Question

Derive the equation W = -Pex . ∆V​

Answer 1

To derive:

Work = -P_(external) × ∆V

Derivation:

Let's consider that a gas is present in a cylinder covered by a piston such that the gas is expanding against an external pressure "P_(ext)":

• We shall also considered that the cross-sectional area of the cylinder is "A"

• Due to expansion, the piston moved by a small distance "s".

Now , we know that :

$\sf{ \therefore \: work = force \times displacement}$

$\sf{ = > \: work = (pressure \times area) \times displacement}$

$\sf{ = > \: work = (P_{ext} \times A) \times s}$

$\sf{ = > \: work = P_{ext} \times (A \times s)}$

Now , (A × s) refers to the volume change denoted as ∆V.

$\sf{ = > \: work = P_{ext} \times (\Delta V)}$

$\boxed{ \sf{ = > \: work = P_{ext} \times \Delta V}}$

Now , when a gas is expanding, work is done by the system and hence work is positive. So , a - ve sign is put in front of the equation to satisfy the real values.

$\boxed{ \sf{ = > \: work =- P_{ext} \times \Delta V}}$

[Hence proved]