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The differential form of internal energy is the following $$dE = TdS - PdV + \mu dN$$

But I naively think that this is inconsistent. Because when I do the following calculation I found it different. $$E = W + Q$$ $$dE = dQ + dW$$ $$dE = TdS - PdV - VdP$$

So where does the other term come from? Furthermore how the last term disapear?

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  • $\begingroup$ Does this help? en.wikipedia.org/wiki/… $\endgroup$
    – Charlie
    Commented Nov 19, 2020 at 15:47
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    $\begingroup$ Since when is W equal to PV???? $\endgroup$
    – Chet Miller
    Commented Nov 19, 2020 at 16:38
  • $\begingroup$ I read the article, and I understand the concept of true differential concept. But I am still confused @Charlie. So can we say then that W is path dependent and is not a true differencial, so it represent only the infinitesimal change? $\endgroup$
    – malachi
    Commented Nov 19, 2020 at 16:49
  • $\begingroup$ Oh, you are right @ChetMiller ! Thanks, it is equal to $W = - \int p dV$. $\endgroup$
    – malachi
    Commented Nov 19, 2020 at 16:53

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The first law is given by $$dU=\delta Q+\delta W$$ or with help of $\delta W=-pdV$ and $\delta Q=TdS$, $$dU=TdS-pdV$$

which is for system where number of particles remain same.

If you add a particle to a system, then the internal energy will change by an amount which is called chemical potential $\mu$. Thus the first and second law of thermodynamics expressed above must, in the case of changing numbers of particle, be modified to contain an extra term, so that $$dU=TdS-pdV+\mu dN$$

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  • $\begingroup$ Thanks, that is really a clear answer and explanation. $\endgroup$
    – malachi
    Commented Nov 20, 2020 at 18:09

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