Kinetic extensions of the nucleation theorem (KNT) are derived using the law of mass action and detailed balance. Results are obtained for the first- and higher-order derivatives of the nucleation rate, J, with change in supersaturation, S, in terms of the cumulants, kappa(n), of a molecular distribution of reciprocal equilibrium cluster growth rates. At constant temperature we find d ln J/d ln S=kappa(1)+1, an exact formulation of the nucleation theorem in terms of nucleation rate, and the extension d(n) ln J/d(ln S)(n)=(-1)(n+1)kappa(n) for the higher-order derivatives (ngreater than or equal to2). The case n=2 is related to the Kelvin relation. Analysis of recent water vapor nucleation rates [Wolk and Strey, J. Phys. Chem. B 105, 11683 (2001)] provides molecular-based estimates for kappa(1) and kappa(2) suitable for comparison with the predictions of classical nucleation theory. The KNT is applied to ion-induced nucleation from the gas phase, by a sequence of reversible chemical reactions, and extensions to multistep kinetics and multicomponent nucleation are presented. Nucleation theorems enable one to deduce molecular-level properties directly from macroscopic rate measurements. Here we show these properties are not those of a single cluster, the critical nucleus, as approximate forms of the theorems would suggest, but instead are averages over a weighted distribution of clusters near critical size. (C) 2003 American Institute of Physics.