At constant pressure $p=\text{const.}$, change of internal energy equals heat subtract work(PV). So, $q=\Delta E - pV$. So is the the change of enthalpy. So we say at constant pressure, the change of enthalpy equals the change of heat. But when the volume is constant $V = \text{const.}$, no work is done. $\Delta E= q$, $\Delta H = \Delta q$. So why not in this situation too?
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At constant volume, the pressure is not necessarily constant. The pressure will usually increase when the system is heated. This makes the $PV$ term not constant. For this reason, you get:
$$\Delta H = \Delta (E + PV) $$ $$\Delta H = \Delta E + \Delta (PV) $$ $$\Delta H = \Delta E + V\Delta P $$
Now heat is just $\Delta E$, because there is no work. So,
$$\Delta H = q+ V \Delta P $$
