Important: Positive work means work done by an external force on a system.

For a mechanical system, the work done by an external force on the system is:

W = ∫ F_ext dx

For a gas, F_ext = - F_gas = - P A. Here P is the pressure exerted by the gas.

Basic Facts about Pressure:

1. Pressure = force/area
2. Units of pressure: 1 pascal ≡ 1 Pa = 1 N/m²
3. Atmospheric pressure: 1 atm ≈ 10⁵ Pa = 10² kPa ≈ 14.7 lbs/sq. in.

For the work done on a gas, we have:

W = - ∫_vi^v_f P A dx = - ∫_vi^v_f P d(Ax) = - ∫_vi^v_f P dV.

Meaning of the minus sign:

Positive work is done on the gas when the gas is compressed;
negative work is done on the gas when the gas expands.
zero work is done on the gas when the gas volume is fixed.

Geometrically, the work done on a gas is the negative of the area under the curve traced out by the process in the P-V plane (see transparency).