Enthalpy and Internal Energy in Constant Volume Calorimeter
The definition of enthalpy is
$$H = U + pV$$

which gives

$$\begin{align}H_2 - H_1 &amp;= (U_2 + p_2V_2) - (U_1 + p_1V_1) \\
\Delta H &amp;= \Delta U + \Delta(pV)\end{align}$$

The equation you cited is only true at constant pressure: if $p_1 = p_2 = p$,

$$\begin{align}
H_2 - H_1 &amp;= (U_2 + pV_2) - (U_1 + pV_1) \\
\Delta H &amp;= \Delta U + p(V_2 - V_1) \\
&amp;= \Delta U + p\Delta V
\end{align}$$

The condition for that equation is that the pressure is constant. That equation is not applicable for constant-volume (isochoric) processes unless the process is also constant-pressure (isobaric).

However, if you do have a process that is both isobaric and isochoric, you are right to say that $\Delta H = \Delta U$.