## 3 Methods
In this work, we mainly focus on heat release during the formation of hydrates. Unlike many other models describing enthalpy changes associated with hydrate formation and melting, our residual thermodynamic model is directly related to the free energy changes as illustrated by Equation (1) below. The heat transport kinetics will be implicitly coupled to Equation (1), and a trivial application of statistical mechanics will prove that using Equation (1) for enthalpy will provide a consistent coupling between free energy and enthalpy changes. The residual, or configurational partition function, will be directly linked to the phase structure. Consistent descriptions of enthalpy and free energy will be needed in order to provide the correct entropy change for the phase transition; to the best of our knowledge, Equation (5) is the only enthalpy model in the available literature that satisfies this requirement. Equation (1) below is a fundamental classical thermodynamics relationship; its derivation is available in any textbook and requires no further explanations. \[\frac{\partial\Big{[}\frac{\Delta G^{Total}}{RT}\Big{]}_{P}\overset{\rightarrow} {N}}{\partial T}=-\left[\frac{\Delta H^{Total}}{RT^{2}}\right] \tag{1}\]  
where \(G\) is free energy and H is enthalpy. The \(\Delta\) symbol is the change in free energy and enthalpy, respectively. The subscripts on the left brackets denote constant pressure and mol numbers. The free energy change related to the formation of hydrate on the interface between a separate hydrate former phase and liquid water can be expressed as  
\[\Delta G^{(H)}=\left[ \begin{array}{c}x_{H_{\text{O}}}^{H}(u_{H_{\text{O}}}^ {H}(T,P,\overset{\rightarrow}{x}^{H})-u_{H_{\text{O}}}^{water}(T,P,\overset{ \rightarrow}{x}))\\ +\sum\limits_{j}x_{j}^{H}(u_{j}^{H}(T,P,\overset{\rightarrow}{x}^{H})-\mu_{ j}^{gas}(T,P,\overset{\rightarrow}{y}^{gas}))\end{array} \right] \tag{2}\]  
where \(\mu\) denotes chemical potential. Subscripts H\({}_{2}\)O and \(j\) denote water and hydrate formers, respectively. Superscripts \(H\), water, and gas stand for hydrate, liquid water, and gas phases, respectively. \(x\) is the corresponding mole fraction in either the liquid or hydrate phase (superscript H), and \(y\) is the mol fraction in the hydrate former phase. \(T\) and \(P\); are temperature and pressure. Due to the implicit and consistent coupling between free energy (a function that determines phase stability) and enthalpy, it makes sense to relate the changes in enthalpy to the phase stability boundaries. This connection is illustrated by the link between the pressure-temperature hydrate phase stability boundaries in Figure 1. The validity and accuracy of the calculations involved in the construction of Figure 1 have been verified through comparison with experimental data in many of our previous papers. For all the practical purposes, this means that the chemical potentials and free energies involved in Equation (2) have also been confirmed. Figure 1 is actually constructed so as to ensure that the free energy change given by Equation (2) is equal to zero. In summary, the treatment of enthalpy changes in this work is entirely different from that used in many other enthalpy models. All our calculations are based on residual thermodynamics and the link through Equations (1) and (2). For this same reason, we do not refer to many publications from other research groups. There are certainly many high-quality publications dedicated to calculations of enthalpy changes in hydrate phase transitions. However, since they are based on very different thermodynamic platforms and reference systems, a review of these models would require too much space and will be of limited value for the main focus of this paper. The liquid water chemical potential has been calculated using the symmetric excess conventions as described in [PERSON] et al. [29] and [PERSON] [30]. Equations (1) and (2) are coupled to an implicit equation for mass transport flux and thermodynamic control through Equation (22) in [PERSON] et al. [29].