Constant Temperature

A constant temperature process is an isothermal process. An example is when a gas in a container that is immersed in a constant-temperature bath is allowed to expand slowly, or is compressed slowly.

At constant temperature there is no change in internal energy.

DU = 0

Apply the First Law:

Q = W

The P-V diagram for this process follows an isotherm, a line of constant temperature.

For an ideal gas at constant temperature, the pressure is inversely proportional to the volume. This makes it straight-forward to do an integration to find that:
Q = W = nRT ln (
Vf
Vi
)