Why is the change in enthalpy computation for a change of state done like this? Suppose that a system whose initial state is Pressure = $P_1$, Volume = $V_1$; and it is taken through a process after which final pressure and volume are $P_2$ and $V_2$ respectively. Now change in enthalpy is defined as $\Delta H= \Delta U +P\Delta V+V\Delta P$The thing which I am confused with is which pressure and which volume do we hold constant for their respective parts in the calculation? Till now I had dealt only with problems involving the $P\Delta V$ part, and it was calculated as $P_2(V_2-V_1)$, same as irreversible work. But when I came across a problem in which Pressure also varied with volume, the second part needed to be taken as $V_1(P_2-P_1)$.Why do we take $P_2$ as constant in one case and $V_1$ in the other?One reason which I thought of was that in an irreversible process, The pressure is first changed by a finite amount and then the volume varies. Thus the pressure throughout the process was $P_2$. This can be a rationale for the $P\Delta V$ part, but I am not sure if it is reasonable to say that the volume at start of the process was $V_1$, which resulted in this method of calculation. So.. what is the reason?
You're confused because the equation is incorrect.  It should read $$\Delta H=\Delta U+\Delta (PV)$$If you insist on writing it the way you did, then P should be $\frac{P_1+P_2}{2}$ and V should be $\frac{V_1+V_2}{2}$.  That will give the same result as the first equation I wrote.