We present a model of insoluble particle activation based on a modification of the Köhler equation in which we introduce a term based on the activity of water adsorbed on the particle surface. We illustrate the model by application to activation data from black carbon (BC) particles. We parameterize the model using a free energy of adsorption that reflects the relative affinity for water vapor adsorbing on either the BC surface or the adsorbed water layer. This enables the parameterization of either chemically modified (hydrophilic) or graphitic (hydrophobic) BC. Several features of a suite of carbon activation data are captured by the model. In particular, upper and lower bounding curves are predicted for activation supersaturation as a function of diameter. We show a large body of recent activation data that all fall within these bounds. The model also predicts that activation of BC aerosol leads to activation diameters from 3 to 10 times smaller than activation of soluble particles of identical dry diameter. The activation of smaller particles may be expected to impact the size distribution of resulting cloud droplets and thus the aerosol first indirect effect on climate. Finally, we compare activation of BC aerosol as calculated with this model to activation of mixed particles of BC and ammonium sulfate. It is shown that for some range of the adsorption free energy a hydrophilic BC aerosol is predicted to activate at lower supersaturation than comparable mixed aerosol of low mass fraction in the soluble component, indicating the utility of a model of mixed particle activation based on the adsorption of water to form an interfacial solution.