thermochemistry problem confusion I can understand upto where the ideal gas equation is used to calculate volume of steam. But on trying to find out $ \Delta V$ , i saw that there are no gases on the reactant side so then $P\Delta V=(\Delta n)\times R\times T$ where $\Delta n$ would be $1.39$. Then $\Delta V$ is simply the volume of the steam i.e., $42.51$. But when calculating the work a value of $42.8$ has been used for $\mathrm dV$. Why is that done? Secondly, there seems to be two different values for $\Delta$H. First they have written it to be $\pu{12470.6 cal}$. and then while calculating $\Delta U$ they have used the value of $\pu{13500 calories}$. (That value seems to have come from the given data: If we multiply $9.72\times 10^3$ (molar $\Delta H_{vap}$) by $1.39$ (number of moles) we get something around $13500$). How did those two different values appear? I also want to tell here that this is not a hw question for me, as you can see this is a solved example, so you can consider it as my attempt to understand thermochemistry better
The $\Delta V$ is supposed to be the volume of the vapor minus the volume of the liquid.  The volume of the liquid is only 0.025 liters, which they neglect.  

I don't know where they get the 12470.6 from.  I'm guessing this is an error.

(0.082)(373)(25)/18=42.48. You solve it basically the way you said. 

Δ(PV)=42.48 liter−atm =1029 cal=42.48 liter-at = 1029 cal

So, ΔU=13500−1029=12471 cal