st               nd
     1  law and 2  law in a simple system
      st      dU Q PdV             dU TdS PdV
     1  law:
                                            1       P
       nd                             dS  dU dV
     2  law:        QTdS
                                            T       T
     The functions U(S, V, N) and S(U,V, N) are called 
     fundamental equations of a system. Each one of them 
     contains full information about a system.
     Generally dU TdS ydx               energy representation
                                   i  i
                               i
                       1          y
                 dS  dU  i dx           entropy representation
                      T           T    i
                                i
   Equations of state
    The intensive variables in the fundamental 
     equations written as functions of the extensive 
     variables (for fixed mole numbers):
     dU TdS PdV   T T(S,V)   P  P(S,V)
     dS 1 dU  PdV  1 1(S,V)   P P(S,V)
         T    T       T  T       T   T
    Generally y y (x ,x ,...,x ,...)
               i  i 1  2   i
      Chemical potential and partial molar quantities
       Chemical potential m for the component i
                                    i
          U                  S          
                                       i
          N           i        N          T
            i S,V,...           i S,V,...
       Quasi-static chemical work
         W  dN                  dU TdS PdV              dN
             c     i i                                    i i
                   i                                        i
       The partial molar quantity x (x is an extensive function) 
         associated with the component i (when T, P are constant)
          x                                               V            
                         x        partial molar volume                    V
          N              i                                   N              i
            i T,P,N ( ji)                                     i T,P,N ( ji)
                    j                                                    j
     Euler relation
      U and S are both homogeneous first order 
         functions of extensive parameters
         U(X1,X2,...,Xi,...) U(X1,X2,...,Xi,...)            l is a constant
         U(x ,x ,...,x ,...)   U(x,x ,...,x ,...)
                1  2     i             1    2       i
                                          
                 U(x,x ,...,x ,...) (x )           U(x,x ,...,x ,...)
        U              1    2       i          i             1    2       i    x
                         (x )                               (x )             i
               i               i                       i               i
                              U(x,x ,...,x ,...)
         Let l = 1 U               1   2     i    x      y x
                                     (x )          i    i i
                            i            i                i
                              UTS PV           N S1UPV                   i N
         Simple systems                        i i                                 i
                                                i             T      T       i  T
           Gibbs-Duhem relation
            UTS PVN  dUd(TS) d(PV)d(iNi)
                                                             i     i
                                                    i                                                                                          i
            1st law of TD: dU TdS  PdV dN
                                                                                                             i        i
                                                                                                    i
            TdS PdV idNi d(TS) d(PV)d(iNi)
                                                i                                                                      i
            SdT  VdPNidi 0
                                               i                                                           in simple systems
            Ud(1)Vd(P) Nd(i)0
                      T                     T                        i       T
                                                            i
             In a single component simple system: d  sdT vdP