Question Video: Knowing the Approximate Lifetime of an Excited Electron Physics

Video Transcript

Which of the following is the closest value to the approximate typical lifetime of an excited electron in an atom? (A) 0.1 nanoseconds, (B) 10 nanoseconds, (C) one microsecond, (D) 10 microseconds, (E) 0.1 milliseconds.

In this question, when we talk about the typical lifetime of an excited electron in an atom, we’re imagining different energy levels within this atom. We can call them 𝐸 one and 𝐸 two. And saying that there’s an electron in this higher energy state, therefore, the electron is excited. Now without doing anything to it, eventually, this electron will decay back down to 𝐸 one. And this action is spontaneous; we can’t predict exactly when it will happen. That’s said, there is a fairly typical amount of time that passes before this decay event occurs. That time is approximately 10 to the negative eighth seconds, a very small amount of time.

In this question, we want to pick which one of our answer options is closest to this typical lifetime of an excited electron. To figure this out, let’s look at this number here a bit more closely. An equivalent way to write 10 to the negative eighth seconds is one times 10 to the negative eighth seconds. As it is, this number is written in scientific notation, but like any number written in this form, we can also express this as a decimal number.

To do that, we would start with our leading value, the one, with a decimal point immediately to its right. And then, because we multiply this value by 10 to the negative eighth, we would move this decimal point eight places to the left. So far, we can see we’ve moved three spots. So here’s four, then five, then six, then seven, and then eight. This is where our decimal point ends up and we’ll fill in the blank places with zeros. And so then, we have one, two, three, four, five, six, seven of those zeros. So then, the value one times 10 to the negative eighth seconds written in decimal form is equal to 0.00000001 second.

Now, we do all this because now we can consider the conversions between milliseconds and seconds, microseconds and seconds, and nanoseconds and seconds, respectively. We can recall that 1000 milliseconds or 10 to the third milliseconds equals one second, while it’s 10 to the sixth or a million microseconds that’s equivalent to one second. And 10 to the ninth or a billion nanoseconds is equivalent to one second of time. So, considering this decimal form of our typical lifetime of an excited electron, let’s see what this time is written in units of milliseconds and then microseconds and then nanoseconds.

Considering milliseconds first, we can multiply the value by 1000 to express it in milliseconds. If we do that, we get this result here. Notice that there are three fewer zeroes to the right of our decimal point. But then, considering this result in milliseconds, we see that answer option (E) has a time also in this unit, but it has a value of 0.1 milliseconds. We see that doesn’t agree with the actual typical lifetime expressed in this unit. So, we’ll cross off option (E).

Now, let’s think about this number, this typical lifetime of an excited electron in microseconds, where one million microseconds equals one second. Multiplying our time in seconds by a million microseconds per second, we come up with this result, 0.01 microseconds. But then, looking at answer options (D) and (C), which offer choices in this unit, we can see they also don’t agree with this value we’re calculating. So, we’ll cross those out.

Finally, let’s convert our time to be expressed in units of nanoseconds. To do this, we multiply our original time value in seconds by one billion nanoseconds per second. And when we do that, the decimal point shifts nine spots to the right. And we get this result here, 10 nanoseconds. Looking at our remaining answer choices, we see this agrees with option (B). And so, this is our choice for the closest value to the approximate typical lifetime of an excited electron in an atom.