### Video Transcript

If the electron had a mass that was twice its actual mass, according to the Bohr model of the atom, by what factor would its Bohr radius change?

Letβs start by recalling that the Bohr model is a simplified model of the atom that describes electrons as occupying circular orbits around atomic nuclei. The Bohr model enables us to predict several things about how electrons behave in atoms, including the radius of the orbit which our given electron occupies. Specifically, the Bohr model tells us that the orbital radius of an electron in a hydrogen atom is given by this equation.

Now an important feature of the Bohr model is that it tells us the electrons in atoms can only have certain specific amounts of energy, and we use the symbol π to denote the energy level of an electron. π can take positive whole number values. So the smallest value it can take is one. And this denotes the lowest energy level that an electron can occupy. An electron in the π equals one energy level has the smallest amount of energy that itβs possible for an electron in that atom to have. And it occupies the closest possible orbit to the nucleus. An electron in the π equals two energy level would have more energy and orbit slightly further away from the nucleus than the π equals one electron.

So in this equation, π π refers to the orbital radius of an electron in energy level π. So that means we can use this equation to calculate the orbital radius of an electron in any energy level we choose. All the other symbols in this equation are constants. 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.

Now this question is about the Bohr radius. In order to answer this question, we need to know that the Bohr radius is just another name for the orbital radius of an electron in the π equals one energy level of a hydrogen atom. That means we can use this equation to come up with an expression for the Bohr radius. All we have to do is substitute in π equals one. When we do this, we also change this subscript π to one. So on the left side of the equation, we have π one, the orbital radius of an electron in the π equals one energy level. And on the right side of the equation, we have four ππ naught β bar squared times one squared divided by the mass of the electron multiplied by the charge of the electron squared.

We can simplify this straightaway by noticing that one squared is of course one and thereβs no need to write down a factor of one. We can just say π one is equal to four ππ naught times β bar squared over π e times π e squared. This is an expression for the Bohr radius. And here we can note that the Bohr radius is often denoted π naught instead of π one.

Now that weβve obtained this expression for the Bohr radius, answering the question is a lot more straightforward. The question asks us how the Bohr radius, thatβs this, would change if the electron had a mass that was twice its actual mass. One of the ways we could answer this question would be to look up the values of all of the constants used in our expression and substitute them in. We could type this into our calculator, and it would tell us the actual value of the Bohr radius, which happens to be about 5.29 times 10 to the power of negative 11 meters. We could then do the same calculation but this time doubling the value for the mass of the electron. We could then compare the results to the actual Bohr radius and see how much it changed.

However, there is a much easier way of answering this question that doesnβt require us to look up the values of all of these constants, and that is to use an algebraic approach. If this is the Bohr radius of a regular electron with a regular mass, then we can calculate the effect of doubling the electronβs mass simply by replacing π e, the mass of an electron, with two π e. So we could say that the Bohr radius for an electron with double mass is equal to four ππ naught β bar squared divided by two π e times π e squared.

What we want to do now is write this expression for the Bohr radius for an electron with double mass in terms of this expression for the actual Bohr radius. We can do this by taking out this factor of two from the denominator and writing our expression like this. We can see that this part of our expression is equal to the Bohr radius for regular electron. But by doubling the electron mass, we essentially introduced this extra factor of a half. This means that by doubling the electronβs mass, we halve its Bohr radius. In other words, its Bohr radius changes by a factor of a half.