# How much energy Maxwell's demon will earn?

Suppose we have one mole of one-atom ideal gas at temperature $T$.

Suppose Maxwell's daemon has separated molecules into two sections, one with speed below mean and another with speed above mean.

How much energy will the daemon gain?

UPDATE

Yes probably this is the same question as how much energy real daemon should spend or how much energy it would be possible to get from the two subvolumes later, with heat machine.

If these values are different, then please explain.

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The answer is directly related to variance of speed of the particles. Now, what is the common variance of speed of particles in your average gas? How does it vary depending on which factors? I'd love to see the answers. But I can bet a mole of gas at temperature T sitting there for hours will yield much less energy than one that still has temperature T but is a mixture of two portions, one of 0.5T, the other 1.5T, freshly mixed. –  SF. Dec 15 '12 at 21:39
Isn't speed variance fully defined by saying gas is ideal? Isn't it Maxwell distributed? –  Suzan Cioc Dec 16 '12 at 10:35
yes, true. I was much more interested in real gasses... –  SF. Dec 16 '12 at 12:29
What do you mean by how much energy does the daemon gain? Do you mean the work that can be extracted? Any increase in internal energy? –  namehere Dec 16 '12 at 12:54
Internal energy won't change if I am not mistake. So, the question is about how many work. –  Suzan Cioc Dec 16 '12 at 17:00

It depends on the amount of gas and the initial temperature, and so on, but there is another question behind that, and it's if demon were possible, then after some separation you could convert the heat energy in work and restart the process 'creating energy'. The main flaw is to suppose that demon knows speed and position of particles, that information, is energy, and perhaps you would need more energy to get that information than the energy you could 'generate' by having it.

EDIT

It seems an interesting question, I'll just add some thoughts to start an answer's direction that could be finished by other...Supposing a single conversion cycle and that you know gas heat capacity, think about this simple linealization

$C$ = heat capacity
$Q$ = heat
$T$ = temperature

$\Delta$$Q = C\Delta$$T$

So we now discuss about $\Delta$$T Would it be \Delta$$T>T$?

$\Delta$$T=T? \Delta$$T<T$?

Besides other details.. (for example the demon can only open or close a door and a near zero temperature particles won't cross the door (no matter which side start), and the gas structure -would it be monoathomic? -molecular? if the second type so it will undergo rotational and vibrational motions that contribute to temperature, and aren't selectable through a door, etc..)

We could suppose a Maxwell-Boltzmann speed distribution and that could lead us to approximate the maximum temperature departure from mean speeds, here a wikipedia graph. That could be a path for calculate $\Delta$$T. But even Maxwell-Boltzmann distribution will be a bad aproximation when we can manipulate single particles, a temperature could be made of a very asymetric distribution of speeds at that scale and could be a single particle with a big departure from median speed, take that extreme case of a single particle at a very high speed compared with all the others being slower, then once separated it would mean a high temperature for one side and a low for the other, but being just one particle you will not get much energy of it because of the low mass; so temperature separation is not enough (at least you could repeat the process in which case you could get energy indefinitely as I point on first answer). Finally The demon should have a "set point", it's a value of temperature that let him decide when to open or close the door, the temperature separation will depend on it, I think normally it's supposed to be mean speed, then perhaps it could be adjusted to be \Delta$$T$= $\frac{T}{2}$, then that's another parameter from which it depends

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The thermodynamic demon is named 'demon' precisely for the reason that it is has knowledge and ability inaccessible to 'mere mortals' - it's the same class of objects as a train moving faster than speed of light, a tool in thought experiment. –  SF. Dec 15 '12 at 21:32
Initial temperature is $T$ and initial amount of gas is one mole. –  Suzan Cioc Dec 16 '12 at 10:36
@SF The daemon has indeed been built in the laboratory so indeed does not have any special abilities! See arxiv.org/abs/1009.5287 the daemon wins in the short term but ultimately cannot overcome the second law owing to Landauer's principle. See cc.gatech.edu/computing/nano/documents/… . –  WetSavannaAnimal aka Rod Vance Nov 19 '13 at 10:45
@WetSavannaAnimalakaRodVance: Well, I guess it's just as useful as physical Turing Machine –  SF. Nov 19 '13 at 11:03