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In our thermodynamics lecture course we have defined $U$ so many different ways I am having a bit of trouble keeping up.

In our last lectures we were using $\text dU = T\text ds - P\text dv$, recently we were introduced to the idea of Gibbs free energy and chemical potentials. All the fundamental thermodynamic quantities were then re written in terms of the Gibbs free energy. Now in our lectures we are using $\text dU = T\text ds - P\text dv + \sum_i\mu_i \text dn_i$.

I don't understand how the internal energy can equal both expressions at the same time since they are identical except one has the extra chemical potential term added to it. Is the second version of internal energy just a more general version?

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    $\begingroup$ Shouldn't the left sides of those equations be $\text dU$? Also, think of what happens if each $\text dn_i$ is $0$. Also also, what other "so many" ways are you referring to? $\endgroup$ Commented Oct 30, 2019 at 11:46
  • $\begingroup$ Ah ok so the second version is the general version and its the same as the previous one in the case where there is no chemical reactions on going. Also made the edit $\endgroup$ Commented Oct 30, 2019 at 11:49
  • $\begingroup$ Well it doesn't have to just be chemical reactions, but yes. $\endgroup$ Commented Oct 30, 2019 at 11:50

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In our thermodynamics lecture course we have defined 𝑈 so many different ways I am having a bit of trouble keeping up.

I don't think you should view different equations involving internal energy as different definitions of internal energy per se. The internal energy of a system is defined as the sum of its kinetic and potential energy at the microscopic level. There are many equations that show the relationship between internal energy and other system properties under various conditions. Examples are the equations for thermodynamic potentials that relate the internal energy of a system to the properties of enthalpy, Helmholtz free energy, and Gibbs free energy. But one shouldn't consider these equations, as well as the ones you gave, as being different definitions of internal energy.

Hope this helps.

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