Observation 1: A discrete line spectrum was generated

As we've discussed before, discrete line spectrum indicates that only lights with certain wavelengthes are generated, for example, lights with wavelengthes 400nm and 500nm are present, while any light with a wavelength in between 400nm and 500nm is missing.

Therefore, we would say that electron transit in hydrogen atoms could only emit lights with specific wavelengthes. What can we further deduce from this?

First of all, as we all know that energy of photons is proportional to frequencies of lights and inversely proportional to their wavelengthes, we could safely say that in a discrete line spectrum, only lights with certain energy are emitted.

On the other hand, light is produced by electron transiting from a higher energy state, or energy level, to a lower energy level, whereas the extra energy is released in the form of lights, the energy of light is just the difference in energy of the two energy levels of electrons.

So if we put these two facts together, we could reach a conclusion that: electrons can only change their energy levels by a specific amount of energy, or energy levels only exist by certain amount of energy difference.

This is pretty much the same as travelling in an elevator. Let's imagine electrons are living in a five-storey building with an elevator carrying them between different floors. If electrons go to storey 1 from a higher storey, they would lose certain amount of energy in the form of lights.

By taking an elevator, electrons could go to any floor but they can't stop in between the storeys. As a result, the lights emitted would carry energy corresponding to the energy difference between 4 storeys (from storey 5 to 1), 3 storeys (from storey 4 to 1), 2 storeys (from storey 3 to 1), and 1 storey (from storey 2 to 1). However, no light could carry energy of fractional storeys. We won't see lights with energy of 1.5 storeys.

So ultimately, we get a discrete line emission spectrum with three lines, i.e. the lights generated when electrons move from storey 5 to 1, from 4 to 1, from 3 to 1, and from 2 to 1.

Now we could conclude that the reason we observed a discrete line spectrum instead of a continuous spectrum is that electrons can only be at energy levels with specific energy, or we'll say energy levels are discrete, just like storeys in a building.

Observation 2: The lines in the spectrum get closer from violet to red

If we examine the discrete line spectrum above carefully, we would notice that the violet line and the blue line are closer, as compared to the blue line and the green line, which is again closer than the green line and the red line. This tells us how the energy levels are located with in the hydrogen atoms.

Again, we remember the fact that wavelengthes of lights are proportional to their energy. So we could translate the observation to the energy differences between the lights are getting smaller when the lights are having higher energy.

Let's consider our five-storey building again. As we know violet is the most energetic light as compared to other colors, we would assume it is generated when electrons transit to storey 1 from storey 5. Similarly, the blue light is generated by electron transit from storey 4 to 1, the green light is from storey 3 to 1, and the red light is from storey 2 to 1.

We'll begin with the green light (storey 3 to 1) and the red light (storey 2 to 1). So we notice that the energy difference between the green light and the red light is just the energy difference between storey 3 and 2, isn't it?

We leave it to you to figure out why we say the energy difference between the blue light and the green light is the energy difference between storey 4 and 3, and that between the violet light and the blue light is for storey 5 and 4.

The table below summarizes the energy differences between adjacent storeys and the respective lights.

Light	Storey	Energy difference
Violet and blue	5 and 4	Smallest
Blue and green	4 and 3	Larger
Green and red	3 and 2	Largest

It looks like we've found something. The energy difference gets smaller for higher storeys.

We may conclude that energy levels converge, or get closer at higher energy.

Energy levels in hydrogen atoms

With the observations and their implications, we're now able to draft a simple diagram to show the energy levels in hydrogen atoms.

Starting from energy level 1 (\(E_1\)) which is the nearest to the nucleus, there're multiple energy levels with different energy. Energy levels would converge at higher energy, until they "touch each other", or become continuous, at the convergence limit. Since there is no more distinct energy level above the convergence limit, we could consider it as the border of the atom.

Do take note that there are more than 6 energy levels in hydrogen atoms. But we only draw 6 energy levels for convention.

Secondly, this diagram is very simplified, and electrons may transit between different energy levels in all possible ways, not only the ones shown by the arrows in the diagram. Therefore, the actual experimental result would show a much more complicated discrete line spectrum. Nevertheless, all our discussions here are valid despite of the complexity of the real scenario.

A third question you may ask is why electrons move to \(E_2\) instead of \(E_1\). We'll explain that in a while.