Work done by a gas during a quasi-static volume change

A volume change, i.e. the contraction or expansion of a container, is a mechanical change — and therefore any energy transferred due to this process will be work. The amount of work due to an infinitesimal quasistatic volume change is

đ 𝑊 = - 𝑝 𝑑 𝑉

where $đ 𝑊 < 0$ for an expansion and $đ 𝑊 > 0$ for a compression.

Derivation

Consider a container of an ideal gas expanding in one direction. The gas therefore exerts a force $⃗ 𝐅$ due to pressure, resulting in a displacement $⃗ 𝐬$ in that one direction. The total volume change will be $⃗ 𝐀 \cdot ⃗ 𝐬$ , where $⃗ 𝐀$ is the normal area vector:

⃗ 𝐅 = 𝑝 ⃗ 𝐀 đ 𝑊 = - ⃗ 𝐅 \cdot 𝑑 ⃗ 𝐬 = - 𝑝 ⃗ 𝐀 \cdot 𝑑 ⃗ 𝐬 = - 𝑝 𝑑 𝑉

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