### Video Transcript

Use the formula π π equals four
ππ naught β bar squared π squared over π e π e squared, where π is the orbital
radius of an electron in energy level π of a hydrogen atom, π naught is the
permittivity of free space, β bar is the reduced Planck constant, π e is the mass
of the electron, and π e is the charge of the electron, to calculate the orbital
radius of an electron that is in energy level π equals two of a hydrogen atom. Use a value of 8.85 times 10 to the
negative 12 farads per meter for the permittivity of free space, 1.05 times 10 to
the negative 34 joule seconds for the reduced Planck constant, 9.11 times 10 to the
negative 31 kilograms for the rest mass of an electron, and negative 1.60 times 10
to the negative 19 coulombs for the charge of an electron. Give your answer to three
significant figures.

Okay, so this seems like a pretty
long question. But actually, all weβre being asked
to do is this bit: Calculate the orbital radius of an electron that is in energy
level π equals two of a hydrogen atom. The rest of the question just tells
us how we can do this. So weβre told we can use this
formula. And this part of the question
defines what all the quantities in this formula are. And this last part of the question
tells us the values of the constants in the equation.

We can recall that this formula
which weβve been given is derived from the Bohr model of the atom, which describes
atoms as consisting of a positively charged nucleus with electrons making circular
orbits around it. Now, the Bohr model has some
limitations, but itβs still pretty accurate when describing systems with one
electron such as the hydrogen atom in this question.

Now, in this question, weβre told
that the electron occupies energy level π equals two. We can recall that π is the
principal quantum number of an electron in an atom. π takes whole number values, which
describe the energy level that an electron has. The lowest value that π can take
is one, which would describe an electron in the lowest possible energy state of an
atom. In the Bohr model, this would
describe an electron in the innermost orbital around the nucleus.

However, in this question, weβre
told that our electron is in energy level π equals two, which means that the
electron occupies the next orbital out. The orbital radius of an electron
is simply the radius of the circular path that it follows around the nucleus. And as weβve been told in the
question, we can calculate the orbital radius of an electron using this
equation.

One interesting thing to note about
this equation is that it only actually contains two variables, the orbital radius
and the principal quantum number. This means that according to the
Bohr model, the orbital radius of an electron is proportional to the square of its
principal quantum number. Now, we want to calculate the
orbital radius of an electron in energy level π equals two. In other words, weβre looking to
find π two. To find this, we simply substitute
two in place of π in this equation, which gives us four ππ naught β bar squared
two squared over π e π e squared, where weβve been told π naught is the
permittivity of free space. β bar is the reduced Planck
constant. π e is the mass of an
electron. And π e is the charge of an
electron.

Since weβre told the values of all
of these quantities in the question, all we need to do now is substitute these
values in and calculate the answer, which gives us this expression. And if we plug all of this into our
calculators, it gives us an answer of 2.10 times 10 to the power of negative 10
meters, which is equivalent to 0.210 nanometers. And this is the final answer to our
question. The orbital radius of an electron
in energy level π equals two of a hydrogen atom is 0.210 nanometers.