Calculate the longest and shortest wavelengths of light emitted by electrons in the hydrogen ato

Calculate the longest and shortest wavelengths of light...

Key Concepts

Quantum Energy Levels

Hydrogen atoms have quantized energy levels determined by quantum mechanics. Electrons in an atom can only occupy specific energy states, and their allowed levels are quantified by the principal quantum number. Transitions between these levels result in the absorption or emission of energy in the form of photons.

Electron Transitions

When electrons change energy levels, they emit or absorb photons whose energy corresponds exactly to the difference between the initial and final states. In the context of emission spectra, the electron transitions from a higher energy level to a lower one, releasing a photon with a specific wavelength.

Rydberg Formula

The Rydberg formula is used to calculate the wavelengths of light emitted or absorbed by hydrogen atoms during electron transitions between different energy levels. It relates the inverse of the wavelength to the difference in the reciprocals of the squares of the quantum numbers for the two levels involved in the transition.

Wavelength and Energy Relationship

The energy of a photon is inversely related to its wavelength; a greater energy difference between the initial and final states results in a shorter wavelength photon, whereas a smaller energy difference yields a longer wavelength. This relationship is fundamental when analyzing and categorizing the lines in an emission spectrum.

Transcript

00:01 Okay, so let's begin to start answering this question by knowing this equation that energy equals h over lambda c.

00:11 So since we can see that wavelength is in the denominator, as wavelength increases, energy decreases.

00:21 Okay? and as wavelength decreases, energy increases.

00:28 So basically, if this is our atom and we're looking at the energy levels around it, a drop between 1 is going to be a lot smaller in energy than a drop between several levels, let's say.

00:45 So going off of that, we can kind of assume that we're starting at n -equals 6, ending in n -equals 1 will probably be very high in energy because it's a big drop between levels.

01:03 So that should equal the smallest wavelength.

01:19 So now let's go ahead and solve it.

01:25 So we're going to equate those two equations to get the delta e's.

01:34 So it's going to look like this is n -final squared, minus 1 over an initial squared.

01:59 Okay, so let's try to plug in n -equals 1 for final and equals 6 for initial.

02:05 So that's going to give us, and i'm going to quickly rearrange this equation to get lambda equals hc over this constant, 1 over 1 squared is just 1, and 1 over an initial is 1 over 6 squared.

02:28 Alright, so now let's plug that into calculators.

02:34 And i got a wavelength of lambda equals 9 .39 times 10 to the negative 8.

02:43 If you got a negative wavelength, don't worry about that.

02:45 Just go ahead and drop the negative because it doesn't matter here...

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