This paper presents a derivation of an equality relating the free energy difference between two equilibrium ensembles of a system and an ensemble average of the work required to switch between these two configurations. The result is shown to hold under the assumption that the system's dynamics are Markovian and microscopically reversible. The equality is given by: $$ \overline{{e^{-\beta W}}}=e^{-\beta\Delta F} $$ where $ \beta=1/k_{B}T $, $ k_{B} $ is the Boltzmann constant, and T is the temperature of the heat bath. The overbar indicates an ensemble average over all possible paths through phase space, given an equilibrium initial state, and the value of the control parameter at all times during the switching process. This equality generalizes several relations used to calculate free energy differences in computer simulations. In the limit of an infinitely long switching process, Eq. (1) becomes equivalent to thermodynamic integration, which assumes that the system is always in equilibrium, the switching process is reversible, and no energy is dissipated. In the limit of infinitely fast switching, Eq. (1) is equivalent to: $$ \left\langle e^{-\beta W}\right\rangle_{0}=e^{-\beta\Delta F} $$ where the angled brackets indicate an equilibrium average with $ \lambda $ fixed in its initial value. Thermodynamic integration and perturbation methods suffer from systematic errors because the simulated systems are never truly in equilibrium. Calculations based on Eq. (1) only require that the initial ensemble is in equilibrium, which is computationally feasible. Although this reduces systematic errors, it increases statistical errors, which may limit its applicability. Previously, this relation has been derived for a Hamiltonian system weakly coupled to a heat bath and based on a master equation approach. In this paper, it is shown that this equality directly follows if we assume that the dynamics of the system are Markovian and microscopically reversible. The Markovian condition ensures that the system is memoryless, and the property of microscopic reversibility ensures that the system is time reversible and that the equilibrium probability distributions are correctly given.This paper presents a derivation of an equality relating the free energy difference between two equilibrium ensembles of a system and an ensemble average of the work required to switch between these two configurations. The result is shown to hold under the assumption that the system's dynamics are Markovian and microscopically reversible. The equality is given by: $$ \overline{{e^{-\beta W}}}=e^{-\beta\Delta F} $$ where $ \beta=1/k_{B}T $, $ k_{B} $ is the Boltzmann constant, and T is the temperature of the heat bath. The overbar indicates an ensemble average over all possible paths through phase space, given an equilibrium initial state, and the value of the control parameter at all times during the switching process. This equality generalizes several relations used to calculate free energy differences in computer simulations. In the limit of an infinitely long switching process, Eq. (1) becomes equivalent to thermodynamic integration, which assumes that the system is always in equilibrium, the switching process is reversible, and no energy is dissipated. In the limit of infinitely fast switching, Eq. (1) is equivalent to: $$ \left\langle e^{-\beta W}\right\rangle_{0}=e^{-\beta\Delta F} $$ where the angled brackets indicate an equilibrium average with $ \lambda $ fixed in its initial value. Thermodynamic integration and perturbation methods suffer from systematic errors because the simulated systems are never truly in equilibrium. Calculations based on Eq. (1) only require that the initial ensemble is in equilibrium, which is computationally feasible. Although this reduces systematic errors, it increases statistical errors, which may limit its applicability. Previously, this relation has been derived for a Hamiltonian system weakly coupled to a heat bath and based on a master equation approach. In this paper, it is shown that this equality directly follows if we assume that the dynamics of the system are Markovian and microscopically reversible. The Markovian condition ensures that the system is memoryless, and the property of microscopic reversibility ensures that the system is time reversible and that the equilibrium probability distributions are correctly given.