How do you generalize #Delta(PV)# from #d(PV)#?

I thought this was cool when I found it in my book---it is indeed a common error to make!

1 Answer
Truong-Son N.
Dec 21, 2017

In fact, it is not that obvious. Oftentimes when we see a differential like #d(PV)#, we might go to the product rule:

#d(PV) = PdV + VdP#

which is fine for differentials. With #Delta# changes though, it is not quite that simple. In other words,

#Delta(PV) ne P_1DeltaV + V_1DeltaP#.

The way my book does it (Physical Chemistry, Levine, pg. 52) is quite ingenious, actually. By definition:

#Delta(PV) = P_2V_2 - P_1V_1#

But here is a way to prove what #Delta(PV)# actually is. For any size of #Delta#,

#Delta(PV) = P_2V_2 - P_1V_1#

#= (P_2 - P_1 + P_1)(V_2 - V_1 + V_1) - P_1V_1#

#= (P_1 + DeltaP)(V_1 + DeltaV) - P_1V_1#

Distribute to get:

#= cancel(P_1V_1) + P_1DeltaV + V_1DeltaP + DeltaPDeltaV cancel(- P_1V_1)#

Therefore, for any size of #Delta#:

#color(blue)(barul|stackrel(" ")(" "Delta(PV) = P_1DeltaV + V_1DeltaP + DeltaPDeltaV" ")|)#

For differentials we do not have to worry about this because the product of two differentials is small, i.e. #dPdV ~~ 0#. But when the change is not small, #DeltaPDeltaV# is also not small.

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