*2.2. Water Activity Coe*ffi*cient*

Water activity coefficients of aqueous [DBNH][OAc] solutions were calculated using Equations (1) and (2). Gmehling et al. [21] derived Equation (1) for the solubility of an organic solute in a solvent starting from the isofugacity condition. It can be used to estimate the activity coefficient of a sub-cooled liquid solvent as a function of the enthalpy of fusion, heat capacity difference between liquid and solid phase, and melting point of the solvent. When the system is close to its melting point, the last two terms of Equation (1) can be neglected and the simplified, Equation (2) is obtained. In the remainder of the article, Equation (1) is referred to as the activity coefficient equation and Equation (2) as the simplified activity coefficient equation.

$$
\ln \mathbf{x}^L \boldsymbol{\gamma}^L = -\frac{\Delta \mathbf{l}\_m}{RT} \left( \mathbf{1} - \frac{T}{T\_m} \right) + \frac{\Delta \mathbf{c}\_p}{RT} \left( T\_m - T \right) - \frac{\Delta \mathbf{c}\_p}{R} \ln \left( \frac{T\_m}{T} \right) \tag{1}
$$

$$
\ln \mathbf{x}^L \boldsymbol{\gamma}^L = -\frac{\Delta l\_m}{RT} \left( \mathbf{1} - \frac{T}{T\_m} \right) \tag{2}
$$


Heat capacity difference, Δ*cp*, is a function of temperature, and its value changes significantly when the temperature of a system is significantly lower than its melting point. Equation (3), previously reported by Sippola and Taskinen [22], was used to calculate the heat capacity change of water at its freezing point.

$$
\Delta c\_p = -19656.303 + 98.468097 \cdot \left( T/[\text{K}] \right) + 234320880 \cdot \left( T/[\text{K}] \right)^{-2} - 0.1386227 \cdot \left( T/[\text{K}] \right)^2,\tag{3}
$$

$$
237 \text{ K} \le T \le 273.15 \text{ K}.$$
