An electron in an Li²⁺ ion moves
from the 𝑛 equals two energy level to the 𝑛 equals one energy level. Using the Bohr model, calculate, to
four significant figures, the energy of the photon produced by this transition.
Li²⁺ is lithium two plus. Lithium has an atomic number of
three. So, the lithium nucleus contains
three protons. We can take away the charge two
plus from the number of protons to work out the number of electrons. In this case, we have a single
electron. For this question, we don’t need to
worry about the neutrons. In the Bohr model, the only thing
that matters is the charges of the nucleus and the electron.
In the Bohr model, the nucleus is
surrounded by concentric shells. Electrons are said to occupy these
shells at fixed restricted distances from the nucleus. Because of the way this system is
sometimes drawn, the Bohr model is sometimes called the planetary model. However, electrons aren’t like
planets. They’re more like waves than
particles. And the shells are not rings; they
are surfaces of spheres.
Each shell is assigned a number,
with the number one given to the shell closest to the nucleus. The lower the value of 𝑛, the more
stable the electron will be if inside that shell. Each shell has an energy, but the
zero point is placed where the electron is infinitely far from the nucleus. This is the equivalent of a shell
number of infinity. And it’s the equivalent, in this
case, of having a lithium three plus nucleus and a completely separated
At the other end of the spectrum,
we have the inner shell, where 𝑛 equals one, where the energy is considered
negative. When an electron moves from a high
energy level to a low energy level, the difference in energy is released in the form
of a photon. In this question, we’re being asked
to work out the energy of the photon released when an electron moves from the 𝑛
equals two energy level to the 𝑛 equals one energy level.
So, we’re going from the first
excited state of lithium two plus to the ground state, releasing a photon of energy
Eph. Before we go any further, we’re
going to need to work out the energy of a shell based on its shell number. This is the equation from the Bohr
model that tells you the energy of a shell based on the shell number. 𝑍 is equal to the number of
protons in the nucleus, while 𝑛 is equal to the shell number.
But what about this term here? Well, the full expression for the
energy of any given shell is quite a complex combination of many constants of
nature. In this expression, we’re combining
the Coulomb constant, the elementary charge, the electron mass, and the reduced
Planck constant. But this is a constant times, a
constant divided by another constant. It all resolves to a constant. And that constant has a value of
about 13.6 electron volts.
If you take the values of the
constants in their SI units, you’ll get an answer in joules. And you can convert to electron
volts by dividing the value in joules by the elementary charge. Thankfully, all that work has been
done, so we have our single condensed constant term of negative 13.6057 electron
volts. Now, all we need to do is work out
the values for 𝐸 two and 𝐸 one and work out the difference and calculate the
energy of the released Photon.
The energy of our core shell is
equal to minus 13.6057 electron volts multiplied by three squared over one
squared. 𝑍 is the number of protons in the
lithium nucleus, which is three. And 𝑛 equals one. This evaluates to minus 122.451
electron volts. The value of our second energy
level, 𝐸 two, is equal to negative 13.6057 electron volts multiplied by three
squared divided by two squared, which gives us negative 30.6128 electron volts.
Now, we can put those values into
our diagram and work out the difference in energy. Our difference in energy is the
final energy minus the initial energy, which is equal to negative 122.451 electron
volts minus negative 30.6128 electron volts, giving us a change in energy of minus
91.8382 electron volts. So, this is the amount of energy
that the electron has lost, becoming more stable. So, the energy of the photon is the
negative of this energy, the exact opposite. Because energy can neither be
created nor destroyed. Giving us an energy for the photon
of positive 91.8382 electron volts.
We can also express this value in
joules, which is 1.47141 times 10 to the minus 17 joules. We convert from electron volts to
joules by multiplying by 1.60218 times 10 to the minus 19 joules per electron
volt. The question asks for our answer to
be to four significant figures, which is 91.84 electron volts, or 1.471 times 10 to
the minus 17 joules. So, we’ve worked out that when an
electron moves from the 𝑛 equals two energy level to the 𝑛 equals one energy
level, in the Bohr model, the energy of the photon produced is 1.471 times 10 to the
minus 17 joules.