The lifting condensation level (LCL) is the height at which a lifted parcel of air reaches saturation and forms a cloud. Over nearly 200 years, many different expressions have been proposed for this height. Some of those expressions are analytic and some are complicated nonlinear equations that must be solved using iterative numerical methods. None of them, however, are exact or dependent in an obvious way on the fundamental parameters of the atmosphere.
Here, the exact, explicit, and analytic expression for the LCL is derived. This expression depends only on fundamental physical parameters of the atmosphere. This expression can be generalized to give the lifting deposition level (LDL; where the air parcel saturates with respect to ice). It can also be generalized to atmospheres on other planets with a condensible gas.
For surface air with a relative humidity of 50 percent, the heights of the lifting condensation level (LCL; where saturation with respect to liquid water occurs), the lifting deposition level (LDL; where saturation with respect to solid ice occurs), and the lifting freezing level (LFL; where aerosols freeze homogeneously).