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Carnot Cycle

How do you increase a system's entropy without increasing the temperature? If entropy is the degree of disorder, won't more "disorderliness" means high kinetic energy, and the average energy of the particle (The temperature) will be higher?

or another way of saying is how to increase the heat energy without increasing temperature (since dQ=TdS)? Are we considering the potential energy between particles? I thought in the case of thermal physics, we would just consider elastic collision for every particle in the system?

And if this is the reason for temperature's independence of heat energy, can this be say so for entropy as well?

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    $\begingroup$ Increase the volume. The number of possible distributions grows as the volume to the power of the number of particles. $\endgroup$ – Pieter Oct 14 '17 at 9:26
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For a monatomic ideal gas,

$$ U = \frac{3}{2}NkT $$

so constant temperature is equivalent to constant internal energy. From the 1st law of thermodynamics (assuming constant particle number), $$ dU = TdS - pdV$$

Letting $dU=0$, we see that

$$ T dS = p dV$$

So we can increase the entropy by increasing the volume at constant temperature (isothermal expansion). We can go even further than that, because $$ pV = NkT$$ so $$ dS = \frac{p}{T}dV = Nk \frac{dV}{V}$$ and therefore $$ \Delta S = S_2 - S_1 = Nk \ln\left(\frac{V_2}{V_1}\right)$$

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One example of increasing entropy without increasing temperature is phase transformation of water. From saturated liquid liquid to saturated vapor point, heat is added to the system but temperature remains constant. The entropy of the system increases.

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