Video: Determining the Reservoir Temperatures of a Carnot Engine

A Carnot engine has an efficiency of 0.500. The temperature of the engine’s cold reservoir changes and its efficiency drops to 0.450. The initial temperature of the cold reservoir is 30.0°C. The temperature of the hot reservoir does not change. What is the temperature of the hot reservoir? What is the temperature of the cold reservoir after its temperature changes?

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Video Transcript

A Carnot engine has an efficiency of 0.500. The temperature of the engine’s cold reservoir changes and its efficiency drops to 0.450. The initial temperature of the cold reservoir is 30.0 degrees Celsius. The temperature of the hot reservoir does not change. What is the temperature of the hot reservoir? What is the temperature of the cold reservoir after its temperature changes?

All right, in this question, we’ve got a Carnot engine which initially has an efficiency of 0.500. We’re told that the temperature of the cold reservoir changes and its efficiency drops to 0.450. We’re also told the initial temperature of the cold reservoir, which is 30.0 degrees Celsius. And finally, we’re told that the temperature of the hot reservoir does not change. We need to find the temperature of the hot reservoir and the temperature of the cold reservoir after its temperature changes.

In order to answer this question, let’s first recall how to calculate the efficiency of a Carnot engine. The efficiency, 𝜂, is given by 𝜂 is equal to one minus 𝑇 sub 𝐶 over 𝑇 sub 𝐻, where 𝑇 sub 𝐶 is the temperature of the cold reservoir and 𝑇 sub 𝐻 is the temperature of the hot reservoir.

So we can use this equation because we already know the efficiency before the temperature of the cold reservoir changes, as well as that temperature, the temperature of the cold reservoir before it changes. We can use this to calculate the temperature of the hot reservoir.

Substituting in our values, we find that 0.500, the efficiency, is equal to one minus 30.0 plus 273 kelvin — that’s 𝑇 sub 𝐶, which is 30 degrees Celsius in kelvin — divided by 𝑇 sub 𝐻, which is what we’re trying to find out. And it’s important by the way to convert all temperatures into kelvin because otherwise the efficiency formula does not work.

Now with a bit of rearranging, this equation will give us what 𝑇 sub 𝐻 is. Evaluating this fraction gives us 𝑇 sub 𝐻 as being 606 kelvin. We can convert this back to degrees Celsius by subtracting 273 from it. And so we find that the temperature of the hot reservoir, 𝑇 sub 𝐻, is equal to 333 degrees Celsius.

Now let’s move on to the next bit. We need to find the temperature of the cold reservoir after its temperature has changed. Well, we know in this case that the temperature of the hot reservoir does not change. We’ve been told that in the question. Therefore, 𝑇 sub 𝐻 is still 333 degrees Celsius.

And we’ve also been told that, after the temperature of the cold reservoir changes, the efficiency drops to 0.450. What we want to find out is 𝑇 sub 𝐶 superscript new. That’s the new temperature of the cold reservoir. And we can do this by using this equation once again.

We can substitute in the values of 𝜂 and 𝑇 sub 𝐻, remembering to convert 𝑇 sub 𝐻 into kelvin again, which it was 606 kelvin, to give us an equation which we can use to calculate 𝑇 sub 𝐶 superscript new.

Rearranging gives us 𝑇 sub 𝐶 superscript new is equal to 606 kelvin multiplied by one minus 0.450. And we can evaluate this to give us a value for 𝑇 sub 𝐶 superscript new of 333.3 kelvin. And we can convert this to degrees Celsius by subtracting 273 again, which gives us a value of 60.3 degrees Celsius. And so our final answer for this part of the question is that the temperature of the cold reservoir after its temperature changes is 60.3 degrees Celsius.

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