In prior work on a thermodynamic and stochastic theory of chemical systems far from equilibrium, the excess work (a Lyapunov function) was shown to predict relative stability of stationary states in reaction-diffusion systems with multiple stationary states. This theory predicts equistability when the excess work from one stationary state to the stable inhomogeneous concentration profile separating the two stable stationary states equals the excess work from the other stable stationary state to that profile. Here we prove that any Lyapunov function of the deterministic reaction-diffusion equations of a given form can be used to predict equistability. Further, we show that the spatial derivative of any Lyapunov function for these equations, which is simpler to calculate, can also be used to predict relative stability.