The Partial Molar Equation

This equation is simply a quite general first-order Taylor series. It states that the differential change of any variable is the sum of the product of its partial derivatives times the differential changes in the independent variables. It is slightly modified from the Taylor series because we have held Tand P constant, thus eliminating the terms in dT and dP. However, we see from it that the derivatives that appear on the right are the partial molar derivatives of Y. For example, if we let T be volume, then 

FIGURE 6.1 If we add species a and b at constant rates and run the mixer vigorously, then the composition of the fluid in the tank will remain constant, while its volume increases. 

Here we did the additions by a special path, which allowed us to perform the integrations. However, V is a state function, dependent only on P, T, and the various Thus, this relation is tme, no matter what path we follow. For any extensive property Y follows that 

We can divide both sides of Eq. 6.4 by the total number of moles n-r, which changes the 7 to a y and changes the number of mols of each species to the mol fraction of that species the result is 

Equation 6.4 and its alternative form Eq. 6.5 have no common names. A good name for them is the partial molar 

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The partial molar equation shows a unique and important relation between the partial molar properties in a mixture. When the differential of the partial molar equation is applied to the Gibbs energy, the result is the Gibbs-Duhem equation, which we will use in Chapter 9. 

The method of tangent intercepts is shown in the main text, based on the partial molar equation, Eq. 6.5. It may also be shown purely geometrically as follows. Figure 6.13 is the same as Figure 6.6, but the point of tangency has been moved to move points b and d further apart, points c, d, and e have been added, and unnecessary text has been deleted. 

The fact that there is no volume change or heat effect on mixing for an ideal solution does not mean that there is no entropy change on mixing. Mixing is always irreversible (an increase in disorder), so ideal solutions have greater entropies than the same species would have if they existed in the pure state, unmixed. If we substitute Eq. 7.26 twice in the partial molar equation (Eq. 6.5) and simplify, we find that for an ideal solution... 

It is shown in Chapter 6 that for any partial molar property (for example, q where Q is any extensive property) of any mixture with any number of species in one phase the partial molar equation requires that... 

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... 

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... 

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... 

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

When M = the requited derivatives are given by equations 62 and 63. Moreover, the derivative on the left side of equation 64 defines the partial molar Gibbs energy, G. Therefore,... 

Equation 163, written as = G- /-RT, clearly shows that In ( ) " is a partial molar property with respect to G /KT. MultipHcation of equation 175 by n and differentiation with respect to at constant T, P, and in accord with equation 116 yields, after reduction, equation 179 (constant T,x), where is the partial molar compressibiUty factor. This equation is the partial-property analogue of equation 178. 

If only one mole of the gas is considered, the quantity G is called the chemical potential, /x, or the partial molar free energy. In this case n = 1, and equation 20.197 becomes... 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

Equation (5.16) can be integrated. We expect the partial molar properties to be functions of composition, and of temperature and pressure. For a system at constant temperature and constant pressure, the partial molar properties would be functions only of composition. We will start with an infinitesimal quantity of material, with the composition fixed by the initial amounts of each component present, and then increase the amounts of each component but always in that same fixed ratio so that the composition stays constant. When we do this. Z, stays constant, and the integration of equation... 

By either a direct integration in which Z is held constant, or by using Euler s theorem, we have accomplished the integration of equation (5.16), and are now prepared to understand the physical significance of the partial molar property. For a one-component system, Z = nZ, , where Zm is the molar property. Thus, Zm is the contribution to Z for a mole of substance, and the total Z is the molar Zm multiplied by the number of moles. For a two-component system, equation (5.17) gives... 

First, we note that all of the thermodynamic equations that we have derived for the total extensive variables apply to the partial molar properties. Thus, if... 

Before leaving our discussion of partial molar properties, we want to emphasize that only the partial molar Gibbs free energy is equal to n,-. The chemical potential can be written as (cM/<9 ,)rv or (dH/dnj)s p H partial molar quantities for fi, into equations such as those given above. 

Equation (5.23) is known as the Gibbs-Duhem equation. It relates the partial molar properties of the components in a mixture. Equation (5.23) can be used to calculate one partial molar property from the other. For example, solving for dZ gives... 

The volume of a solution is sometimes expressed as a function of composition and the partial molar volume is then obtained by differentiation. For example, Klotz and Rosenburg2 have expressed the volume of aqueous sodium chloride solutions at 298.15 K and ambient pressure as a function of the molality m of the solution by the equation ... 

In the case of solutions, lnF] would be determined from equation (6.85) using the partial molar volume of the solvent, V, in the solution, rather than the molar volume of the pure solvent. 

From the Debye-Hiickel expressions for lny , one can derive equations to calculate other thermodynamic properties. For example L2, the relative partial molar enthalpy,q and V2, the partial molar volume are related to j by the equations... 

In Chapter 5, we defined the partial molar property Z, and described how it could be used to determine the total thermodynamic property through the equation... 

Writing Equation 6.11 for the partial molar free energy of Component i in a mixture ... 

If the partial molar heat capacities are substantially constant over the temperature range of interest, this equation may be solved to determine the relationship between the temperature and the fraction conversion. 

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