Partial molar derivative

Note that the partial molar derivatives may also be taken under conditions other than constant p and T. 

The variations (d/drij) with respect to chemical species (usually at constant T, P) are referred to as partial molar quantities. For any molar extensive property X, the partial molar derivative Xt with respect to chemical component i is given by... 

The chemical potential /i, of a species i is the partial molar derivative of the Gibbs energy G, enthalpy H, Helmholtz eneigy A, or internal energy U of substance i ... 

The activity-coefficient equation is obtained from by taking the partial molar derivative of (ng ) and setting to zero the sum of the derivatives of In yj, on account of the Gibbs-Duhem equation, leaving... 

The activity coefficient is obtained by taking partial molar derivatives of... 

Note that the temperature and pressure (as well as certain mole numbers) are being held constant in the partial molar derivative. Later in this chapter we will be concerned with similar derivatives in which T and/or P are not held constant such derivatives are not partial molar quantities ... 

The derivative operator appearing in (3.4.5) is called the partial molar derivative, and the quantity F,- defined by (3.4.5) is called the partial molar F for component i. It is the partial molar property that can always be mole-fraction averaged to obtain the mixture property F. Note, however, that F is itself a property of the mixture, not a property of pure i partial molar properties depend on temperature, pressure, and composition. We emphasize that the definition (3.4.5) demands that F be extensive and that the properties held fixed can only be temperature, pressure, and all other mole numbers except N,. Partial molar properties are intensive state functions they may be either measurable or conceptual depending on the identity of F. 

The partial derivative is a linear operator therefore, the partial molar derivative (3.4.5) may be applied to all those expressions given in 3.2, producing partial molar versions of the fundamental equations. In particular, when we apply the partial molar derivative to the integrated forms (3.2.29)-(3.2.31) of the fundamental equations, we obtain the following important relations among partial molar properties ... 

For mixtures, this derivative must be evaluated with all mole numbers fixed, and we remind ourselves of that by the subscript N. Now apply the partial molar derivative in (3.4.5) to both sides of (3.3.32) we obtain... 

To obtain the partial molar properties of ideal-gas mixtures we apply the partial molar derivative (3.4.5) either to the ideal-gas law, to obtain the partial molar volume, or to the general expression (4.1.15), to obtain other properties. The generic expression (4.1.15) yields... 

First-law properties. The partial molar volume can be found by appl)fing the partial molar derivative (3.4.5) to the equation of state (4.1.20) the result is... 

We caution that (4.2.31) cannot be derived in a simple way by applying the partial molar derivative to the difference in residual Gibbs energies given in (4.2.30). The difficulty is that the partial molar derivative imposes a fixed pressure, but when the Ihs of (4.2.30), g (T, V, x ), is changed at fixed pressure, the mixture and ideal-gas volumes are no longer the same. Consequently, the isobaric derivative of the Ihs of (4.2.30) is not an isometric residual property in particular, it is not the Ihs of (4.2.31). 

The strategy for devising models for activity coefficients is based on modeling g, rather than modeling the y, directly. With a functional form adopted for the corresponding expressions for the y, can be obtained by applying the partial molar derivative in (5.4.10). In addition, if the model parameters are known functions of T and P, then expressions for and can be obtained from (5.2.11) and (5.2.12). This would enable us to obtain the T and P effects on the y, from (5.4.16) and (5.4.17). 

Applying the partial molar derivative in (5.4.10) to (5.6.24) provides Wilson s expressions for the activity coefficients... 

Here we show PROPERTY in capitals and its partial molar derivative, properin lowercase letters to emphasize that the derivative is normally taken of an extensive property, such as the enthalpy of a system, but the resulting (properis intensive, for example, enthalpy per mol Because a partial molar property is the derivative of an extensive property with respect to number of mols it is an intensive property itself. Partial molar values normally exist only for extensive properties (V, U, H, S, A, G). They do not exist for intensive properties (T, P, viscosity, density, refractive index, all specific or per unit mass properties). There is no meaning to the terms partial molar temperature (degrees per mol at constant T ) or partial molar specific volume (cubic feet per mol per mol ). 

In Chapters 7 and 9 we will use the partial molar derivative of the compressibility factor z, which is an intensive, dimensionless quantity. This usage seems to contradict the previous paragraph. However, if we define an extensive property Z=nz and insert its value in Eq. 4.18 we will find that Zi, is perfectly well behaved and has the right dimensions. This procedure is also sometimes used for other intensive... 

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... 

In principle. Baa, Bbb, and Bab can be calculated from molecular theories. In practice, that has proven difficult, so that while molecular interaction theory provides some information, we currently use correlations like the little EOS (Eqs. 2.48,2.49 and 2.50) to estimate Baa and Bbb, and use the semitheoretical or empirical mixing rules described below to estimate Bq. Equation 9.14 assumes that while B of the mixture depends on vapor composition (so that we can find its needed partial molar derivative), Baa, Bbb, and Bab depend only on the properties of the pure species and the temperature. 

By the normal methods of forming partial molar derivatives (see Problem 9.36) we find that that... 

If this is to be an equilibrium change then d G + PE)system must be zero. We know that the derivatives are partial molar derivatives (see Chapter 6), and that... 

This is a rare example of a partial molar derivative of an intensive property z,. However, we see that it is the logical result of representing an extensive property V by the number of mols and a set of intensive properties. This causes no difficulties, and the resulting equation is widely used. 

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