Question

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!

Answer 1

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.