One definition of the chemical potential is the change of internal energy of the system with respect to a particle of the added substance with the system entropy, molar volume, and all other species particle numbers remaining constant.
A thought experiment: Let's suppose I have a completely rigid container with 1 mole of argon in it with an infinite amount of insulation surrounding the container. Therefore, I have zero heat transfer and the molar volume is constant. I admit one atom of argon into the container in a reversible manner (same P and T). The pressure of the system must increase differentially according to the ideal gas law. Hence the entropy will decrease differentially except that it must stay constant and so temperature will have to increase for this process to be isentropic. But for an ideal gas, internal energy depends only on temperature. So dU/dn is positive. Yet, when I look up values for argon as a gas at STP, its value is 0 (like all the elements in their standard states). Where is my reasoning wrong?