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Consider what happens to the total energy of the gas as it shrinks. By the virial theorem $K=-P/2, w$e can write the total energy $E$ of the gas cloud as $E=K+P=-P/2+P=P/2$. Additionally, according to equation (2) from the webpage, $P\sim -N^2/V^{1/3}$, so $E\sim -N^2/V^{1/3}$. Thus, as volume the volume decreases, the total energy becomes more and more ...

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There is also, a gas of photons in the right side. It was trapped there when you assembled your box, and since you are assuming a perfectly zero emissivity, these photons must be perfectly reflected from all surfaces. That means they are blue shifted if the wall moves toward the right and red-shifted if the wall moves toward the left. Result: If you ...

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The Baez article is strongly misleading in that it applies simplified concepts and arguments appropriate for gases with short-range interactions to a system with many particles interacting purely gravitationally. Systems where short-range forces dominate (Lenard-Jones, van der Waals forces... ) such as rarified gases can be described in such simplified ...

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You are correct in your proof. However the only situation in which both inequalities are satisfied is when the latter one is equal to zero. In this way you are saying that it is only zero for the reversible path. If the Clausius statement is violated then the limit of the composite system of Carnot engines is violated. This then comes back to why is the ...

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In your case, heat conduction is considered to be quasi-static for the subsystems. Let $\Delta Q$ be positive number, energy transferred as heat. Then, change of entropy $S_2$ of the subsystem 2 is $\Delta Q/T_2$, change of entropy $S_1$ of the subsystem 1 is $-\Delta Q/T_1$. For the whole system containing both subsystems, the change of entropy is $$... 1 The equation:$$ dS = \frac{dQ}{T} $$only applies to reversible processes. For an irreversible process dS \gt dQ/T. To see this start with the expression for the change in internal energy:$$ dU = dQ - dW  The internal energy is a state function, so this equation always applies whether the process is reversible or irreversible. So for a reversible ...

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The entropy of a gas does not simply depend on the number of ways to arrange the particles that make up the gas but also on the number of ways of distributing the available energy between those particles. In an adiabatic expansion there is no heat transfer but he gas does do work on its surroundings. This reduces the internal, and so reduces then number of ...

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