Carnot Cycle with negative temperatures

[Thomas Brady]

Homework Statement

Heat engines at negative temperatures. Consider using two heat reservoirs to run an engine (analogous to the Carnot cycle of chapter 3), but specify that both temperatures, T_hot and T_cold, are negative temperatures. The engine is to run reversibly.
(a) If the engine is to do a positive net amount of work, from which reservoir must the energy be extracted?

(b) Under the same conditions, what is the engine’s efficiency?

(c) If you drop the requirment of reversibility, what is the maximum efficiency and how would you achieve it?

Homework Equations

ΔS = Q/T

η = W_in/Q_out = (T_hot - T_cold)/T_hot

The Attempt at a Solution

So for (a) I'm pretty sure the reservoir with a lower magnitude negative temperature is the one from which energy must be extracted by how negative temperatures work to my knowledge. I'm not sure what this means for the efficiency. If T_hot is the temperature from which energy is extracted then the typical expression for max efficiency would have a negative result which I don't think is right. Do I need to get a different expression for efficiency? Also wouldn't the maximum efficiency achievable for part (c) still be the same as the efficiency of the reversible engine? Thanks for the help.

P.S. Sorry I am a novice and am having trouble with subscript

 

[Twigg]

What does entropy balance tell you if the system is reversible? If it's irreversible? Remember that you have heat enter/exiting the system at two reservoirs at different temperatures, and that the full equation for entropy balance is $\Delta S = \sum \frac{Q}{T} + \sigma$, where $\sigma$ is how much entropy is generated by irreversibilities. Use that information and solve for the efficiency as a function of $\sigma$.

Also wouldn't the maximum efficiency achievable for part (c) still be the same as the efficiency of the reversible engine?

Nope! Negative temperatures do weird things, as you'll see.

 

[rude man]

No such thing as negative K temperatures so I wouldn't pursue it.

 

[TSny]

No such thing as negative K temperatures so I wouldn't pursue it.

Uh Oh... duck

 

[DrClaude]

No such thing as negative K temperatures so I wouldn't pursue it.

What do you mean?

 

[rude man]

What do you mean?

I mean tthat after you get to zero K, which is absolute zero, you just don't want to get any colder!

 

[rude man]

Uh Oh... duck

Did I get another one wrong:

 

[TSny]

I mean tthat after you get to zero K, which is absolute zero, you just don't want to get any colder!

That's true in the sense that if you could extract enough energy to get a system to 0 K, you cannot extract more heat and get to negative temperatures.

But there are special systems for which you can imagine adding a finite amount of heat until the temperature becomes infinite, and then add even more energy to get to negative temperatures! Thus, negative temperatures are "hotter" than any positive temperature.

This craziness is a result of the way temperature is defined in thermodynamics in terms of heat and entropy. Before I say too much that is wrong, I will refer you to https://en.wikipedia.org/wiki/Negative_temperature.

 

[rude man]

That's true in the sense that if you could extract enough energy to get a system to 0 K, you cannot extract more heat and get to negative temperatures.

But there are special systems for which you can imagine adding a finite amount of heat until the temperature becomes infinite, and then add even more energy to get to negative temperatures! Thus, negative temperatures are "hotter" than any positive temperature.

This craziness is a result of the way temperature is defined in thermodynamics in terms of heat and entropy. Before I say too much that is wrong, I will refer you to https://en.wikipedia.org/wiki/Negative_temperature.

One lives and one learns ...
What goes around, comes around?