Ladders and Tramlines: Electrons as waves and particles

What’s an eV?

eV stands for electron volt – that is, the amount of energy needed to move one electron through a potential difference of one volt. If that doesn’t mean much to you, don’t worry about it; the point is that it’s a small unit of energy, convenient for the scale of energy levels in an atom. To give you an idea of just how small an eV is…a 100 watt light bulb is putting out 100J of energy per second. That’s about 625,000,000,000,000,000,000eV per second!You may also have run across KeV (kilo-eV; that’s one thousand eV) and Mev (mega-eV, which is one million eV) in the med phys course.

Put another way, if we shine light on potassium, for example, photoelectric emission only happens if the energy of the incident light-bullets (or PHOTONS) is greater than 2.1eV – the work function energy of K Anything less, the electrons in the metal just get agitated.

Converting from eV to J:

2.1 x 1.6 x 10-19J = 3.36 x 10-19J is the minimum energy a light-bullet has to have to hook an electron out of the potassium surface. If the photons have more energy than this, the extra is picked up as the KE of emitted electrons.

Given h = 6.63 x 10-34 Js and E = hf = hc/λ, this corresponds to a wavelength of 592nm – a yellow-green colour.

Electrons in atoms

We know that electrons don’t just wander about – instead they exist in well-defined energy levels, much like the rungs of a ladder define how much gravitational potential energy it takes to climb them. But, the rungs aren’t evenly spaced. The ones near the nucleus are bigger jumps than the ones further away. For hydrogen the biggest jump an electron needs to get out is 13.6eV  – the first ionisation energy of hydrogen. If given 13.5eV, the electron just sits there, vibrating mutinously. To get it out it’ll have to get EXACTLY 13.6eV of energy.

These rungs of energy are a little bit like tramlines, the electrons almost run on rails. Why do electrons exist in these ‘tramlines in the sky’?

Because the wavelength of the electron has to fit inside the orbital like a curved guitar string, setting up a standing wave around the orbital. Any old orbital won’t do – only those allowing a whole number of wavelengths to fit is allowed. If you look carefully at the picture, you can see the circular standing waves on the loop of wire which is being driven by a small oscillator box at the bottom This is exactly how the electrons behave when orbiting a nucleus.

Here’s the ‘energy ladder’ for hydrogen. Remember, no intermediate steps are allowed. Each step corresponds to the emission of a photon having the same energy as the ‘jump’ between the rungs. Those falling to n=1 are in the UV, (Lyman series) those falling to n=2 are the Balmer series, shorter, visible wavelengths. n=1 is sometimes called the GROUND STATE and in books is sometimes given a – sign denoting that energy is required to be given to the electron (think of it stuck down a well and has to climb out).

Lyman is misspelled...
Lyman is misspelled...

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