Isothermal process

An isothermal process is carried out at constant temperature. To keep the temperature fixed while the gas expands or is compressed, it must remain in thermal contact with a large heat reservoir and change slowly enough to stay in step with it. For an ideal gas the internal energy depends only on temperature, so along an isotherm ΔU = 0 and the first law collapses to Q = W: all the work done by the gas is supplied as heat from the reservoir.

The work, and hence the heat, is W = Q = nRT ln(V₂/V₁), positive for expansion and negative for compression. On a pressure–volume diagram the path is the hyperbola PV = nRT = const, the curve of Boyle's law. Of the two limiting processes, the isothermal extracts the most heat from a reservoir per unit expansion.

Isothermal steps form two of the four legs of the ideal Carnot cycle, where heat is taken in isothermally from a hot reservoir and rejected isothermally to a cold one. Real isothermal changes are an idealisation approached only in the limit of infinitely slow, reversible operation.

Central to Carnot's 1824 analysis of the ideal heat engine and drawn explicitly once Clapeyron plotted the isotherm as a hyperbola; the logarithmic work formula follows directly from Boyle's seventeenth-century law combined with the ideal gas law.