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I was thinking about the question in the title. I found the following thread, and some of the answers are making my head spin!

Is it possible to focus the radiation from a black body to make something hotter than that black body?

I am yet to delve into the world of optics (though I hope to soon) so I was hoping I could prove this proposition just using the laws of thermodynamics. However, nobody in the thread above posted the proof that I came up with.

This takes the form of a proof by contradiction. I construct a theoretical system in which the above process occurs, then analyse it to prove that it breaks the 2nd law of thermodynamics. The system is as follows:

  1. A light source with a temperature of 6000K (sufficiently large to be considered a thermal reservoir)
  2. An object whose initial temperature is 1000K

Proposed Process: The light from the source is focused onto the object. The object is heated to a final temperature of 7000K. For simplicity, assume no heat is transferred from the object to its surroundings during this process.

  • Analysis: The change in entropy of the hot reservoir is calculated by the following equation: $$\Delta S = -Q / T(\mathrm{Hot})$$ (Equation taken from [1], please advise me if I have not applied this correctly)
  • The change in entropy of the object being heated is calculated with: $$\Delta S = m C_p \ln (T(\mathrm{Hot}) / T(\mathrm{Cold}))$$ (Equation taken from [2], please advise me if I have not applied this correctly)

Substituting the conditions of my system into the above equation with a value of 100 kJ for $Q$ and $C_p = 1\:\mathrm J / ( \mathrm{kg K} )$ and $m = 1\:\mathrm{kg}$ we get the following result: \begin{align} \Delta S (\mathrm{Total}) &= \Delta S (\mathrm{Hot Reservoir}) + \Delta S (\mathrm{Object}), \\ \Delta S (\mathrm{Total}) &= - 100,000\:\mathrm{J} / 6000\:\mathrm{K} + \ln (7000\:\mathrm K / 1000\:\mathrm K) \times 1\:\mathrm J / ( \mathrm{kg K} ) \times 1\:\mathrm{kg} \\ \Delta S (\mathrm{Total}) & = -14.7 \:\mathrm J /\mathrm K \end{align}

The second law of thermodynamics states that the change in entropy for a real system must be greater than zero. Therefore, it is impossible to build a device that will focus the light from the source in such a way that it achieves the above.

Thus we have found a counterexample for this class of device, proving it cannot exist.

I realise that is a slightly different question to that in the other thread, as it was asking about the physical reasons why building such a device is not possible. This required the use of the aforementioned head spinning optics!

Please would you advise me if you agree with my proof?

References

  1. http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node41.html

  2. http://www.chem.arizona.edu/~salzmanr/480a/480ants/2ndlawap/2ndlawap.html

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    $\begingroup$ Try the Clausius formulation of the second law: "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time". That's all you need in one sentence. $\endgroup$ – CuriousOne Dec 22 '14 at 13:00
  • $\begingroup$ Thanks for your reply, yes this does disprove this concept in one sentence! Not sure why I didn't think of this, it seems so obvious now.. $\endgroup$ – Appguy1 Dec 22 '14 at 13:30
  • $\begingroup$ Also I realise my example above also "proves" that an object cannot be heated to 5000 K.. Might have to give that some further thought! $\endgroup$ – Appguy1 Dec 22 '14 at 13:41
  • $\begingroup$ Ah, it is because I just made up the final temperature. When I actually calculate it using the first law (Q = mCPdeltaT) T(Final) = 101,000 K. In this case the Delta S is positive. However, if I use 7000 K as my final temperature, the Delta S is still very slightly positive. Something is not quite right with my working.. $\endgroup$ – Appguy1 Dec 22 '14 at 13:59

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