We are interested in the modeling of liquid-vapor-gas flows with interfaces and phase transition. We describe these flows by a hyperbolic single-velocity three-component compressible flow model with instantaneous pressure relaxation, which is composed of the phasic mass and total energy equations, two volume fraction equations, and the mixture momentum equation. The model includes thermal relaxation terms to account for heat transfer, and chemical relaxation terms to describe mass transfer between the liquid and vapor phases of the species that may undergo transition. To numerically solve the model system we use a fractional step method where we alternate between the solution of the homogeneous system via finite volume HLLC-type schemes and the solution of systems of ordinary differential equations that take into account the relaxation source terms. In this work we propose a novel numerical procedure for chemical relaxation that can efficiently describe arbitrary-rate mass transfer, both slow finite-rate processes and stiff instantaneous ones. The technique extends the idea that we have proposed previously for two-phase flows, which consists in describing the relaxation process by a system of ordinary differential equations that admits an analytical semi-exact exponential solution. The resulting numerical procedure is easily applicable to an arbitrary equation of state and it has relatively limited computational cost since it uses simple explicit formulas. Several numerical tests are presented to show the effectiveness of the proposed technique, including a two-dimensional simulation of an underwater explosion.