A Carnot engine operates between temperatures of 5℃ and 500℃. The output is used to run a Carnot refrigerator operating between -5℃ and 25℃. How many joules of heat energy does the refrigerator exhaust into the room for each joule of heat energy used by the heat engine?

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A spacecraft's carne heat engine is designed to work between 100 °C and 600 °C to generate energy. This energy powers a Carno cooling system that regulates the spacecraft's internal temperature. The cooling system works between negative °C and 20 °C determine in jus the ratio of heat energy discharged by the cooling system to the energy consumed by the heat engine. So that's our end goal is to determine the ratio of heat energy discharge by the cooling system to the energy consumed by the heat engine in units of jewels. OK. So we're given some multiple choice answers. Let's read them off to see what our final answer might be. A is 5.4 B is 4.2 C is 5.6 and D is 4.4. OK. So first off, let us write down all of our known variables for the engine and for the cooling system. So for the temperature of the hot reservoir for the engine. Let's call it T subscript capital. He is equal to 600 °C. But let's quickly convert that to Kelvin. And to do that, all the do is take 600 plus 273. And when we add those together, we should get 873. Kelvin and the temperature at the cold reservoir for the engine. Let's call it T ce is equal to 100 °C. And let's convert that to Kelvin like we just did. So 100 plus 273 equals 373 Calvin. OK. So the cooling system temperature at the hot reservoir. So considering the cooling system, so the temperature at the hot reservoir, so let's call it THS is equal to 20 °C. So let's convert that to Kelvin really quick. So 20 plus 2 73 is equal to 93 Kelvin. And then the temperature at the cold reservoir or the cooling system, let's call it T subscript CS is equal to negative 10 °C. So let's convert that Kelvin. So negative 10 plus 2 73 is equal to 2 63 Havin. OK. So at this stage, we now need to find the ratio of Q Hs. So cute HS divided by qhe where Qhs is the heat energy exhausted by the cooling system. And qhe is the heat energy consumed by the heat in the engine. OK. So we also need to recall that the engine and the cooling system both operate at a maximum corno efficiency the work done by the engine. So the work output of the engine is equal to the work input of the cooling system. So we can write that w so the work output of the engine is equal to the work input of the cooling system. So in order to find the expression for the work done by the engine, we need to recall and use the equation for the thermal efficiency of a heat engine operating between the hot reservoir and the cold reservoir. And the equation for that is a ac so that the efficiency of a car engine is equal to one minus the temperature of the cold reservoir for the engine divided by the temperature of the hot reservoir of the engine is equal to the work output of the engine divided by the heat energy for the hot reservoir of the engine. So we can rearrange this equation to solve for the work output of the engine using a little bit of algebra. So the work output of the engine is equal to the heat energy of the hot reservoir multiplied by one minus one minus the temperature of the cold reservoir of the engine divided by the temperature of the hot reservoir of the engine. OK. So this equation becomes the work input of the cooling system, the coefficient of a performance for a cooling system is KC or C represents car is equal to the temperature of the cold reservoir of the system. Divided by the temperature of the hot reservoir or the cooling system minus the temperature of the cold reservoir of the cooling system is equal to the heat energy of the cold reservoir. For the SS for the cooling system divided by the work input of the system. Now we need to rearrange this to solve for the energy the heat energy of the cold reservoir for the cooling system, which is written as Q CS. So the heat energy of the cooling system or the, the heat energy for the cold reservoir for the cooling system is equal to the work input in the cooling system multiplied by the temperature of the cold reservoir for the cooling system divided by the temperature of the hot reservoir of the cooling system minus the temperature of the cold reservoir of the cooling system. So now we need to find the heat exhausted by the cooling system. So the heat exhausted by the cooling system is Q subscript HS is equal to the work input of the system plus the work input of the system multiplied by the temperature of the cold reservoir of the system divided by the temperature of the hot reservoir of the system minus the temperature of the cold reservoir of the system. OK. So now you can write that that Qhs is equal to Wnwins multiplied by one plus T CS divided by TH S minus T CS since though. So now moving along, since the work output of the engine is equal to the work input of the cooling system, we can then go on to write that the heat energy of the cooling system is equal to the work output of the engine multiplied by one plus the temperature of the cold system divided by the temperature of the hot system minus the temperature of the cold system. So, so take a breather, we're almost there. We're chugging right along here. A now we need to substitute the expression for the work output engine from equation one into the following expression which let's go back up here for a second. So equation one states that the output the work output of the engine is equal to the heat energy of the engine multiplied by one minus the temperature of the cold reservoir for the engine divided by the temperature of the hot reservoir for the engine. OK. That was equation one. OK. So now we need to substitute the expression for the work output of the engine from equation one into the following expression. So Q so the heat energy of the system is equal to the heat energy of the, of the engine multiplied by one minus the temperature of the cold reservoir for the engine divided by the temperature of the hot reservoir of the engine multiplied by one plus the temperature of the cold reservoir for the cooling system divided by the temperature of the hot reservoir for the for the cooling system minus the temperature of the cold reservoir for the cooling system. Ok. So now we need to rearrange this equation to solve for the heat energy of the system divided by the heat energy of the engine. So let's do that. So the heat energy of the cooling system divided by the heat energy of the engine is equal to one minus the temperature of the cool old reservoir for the engine divided by the temperature of the hot reservoir of the engine multiplied by one plus the temperature of the cold reservoir for the system divided by the temperature of the hot reservoir of the system minus the temperature of the cold reservoir of the system. So now we need to simplify. So Q Hs divided by Qhe is equal to one minus T ce divided by the multiplied by THS divided by T Hs minus T CS. So now at this stage, we can plug in all of our known variables and solve for our final answers. So Qhs divided by qhe is equal to one minus three. Kelvin divided by 873. Kelvin multiplied by 293 Calvin divided by 293 Elvin minus 263 Calvin. And when we plug that into a calculator we should get 5.59 which rounds to 5.6 which is our final answer and don't forget that it's in units of jewels. So let's look at our multiple choice answers to see what the final answer should be. So out of the multiple choice answers, the correct answer is the letter C 5.6. Thank you so much for watching. Hopefully that helped and I can wait to see you in the next video. Bye.