## Integrals [ENDORSED]

$w=-P\Delta V$
and
$w=-\int_{V_{1}}^{V_{2}}PdV=-nRTln\frac{V_{2}}{V_{1}}$

Lucia H 2L
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### Integrals

Can someone explain why we keep finding derivatives and integrals when deriving work equations?

I understand that reversible isothermal expansion involves changing energy at infinitesimal increments but that is about it.

Vincent Tse 1K
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### Re: Integrals

Not sure I entirely understand your question, but here's my take:

Derivatives are super small parts of something. Like a "small piece of X". It's helpful to use derivatives in the sciences to refer to an infinitesimally small part of something that we don't know much about, or something that keeps changing.

Integrals are the opposite; they are summing up many tiny small parts of something (the differentials/derivatives) to make up something whole.

Example: w = -PV. If you want to find the work done at from volume V1 expanding to V2, then you sum up "all the small pieces of V" together from the bounds V1 to V2 to find the total amount.

Swetha Sundaram 1E
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### Re: Integrals

As Vincent said above, I would agree in saying that integrals are used because they represent the value of the sum between two values.

Chem_Mod
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### Re: Integrals  [ENDORSED]

You have the main idea and Vincent's answer is good.

A small volume change is represented by dV where d is the infinitely small change (derivative) in the volume.
To get the total volume change we need to add up (sum) all the very small changes (dV).

An infinite sum is represented by an integral. Remember in class I made the joke: $\int um$ for Sum

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