This chapter begins with a brief description of the continuous growth model of solidification. One effect that becomes important in rapid solidification is that the local composition and undercooling of the interface are dominated by the rate of attachment of atoms from the liquid to the solid, which sets the velocity scale of the advancing interface. The chapter utilizes results from the asymptotic analysis to derive the form of the non-equilibrium partition coefficient k(v0) needed to compute solute partitioning and interface undercooling in the continuous growth model model. It aims to derive the kinetic undercooling at the interface for an alloy whose phases are described by Henry's law.