Consider the following four quantum jumps in a hydrogen atom: A. ni=2, nf=6 C. ni=7, nf=8 B. ni=

Consider the following four quantum jumps in a hydrogen...

Key Concepts

Hydrogen Atom Energy Levels

In the hydrogen atom, electrons occupy quantized energy levels that are determined by the principal quantum number n. The energy associated with each level is given by the Bohr model, and transitions between these levels involve discrete changes in energy. This quantization is fundamental to understanding phenomena such as spectral lines and photon emission or absorption.

Quantum Transitions

Quantum transitions refer to the changes that occur when an electron moves from one energy level to another within an atom. These jumps can result in either the absorption or emission of a photon depending on whether the electron moves to a higher or lower energy level. The specific energy difference between the initial and final states determines the frequency, and therefore the wavelength, of the associated photon.

Photon Energy-Wavelength Relationship

The energy of a photon is directly related to its frequency and inversely to its wavelength, as expressed in the equation E = hc/?. This relationship implies that larger energy differences between quantum levels result in photons with shorter wavelengths, while smaller energy differences correspond to longer wavelengths. This is crucial for analyzing which transitions produce the highest or lowest energy (and thus the shortest or longest wavelength) photons.

Rydberg Formula

The Rydberg formula provides a method to calculate the wavelengths of photons involved in transitions between energy levels in hydrogen. The formula, 1/? = R(1/n_f^2 - 1/n_i^2), captures the inverse proportionality between the wavelength and the difference of the squares of the reciprocals of the quantum numbers. This tool is essential for determining the specific wavelength values corresponding to transitions between any two given energy levels.

Transcript

00:02 So this is a question that is asking about a hydrogen atom absorbing or emitting a photon in certain transitions and we are to find the longest wavelength photon and the shortest wavelength photon that can be emitted or absorbed in these transitions and we're supposed to give those two wavelengths.

00:25 So we'll proceed as follows.

00:26 First we're going to calculate the energy, the largest energy and the smallest energy of the photon and from that then we can calculate the longest wavelength and the shortest wavelength.

00:41 Now to see which energies we need to actually calculate because we don't need to calculate them all, let's just draw a qualitative diagram of the energy levels involved here.

00:53 Let's start at the n is equal to 2 energy level.

00:56 I'll draw that like that and this is n is equal to 2 and then let's say that this up here is the n is equal to 3 energy level.

01:05 Now the important point i want to draw in this diagram is that as you go up in energy levels the spacing gets smaller.

01:15 So i'm not going to make this perfectly proportional to the actual energy levels but i'm going to show that the n is equal to 4 level is closer to the n is equal to 3 level than 3 is to 2.

01:26 Similarly if we go up to the next highest one it's going to be even closer.

01:32 5 is going to be closer to 4 than 4 is to 3 and then if we go up to 6 it's going to be closer to 5 than 5 is to 4 and then we can go up to n is equal to 7.

01:54 So we can just use a diagram like this to kind of figure out which energies we actually need to calculate because they only want to know what the longest wavelength and the shortest wavelength which is going to correspond to the smallest energy and the largest energy respectively.

02:18 We'll look at the different energies here.

02:24 It says that in choice a we're going from n is equal to 2 to n is equal to 6.

02:37 So that would be like this 2 to 6 and if you also notice choice b is an emission it's going from 6 to 2 so the the wavelength involved there is going to be the same as the wavelength involved in the absorption for a is going to be the emission wavelength in d because you're going back from 6 to 2.

03:04 But let's look at the other energies here.

03:07 For choice b we're going from 2 to 8 so that's a bigger energy jump than the other one.

03:15 Going from 2 to 8 is more energy than going from 2 to 6 and so 2 to 8 might be the largest energy and in fact it is because the last one is choice c which is going from 7 to 8.

03:32 So the largest energy is going from 2 to 8 and the smallest energy is going from 7 to 8.

03:41 So we're really going to need to calculate those two energies because then we're going to calculate from them the two wavelengths.

03:50 So we will say that the energy of the energy levels is given by this formula.

03:59 It's a negative 13 .6 electron volts times 1 over n squared where n is the quantum number associated with that energy level.

04:16 So therefore the difference in energy is just going to be subtracting those energies from those two energy levels and what you end up getting is a negative 13 .6 electron volts times 1 over n final squared minus 1 over n initial squared.

04:49 And so let us calculate then the energy differences for the largest energy difference and the smallest energy difference.

04:58 The largest energy difference is going to be as we said for b.

05:05 So the energy difference for transition b is going to be a negative 13 .6 electron volts and for b the final energy level is 8.

05:24 So this is 1 over 8 squared and then the initial energy level was 2.

05:30 This is 1 over 2 squared and so if we take 1 divided by 8 squared minus 1 divided by 2 squared you get a negative 0 .234375 and i'm going to multiply that times a negative 13 .6 and we're going to get that the energy difference the largest energy difference we're going to have in these four transitions is going to be this one...

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