Consider a system which is characterized by the extensive variables $(U,V,N_1,…,N_m)$. For a quasistatic process which occurs in contact with some pressure reservoir and where the $N_i$ are constant, one has
$$dU = TdS -PdV \implies TdS = dQ,$$
where the implication follows from the First Law. Defining the relevant Legendre transform for constant pressure processes (the enthalpy, $H = U+PV$) leads to
$$dH = TdS +VdP = TdS = dQ,$$
where we have used $dP = 0$ for this quasistatic process occurring while in contact with a pressure reservoir.
Now I emphasize that this development has depended intimately on $dN_i = 0$ throughout. Nevertheless, in his Chapter 6.4, Callen uses $dH = dQ$ even for processes involving chemical reactions (but which are otherwise closed), wherein the $dN_i$ are certainly not constant in general. He provides no justification for using this though, so I am hoping someone can clear that up.
Best Answer
Writing dQ=TdS is the part of your analysis that is incorrect. It should read $$dQ=TdS-Td\sigma$$, where $d\sigma$ is the differential entropy generated due to irreversibility of the chemical reaction. So, $$Td\sigma=-\sum{\mu_j}dN_j$$which captures the entropy generation due to the spontaneous reaction.