### 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.