Why does U = T S − P V + ∑ i μ i N i

Question

Why does   U  =  T  S  −  P  V  +      ∑    i        μ    i        N    i

arbustekp · Accepted Answer

I&#039;m going to suppress the sum over different chemical potentials for simplicity as it doesn&#039;t materially affect the argument. If we write U as a function of S,V,N, then extensivity of U is mathematically defined as follows  U  (  λ  S  ,  λ  V  ,  λ  N  )  =  λ  U  (  S  ,  V  ,  N  )Physically, this is saying that if you scale the quanities that characterize your physical system by a certain amount, then the energy scales by the same amount. The mathematical terminology for this is that U is a homogeneous function of degree 1 in S, V, and N. Now, there is a theorem on homogeneous functions called Euler&#039;s homogeneous function theorem which (up to a technical assumption or two) states that a function   f  :            R        n    →      R   is homogeneous of degree k&amp;gt;0, namely  f  (  λ  x  )  =      λ    k    f  (  x  )if and only if  k  f  (  x  )  =  x  ⋅  ∇  f  (  x  )Since the energy is a homogeneous function of degree 1, this theorem tells us that                    U        (        S        ,        V        ,        N        )                            =        (        S        ,        V        ,        N        )        ⋅        ∇        U        (        S        ,        V        ,        N        )                                          =        S                              (                                          ∂                U                                            ∂                S                                      )                                V            ,            N                          +        V                              (                                          ∂                U                                            ∂                V                                      )                                S            ,            N                          +        N                              (                                          ∂                U                                            ∂                N                                      )                                S            ,            V                              On the other hand, the fundamental thermodynamic relation you wrote (as the first equation) above allows us to identify                                          (                                          ∂                U                                            ∂                S                                      )                                V            ,            N                                              =        T                                                  (                                          ∂                U                                            ∂                V                                      )                                S            ,            N                                              =        −        P                                                  (                                          ∂                U                                            ∂                N                                      )                                S            ,            V                                              =        μ            so that we get the desired result:  U  (  S  ,  V  N  )  =  T  S  −  P  V  +  μ  N

Why does U=TS−PV+sum_i μ_iNi?

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