Equation Shreder

Equation (3.19) is the differential form of the Le Chatelier-Shreder s equation. We can obtain the integral form of this equation after integration in the range from the activity ay A at temperature T to the activity ayA = (the pure solvent) at temperature rtus, a-In the simplified case, we assume that the difference between the heat capacity in the liquid and the solid state of the component considered, Ag/iC, does not depend on temperature. We then get... 

However, in the calculation of the phase diagrams, i.e. the dependences of the activity of components in their saturated solution on temperature, we use the Le Chatelier-Shreder s equation in the form explicit for temperature... 

If the liquid phase behaves like a simple regular solution, for the dependence of the liquidus temperature on composition (the Le Chatelier-Shreder s equation (3.22)) we get... 

In the application of the model to calculate the phase diagrams, the liquidus temperature of component i, is calculated using the values of the enthalpy and temperature of fusion according to the simplified and adapted Le Chatelier-Shreder equation... 

For solutions being ionic in character, the processing of the Le Chatelier-Shreder s equation in the vicinity of the melting point of the component A must be slightly different... 

Let us consider the dissolution of the admixture X in the dystectically melting solvent AB. The course of the liquidus curve of substance AB in the AB-X system (Figure 3.48) is described by the Le Chatelier-Shreder equation, which in the limit for Xr(AB) -> 1 we can express in the differential form... 

For the equilibrium solidus-liquidus of the component i in a simple eutectic system in which no compounds are formed and the solubility of the components in the solid state does not exist, the differential form of the Le Chatelier-Shreder s equation holds... 

Fellner and Matiasovsky (1974) derived an equation for cryoscopy in the eutectic mixtures, which was not found to be analogous with the Le Chatelier-Shreder s equation. 

The enthalpy or entropy balance (the latter being obtained using the LeChatelier-Shreder s equation applied to both components) yields the following relation for the heat of fusion of the eutectic mixture... 

In the calculation of the phase diagram, the equilibrium molar fraction is inserted into LeChatelier-Shreder s equation and the temperature of primary crystallization of every constituent is then calculated. For the optimized phase diagram, the Gibbs energy of mixing of the system is calculated using the condition... 

The Le Chatelier-Shreder equation presents the thermal dependence of the liquidus in the pre-eutectic region of the phase diagram with the simple eutectic. The same equation expresses the thermal dependence of the solubility of substance B in the solvent A . This equation allows us to draw some conclusions concerned with the interactions in eutectic systems, and the behaviour of the substance s solubility during variation of the solution temperature. The most general of these conclusions are as follows ... 

The calculations show, however, that the solubilities of metal-oxides in the melts based on alkali-metal halides are considerably lower than the values predicted by the Le Chatelier-Shreder equation, i.e. the properties of the oxide solutions are characterized by negative deviations from ideality. This is caused by the fact that the sum of the interaction energies of the dissolved substance-dissolved substance system (non-dissociated oxide, Me2+-02 )... 

As seen from equation (3.7.32), this coefficient is dependent both on the melt temperature and on the melting point of the metal-oxide. In contrast with the derivative from equation (3.7.31), the value of the thermal coefficient of solubility is positive, and consequently the Le Chatelier-Shreder equation predicts an increase in the solubility of any substance together with elevation of the solvent temperature. The data on the oxide solubilities obtained for the molten CsCl-KCl-NaCl eutectic at different temperatures allow us to verify the formulated conclusions. For this purpose, we write equation (3.7.32) in the form of finite differences, taking into account the fact that pPMe0 = — log PMeQ ... 

Since the dependence pPMe0 - T agrees with these criteria, it is most expedient to attribute the derivative to a point from the central part of the 600-700 °C segment, i.e. 650 °C (923 K). The values of relative thermal coefficients of solubility obtained using the Le Chatelier-Shreder equation and the experimentally obtained coefficients are collected in Table 3.7.12. 

A comparison of the experimental and the predicted data shows that the changes (increase) in solubilities of most of the metal-oxides studied with the elevation of the melt temperature is in good agreement with the Le Chatelier-Shreder equation. The estimation of the change in the oxide solubility in the vicinity of the experimental temperature can be made successfully using equation (3.7.32). 

Le Chatelier-Shreder equation, which is valid for the case of ideal solutions, i.e. for the solutions, where the interactions between the dissolved particles and between the solvent constituents are close to that between the solvent and the dissolved substance. 

While discussing the obtained slope values it should be noted that the theoretical slope value calculated according the Le Chatelier-Shreder equation (3.7.19) considered above is approximately equal to 7300. This value is practically coincident with the slope in equation (3.7.55), whereas the slope in equation (3.7.56) deviates from the theoretical one by less than 20%. This seems to be explained by the fact that the chemical properties of magnesium (and Mg2+ ion) resemble those of lithium rather than the properties of the other alkaline-earth metals (it is the so-called diagonal periodicity). Therefore, the properties of MgO and Li20 in the molten KCl-LiCl eutectic are close to ideal, and the Le Chatelier-Shreder equation is correct just for such solutions. At the same time, the chemical properties of magnesium and lithium cations differ essentially from those of Na, K and Cs. 

Nan and Delimarsky investigated solubilities of acidic metal oxides (M0O3, WO3, Ti02) in molten borax at 900 C by a similar method. The control of saturation was performed by a gravimetric analysis. Solubilities of the oxides deviated from those predicted by Shreder s equation, because of chemical interactions between the substances dissolved and the solvent. Oxide solubilities increased with the reduction of melting point of oxide. Solubilities were 66.1 mol% for M0O3, 63.2 mol% for WO3 and 21.2 mol% for Ti03. 

We have studied oxide solubilities in molten CsCl-KCl-NaCl at 600 and 700 C in order to estimate oxide solubility ehanges with temperature. Oxide solubilities were lower than those predicted by Shreder s equation but estimations of their thermal ehanges usually were in a good agreement with those calculated from Shreder s equation. Oxides solubilities (in molar fractions) in KCl-NaCl and CsCl-KCl-NaCI have been found to be elose at the same temperature. 

The pP value obtained aeeording the above equations ean be eorreeted for temperature if it differs from 700 C. It has been shown that at relatively small temperature ehanges oxide solubility is elose to value predieted by the Shreder s equation. This allows one to estimate the solubility products at different temperatures using the following equation ... 

Shreder in his description of a solvent by the equation, logx = - [Tf ,eitj -... 

T]/(R T(meit)T) since then known as the LeChatelier-Shreder equation. It was based on the assumption that the molar heat of dissolution over the whole range of temperatures and compositions is constant and equals to the molar heat of fusion of the pure component, i.e., showing the relationship AH <, //r AH(, eit/... 

The obtained results demonstrate that increase in the melt temperature causes the increase of the oxide solubility (pK Meo) l>y 0.3-0.4. These data show the metal oxide solubilities in chloride melts of different cation compositions. Practically the same oxoacidic properties at the same temperature are approximately the same if the solubilities are expressed in molar fractions. Although oxide solubilities in the studied melt are lower than those predicted by the Shreder equation ... 

T dependence in CsCl-KCl-NaCI eutectic is constracted. The slope of this dependence is in a good agreement with the value predicted by the Shnedei s equation that extends the range of use of the Shreder s equation for predictions of metal oxide solubilities in the molten halides at temperatures of the order of 500°C. 

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