Partial molar value

In some cases, reported data do not satisfy a consistency check, but these may be the only available data. In that case, it may be possible to smooth the data in order to obtain a set of partial molar quantities that is thermodynamically consistent. The procedure is simply to reconstmct the total molar property by a weighted mole fraction average of the n measured partial molar values and then recalculate normalised partial molar quantities. The new set should always be consistent. 

Any physical interpretation of a partial molar quantity must be consistent with its definition. It is simply the change of the property of the phase with a change of the number of moles of one component keeping the mole numbers of all the other components constant, in addition to the temperature and pressure. It is a property of the phase and not of the particular component. One physical concept of a partial molar quantities may be obtained by considering an infinite quantity of the phase. Then, the finite change of the property on the addition of 1 mole of the particular component of this infinite quantity of solution at constant temperature and pressure is numerically equal to the partial molar value of the property with respect to the component. 

The dependence of XE on mole fraction can be analysed to obtain the dependence on mole fraction of the relative partial molar quantities, Xx — X and X2 — X2, which are calculated from the intercepts on the and X2-axes of the tangent to the XE -curve. Consequently the XE -data must be precise if these partial molar values are to be meaningful. For example, one of the features of some aqueous mixtures is a minimum in the value of V2 — V2 at low values of x2. This means that the VE -curve has a point of inflexion at this mole fraction. 

In case of a solution, like the FeO-MnO solid solution, the integral molar free energy of mixing would be related to the partial molar values, as per their definition, in the following manner ... 

Partial Molar Quantities. — The thermodynamic functions, such as heat content, free energy, etc., encountered in electrochemistry have the property of depending on the temperature, pressure and volume, i.e., the state of the system, and on the amounts of the various constituents present. For a given mass, the temperature, pressure and volume are not independent variables, and so it is, in general, sufficient to express the function in terms of two of these factors, e.g., temperature and pressure. If X represents any such extensive property, i.e., one whose magnitude is determined by the state of the system and the amounts, e.g., number of moles, of the constituents, then the partial molar value of that property, for any constituent i of the system, is defined by... 

Since the partial molar value (with respect to i) of any extensive property X is defined as (5//3N ) equation (6) shows that is the partial... 

One significant aspect of partial molar properties is that represented by equation (26.6). If Qi is the partial molar value of any property in a system containing n - moles of the constituent i, then the total value O for the system is given by the sum of all the terms. For a system consisting of a single, pure substance the partial molar property is identical with the ordinary molar value. This result has often been used in the earlier treatment. 

In order to indicate the fact that the value of G as given by equation (42.1) applies to the constituent 2, i.e., the solute, a subscript 2 is sometimes included. However, this is usually omitted, for in the great majority of cases it is understood that the apparent molar property refers to the solute. It i.s seen from equation (42.1) that o is the apparent contribution of 1 mole of the component 2 to the property G of the mixture. If the particular property were strictly additive for the two components, e.g., volume and heat content for ideal gas and liquid solutions, the value of 4>q would be equal to the actual molar contribution, and hence also to the partial molar value. For nonideal systems, however, the quantities are all different. 

For the case of nonideal solutions, show that the expressions for the enthalpy balances are of the same form as those given by Eqs. (2-33) through (2-37) except for the fact that the ideal solution enthalpies of the pure components hjit Hfi are replaced by their partial molar values, Hyi or the virtual values of their partial molar enthalpies hjit H i see Chap. 14. 

At partial molar value Ax = (dA/dn p r (, energy units per mole... 

The partial molar values per se are intensive properties as they do not depend on the total amount of solution and may be both positive and negative. If the solution pressure and temperature do not change, any of its extensive property is a function only of its composition ... 

If the amounts dN, dN, dN, ..., have the same proportions as in the initial solution, the composition of the latter does not change. This is the same as mixing two solutions of the same composition the solution amount increases, and its composition does not. In this case the very partial molar values g. also do not change, and "LNSg. in the second addend in Equation (1.11) are equal to 0. Then... 

