A refrigerator has a coefficient of performance of 2.25, runs on an input of 135 W of electrical power, and keeps its inside compartment at 5°C. If you put a dozen 1.0-L plastic bottles of water at 31°C into this refrigerator, how long will it take for them to be cooled down to 5°C? (Ignore any heat that leaves the plastic.)

Video transcript

Welcome back everybody. We are told that we put 500 g of water into an ice cube tray. Now the ice maker has a coefficient of performance of three and it uses 240 watts of electrical power. And we are tasked with finding how long does it take for the water to freeze into ice? Well, we're given a coefficient of efficiency here. So let's work with that. The formula or this is that the coefficient of performance? My apologies, is equal to the absolute value of Q. C. Divided by the absolute value of W or work. We don't have any of these terms. So we're going to have to find these terms here. Let's start out with our Q. C. This is just going to be equal to our mass times our latent heat of fusion of water. So let's go ahead and plug in. Those terms here are mass is going to be 500 g but we do need it in terms of kg. So I'm gonna multiply this by 10 to the negative third. And then our latent heat of fusion of water is 3.33 times 10 to the fifth, joules per kilogram. Now these kilogram units are going to cancel out and if we multiply straight across we get that are absolute value Q. C. Is 1.67 times 10 to the fifth tools. Great. So we've dealt with this numerator this term here but now we need to deal with work. So but we also need to incorporate time somehow. So here's what I'm gonna do. I see that we're given power and we can use the formula for power to find our time. The formula for power is simply work over time. I multiply both sides by time. Time on this side is going to cancel out. And we are given that the work is equal to power times time. Power and time are both going to be positive. So this is equal to the absolute value of power times time. So let's go ahead and sub this back into our formula right here we get that K is equal to our absolute value of Q C. All divided by our power times time. Now time is what we are looking for. So I need to isolate that term. I'm actually going to multiply both sides by T over K over a little bit of algebra here, these T terms are going to cancel out on the right. These K terms are going to cancel out on the left and we are left with the time is equal to our absolute value of Q C. All divided by power times our coefficient of performance. We know all this. So let's go ahead and plug in our terms here, our absolute value Qc Is 1.6, 7 times 10 to the 5th. Our power is 240 watts in our coefficient of performance is three. And when you plug all of this into our calculator, we get that our time it takes to freeze water into ice is 232 seconds at the specific ice maker, giving us a final answer choice of a thank you all so much for watching. Hope this video helped. We will see you all in the next one.