### Video Transcript

In this video, our topic is units of energy. Energy, which is the capacity to do work, can be expressed in all different kinds of units. We see just some of the examples of those units here. And in this lesson, we’ll be focusing on a specific one, kilowatt-hours.

At first glance, this unit may not seem to be a unit of energy. After all, we see the word watt in this name. And isn’t watt, abbreviated capital W, the SI base unit for power? Well, indeed it is. But despite this, a kilowatt-hour really is an amount of energy and not power.

To see that this is so, let’s recall that the SI base unit for energy is the joule, abbreviated capital J. A joule is an amount of energy. And if we divide a joule by one second of time, then that, we can recall, is equal to the SI base unit for power, the watt that we just mentioned. More specifically, one joule of energy transferred over a time interval of one second is equal to one watt of power.

Now, since these ones are implied, we can remove them from this expression and not change its meaning. And then, say that we multiply both sides of this equation by one second of time. We can see that, in that case, the unit of seconds cancels from the left-hand side. And we see that a joule is equal to a watt times second or watt second. Just to be clear, a joule is still a unit of energy, and a watt still indicates power, while second, of course, indicates time.

Describing this generally, we could say that energy is equal to power times time. Which means that if we have some amount of power multiplied by some amount of time, then that product will equal some amount of energy. And that brings us back to this unit, the kilowatt-hour. The kilowatt-hour is the standard unit of electrical energy consumption. This is the typical unit in which electrical companies will bill customers for their electricity usage. Customers will pay so much money per kilowatt-hour of electrical energy used. And now that we see that an amount of power times an amount of time is an amount of energy, it starts to make a bit more sense that a kilowatt-hour really is an amount of energy.

We can consider that in this term, there are basically three parts. First, there’s the part we already noted, which is the power unit, watt. A second part is this prefix kilo-. This prefix indicates 1,000. For example, we talk about a kilometer being 1,000 meters, or a kilogram being 1,000 grams. And in the same way, a kilo- watt or a kilowatt is 1,000 watts.

So, these two parts of our expression together just gives 1,000 watt. And then, this third part, the hour, adds a time component. And this can remind us of our expression up here, where we have an amount of power times an amount of time. An hour is different from a second, but we can convert between the two. And in fact, let’s do just that as a part of showing that a kilowatt-hour really is equal to some amount of energy in joules.

Here’s how we’ll do that. If we were to represent kilowatt-hour in mathematical symbols, it might look like this, a kilowatt multiplied by an hour. And then, as we said, this prefix kilo- refers to 1,000, so our expression becomes 1,000 watts times an hour. And now, we can consider this time unit of an hour and see how many seconds this is equivalent to. We can think about it this way: one hour is equal to 60 minutes of time, and every one of those minutes is equal to 60 seconds. This means that we can substitute 60 seconds of time in for this unit of one minute here.

And so, now, we see that one hour is equal to 60 times 60 seconds. And 60 times 60 seconds is equal to 3,600 seconds. Since one hour is equal to 3,600 seconds, we can take this term and substitute it in for hour in our mathematical expression. Okay, so, one kilowatt-hour is 1,000 watts times 3,600 seconds. And now, if we take the units in this expression and collect them together over to the right, we see that a kilowatt-hour is equal to 1,000 times 3,600 watt seconds.

And then, according to our equation over here, a watt second is defined as a joule, which means we can replace watt times seconds here with the unit joules. What we’re finding is that a kilowatt-hour, this unit of energy, is equal to 1,000 times 3,600 joules. We can abbreviate a kilowatt-hour as one k capital W h. And when we multiply 1,000 by 3,600, we get a result of 3,600,000.

So, we’ve now shown two things. We’ve shown, first of all, that a kilowatt-hour is indeed an amount of energy rather than an amount of power. And we’ve also shown just how much energy in joules a kilowatt-hour is. Let’s record that conversion for safekeeping over here. And then, let’s consider a few variations on this unit of a kilowatt-hour.

We saw, for example, that this prefix kilo- indicates 1,000. But what if that prefix was absent and we just had one watt-hour? How many joules of energy would that be? Looking at this conversion from kilowatt-hours to joules, we can see that if we divide both sides by 1,000, then on the left-hand side, that 1,000 that’s in the denominator and the kilo- that’s in the numerator, which we saw earlier represents 1,000, will cancel one another out.

So, what we find is that one watt-hour is equal to 3,600,000 joules divided by 1,000. And considering our numerator, we know that three of the zeros from it will be effectively canceled out by the three zeros in our denominator of 1,000. So, when we lose those three zeros, we find a result of 3,600 joules. That’s what one watt-hour of energy is equal to.

But then what would happen if instead of starting at kilowatt-hours and getting smaller, we got larger? What if we wanted to replace the prefix kilo- with the prefix mega-? In other words, we’re working with a megawatt-hour. In that case, we could recall that mega-, this prefix, indicates 1,000,000. And so, if we wanted to work with a megawatt-hour instead of one watt-hour, we would multiply by 10 to the sixth or 1,000,000.

