Partial molar

Since we make the simplifying assumption that the partial molar volumes are functions only of temperature, we assume that, for our purposes, pressure has no effect on liquid-liquid equilibria. Therefore, in Equation (23), pressure is not a variable. The activity coefficients depend only on temperature and composition. As for vapor-liquid equilibria, the activity coefficients used here are given by the UNIQUAC equation. Equation (15). ... 

The partial molar enthalpy for every component i is found from an appropriate form of the Gibbs-Helmholtz equation... 

VSTR is useful for estimating partial molar volumes at infinite dilution but is not used here because of Equation (4-17)... 

The heat evolved will now be a differential heat of adsorption, equal at constant volume to Qd or per mole, to qd - AI2, where Ae2 is the change in partial molar energy. It follows that... 

The partial molar entropy of adsorption AI2 may be determined from q j or qsi through Eq. XVII-118, and hence is obtainable either from calorimetric heats plus an adsorption isotherm or from adsorption isotherms at more than one temperature. The integral entropy of adsorption can be obtained from isotherm data at more than one temperature, through Eqs. XVII-110 and XVII-119, in which case complete isotherms are needed. Alternatively, AS2 can be obtained from the calorimetric plus a single complete adsorption isotherm, using Eq. XVII-115. This last approach has been recommended by Jura and Hill [121] as giving more accurate integral entropy values (see also Ref. 124). 

In experimental work it is usually most convenient to regard temperature and pressure as die independent variables, and for this reason the tenn partial molar quantity (denoted by a bar above the quantity) is always restricted to the derivative with respect to Uj holding T, p, and all the other n.j constant. (Thus iX = [right-hand side of equation (A2.1.44) it is apparent that the chemical potential... 

From cross-differentiation identities one can derive some additional Maxwell relations for partial molar quantities ... 

We define a partial molar volume Vi such that V = riiVi -I- U2V2... 

PARTIAL MOLAR EXCESS ENTHALPY AT INFINITE DILUTION OF THIAZOLE IN VARIOUS SOLVENTS AT SIS.IS K... 

Neutron magnetic moment Partial molar quantity A... 

When the film thickens beyond two or three molecular layers, the effect of surface structure is largely smoothed out. It should therefore be possible, as Hill and Halsey have argued, to analyse the isotherm in the multilayer region by reference to surface forces (Chapter 1), the partial molar entropy of the adsorbed film being taken as equal to that of the liquid adsorptive. By application of the 6-12 relation of Chapter 1 (with omission of the r" term as being negligible except at short distances) Hill was able to arrive at the isotherm equation... 

The chemical potential is an example of a partial molar quantity /ij is the partial molar Gibbs free energy with respect to component i. Other partial molar quantities exist and share the following features ... 

We may define, say, partial molar volume, enthalpy, or entropy by analogy with Eq. (8.5) ... 

Partial molar quantities have per mole units, and for Yj this is understood to mean per mole of component i. The value of this coefficient depends on the overall composition of the mixture. Thus Vj o the same for a water-alcohol mixture that is 10% water as for one that is 90% water. 

For a pure component the partial molar quantity is identical to the molar (superscript °) value of the pure substance. Thus for pure component i... 

A useful feature of the partial molar properties is that the property of a mixture (subscript mix) can be written as the sum of the mole-weighted contributions of the partial molar properties of the components ... 

Relationships which exist between ordinary thermodynamic variables also apply to the corresponding partial molar quantities. Two such relationships are... 

As noted above, all of the partial molar quantities are concentration dependent. It is convenient to define a thermodynamic concentration called the activity aj in terms of which the chemical potential is correctly given by the relationship... 

Remember that Vj is the partial molar volume of the solvent. Therefore a completely general relationship between n and the solvent activity is given by... 

This is converted to a partial molar quantity by differentiation ... 

As in osmotic pressure experiments, polymer concentations are usually expressed in mass volume units rather than in the volume fraction units indicated by the Einstein equation. For dilute solutions, however, Eq. (8.100) shows that

partial molar volume of the polymer in solution, and M is the molecular weight of the polymer. Substituting this relationship for (pin Eq. (9.9)gives... 

We divide by Avogadro s number to convert the partial molar Gibbs free energy to a molecular quantity, and the minus sign enters because the force and the gradient are in opposing directions. Recalling the definition of chemical potential [Eq. (8.13)], we write jUj + RT In aj = ii2 + RT In 7jC, where aj... 

From the definition of the partial molar quantities [Eq. (8.8)] we write... 

The Gibbs-Duhem equation also follows from the definition of partial molar quantities nid/Hi + r 2d 2 0. With the Gibbs-Duhem equation, d G/dc2 becomes... 

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