Gibbs and Helmholtz energies of formation of sI clathrate hydrates from CO$_2$, CH$_4$ and water

We determine thermodynamic stability conditions in terms of Helmholtz and Gibbs energies for sI clathrate hydrates with CH$_4$ and CO$_2$ at 278 K. Helmholtz energies are relevant for processing from porous rocks (constant volume), while Gibbs energies are relevant for processing from layers on the ocean floor (constant pressure). We define three steps leading to hydrate formation, and find Helmholtz energy differences from molecular simulations for two of them using grand-canonical Monte Carlo simulations at constant temperature and volume; while the third step was calculated from literature data. The Gibbs energy change for the same steps are also determined. From the variations in the total Helmholtz and Gibbs energies we suggest thermodynamic paths for exchange of CH$_4$ by CO$_2$ in the isothermal hydrate, for constant volume or pressure, respectively. We show how these paths for the mixed hydrate can be understood from single-component occupancy isotherms, where CO$_2$, but not CH$_4$, can distinguish between large and small cages. The strong preference for CH$_4$ for a range of compositions can be explained by these.