The Gibbs-Duhem equations

The energy of a single-phase system is a homogenous function of the first degree in the entropy, volume, and the number of moles of each component. Thus, by Euler s theorem2 

This equation is extremely important (see Section 5.12 for some applications). It is known as the Gibbs-Duhem equation, and such equations as the Duhem-Margules equation may be derived from it. Since no limitation has been put on the type of system considered in the derivation, this equation must be satisfied for every phase in a heterogenous system. We recognize that the convenient independent variables for this equation are the intensive variables the temperature, the pressure, and the chemical potentials. 

From mathematics we know that if w is a homogenous function of the nth degree in x, y, and z, then (dw/dx)y7, (dw/dy)x 2, and (dw/8z)xy are homogenous functions of degree (n — 1) in x, y, and z. Therefore, many other relations can be obtained, some of which are useful, by the application of Euler s theorem to the various partial derivatives of E, H, A, and G. 

The derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations 

In the general case the total mixture property NM is a function of the temperature T and pressure P in addition to the number of moles of its components iVi, N. and Eq.11.4.1 becomes  

This expression represents the Gibbs-Duhem equation and indicates that the intensive properties of the mixture temperature, pressure and partial molar properties, cannot vary independently. Restricted to constant T and P, Eq. 11.6.3 becomes  

We will present two applications of the restricted form of the Gibbs-Duhem equation in Examples 11.3 and 11.4, while Eq. 11.6.3 will be used in Chapter 13 to evaluate the thermodynamic consistency of vapor-liquid equilibrium data. 

The following expressions are proposed for the composition dq) dracy of the partial molar volumes of a binary mixture at a givra temperature and pressure  

Introduction of Eqs (A) and (B) into the restricted form of the Gibbs-Duhem equation yields  

In the same way, the additivity rules for 17 and A can be derived. Any extensive property J of a system follows the additivity rule 

This is true also for the total number of moles, N = or the total mass, M = 

Equation (11.96) shows that if the composition varies, the chemical potentials do not change independently but in a related way. For example, in a system of two constituents, Eq. (11.96), becomes 

If a given variation in composition produces a change in the chemical potential of the first component, then the concomitant change in the chemical potential of the second component djX2 is given by Eq. (11.97). 

By a similar argument it can be shown that the variations with composition of any of the partial molar quantities are related by the equation 

The Gibbs—Duhem equation provides a very useful relationship between the partial molar properties of different species in a mixture. It results from mathematical manipulation of property relations. The approach is similar to that used in Chapter 5 to develop relationships between properties. The reason the Gibbs-Duhem equation is so useful is that it provides constraints between the partial molar properties of different species in a mixture. For example, in a binary mixture, if we know the values for a partial molar 

We begin with Equation (6.17), the definition of a partial molar property  

Equation (6.19) is the Gibbs—Duhem equation. Its straightforward derivation should not overshadow its tremendous utility. 

To see the usefulness of the Gibbs-Duhem equation, lets examine the scenario where we wish to find the partial molar volume of species h in a binary solution when we know the partial molar volume of species a, Va, as a function of composition. If we apply Equation (6.19) to the property volume, we get  

if we have an expression for (or plot of) the partial molar volume of species a vs. mole fraction, we can apply this equation to get the corresponding expression for species b. The expressions for partial molar properties are not independent but rather constrained by the Gibbs-Duhem equation. Such an interelation makes sense from a molecular perspective. The partial molar properties are governed by how a species behaves in the mixture. We expect the partial molar properties of a and b to be related since it is the same 

As shown by the final equation on the right, the equation for V = V(xB) is that of a straight line with slope 

In this section, we wish to derive the Gibbs-Duhem equation, the fundamental relationship between the allowed variations dRt of the intensive properties of a homogeneous (singlephase) system. Paradoxically, this relationship (which underlies the entire theory of phase equilibria to be developed in Chapter 7) is discovered by considering the fundamental nature of extensive properties Xu as well as the intrinsic scaling property of the fundamental equation U = U(S, V, n, n2,. .., nc) that derives from the extensive nature of U and its Gibbs-space arguments. 

