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I have run into an issue while messing around with the fundamental equation for internal energy.

$$dU=TdS-PdV + \sum_{i=1}^N{\mu_idn_i}$$

Using Euler's theorem for homogenous we get.

$$U=TS-PV + \sum_{i=1}^N{\mu_in_i}$$

Now, from the fundamental equation, we know that the entropy derivative of internal energy at constant volume and moles of each species is temperature. So it should be that if I differentiate the Euler equation with respect to entropy at constant volume and moles of each species that it reduces down to temperature.

$$\frac{\partial}{\partial S} U = T + S \frac{\partial T}{\partial S} -P \frac{\partial V}{\partial S}-V \frac{\partial P}{\partial S} + \sum_{i=1}^N \mu_i \frac{\partial n_i}{\partial S} + \sum n_i \frac{\partial \mu_i}{\partial S}$$

But the right side of the equation has a bunch of extra terms. I've tried to make substitutions using Maxwell equations and a couple other things, but I cannot quite figure out how the terms cancel. This leads me to believe that for some reason I don't need to have them in the first place, which I don't understand since I'm under the impression that T, P, V, n, and mu all depend on S. Any insight as to what dots I'm not connecting would be great.

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Congratulations, you've (re)discovered the Gibbs-Duhem relation.

Differentiating $U$ with respect to $S$, we have, as you note,

$$\frac{\partial U}{\partial S} = T + S \frac{\partial T}{\partial S} -P \frac{\partial V}{\partial S}-V \frac{\partial P}{\partial S} + \sum_{i=1}^N \mu_i \frac{\partial n_i}{\partial S} + \sum n_i \frac{\partial \mu_i}{\partial S}.$$

Applying Gibbs-Duhem, we obtain

$$\frac{\partial U}{\partial S} = T -P \frac{\partial V}{\partial S} + \sum_{i=1}^N \mu_i \frac{\partial n_i}{\partial S} .$$

Applying the conditions of constant volume and moles of each species, we recover

$$\left(\frac{\partial U}{\partial S}\right)_{V,n_i} = T ,$$

as expected.

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