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 occurs slowly enough to allow the system to continually adjust to the temperature of the reservoir through heat exchange. An alternative special case in which a system exchanges no heat with its surroundings (Q = 0) is called an adiabatic process. In other words, in an isothermal process, the value ΔT = 0 but Q ≠ 0, while in an adiabatic process, ΔT ≠ 0 but Q = 0.
For the special case of a gas to which Boyle's law applies, the product pV is a constant if the gas is kept at isothermal conditions. The value of the constant is nRT, where n is the number of moles of gas present and R is the ideal gas constant. In other words, the ideal gas law pV = nRT applies. This means that
holds. The family of curves generated by this equation is shown in the graph presented here. Each curve is called an isotherm. Such graphs are termed indicator diagrams and were first used by James Watt and others to monitor the efficiency of engines. The temperature corresponding to each curve in the figure increases from the lower left to the upper right.
In thermodynamics, the work involved when a gas changes from state A to state B is simply
For an isothermal, reversible process, this integral equals the area under the relevant pressure-volume isotherm, and is indicated in yellow in the figure at right for an ideal gas. Again p = nRT / V applies and so
It is also worth noting that, for many systems, if the temperature is held constant then the internal energy of the system also is constant, and so ΔU = 0. From ΔU = Q + W it follows that Q = -W for this same isothermal process. In other words,
If the gas is compressed, meaning ΔV = (final volume - initial volume) < 0, then the natural log term is negative and, therefore, W < 0. That is, during isothermal compression the gas does negative work, or, equivalently, the environment does positive work. Restated, the environment does work on the gas.
If the gas expands, meaning ΔV = (final volume - initial volume) > 0, then the natural log term is positive and, therefore, W > 0. That is, during isothermal expansion the gas does positive work, or, equivalently, the environment does negative work. Restated, the gas does work on the surroundings.
Isothermal processes can occur in any kind of system, including highly-structured machines, and even living cells. Various parts of the cycles of some heat engines are carried out isothermally and may be approximated by a Carnot cycle.