Recall from Section 2.10 that the characteristic feature of extensive properties XL is their uniform scaling with respect to the size of the system, expressible in terms of a multiplicative positive scale factor A. Re-sizing the macroscopic system merely means that all extensive properties are multiplied by the common scale factor A, 

We begin by rewriting the Gibbs fundamental energy equation U = U(S, V, n, n2. nc) in symbolic form, with t = c + 2 extensive arguments 2Q  

The general scaling property of a macroscopic system can then be expressed by the math- 

In the absence of nonp E-work, an extensive property such as the Gibbs energy of a system can be shown to be a function of the partial derivatives  

In this context G itself is often referred to as the integral Gibbs energy. 

For a binary system consisting of the two components A and B the integral Gibbs energy eq. (1.88) is 

By combining the two last equations, the Gibbs-Duhem equation for a binary system at constant T and p is obtained  

In general, for an arbitrary system with i components, the Gibbs-Duhem equation is obtained by combining eq. (1.78) and eq. (1.90)  

We must be aware of one very important relationship between solution components, which is that they are not all independent of one another. This seems reasonable enough qualitatively. You can well imagine that changing the concentration, say, of one component of a binary system would have some effect on the activities and activity coefficients of both components, not just one. These changes can be quantified, and this is a highly useful device, because it is very common to measure the activity of only one component in a binary system as a function of concentration, and then calculate the activity of the other component, instead of measuring it too. We mentioned one way of doing this in 5.8.4, the isopiestic method. 

This relationship was introduced in 4.14.2. Equation (4.72) from that section is 

This looks more useful, but it is difficult to integrate because a plot of JC2A1 versus — In is asymptotic to both axes. If we rewrite the equation with x,y, instead of a and note that 

There is a rather large literature on how to perform this integration graphically and analytically. We will show just one very effective method, introduced by Darken and Gurry (1953, Chapter 10) see also Lupis (1983, Chapter 5). If we define 

Combining this with (10.117) then gives, after some manipulation, 

Not all the chemical potentials (and therefore, the activity coeflicients) in a mixture are independent of each other. They are all related to one another through the Gibbs-Duhem equation. To derive this equation, we start with tlie fundamental equation of thermodynamics for the Gibbs free energy, which can be written as 

This relation is known as the Gibbs-Duliem equation. Dividing both sides of the equation by the total number of moles in the system, we find that the Gibbs-Duhem equation can also be written as 

The Gibbs-Duhem equation provides a relation between the chemical potentials of each of the chemical species in a given phase. At constant temperature and pressure, this relation simplifies to 

The chemical potential can be written in terms of the activity coefficient (see Eq (6.4)). Subsituting this expression into the Gibbs-Duhem equation, at constant temperature and pressure, we find 

Suppose we know the activity coefficient of component 1 71 as a function of composition. Then, we can determine the activity coefficient of component 2 72  

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It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

The pressure at which standard-state fugacities are most conveniently evaluated is suggested by considerations based on the Gibbs-Duhem equation which says that at constant temperature and pressure... 

Equation (16) is a differential equation and applies equally to activity coefficients normalized by the symmetric or unsymme-tric convention. It is only in the integrated form of the Gibbs-Duhem equation that the type of normalization enters as a boundary condition. 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

The special case of equation (A2.1.27) when T and p are constant (dJ= 0, dp = 0) is called the Gibbs-Duhem equation, so equation (A2.1.27) is sometimes called the generalized Gibbs-Duhem equation . 

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... 

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. 

If temperature and pressure are constant, equation 130 reduces to equation 131 (constant T,P) which is a common form of the Gibbs-Duhem equation. 

A great deal of study and research has gone into the development of working equations that can represent the curves of Figure 3. These equations are based on solutions of the Gibbs-Duhem equation ... 

Moreover, the Gibbs/Duhem equation for a solution at given T and P, Eq. (4-52), becomes... 

Moreover, Eq. (4-122), the Gibbs/Duhem equation, may be written for experimental values in a binaiy system as... 

Because experimental measurements are subject to systematic error, sets of values of In y and In yg determined by experiment may not satisfy, that is, may not be consistent with, the Gibbs/Duhem equation. Thus, Eq. (4-289) applied to sets of experimental values becomes a test of the thermodynamic consistency of the data, rather than a valid general relationship. 

