## 4 Results
### Solid Solutions  
#### 4.4.1 Presentation  
This section concerns binary solutions where the two compounds are completely miscible into each other, in solid or in liquid phases, which lead again to a phase transition spread over a temperature range. In a previous paper [16], we have simplified the problem, presenting an ideal solution. The results presented here are issued with this assumption but the general commentaries could be relevant for any non-ideal solution. In Figure 25a we present the phase diagram (at atmospheric pressure) for the ideal mixture of a substance \(A\) (solvent) whose melting temperature when is pure is \(T_{F}^{A}\) = 300 K and a substance \(B\) (solute) whose melting temperature when is pure is \(T_{F}^{B}\) = 320 K. We give results for three initial mass concentrations \(x_{0}\) = 0.2, 0.5 and 0.8. The initial mass concentration \(x_{0}\) is the initial mass fraction of the solute \(B\) and if the initial mass of the two compounds are \(m_{A}^{0}\) and \(m_{B}^{0}\), we have:  
\[x_{0}=\frac{m_{B}^{0}}{m_{A}^{0}+m_{B}^{0}} \tag{20}\]  
The phase diagram Figure 25a presents two curves translating, for a given temperature \(T\), the equilibrium between a liquid solution at \(x^{L}\) and a solid solution at \(x^{S}\) : a _liquidus_ representing the temperatures \(T\) versus \(x^{L}\) and a _solidus_ representing the temperatures \(T\) versus \(x^{S}\). For these ideal solutions, we have established [16] that the variation of the enthalpy, during the heating of a crystallized solution at \(x_{0}\) is given by:  
\[\mathrm{d}h=[c_{PA}^{*}(1-x_{0})+c_{PB}^{*}x_{0}]\mathrm{d}T+\ell_{F}^{A}(T_{ F}^{A})\mathrm{d}X_{A}^{L}+\ell_{F}^{B}(T_{F}^{B})\mathrm{d}X_{B}^{L}\quad \mathrm{when}\quad T_{\rm solidus}\leq T\leq T_{\rm liquidus} \tag{21}\]  
or  
\[\mathrm{d}h=[c_{PA}^{*}(1-x_{0})+c_{PB}^{*}x_{0}]\mathrm{d}T\quad\mathrm{when} \quad T\leq T_{\rm solidus}\quad\mathrm{or}\quad T\geq T_{\rm liquidus} \tag{22}\]  
where \(T_{\rm liquidus}=T(x^{L}=x_{0})\) and \(T_{\rm solidus}=T(x^{S}=x_{0})\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(y_{0}\) **(\%)** & \(T_{\rm liquidus}\) & \(T_{\rm onset}\) & \(T[\chi_{w}^{L}=10\%]\) & \(T[\chi_{w}^{L}=50\%]\) & \(T[\chi_{w}^{L}=90\%]\) & \(\chi_{w}^{L}[T_{\rm onset}]\) **(\%)** \\ \hline
0.001 & \(-\)0.001 & 0.00 &\(\ell_{F}^{A}(T_{F}^{A})\) and \(\ell_{F}^{B}(T_{F}^{B})\) are the mass latent heats of melting for the constituents \(A\) and \(B\) respectively. We have also made the assumptions that \(c_{PA}^{L}=c_{PA}^{S}=c_{PA}^{*}\) and \(c_{PB}^{L}=c_{PB}^{S}=c_{PB}^{*}\) (\({}^{*}\) means pure substance). The mass fractions \(\chi_{A}^{L}\) and \(\chi_{B}^{L}\) are defined by:  
\[\chi_{A}^{L}=\frac{m_{A}^{L}}{m_{A}^{0}+m_{B}^{0}}\qquad\chi_{B}^{L}=\frac{m_{ B}^{L}}{m_{A}^{0}+m_{B}^{0}} \tag{23}\]  
and these values are calculated considering the conservation of masses [16]. The function \(h(T)\) is determined by numerical integration of Equations (21) and (22). In Figure 25b we present the functions \(h(T,x_{0})\) for the three chosen concentrations and \(h(T)\) for the pure substances \(A\) and \(B\). Figure 25: Phase diagram and specific enthalpy for three concentrations for a binary presenting a solid solution. (**a**) Phase diagram; (**b**) Enthalpy. **Remark 2**. : _As for certain figures above, we recall that the abscissa line represents the programmed temperature of the calorimeter plate (time in fact) for the thermograms and the local temperature \(T\) for \(\dfrac{dh}{dT}\)._  
As for the pure substances and the saline solutions, we observe a strong difference between the thermogram and the thermodynamical properties represented by \(\dfrac{\mathrm{d}h}{\mathrm{d}T}\). The actual temperature range \(T_{\mathrm{liquidus}}-T_{\mathrm{solidus}}\) is not at all represented by the width of the thermogram which is wider. These thermograms are greatly enlarged towards the temperatures higher (or very higher) than \(T_{\mathrm{liquidus}}\).