Free electrons that are not part of an atom can have all sorts of different energy levels.
However, when an electron is bound to a nucleus and it's part of an atom, there are certain energy
levels that are just not allowed. This is called having discrete energy levels. I'm going to talk
about what this means without using complicated math. Let's talk about the photons in a rainbow:
The same math applies to both photons and electrons, but photons are easier to visualize.
Photons are the quantum particles that make up electromagnetic radiation. The visible region
of the electromagnetic spectrum is what most people would call a rainbow. Like electrons,
photons travel as waves and can have all sorts of different energy levels associated with them. Each
energy level has a unique wavelength associated with it. Wavelength is the distance between two
neighboring identical points along a wave. Our eyes perceive different wavelengths of photons
as different colors of light. A normal rainbow that's not missing any colors
consists of photons with many different energy levels. This is a continuous spectrum. If instead,
your photons had discrete energy levels, you would not be seeing the full spectrum of colors:
You would be seeing a set of specific colors. This is what discrete energy levels look like. Back to
electrons: Atomic orbitals are regions of space with a high probability of containing electrons.
They come in a lot of different shapes and the shapes get more complicated as the number
of electrons in an atom increase. Energy level diagrams show the discrete energy levels that are
allowed for electrons in a given atom. In general, lower energy electrons are closer to the nucleus
and higher energy electrons are farther away. If you're looking at these images,
it's fairly straightforward to understand what is happening, but it is hard to understand the
Schrödinger Equation and to understand why it's happening. Solving this would require
some advanced calculus. Erwin Schrödinger was an Austrian physicist who was famous for two main
reasons: His cat and receiving the 1933 Nobel Prize in Physics. Schrödinger's equation does
not specifically describe electrons, but it's useful for describing the behavior and energies
of submicroscopic particles in general. Think of it as an analog to Newton's Second Law of Motion:
Force equals mass times acceleration. Newton's equation describes the behaviors of macroscopic
particles, but when you start trying to apply it to the very, very tiny sub-microscopic particles,
it just doesn't work and you need to use Schrödinger's equation instead.
When we're dealing with sub-microscopic particles, our understanding of physics from everyday life
just doesn't apply. There is a concept called wave-particle duality, where sometimes we have
to treat sub-microscopic particles as waves and other times as particles in order to make sense of
their behaviors using math. So in the macroscopic world, a wave is a disturbance that propagates
through space and time. So wave in an ocean isn't the water, but it's the movement of the
water. And in the macroscopic world, a particle is an object: like this baseball is a particle.
However, in the sub-microscopic world, waves and particles are one and the same.
Albert Einstein very eloquently said that this is confusing:
"It seems as though we must sometimes use the one theory and sometimes the other," "while at
times we may use either. We are faced with a new kind of difficulty. We have two contradictory"
"pictures of reality; separately neither of them fully explains the phenomena of light," "but
together they do." In classical mechanics, we have some mathematical equations that work for waves
and different mathematical equations that work for particles. Wave-particle duality is the inability
of classical mechanics, like Newton's Laws of Motion, to fully describe the behaviors
of sub-microscopic particles. If two waves are traveling together, the resultant wave
we would actually be able to observe would be the sum of the individual waves. In this case ,our
individual waves are in-phase, meaning their peaks are all aligned and their valleys are all aligned.
If we find the sum of these two waves, we have a resultant wave with a greater amplitude. If
these were visible light waves, we would perceive this greater amplitude as being brighter light.
This is constructive interference. However, if our two starting waves were out-of-phase
and the peaks in one aligned with the valleys in the other: If we were to sum these together
at each and every point along the wave, we would end up with a resultant wave with an amplitude
of zero. If this was visible light, we would perceive this as the absence of light. This is
destructive interference. These two waves that are out-of-phase literally destroyed each other.
If an electron is bound to a nucleus and part of an atom, it's not traveling in a straight
line but it's orbiting that nucleus. Electrons like orbiting with energy
levels that enable them to constructively interfere with themselves. Let me reword that:
In this diagram, we have a certain orbit. We can calculate this based on circumference of a circle,
2 pi r, where r is the radius. And the way that I drew this, that circumference happens
to be equal to 6 times the wavelength. What that means is after this electron orbits once and it
starts its second orbit, it is actually going to be constructively interfering with itself.
So this is an example of one of the discrete, allowable energy levels. This is going to be
the case whenever you have an integer number of wavelengths as your circumference. So 2 pi r
is circumference of the circle and equal to n times lambda. n is integer number,
meaning counting numbers: one, two, three, four, five, and so on. Lambda is wavelength. So
there would be a series of discrete energy levels that are allowable. You can calculate
them by plugging one, two, three, four, five, six, and so on into n. Now, this
is showing a circumference of six wavelengths. Let's move this electron a little bit farther away
to a location where the circumference is equal to six and a half wavelengths. In the
very first orbit, everything is okay, but that's six and a half wavelengths.
Starting in its second orbit, it's actually destructively interfering with itself.
And because we can treat electrons as waves that means it's destroying itself. The electron
literally cannot be here. This is a forbidden energy level. Thanks for watching Chemistry in a
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