Entropy, chemical potential, temperature

[aaaa202]

For a thermodynamic system there exists a function called entropy S(U,N,V) etc.
We then define for instance temperature as:
1/T = ∂S/∂U
μ = ∂S/∂N
etc.
When taking these partial it is understood that we only take the derivative of S wrt the explicit dependece on U,N etc. right? Because couldn't U carry an N dependence? I mean it does not for me make physical sense that the energy of the system should not be related to the number of particles in it. Actually it seems also a bit weird that there should be an explicit U dependence. Does this come from the fact that we are given the mean value of the internal energy of the system?

 

[DrDu]

When taking these partial it is understood that we only take the derivative of S wrt the explicit dependece on U,N etc. right?

Yes, that is the definition of a partial derivative as opposed, e.g. to a total derivative.

 

[Useful nucleus]

For a thermodynamic system there exists a function called entropy S(U,N,V) etc.
We then define for instance temperature as:
1/T = ∂S/∂U
μ = ∂S/∂N
etc.
When taking these partial it is understood that we only take the derivative of S wrt the explicit dependece on U,N etc. right? Because couldn't U carry an N dependence? I mean it does not for me make physical sense that the energy of the system should not be related to the number of particles in it. Actually it seems also a bit weird that there should be an explicit U dependence. Does this come from the fact that we are given the mean value of the internal energy of the system?

'

In the entropy representation U,V,N are independent variables. On the other hand in the energy representation S,V,N are independent variables. One should not mix the two representations in the same time.