The right-hand side of this equation is exactly the quantity that Eq. (4-289), the Gibbs/Duhem equation, requires to be zero for consistent data. The residual on the left is therefore a direct measure of deviations from the Gibbs/Diihem equation. The extent to which values of this residual fail to scatter about zero measures the departure of the data from consistency with respect to this equation. 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

Worth noting is the fact that Barkers method does not require experimental yf values. Thus the correlating parameters Ot, b, and so on, can be ev uated from a P-X data subset. Common practice now is, in fact, to measure just such data. They are, of course, not subject to a test for consistency by the Gibbs/Duhem equation. The worlds store of X T.E data has been compiled by Gmehling et al. (Vapor-Liquid Lquilibiium Data Collection, Chemistiy Data Series, vol. I, parts 1-8, DECHEMA, Frankfurt am Main, 1979-1990). 

Onee again, integrating as in the Gibbs-Duhem equation, yields... 

The Gibbs-Duhem equation allows the determination of activity coefficients for one component from data for those of other components. 

The Gibbs-Duhem equation is extremely important in solution chemistry and it can be seen from equation 20.171 that it provides a means of determining the activity of one component in a binary solution providing the activity of the other is known. 

But Langmuir s isotherm for the solute entails the generalized form of Raoult s law (Eq. 13) as a necessary thermodynamic consequence. This can best be seen from the Gibbs-Duhem equation,... 

Trustworthy thermodynamic data for metal solutions have been very scarce until recently,25 and even now they are accumulating only slowly because of the severe experimental difficulties associated with their measurement. Thermodynamic activities of the component of a metallic solution may be measured by high-temperature galvanic cells,44 by the measurement of the vapor pressure of the individual components, or by equilibration of the metal system with a mixture of gases able to interact with one of the components in the metal.26 Usually, the activity of only one of the components in a binary metallic solution can be directly measured the activity of the other is calculated via the Gibbs-Duhem equation if the activity of the first has been measured over a sufficiently extensive range of composition. 

B. Constant-Pressure Activity Coefficients and the Gibbs-Duhem Equation.. 158... 

The advantages of constant-pressure activity coefficients also become clear when we try to relate to one another the activity coefficients of all the components in a mixture through the Gibbs-Duhem equation (P6, P7). For variable-pressure activity coefficients at constant temperature we obtain... 

Vapor-liquid equilibrium data are said to be thermodynamically consistent when they satisfy the Gibbs-Duhem equation. When the data satisfy this equation, it is likely, but by no means guaranteed, that they are correct however, if they do not satisfy this equation, it is certain that they are incorrect. 

A consistency test described by Chueh and Muirbrook (C4) extends to isothermal high-pressure data the integral (area) test given by Redlich and Kister (Rl) and Herington (H2) for isothermal low-pressure data. [A similar extension has been given by Thompson and Edmister (T2)]. For a binary system at constant temperature, the Gibbs-Duhem equation is written... 

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... 

An expression for V can be obtained from equation (5.29) by integration of the Gibbs-Duhem equation. Starting with the Gibbs-Duhem equation equation (5.23) applied to volume gives... 

We cannot generalize from this example. In some systems Pj < P 1 while P > P . In other examples Vf- > P . 1 while V < Pm. . The thing that must be true from the Gibbs-Duhem equation is that... 

In summary, in the limit as x2 —> 0 and xi — 1, /i —>.V /f and f2 —> x2A h..x-It can be shown from the Gibbs-Duhem equation that when the solute obeys Henry s law, the solvent obeys Raoult s law, To prove this, we start with the Gibbs-Duhem equation relating the chemical potentials... 

Most of the methods we have described so far give the activity of the solvent. Often the activity of the solute is of equal or greater importance. This is especially true of electrolyte solutions where the activity of the ionic solute is of primary interest, and in Chapter 9, we will describe methods that employ electrochemical cells to obtain ionic activities directly. We will conclude this chapter with a discussion of methods based on the Gibbs-Duhem equation that allow one to calculate activities of one component if the activities of the other are known as a function of composition. 

In equation (5.27), we used the Gibbs-Duhem equation to relate changes in the chemical potentials of the two components in a binary system as the composition is changed at constant temperature and pressure. The relationship is... 

Dividing by the total number of moles, ( ( -fH2), expresses the Gibbs-Duhem equation in terms of mole fractions ... 

The osmotic coefficient 4> and activity coefficient are related in a simple manner through the Gibbs-Duhem equation. We can find the relationship by writing this equation in a form that relates a and 2-... 

For a binary solution containing 2 = m moles of solute and n = 1 /M moles of solvent (with M in kg-mol 1), the Gibbs-Duhem equation becomes... 

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