Why does heat of reaction equal the change in enthalpy for chemical reactions?

Chemists say something like "amount of heat consumed for a chemical reaction equals the change in enthalpy" but I cannot understand why this is the case.

Here is my argument:
Since $H = U +PV$, we have $dH = T dS + V dP + sum_i mu_i dN_i$. If we assume that the heat flow is quasistatic so that we can use $dQ=TdS$, and assuming that $P$ is constant during the reaction so that $dP=0$, we have $dH = dQ + sum_i mu_i dN_i$.

Apparently we have an additional term $sum_i mu_i dN_i$, so that $dH neq dQ$.

Where am I wrong?