Video: Comparing the Mass of an Electron to the Mass of a Proton

The ratio of the mass of a proton to the mass of an electron is 1836. What is the ratio of the mass of the electron to that of the proton? Answer to four significant figures.

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Video Transcript

The ratio of the mass of a proton to the mass of an electron is 1836. What is the ratio of the mass of the electron to that of the proton? Answer to four significant figures.

Okay, so in this question, let’s first of all realize that we’re dealing with the masses of two particles. Firstly, the mass of a proton and, secondly, the mass of an electron. And this question tells us that the ratio of these two masses, the ratio of the mass of a proton to the mass of an electron, is 1836. In other words, if we start out by saying that π‘š subscript 𝑝 is what we’ll call our mass of the proton and π‘š 𝑒 is what we’ll call our mass of the electron, then the ratio of the mass of the proton to the mass of the electron is equal to 1836. That’s what we’ve been told in the question.

And by the way, sometimes a ratio is written as π‘š 𝑝 to π‘š 𝑒, the ratio of the mass of a proton to the mass of an electron. And if we were to write a ratio exactly like this, then we would have to say that this ratio is equal to 1836 to one. Which, in other words, means that the mass of the proton is 1836 lots of the mass of an electron. Regardless of the exact value of the mass of the electron or, for that matter, the mass of the proton. We don’t need to know the individual masses of the electron or the proton. All we care about is that the mass of the proton is 1836 times larger than the mass of the electron. And that’s more clearly seen when we write our ratio like this.

The mass of the proton divided by the mass of the electron is equal to 1836. Therefore, the mass of the proton is 1836 times larger than the mass of the electron. And additionally, we should know that when we’re told ratios in a question, we’re told them as ratios of one quantity, that’s the mass of the proton in this case, to the other quantity. Which, in this case, is the mass of the electron. And in that situation, we write it as the first quantity over the second quantity.

Anyway, so we’ve now realized that the mass of the proton divided by the mass of the electron is equal to 1836. What the question is asking us to do is to find the ratio of the mass of the electron to that of the proton. In other words, the question is wanting us to find the ratio between the mass of the electron and the mass of the proton. And we don’t know what this is, so let’s call that π‘₯. So how are we going to go about doing this? We know π‘š 𝑝 divided by π‘š 𝑒. And we want to find it’s reciprocal, π‘š 𝑒 divided by π‘š 𝑝. Well, to do this, let’s start with this equation here and work our way towards this equation. Where we’ll have π‘š 𝑒 divided by π‘š 𝑝 on one side and everything else on the other. And that everything else will be equal to π‘₯.

So starting here then, let’s start by multiplying both sides of the equation by π‘š subscript 𝑒. Because when we do this, we’ve now got an π‘š subscript 𝑒 in the numerator and denominator on the left-hand side. And so, they cancel. That just leaves us with π‘š subscript 𝑝 on the left-hand side and 1836 times π‘š subscript 𝑒 on the right. Then what we can do is to divide both sides of the equation by π‘š subscript 𝑝. Because when we do, the π‘š subscript 𝑝 on the left-hand side in the numerator cancels with the one in the denominator. And another way to think about this is that π‘š subscript 𝑝 divided by π‘š subscript 𝑝 is just one. Anything divided by itself is one. And on the right-hand side, we’ve got 1836π‘š subscript 𝑒 divided by π‘š subscript 𝑝.

So cleaning everything up, our equation looks a bit like this. Now, all we need to do is to divide both sides of the equation by 1836. Because when we do, we see that on the right-hand side now, we’ve got a factor of 1836 in the numerator and in the denominator. Those cancel because 1836 divided by itself is one. And so, all we’re left with on the right-hand side is π‘š subscript 𝑒 divided by π‘š subscript 𝑝. That’s exactly what we were going for earlier. And on the left-hand side of our equation, we’re left with one divided by 1836. Which means that at this point, if this side of our equation is exactly the same thing as what we were going for here, π‘š subscript 𝑒 divided by π‘š subscript 𝑝, then the other side of our equation, one divided by 1836, must be equal to π‘₯. Which was the ratio of the mass of the electron to that of the proton.

And so, all that’s left for us to do now is to evaluate this fraction and to give our answer to four significant figures as the question asks us to. When we plug our equation into our calculator, we find that one divided by 1836 is equal to 0.000544662 dot dot dot, so on and so forth. But, once again, we need to give our answer to four significant figures. So let’s start counting significant figures. Now, we should realize that none of these leading zeroes are significant. In fact, the very first significant figure starts here. This five is our first significant figure. And so, starting here, that’s our significant figure, this is our second, this is our third, and this is our fourth. Our fourth significant figure is a six.

Now, exactly what happens to that significant figure depends on the next value, which in this case also happens to be a six. But then, because six is larger than five, this means that the significant figure in question is going to round up. It’s going to become a seven. At which point, we’ve now rounded our answer to four significant figures, which means we’ve found the answer to a question. The ratio of the mass of the electron to the mass of the proton, to four significant figures, is 0.0005447.

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