[Schrödinger]

Hey guys

I need some help understanding the concept of enthalpy.
I read that Enthalpy is defined as the heat flow (in or out of the system) during a process at constant pressure.

I also read the following in wikipedia :
As a differential expression, the value of H can be defined as[3]



where

    δ represents the inexact differential,
    U is the internal energy,
    δQ = TdS is the energy added by heating during a reversible process,
    δW = pdV is the work done by the system in a reversible process,
    dS is the increase in entropy (joules per kelvin),
    p is the constant pressure,
    dV is an infinitesimal volume, and
    T is the temperature (kelvins).

But why is this correct?
Remembering that enthalpy is defined at constant pressure, dp=0. So why is the Vdp term still present in the above equation?

[Yggdrasil]

Enthalpy is not defined as the amount of heat flow in/out of a system at constant pressure.  Enthalpy is defined as U + pV.  The fact that ΔH = q for a process at constant pressure is just a useful corollary to this definition.

[Schrödinger]

You said $$ \Delta H /$$ = U + pV, without any other specifications.
But isn't this just q for a process? How is enthalpy different from q then?

[Schrödinger]

I'm sorry. I just found out that I completely misread the definition. 
I have understood that $$ \Delta H = q_{p} /$$  is just a way to calculate $$ \Delta{H} /$$, but is not defined at constant pressure.

Thanks for helping me out

[Juan R.]

For a simple system U = U(S,V,N) but sometimes we are interested in pressure p as variable. Then we want to introduce a new function enthalpy H = H(S,p,N).

The general procedure to change a variable (e.g. V) by its conjugate (e.g. p) is achieved using a Legendre transformation. For the change V--> p this is defined as

H == U + pV

Some texts give a geometrical interpretation of the transformation but I prefer a more direct form to understand it. Just notice that we want to eliminate a term pdV from dU and add a new term Vdp, then we add to U the term pV because

dH = dU + pdV + Vdp

dH = (TdS + mudN -pdV) + pdV + Vdp

dH = TdS + mudN + Vdp

That is we obtained the function H = H(S,p,N)

The advantage of H is that for constant p

dH = DQ

and thus changes in enthalpy are directly related to heat changes of processes at constant p. For instance, chemical reactions under constant atmospheric pressure.