Ideal gas - isothermal process function of volume


An isothermal process is a change of a system, in which the temperature remains constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir (heat bath), and the change in the system will occur slowly enough to allow the system to continue to adjust to the temperature of the reservoir through heat exchange. In contrast, an adiabatic process is where a system exchanges no heat with its surroundings (Q = 0). In other words, in an isothermal process, the value ΔT = 0 and therefore the change in internal energy ΔU = 0 (only for an ideal gas) but Q ≠ 0, while in an adiabatic process, ΔT ≠ 0 but Q = 0.
For an isothermal, reversible process, this integral equals the area under the relevant pressure-volume isotherm, and is indicated in purple in Figure 2 for an ideal gas. Again, p = nRT/V applies and with T being constant (as this is an isothermal process), the expression for work becomes as shown here.

By convention, work is defined as the work on the system by its surroundings. If, for example, the system is compressed, then the work is positive and the internal energy of the system increases. Conversely, if the system expands, it does work on the surroundings and the internal energy of the system decreases.

Related formulas


Wwork (J)
nnumber of moles (mole)
Rmolar gas constant
TTemperature (K)
V2volume at state 2 (m3)
V1volume at state 1 (m3)