But then, since this is an equation, anything we do to one side, we have to do to the other side as well. So, we multiply both sides by 10 raised to the sixth power. That’s 1,000,000. On the left-hand side, having 1,000,000 watt-hours is the same as having one megawatt-hour. We write that like this: one capital M capital W h. Then, over on the right, if we multiply these two numbers together, we get this result, which is 3,600,000,000 joules.

But there’s really no reason to stop there. Why not exceed mega- and consider a gigawatt-hour or even a terawatt-hour? We can recall that the prefix giga- refers to 1,000,000,000 — that’s 10 to the ninth — and tera- refers to 1,000,000,000,000, 10 to the 12th. So, if we start out once more with a single watt-hour, which we know is equal to 3,600 joules, then we can turn that into one gigawatt-hour, 1,000,000,000 watt-hours, by multiplying both sides of the equation by 10 to the ninth.

So, one gigawatt-hour is this many joules. That’s 3,600,000,000,000 of them. And a terawatt-hour is even more. To get that, we can multiply one watt-hour and 3,600 joules by 10 to the 12th, 1,000,000,000,000. And we see that one terawatt-hour is this many joules. That’s 3,600,000,000,000,000 joules. That’s quite a lot. Knowing what we know now about kilowatt-hours, let’s get a bit of practice through an example exercise.

Which of the following is the correct definition of a watt-hour? (A) A watt-hour is the amount of time it takes for a one-watt device to transfer one joule of energy. (B) A watt-hour is the amount of energy that an electrical device transfers in one hour. (C) A watt-hour is the amount of time it takes for a process to increase in power from zero watts to one watt. (D) A watt-hour is a measure of the power of a process that is equivalent to the transfer of one joule of energy in one hour. (E) A watt-hour is the amount of energy transferred by a process that has a power of one watt and acts for one hour.

Okay, so, of these five definitions, we want to pick which one is the correct definition of a watt-hour. To get started on our answer, we can recall that a watt is the SI unit for power. A watt, abbreviated with a capital W, is defined as one joule of energy transferred over a time interval of one second. We can say then that a watt is a joule per second. So, that’s what a watt is.

But in this question, we’re interested to know about a watt-hour. A watt-hour is a watt, an amount of power, multiplied by an hour, which is an amount of time. To understand better what power times time is in general, we can multiply both sides of our equation for a watt by one second of time. When we do this, we see that the units of second cancel out on the right. And we find that a second times a watt — or in other words, a watt second — is equal to a joule. And so, that’s an amount of energy.

Now, our question isn’t asking about a watt second, but it is asking about an amount of power over an amount of time. And so, even though we don’t have exactly the right units here — we have watt seconds instead of watt-hours — we know that watt-hours will be equal to some amount of energy in units of joules. Knowing that fact goes a long way to helping us give the correct definition of a watt-hour. We know it’s an amount of energy, so any of these candidate definitions that don’t describe a watt-hour in that way can’t be correct.

For example, option (A) says that a watt-hour is an amount of time, so that’s not correct. Option (C) also says that a watt-hour is an amount of time, so we’ll cross that off our list as well. And option (D) says that a watt-hour is a measure of power. But again, because we now know that a watt-hour is actually an amount of energy, we can say right away that option (D) can’t be the correct definition either.

So, that leaves these last two options. Let’s review them in order. Option (B) says that a watt-hour is the amount of energy that an electrical device transfers in one hour. This option seems promising because it defines a watt-hour as an amount of energy. But let’s think about what it means, that it’s purportedly an amount of energy that’s transferred in one hour. What this definition is saying, then, is that we’re talking about an amount of energy but not just energy by itself. It’s energy that’s transferred over a time interval of one hour. In other words, it’s really describing an energy per an amount of time.

But from our earlier definition, we can recall that an energy divided by an amount of time or an energy transferred over some amount of time is equal to some amount of power. So, actually, option (B) is saying that a watt-hour is an amount of energy per time or, in other words, it’s an amount of power. But power and energy are different. And we know that a watt-hour is actually an amount of energy. So therefore, option (B) is also off our list.

The last option defines a watt-hour as an amount of energy transferred by a process that has a power of one watt and acts for one hour. So, if we imagine a process with the power of one watt, if this process acts for an hour of time, then we would multiply a watt by an hour. When we’ve done that, we have a watt-hour, a power multiplied by an amount of time. And our equation over here tells us that, in general, if we multiply an amount of power by an amount of time, we get an amount of energy. So, truly, this option defines watt-hour as an amount of energy, specifically as the energy transferred by a process with a power of one watt that acts for one hour of time. This is the correct definition of a watt-hour.

Let’s summarize now what we’ve learned about units of energy. Starting off, we learned that of many possible energy units, the kilowatt-hour, abbreviated k capital W h, is the standard unit used by companies that sell us electrical power. We learned further that one kilowatt-hour is equivalent to 3,600,000 joules. And lastly, we’ve learned conversions between various numbers of watt-hours and their equivalent in joules. One watt-hour is 3,600 joules. A megawatt-hour is 3,600,000,000 joules. A gigawatt-hour is 3,600,000,000,000 joules. And one terawatt-hour is 3,600,000,000,000,000 of them. This is a summary of units of energy.