Why isn't d H = Q + ∑ i μ i d N i (constant pressure)

Question

Why isn&#039;t    d  H  =  Q  +      ∑    i        μ    i    d      N    i   (constant pressure)

ce1ret3i · Accepted Answer

The chemical potential   d      μ    i   is defined as the partial molar free energy:      μ    i    =            (                        ∂          G                          ∂                      N            i                              )              T      ,      P      But, as a result of this and the other basic relationships, it is also given by:      μ    i    =            (                        ∂          U                          ∂                      N            i                              )              S      ,      V        =            (                        ∂          H                          ∂                      N            i                              )              S      ,      P      So,  d  H  =  T  d  S  +  V  d  P  +  ∑            μ      i        d          N      i      and, at constant pressure,  d  H  =  d  U  +  P  d  V  =  T  d  S  +  ∑            μ      i        d          N      i      But, dQ is not equal to TdS unless the process is reversible.Also, please don&#039;t mix differentials up with finite changes. A finite change should be represented by a   Δ. And Q is not equal to   T  Δ  S unless the temperature is constant (and, again, unless the process is reversible).Also,  Δ  H  =  Qonly applies to a closed system (i.e., no mass entering or leaving). Obviously, if mass is entering or leaving, it does so with its own enthalpy (and other properties), and this changes the enthalpy of the system.

Why isn't dH=Q+sum_i mu_i dN_i

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