Video Transcript
What is nine times 10 to the power of seven joules in kilowatt-hours?
Now we’ve been asked to convert this quantity in joules to kilowatt-hours. So first of all, we need to know how joules link to kilowatt-hours. To do this, we can recall that power is defined as the energy transferred per unit time. And the standard units for each one of these quantities is watts for power, joules for energy, and seconds for time. So this looks like we’re nearly there. We’ve got joules, and we’ve got something that looks like watts. And if we wanted to, we can relate seconds to hours as well.
Now, what we can do is to rearrange this equation by multiplying both sides by 𝑡. What that tells us is that power multiplied by time is equal to the energy transferred. And we can substitute in all the units because the unit of power, one watt, multiplied by the unit of time, one second, is equal to the unit of energy, one joule.
Now, one watt times one second is the same as one watt-second. However, in this question, we want kilowatt-hours. So we need to link watts to kilowatts and seconds to hours. We can first recall that one kilowatt is equal to 1000 watts. That’s what the prefix kilo means. It means 1000. So what we can do here is to multiply both sides of the equation by 1000.
So now that we’ve done that, we can take this 1000 and multiply it by the one watt. This way, we get 1000 watts. And that is equal to one kilowatt. So we can replace 1000 times one watt with one kilowatt. And the right-hand side can simply be replaced by 1000 joules. Therefore, in other words, one kilowatt times one second, or one kilowatt-second, is the same as 1000 joules.
Now let’s convert seconds to hours. We can recall that one hour has 3600 seconds in it, because every hour has 60 minutes in it and every minute has 60 seconds in it. So every hour has 60 times 60 seconds in it. And 60 times 60 is 3600. So what we can do is to multiply this equation now by 3600 on both sides. What that leaves us with is that 3600 times this part, which is one kilowatt, times one second is equal to 3600 times this part, which is 1000 joules.
Now what we can do is to multiply the 3600 by one second, which will give us the following: One kilowatt times 3600 times one second is equal to 3600 times 1000 joules. But this bracket here, 3600 lots of one second, is the same as one hour because that’s 3600 seconds. So we can replace all the stuff in the parentheses with one hour. And hence we’ve got one kilowatt times one hour on the left-hand side. And that is equal to one kilowatt-hour. And hence we’ve arrived at the units given to us in the question.
So now let’s simplify the right-hand side: 3600 times 1000 joules. Well, 3600 times 1000 is the same as 3.6 times 10 to the power of six. And as we said earlier, the left-hand side is equal to one kilowatt-hour. Therefore, one kilowatt-hour is equivalent to 3.6 times 10 to the power of six joules. And remember, both of these are units of energy, because on the right-hand side, we have joules, which is already a unit of energy, and on the left we’ve got a unit of power, kilowatts, multiplied by a unit of time, hours. And this product between the units of power and time gives us overall a unit of energy.
So to recap, one kilowatt-hour is equal to 3.6 times 10 to the power of six joules. So then how many kilowatt-hours are there in nine times 10 to the power of seven joules? Well, in nine times 10 to the power of seven joules, there are this many kilowatt-hours. In other words, this is the amount of energy in joules that we want to find in kilowatt-hours. And this is the amount of energy in one kilowatt-hour.
So dividing nine times 10 to the power of seven by 3.6 times 10 to the power of six will give us the number of kilowatt-hours in nine times 10 to the power of seven joules. So when we evaluate the fraction, we find that this is equal to 25. And, therefore, we’ve reached our final answer. Nine times 10 to the power of seven joules is equal to 25 kilowatt-hours.
