Lawrence F. Beste E. I. du Punt de Nemours 8 Co. W m i n g t u n , Delaware
A Geometric Approach to the Gibbs-Duhem Equation
Nyburg and Halliwelll have made a valuable contribution to the teaching of chemistry with their geometric approach to extensive properties. We shall show that their method may he profitably extended to explain the application of the calculus to such problems. In particular, we shall use geometry to illustrate the derivation and application of the GihhsDnhem equation. We consider an extensive property, such as the volume V of a system of two components a t some constant temperature and pressure. I n Figure 1 the volume is plotted vertically; the number of moles of the two components, nl and n,, are plotted on the horizontal axes. As shown by Nyburg and Halliwell, the surface representing V is generated by a series of straight lines through the origin. The surface itself may he curved as shown in Figure 1. The point A represents V I O ,the volume of one mole of pure component 1; B represents V z O .
words, the slope is the partial molal volume of component 1, in terms of the calculus it is (dV/dnl),,. We can now stat,e the following:
v,;
In general, of c o u r s e , A ~ will not be linear. However, it is shown in elementary calculus that equation (1) becomes true in the limit as the change inn, (and hence in V ) becomes infinitesimally small; B-A' then hecomes d V . I n a similar way V increases from B to C as n2 is increased by an amount dn2. The total change in V can now be given as:
I n equation ( 2 ) dV is the total differential of V . The above exposition is not a. rigorous derivation of d V , but it does provide an easy way to visualize dV and see what it means. Relations Between Partial Molal Quantities
Figure 1 .
Volume of a system of tuocomponentr
We now ask how the volume changes when nl and nl are changed arbitrarily by an infinitesimal amount: that is, when n, is changed by an amount dn,, and n2by an amount dnz. This process is illustrated in Figure 2. We start with the volume V Aa t point A. For the sake of brevity we shall say the volume equals A a t point A, that is, the volume is given by the vertical distance of point A above the basal plane. As n, is increased by an amount anl, the volume increases to B. Thus, the increase in volume is B-A or B-A'. We suppose for a moment that is a straight line. The slope of A Tis the late at which the volume changes with respect to nl when n~ is held constant. In other N~suno,5. C., AND HALLIWELL, H. F., J. CHEM.EDUC., 38, 123 (1961). Note that on page 125 n, and n2 were interchanged through B typographical error [see J. CHEM.EDUC.,38, 329 (1961)l.
All the straight lines which form the volume surface in Figure 1 are lines of constant composition; that is, along any one line the ratio nl/n2 is constant. How can we write the equation of one of these lines? We lirst note that as we proceed out from the origin along any line of constant composition we merely add more and more of the Jame subdance. For example if we start with ethanol and water in the molar ratio 0.6/0.4and add more ethanol and water in the same ratio, we will remain on the same line of constant composition. Every sample of such s, mixture must have the same intensive properties as every other sample. I t follows that the partial molal quantities will remain constant. This being so, we may integrate equation (2) and
Figure 2.
Change of volume with increase of"1 ond na.
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free energy of the system minus the free energy of the components. Since AF is zero for either component, the AF surface includes the nl and n2 axes (Fig. 3). Since Figure 3 applies to a realizable mixture, the value of AF is everywhere negative or zero. We plot the negative of AF in the figure, because the eye would be confused by a surface falling below the basal plane. The partial molal free energy (or chemical potential) of component 1 is p1 corresponding to Vl in the volume case. Similarly, p2 corresponds to Vz. Thus, one form of the Gibhs-Duhem equation can he obtained directly from equation (5) as Figure 3.
+
n~dw n&
Free energy of mixing twocomponenh
obtain the familiar result V = ntV1
+~
V S
It is to he noted that equation (3) applies only to one straight line on the volume surface with a constant ratio of nl to n2. If we had the appropriate values of VI and V2we could in theory establish an infinite number of lines and obtain thus the entire surface. It will he better not to perform this labor but to do something else. Imagining that we have established the surface, we may take the opportunity to walk about on it a t will. That is, we need not keep to a line of constant composition hut may let nl and n, vary independently. This will amount to differentiating equation (3) without holding Vl and constant. The result is
v2
dV = n d v ,
+ vldnl + ndVs +
(4)
The dV we obtain in this n a y is the total differential of V and must be identical with dV in equation (2). I t follows that the first and third terms on the righthand side of equation (4) together equal zero. That is,
+
ncdV~ n d v * = 0
=0
(7)
This important equation establishes a relation between the chemical potentials, pl and p2. We shall want to follow this relation across an entire one-mole-total diagram from pure component 1 to pure component 2. But before we do this, we shall put equation (7) into a more convenient form. We change a, and n2 into mole fractions XI and xa by dividing through by (nl n?). Letting XI be the independent variable, we obtain
+
which is more useful after rearranging to
This equation is very useful for qualitative deductions as well as quantitative. For example, consider the case of the ideal solution2 graphed in Figure 4. The equation for component 1is rr, - p,'
=
RT in XI
(9)
(5)
Of course, the intensive properties change only with a change in composition. To make this fact explicit we divide equation (5) through to obtain either of the two forms:
The subscripts in equation (6) remind us that throughout this derivation we have been keeping to a constant pressure and temperature. Equation (6) provides an extremely useful relation hetweeen the two partial molal quantities. For example, if one has found an equation to express the variation of Vl with composition, equation (6) permits one immediately to write down an equation for the variation of V2 with composition. Equation (6) is one of those fundamental relationships which must be true; an apparent violation indicates there is something wrong with the data. The Gibbs-Duhem Equation
To obtain the Gihbs-Duhem equation we merely start with the Gibbs free energy F instead of the volume. Since the absolute value of free energy is an indefinite quantity we are really interested in AF, that is, the 510
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Figure 4.
