today let's work through a practice problem regarding the
Bohr model. so the question asks what is the wavelength of the photon that will
cause an electron in the N equals 4 energy level of a hydrogen atom to jump
to N equals 6. so we've depicted our hydrogen atom
we've got the 1 proton in the nucleus and we have an electron that is
currently sitting in N equals 4 and that arrow signifies that we're jumping up to
N equals 6 so this has happened because that electron has absorbed a photon of a
very particular energy to allow that transition and we want to know what is
the wavelength of that Photon. in order to answer this we will definitely need
this equation that involves the Rydberg constant and the two energy levels just
to be clear you can see two different versions of this so here's the other
version they're actually the same the second one starts with a negative sign
but the order of n F and n I have switched and so it is actually the same
if you do the algebra you can use either of these here we'll use the first one we
also need to know that E equals hv or the energy of a photon is equal to
Planck's constant times nu which we often just write as V and that's equal
to the frequency and we also need to know that C equals lambda nu or the
speed of light equals the wavelength of a photon times the frequency of the
photon so this is all the information we need if you're not sure how to start go
back and check out my tutorial on the Bohr model that should give you all the
tips you need and when you're ready give this a try
so here is the equation that we need to use this is going to tell us the change
in energy of the electron there are some transition that the electron is involved
in and so it goes from one energy level to another it is going to experience a
change in potential energy and this is how we calculate it so ni is the initial
energy level and nf is the final energy level so we need to plug in four and six
because we started at N equals four so that's ni and we ended at N equals six
so that's NF so we plug those in and we're ready to do our calculation if we
go ahead and do the arithmetic very carefully calculate what's inside the
parentheses first and put it into a calculator we get seven point five six
six times ten to the negative 20 joules that is the change in energy of the
electron due to this transition. now that is very useful because due to
conservation of energy the change in energy of the electron is precisely
equal to the energy of the photon that it absorbed so a photon of some energy
struck the electron the electron absorbed that energy and then jumped up
to a higher energy level so now that we know the energy of the photon we want to
get wavelength so we know that energy equals Planck's constant times frequency
and we also know that the speed of light equals wavelength times frequency so we
want wavelength so let's manipulate the second equation to read frequency equals
speed of light over wavelength and now we have another way to express V instead
of V we can write C over lambda so let's rewrite the top equation as e equals HC
over lambda so that equation is going to be useful to us because we have energy
and we have Planck's constant and the speed of light those are constants and
we can now solve for wavelength so let's rearrange a little bit so that we are
solving for wavelength we can just bring wavelength up on the left and bring
energy down on the right and now we have lambda equals HC
over E, or the wavelength of the photon equals Planck's constant times the speed
of light divided by the energy of the photon so we were just deriving this
equation really quickly so that we're not confused about how we're using it
but that is the equation that we're going to use so let's start plugging
things in the wavelength of the photon involved in this transition is going to
be equal to Planck's constant which is six point six to six times 10 to the
negative 34 joules seconds times the speed of light three point zero zero
times ten to the eighth meters per second divided by the energy of the
photon which is equal to the change in energy of the electron those values are
equivalent due to conservation of energy so joules will cancel because we have
joules on top from planks and we have tools on the bottom from energy seconds
will cancel because we have seconds on top from Planck times meters per second
from the speed of light so seconds will cancel and if we plug all of this into a
calculator we get two point six two six times ten to the negative six meters as
the wavelength for the photon that is involved in this transition.