  The stochastic thermodynamics of a dilute, well-stirred mixture of
chemically-reacting species is built on the stochastic trajectories of reaction
events obtained from the Chemical Master Equation. However, when the molecular
populations are large, the discrete Chemical Master Equation can be
approximated with a continuous diffusion process, like the Chemical Langevin
Equation or Low Noise Approximation. In this paper, we investigate to what
extent these diffusion approximations inherit the stochastic thermodynamics of
the Chemical Master Equation. We find that a stochastic-thermodynamic
description is only valid at a detailed-balanced, equilibrium steady state.
Away from equilibrium, where there is no consistent stochastic thermodynamics,
we show that one can still use the diffusive solutions to approximate the
underlying thermodynamics of the Chemical Master Equation.
