### Video Transcript

The diagram shows the binding
energy of each energy level of a hydrogen atom. If an electron is in the ground
state, what energy level would it transition to if it absorbed a photon with a
wavelength of 97.4 nanometers? Use a value of 4.14 times 10 to the
negative 15 electron volt seconds for the value of the Planck constant.

Here, we see several energy levels
available to an electron. And we know the electron begins in
the ground state or energy level one. Say it absorbs a photon. This transfers the energy of the
photon to the electron. And that amount of energy must
match the energy difference between the electron’s initial level and some other
level, causing the electron to transition to that other level. In other words, when this electron
transitions, the difference in binding energy between the electron’s final and
initial levels must be accounted for by the energy of the absorbed photon. We call this energy difference Δ𝐸,
and it equals 𝐸 final minus 𝐸 initial.

Now, we already know 𝐸 initial
from the diagram. So, if we also know the energy of
the photon which accounts for Δ𝐸, we can calculate the binding energy of the level
that the electron will transition to. Then, we can match it to one of
these levels, and we’ll have our answer. Put mathematically, we can solve
this formula for 𝐸 final by adding 𝐸 initial to both sides. So 𝐸 initial cancels out of the
right-hand side. And writing the formula a bit more
neatly, we have that Δ𝐸 plus 𝐸 initial equals 𝐸 final.

Now, we already know 𝐸
initial. And it turns out we also have all
the info we need to calculate the energy of the photon, which, remember, acts as
Δ𝐸. The photon has a wavelength of 97.4
nanometers. And recall that we can relate the
wavelength, 𝜆, of a photon to its energy, 𝐸, using the formula 𝐸 equals ℎ𝑐 over
𝜆, where ℎ is the Planck constant, whose value was given to us in the question
statement, and 𝑐 is the speed of light, 3.0 times 10 to the eight meters per
second.

Now, substituting in the values on
the right-hand side, we have that the photon’s energy equals the Planck constant
times the speed of light divided by the photon’s wavelength. But notice that the wavelength is
currently expressed in nanometers. And to calculate, it should be
written in plain meters. So recall that one nanometer equals
10 to the negative nine meters. And we can make a substitution to
write 𝜆 as 97.4 times 10 to the negative nine meters.

Now, calculating, we found that the
energy of the photon is 12.75 electron volts, which will act as our Δ𝐸 value. So, now that we know Δ𝐸 and 𝐸
initial, we can substitute them into this formula and find the binding energy at the
electron’s final level. Thus, we have 12.75 electron volts
plus negative 13.6 electron volts, which equals negative 0.85 electron volts. According to the diagram, this
value matches the binding energy at energy level four. Thus, if the electron in the ground
state absorbed a photon with a wavelength of 97.4 nanometers, it would transition to
energy level four.