Here, we have to find out the work done in an isothermal expansion of gas. We have, The work done by a gas when it expands by a very small volume, dV under a pressure, P. dW = PdV Hence, the total work done by a gas when the volume of one mole of gas increases from V1 to V2. W =
V2
∫
V2
PdV =
V2
∫
V1
(
RT
V
)dV [Since PV = RT ⇒ P =
RT
V
] Since, the temperature, T is a constant for an isothermal change, W = RT‌
V2
∫
V1
dV
V
= RT[logeV]2VV1 = RT(logeV2−logeV1) = RTloge
V2
V1
But, P1V1 = P2V2 = RT ∴ W = P1V1loge(
V2
V1
) That is, the work done in an isothermal expansion of gas is given by W = P1V1loge