I watched this MIT videolecture (minute 4.12) where there is the relationship:

$$ \left( dU \right)_{S,V} < 0 $$

which states that the internal energy, at constant entropy and volume, is less than zero.

Why? I know that:

$$ T \, dS = dU + P \, dV $$

so, if $ dS=dV=0 $, I get $ dU=0 $.

Thank you in advance.

  • $\begingroup$ It would be helpful if you could provide a bit more context, so that people don't need to watch a 4 minute video to be able to answer. What is the video about? $\endgroup$ – By Symmetry Nov 14 '19 at 9:18
  • $\begingroup$ Hello @BySymmetry, the video is about a basic course on thermodynamics and the goal of the videolecture is the understand when a process is spontaneous. $\endgroup$ – Gennaro Arguzzi Nov 14 '19 at 9:29
  • $\begingroup$ I started watching the video. When it got to the point where he started talking about property changes holding others constant he referred to an isolated system. That raised a red flag to me because there can be no change in internal energy of an isolated system, whether or not changes are spontaneous. That’s when I stopped watching. $\endgroup$ – Bob D Nov 14 '19 at 9:53
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    $\begingroup$ The equation you wrote is valid only if no reaction is occurring. Otherwise, U also changes when the number of moles of the various species changes. $\endgroup$ – Chet Miller Nov 14 '19 at 13:03
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    $\begingroup$ There is an additional term equal to the sum of the chemical potentials of the various species times the change in their number of moles. $\endgroup$ – Chet Miller Nov 14 '19 at 15:05

Take system in thermal equilibrium with its surroundings at temperature T. When change in the system occurs and there is a transfer of energy as heat between the system and the surroundings the Clausius inequality ($dS \ge dq/T$) reads:

$dS - \frac{dq}{T} \ge 0$

At constant volume i.e. in the absence of non-expansion work, $dq_V = dU$ consequently

$dS - \frac{dU}{T} \ge 0$

that can be rewritten as

$TdS \ge dU$

At constant entropy the expression becomes

$dU_{S,V} \le 0$

This relation states that if the entropy and volume of the system are constant, then the internal energy must decrease in a spontaneous change. If the entropy of the system in unchanged during a transformation then there must be an increase in the entropy of the surroundings, which can be achieved by a transfer of heat.

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  • $\begingroup$ Did you watch the video? $\endgroup$ – Bob D Nov 14 '19 at 11:49
  • $\begingroup$ Hello @gioretikto; thank you for your explanation. $\endgroup$ – Gennaro Arguzzi Nov 14 '19 at 12:49

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