Question

Why is pressure always negative in the formula `w = -P DeltaV`?

I understand why pressure is negative when a gas is expanding, but when a gas is being compressed, shouldn't pressure increase?

Answer 1

Pressure is NEVER negative, ever. It is always, always positive (you cannot "un-apply" pressure or impart "negative energy"), and in the case of pressure-volume work, in most cases the external pressure is constant and it is the internal pressure that might change.

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Work is defined with respect to either the system or its surroundings. In your case, since `w = -PDeltaV`, work is defined from the perspective of the system, and the first law of thermodynamics is written:

`DeltaE = q + w = q - PDeltaV`

And for two cases (`DeltaV` is `(+)` or `(-)`), we assign a negative sign to the pressure-volume work equation in order to match the signs.

CASE 1: `DeltaV > 0`

• When the system does pressure-volume work on the surroundings, the system expands, and the work is negative with respect to the system.

EXPANSION work BY the system ON the surroundings:

`underbrace(w)_((-)) = - underbrace(P)_((+))underbrace(DeltaV)_((+))`

Thus, energy is released from the system in this scenario.

CASE 2: `DeltaV < 0`

• When the system has pressure-volume work done upon it, the system compresses, and the work is positive with respect to the system.

COMPRESSION work ON the system BY the surroundings:

`underbrace(w)_((+)) = - underbrace(P)_((+))underbrace(DeltaV)_((-))`

Thus, energy is absorbed into the system in this scenario.

And if you wish to confuse yourself, you could define work from the perspective of the surroundings, and then `w = PDeltaV`, with `DeltaE = q - w` instead... In either case, `P > 0`, always.