Chemical poten1ial.i in on ideal solution.
In equation (9), a0 is the chemical potential of pure component 1. The same equation, with change of subscript, applies to component 2. Equation (8) tells us that where the slope of the curve for one component is negative, the slope for the other is positive; we see in Figure 4 that 6p1/bx1 is everywhere positive and bp2/bxl everywhere negative. Also, in accord with equation (a), the slope of each curve approaches e infinity as the corresponding mole fraction approaches zero. Furthermore, where XI = x2 2 HILDEBRAND, J. H., A N D SCOW,R. L., "The Solubility of Naneleetrolytes," 3rd ed., Reinhold Publishing Carp., New York, 1950, pp. 20-21.
in the center of the plot one slope is exactly the negative of the other. It is still more interesting to consider non-ideal systems containing maxima and minima, such as the one graphed in Figure 5. Equation (8) tells us that where one component has a maximum (sero slope) the other will have a minimum (zero slope). These maxima and minima are usually the result of a large excess enthalpy of mixing. This follows because the partial molal enthalpy is zero for the pure component and tends to be maximum a t infinite dilution. However, a t infinite dilution the entropy dominates the curve as it approaches minus infinity; so a positive enthalpy can dominate only toward the center of the plot.
for the chemical potentials (such as those in Figure 5). This becomes obvious when one recalls the intercept method for computing partial molal quantities from a one-mole-total diagram. As another application of the Gibbs-Duhem equation we cite an example from polymer solution theory. The chemical potential of the solvent is usually expressed by the Flory-Huggins relation3: r
- p10 = R T M l
- ul)
+ (1 - l/z)vl +
(10)
I n this equation U S is the volume fraction of polymer, x is the number of segments in the polymer chain, each of which is the same size as a solvent molecule, and xl is an interaction parameter. The corresponding equation for the polymer is r p - pz9 = RT[lnu2 - ( z - 1)(1 - v2) x,x(l - ad1] (11) Fairly recently, Tompa4 brought equation (10) in better line with most data by adding an empirical term. His expression is
+
PI
- #lo = RT[ln(l
- ud
+ ( 1 - l/z)u2 + x,uzP+ x*v?l
(12)
One might thinkat first blush that equation (11) should be similarly expanded by addition of X2x(1-~2)S, but the Gibbs-Dnhem eqnation tells us this is not so. If xl and x2 are to havt the same valnes as in equation (12), then the proper expression for the polymer is Figure 5.
Chemical potentioh in o mixture exhibiting phase seporolion.
A diagram such as Figure 5 warns us to expect phase separation. Component 1 can find two compositions (A and B) in which it has exactly the same chemical potential; a t these same compositions component 2 also has equal chemical potentials. Any composition falling in the region where the curves are dashed will separate into two phases having compositions A and R. Phase separation indicates that the total free energy of the system is higher between compositions A and B than it is a t A (or B). We see this condition in Figure 3, which is drawn for a system exhibiting phase separation. We follow the l i e across the surface (one-mole-total diagram) and find, starting from either axis, that the free energy first decreases, passes through a valley, goes over a hill and through another valley, and finally increases to the other axis. The two inflection points in the A F curve fall a t exactly the same compositions as the maxima and minima in the corresponding curves
The most convenient form of the Gibbs-Duhem equation for checking the correspondence of equations (12) and (13) is
Conclusions
Application of the calculus to extensive and intensive properties can be made clear and easy to remember by means of a geometric approach. The Gibbs-Duhem equation, an important and useful equation, is amenable to such treatment. a FLORY, P. J., "Principles of Polymer Chemistry," Cornell University Press, Ithaca, N. Y., 1953, pp. 495-518. ' TOMPA, H., "Changements de phases," Socidtd de ehimie physique, Paris, 1952, pp. 163-5.
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