## Thermodynamic Evidence for [PERSON]'s Rule  
### Modeling pT-Lines  
We start by a discussion on the validity of Equation (1) for modeling pT-lines of vapor-liquid and vapor-sorbed equilibria. For such equilibria in contact with a reservoir at given \(p\) and \(T\), the Gibbs free energy per mole, or chemical potential \(g\) (J/mol), of the molecules is equal for both phases [17]; that is, \(g_{v}=g_{l}\) for vapor-liquid and \(g_{v}=g_{s}\) for vapor-sorbed equilibria. With expressions for \(g\)(\(p\),\(T\)) of the relevant phases, one can thus arrive at formulas for pT-lines. When the phases are ideal gases or diluted solutions, or when \(g\) of one of the phases can be considered constant, one thus obtains the Van't [PERSON] equation [17]. In our case, however, simple analytic expressions for \(g\)(\(p\),\(T\)) are not available for the liquid and sorbed states, not even for van der Waals fluids [18]. Instead, one may start with the differential relations \(dg_{v}=dg_{l}\) for vapor-liquid and \(dg_{v}=dg_{s}\) for vapor-sorbed equilibria. Then, using the thermodynamic identity \(dg=-sdT+vdp\) for e.g., both vapor and liquid, one obtains:  
\[\frac{dp}{dT}=\frac{s_{v}-s_{l}}{v_{v}-v_{l}}\equiv\frac{\Delta s}{\Delta v}= \frac{\Delta h}{T\Delta v}. \tag{3}\]  
This is the Clausius-Clapeyron equation [17]. From here, the common way forward is to assume that far below the critical point \(T<<T_{c}\), the evaporation heat \(\Delta h\) is independent of \(T\), that the vapor is an ideal gas, and that the molar volume of the liquid can be ignored. Then, \(\Delta v\cong v_{v}=p/RT\) and Equation (3) can be integrated, giving:  
\[\ln\frac{p}{p_{0}}=\frac{\Delta h_{0}}{R}\bigg{(}\frac{1}{T_{0}}-\frac{1}{T} \bigg{)}\equiv\frac{\Delta s_{0}}{R}-\frac{\Delta h_{0}}{RT}. \tag{4}\]  
Here, \(p_{0}\), \(T_{0}\), \(\Delta h_{0}\), and \(\Delta s_{0}\) are values at any reference point on the pT-line. As \(p_{0}\) and \(T_{0}\) are related by the pT-line, and because \(\Delta s_{0}=\Delta h_{0}/T_{0}\), the pT-line has only two free parameters. We thus obtain Equation (1). We note that for salt hydrate equilibria in the range \(T=20\)-\(90\)\({}^{\circ}\)C, vapor pressures are usually in the order of \(p_{0}=10\) mbar, so that \(\Delta h_{0}\) and \(\Delta s_{0}\) will actually be fitted around 10 mbar rather than at 1 atm. Using Equation (1) with, e.g., \(p_{0}\)\({}^{\prime}=1\) atm instead of \(p_{0}=10\) mbar, is then a matter of adding a constant offset; that is, with \(\Delta s(p_{0}\)\({}^{\prime})\equiv\Delta s(p_{0})+R\ln(p_{0}/p_{0}\)\({}^{\prime})\) and \(\Delta h(p_{0}\)\({}^{\prime})\cong\Delta h(p_{0})\) by assumption, we arrive at Equation (4) with \(p_{0}\)\({}^{\prime}\). How good, though, is the assumption that \(\Delta h\) is independent of \(T\)? This assumption seems suspicious because \(\Delta h\) vanishes towards the critical point \(T_{c}\). For vapor-liquid transitions, the van der Waals equation provides evidence that \(\Delta h\) is indeed independent of \(T\). The van der Waals equation of state reads [19]  
\[\bigg{(}\frac{p}{p_{c}}+\frac{3}{\left(v/v_{c}\right)^{2}}\bigg{)}\bigg{(} \frac{v}{v_{c}}-\frac{1}{3}\bigg{)}\frac{Q}{V}=\frac{8}{3}\frac{T}{T_{c}}. \tag{5}\]  
Here the critical pressure, molar volume and temperature \(p_{c}\), \(v_{c}\), and \(T_{c}\) are related by \(p_{c}v_{c}=3 RT_{c}/8\), so that Equation (5) has only two independent parameters. For the van der Waals equation, the latent heat \(\Delta h\) at \(T<<T_{c}\) approaches \(\Delta h=9p_{c}v_{c}=27 RT_{c}/8\)[20; 21] and deviates by only about 5% from this value for \(T/T_{c}<0.5\). This is illustrated by Figure 3, showing the latent heat \(\Delta h/RT_{c}\) (orange) as well as the slope of the pT-line \((1/T_{c})\)d(ln\(p\))/d(\(-1/T\)) (blue). These quantities are approximately equal if \(\Delta h/R\cong\) d(ln\(p\))/d(\(-1/T\)), according to Equation (4). We note that Figure 3 as well as the following Figures 4 and 5 were made using published tables [21].