The key quantity in the study of rates of activated processes by the quantum transition-state theory based on the Feynman path-integral formulation is a free-energy barrier associated to a reaction coordinate. The free-energy barrier represents the reversible work done against the quantum potential of mean force acting on; thermal path whose centroid (center of mass) is held fixed, along a reaction coordinate defined by the centroid. A reversible thermodynamic cycle leads to a simple method to calculate this barrier by thermodynamic integration. The capability of the method is demonstrated in three models: a flux of protons impinging on a symmetric Eckart barrier; a particle in a double-well potential; and a point defect in a silicon lattice. Analysis of the temperature dependence of the free-energy barrier shows a crossover from a high-temperature regime, where the potential energy increment gives a good:approximation to the barrier, to a low-temperature one, where the barrier is close to the difference between potential and kinetic energy increments. Each regime displays a characteristic ratio between the kinetic energy and the heat exchanged as the system moves reversibly along the reaction coordinate. (C) 1997 American Institute of Physics.