# Internal energy at constant entropy and volume is less than zero - Explanation

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 $$dU=0$$.

• 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? – By Symmetry Nov 14 '19 at 9:18
• 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. – Gennaro Arguzzi Nov 14 '19 at 9:29
• 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. – Bob D Nov 14 '19 at 9:53
• 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. – Chet Miller Nov 14 '19 at 13:03
• 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. – 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.

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