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.

ΔE_int = 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:

P

=

nRT V

, so:

W

=

∫

P dV

= nRT

∫

1 V

dV

The integral of 1/V is ln(V), and ln(A)-ln(B) = ln(A/B).

Therefore:     Q = W

= nRT ln

(

Vf Vi

)