Integrating Equation (1.12), on condition of constancy median partial molar values g we obtain Equation... 

Which is called the rsf Gibbs-Duhem equation. It shows that any extensive property g, for instance volume, heat capacity or energy of the solution, may be determined from its composition, if corresponding median partial molar values of its components are known. For instance, if the mole amount of individual components in the composition of a solution and their partial molar volumes are known, then the volume of the entire solution will be equal to the sum of their products. 

It describes the association between partial molar values of different components in one solution at constant temperature and pressure. It follows from it that if content of only one component changes, the partial molar values of all components change, but so that... 

In other words, an increase in an extensive parameter of a solution due to addition of a component i is compensated by a decrease of median partial molar values of the remaining components in its composition. 

Equations (1.13), (1.14), (1.16) and (1.17) are very important for further understanding of the effect of change in water composition of solutions on their extensive properties, first of all on volume and energy. In this connection very significant are median partial molar values of the components in water solutions, which are determined experimentally in pure solutions of individual components at an increase of concentration by 1 mole. Ihe obtained results may be found in reference literature (Naumov et ah, 1971 Wagman et ah, 1982 CRC Handbook of Chemistry and Physics, 2004-2005, etc.). 

The thermodynamic analysis of solutions is facilitated by the introduction of quantities that measure how the extensive thermodynamic quantities (V, E, H, G,. ..) of the system depend on the state variables T, P, and nj. This leads to the definition of partial molar quantities where, if we let Y be any extensive thermodynamic property, we can define the partial molar value of Y for the ith component as ... 

ELDAR contains data for more than 2000 electrolytes in more than 750 different solvents with a total of 56,000 chemical systems, 15,000 hterature references, 45,730 data tables, and 595,000 data points. ELDAR contains data on physical properties such as densities, dielectric coefficients, thermal expansion, compressibihty, p-V-T data, state diagrams and critical data. The thermodynamic properties include solvation and dilution heats, phase transition values (enthalpies, entropies and Gibbs free energies), phase equilibrium data, solubilities, vapor pressures, solvation data, standard and reference values, activities and activity coefficients, excess values, osmotic coefficients, specific heats, partial molar values and apparent partial molar values. Transport properties such as electrical conductivities, transference numbers, single ion conductivities, viscosities, thermal conductivities, and diffusion coefficients are also included. 

Let E stand for any of the extensive properties, such as internal energy, enthalpy, etc. Then the partial molar value of the property E, for the fth substance, is defined by... 

It may be noted that the partial molar value of ( is identical with the chemical potential. This arises from the fact that in the equation (2-61)... 

Nevertheless, the partial molar values play an important role. The composition w enters the configurational partial molar entropy, and the partial molar enthalpy of oxygen should be rather independent of w and the model adopted. Experimental data at constant w (like in fig. 52) across the homogeneity range give indeed only modestly varying (between 160 and 172 kJ/mol) whereas changes markedly (between some 100 and 210 J/mol K)... 

This gives us the differential of the affinity in intensive variables (partial molar values) and extent ... 

As this property is extensive, it has eorresponding partial molar values, relating to each eomponent of the solution, defined by ... 

Equation (13.25) does not include the volumes of the micellar or aqueous phase, which can be obtained from the partial molar values of the surfactant. A better expression is... 

Jj partial molar value of the variable J relative to component i... 

Thermodynamic Properties of the Mixtures In those cases where F represents a molar thermodynamic property of the binary mixture of water and the cosolvent, the partial molar quantities of the components are of interest. Differentiation of Equation 3.43 with respect to the mole fractions yield the partial molar excess values. The excess partial molar value for water is ... 

Similarly, toe excess partial molar value for toe cosolvent starts with toe second-... 

For the chemical potential (and for all the partial molar values) of a structure element, we again find the same difficulty as encountered in the case of ions. Indeed, it is impossible to maintain the quantities of elements for all types of structure elements of a solid, except for one of them because, in addition, the electrical neutrality but also the ratios between sites must be preserved in a transformation. In order to circumvent the difficulty, we use the same approach as with the ions,... 

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