## 4 Results
### Case of Small Concentrations  
#### 4.3.1 Presentation  
As a rule of thumb, it is very difficult to find very pure materials. For instance the PCM based on alkanes are solutions whose the highest purity is commonly between 98% and 99.5%. In addition, in cases where PCMs are dispersed there may be some slight dissolution in the PCM of the dispersant or microencapsulation products. This section is focused on the small solute concentrations. The main results of the preceding section on saline solutions can be applicable, but consequences adapted to these small solute concentrations will be highlighted. Generally the total amount of impurities is given in mass. However, the nature and quantities of each impurities (solutes) should be known because it is the sum of the molar fractions (of osmolar fractions in case of ionisation of the solutes in water) which is needed to calculate the cryoscopic lowering. The cryoscopic lowering is the difference between the melting temperature of the pure substance \(T_{F,w}\) and the equilibrium temperature \(T_{\mathrm{liquidus}}\) at the molar concentration \(y_{0}=\sum_{i}y_{i}\) where the \(y_{i}\) are molar fractions (or osmolar fractions) of each impurity \(i\). For the very low concentrations, the cryoscopic lowering is given by the [PERSON]'s law:  
\[T_{F,w}-T_{\mathrm{liquidus}}=\frac{\widehat{R}}{\widehat{M}}\frac{{T_{F,w}} ^{2}}{\ell_{F,w}}y_{0} \tag{19}\]  
with \(\widehat{R}=8.31\,\mathrm{J}\,\mathrm{K}^{-1}\,\mathrm{mol}^{-1}\) the ideal gas constant and \(\widehat{M}\) the molar mass (in kg) of the solvent (water). In such a case, where the amount of impurities is only of the order of a few percents, there are already important modifications of the shapes of the thermograms which are now going to be analyzed and the consequences on the phase changes evaluated. In Figure 23 we present, for solutions with \(y_{0}\leq 1\%\), the values of the enthalpies \(h\) calculated from Equation (18) with \(T_{\mathrm{liquidus}}\) calculated from the Raoult'law Equation (19). To anticipate the results presented below, we made the difference between the concentrations extremely low (\(y_{0}=0.001\%\) or \(0.01\%\)) in Figure 23a and the others (\(0.1\%\leq y_{0}\leq 1\%\)) in Figure 23b. Figure 22: Thermograms for an aqueous solution at 2% for a cooling at \(\beta=-5\) K/min (blue) and for a heating at \(\beta=5\) K/min (red) versus \(T_{\mathrm{plt}}\) and \(\frac{\mathrm{d}h}{\mathrm{d}T}\) (black) versus the local temperature \(T\). Applying our model with the value of \(h\) calculated for Figure 23, we present in Figure 24 the thermograms calculated for melting at 0.5 K/min. In Figure 24a, we present the thermograms of pure water and the thermograms for very dilute solution (\(y_{0}\) = 0.001% and 0.01%). For the melting of pure ice in the presence of these solutions of extreme purity, the thermograms have a shape useable for pure water except the rounded corner of the left foot of the peak. However, this figure, despite this deformation gives the same \(T_{\rm onset}\) that for the pure substance. Figure 23: Examples of the specific enthalpy for aqueous saline solutions for very small concentrations (in percentage). (**a**) Very dilute cases; (**b**) Classical dilute cases. However, in the second Figure 24b the concentration \(y_{0}\) are sufficiently high to involve important modifications of the shapes of the thermograms. In this case, as it has been presented in Figure 20, the \(T_{\text{onset}}\) classically determined has no significance and is different from the desired temperature \(T_{\text{liquidus}}\). However, taking into account the smallness of the concentrations, the difference \(T_{\text{liquidus}}-T_{\text{onset}}\) is relatively small, of the order of the accuracy due to the calibration. For example, for a concentration of \(y_{0}=0.1\%\), this difference is 0.2 K and for a concentration of \(y_{0}=0.5\%\) this difference is 0.9 K, the differences with the temperature of melting of the pure substance \(T_{F,w}-T_{\text{onset}}\) being 0.3 K and 1.4 K respectively.