S. C. Nyburg and H. F. Halliwell
Colleae of North StafforGhire Keele, Staffordshire, England University
II
A ~eimetricApproach to Extensive Properties
I t is in~porta.ntthat ~t~udents of chemist,ry should he aware of t,he value of thermodynamics as an essential t,ool in their professional equipment. Unfortunat,ely t,his is often not the case, and the reason seems t,o be that on!y a few are easily able to bridge the gap bet,ween tthe behaviour of materials in the laboralory and the mathematical formulation of the problem. I'articularly difficult is that part of thermodynamics which deals wit,ll the way in which extensive pr~pert~ies of mixtures vary with change in composition. Here, wen at the outset, students are confronted midh proofs of relationships which involve the use of the partial differeniial calculus (side the traditional approach to partial molal quantities) and they find difficulty in appreciating the significance of the algebraic symbolism. We suggest that these difficulties are avoided if a geometric approach is used. There is nothing essentially new involved in presenting thermodynamic properties geometrically. The method was used freely by Gibbs ( 1 ) and by Bakhuis Roozehoom (2) to name but two pioneers, but there has been a not,iceable tendency away from geometry in favour of algebra. Moreover t,here apparently have been 110 serious attempts to present extensive, as opposed t,o intensive, properties geometrically. Graphical Representation of Extensive Properties
An ext,ensive property X of a system is one which increases X-fold when X such identical systems are put together under the same intensive conditions; examples are volume V, internal energy E, enthalpy H, Gibhs free energy F, entropy S, heat capacity C. Composition must be expressed in moles, not in percentage composition by weight, if any attempt a t theoretical int,erpretat,ion of t,he extensive property is t,o be possible. We restrict our attentiou here to systems of two romponents, and shall designate by n~and n2the numher of moles of each present. Let us consider first the total volume of a system conlposed of two completely miscible liquids; partial miscibility is dealt with later. Provided n, and n, are the only independcnt variables, i.e. temperature and pressure are kept constant, we can represent the relation between say, total volume V, n,, and nz by means of a three-dimensional diagram in which V is the vertical ordinate and nl and n2arehorizontal (Figs. 1 and 2). We notice immediately some important features. In Figure 1 any line OB lying in the n,-nr plane represent,~mixtures of the same composition because, hy similar triangles, nl/nr is constant for all points on OR. Let A represent a mixture of composition nr nr and .AC its volume. Let B represent X such mixt,ures put together and BD the new total volume.
From the definition of an extensive property, BDIAC = A. But OB/OA = h.n?/nl = A. Accordingly OCD is a straight line. I t follows therefore that, in Figure 2, the surface representing the uariatiun of an extensive propertu ciith composi!ion i s generated by straight lines t h r o u ~ h the origin. This is equivalenl
Figure 1. Comhucfion of mold diogram.
to saying that a t constant temperature and pressure, X is a homogeneous function of the first degree in n, and 112, but the geometric statement, is easier to comprehend. It should be pointed out here that this surface is not in general a plane although it may closely approximate to it in certain cases. Thus where X is volume it is very rare for the surface to deviate markedly from a plane; volume contraction or expansion of liquid mixtures is rarely more than a few per cent of the total volume although theoretically important nevertheless. Where X is related to the energy of the system, marked deviations from a planar surface are common and we shall cite several examples.
Figure 2. Stmight line generators of molal diogram.
We shall refer to such three-dimensional representations of extensive properties in terms of molal composi tion as molal diagrams. Volume 38, Number 3, Morch 196 1
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The second feature to he noticed on the molal diagram, Figure 2, is that the straight lines OA and OE represent respectively the value of X for varying amounts of the pure components, the points P and R giving the values of X for one mole of each pure component. Third, it is readily seen that the surface OAFE is cut by a vertical plane n, n2 = 1in a curve PQR which renresents t,he value of X for one-mole-total mixtures. If n, and n2 are plotted on the same scale, this cut lies a t 45' to the n,- and nt-axes. The abscissa on this vertical cut reads directly in mole fraction, x.
+
Some Related Two-Dimension01 Diagrams
There are two kinds of graph commonly used to represent the dependence of an extensive property on molal composit.ion. One type, the constant nl (or n2) diagram shows the way in which the property varies when a fixed amount of one component, is added to increasing amounts of the ot,her. Figure 3a shows
1
2
3
4
5
moles water, n, + (0)
0
1 2 4 moles sulfuric acid, n, --r
(b)
Figure 3. Heat capocity conrtont-n diagrams fw H~SOI, H 1 0 syltem: (01 mmtm-one-mole sulfuric acid, and [bl comtant-one-mole water.
(5) the heat capacity a t 25'C of mixtures made from I mole sulfuric acid with increasing quantities of water. There is, of course, a complementary diagram (Fig. :ib) fihowing t,he heat capacity of mixtures made from one mole of water wit,h increasing quantities of sulfuric acid. (For aqueous solutions it is more customary to plot the curve for 1000 g, i.e., 55.5 moles water, with increasing quantities of solute.) The second type is theone-mole-total diagram, showing the value of the extensive property for those mixtures in which nl n2 = 1. Figure 4 shows the diagram corresponding to Figures 3a and 35 for the heat capacity of the water-sulfuric acid system. These two types of diagram are simply special vertical cuts Figure 4. One-mole-total diagram through the surface of for the heat capacity of the H2SO1, a single molal diagram H 2 0 gystem.
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and Figure 5 shows the relationship between them. This important interconnection is more easily comprehended by the use of a model (Fig. 6) in whirh the straight line generators are rays of light from a point. source. The molal surface is generated from t,he onemole-total diagram which is represented by a curve of the extensive property drawn to scale in black ink on a sheet of glass. Alternat,ively a silhouette of the onemole-total diagram ran he u e d . The inset is a photograph of the model self-illuminated by a 6-v, 6-w automobile lamp. Clearly the twodimensional one-mole-total diagram governs all the features of the molal surface since the latter is completely generated from the former. Partial Molal Quantities
I
I 0
Figure 5. Cuh corresponding to Figure, 3 ond 4 en a mold diagram.
(1) Molecular interactzon. Variat,ions in X are a measure of the mutual interaction between the component molecules and in general t,he manner in which this interaction affects X will depend upon the ratios of the different molecules present, i.e. upon the composition. To check any model of the system one needs to estimate what rhange in X per mole there will be for each of the component,^ added in turn hut with the total composition unchanged. The student needs to realize the importance of t,hese changes in X which are the partial molal quant,ities of the component,s a t that composition and, measured this way, would require an infinite amount of mixhre. (g) The molal diagram. The effect of the addition of component 1 will he revealed on a constant-n2 diagram (Fig. 3a), and vice versa (Fig. 3b). Since an infinit,e amount of mixture is ruled out, we use the slope of X with n, a t the required composition to give the part,ial molal quantity of component 1 and again, vice versa,
(5) The method of intercepts. The usual proofs that the appropriate intercepts on a one-mole-total diagram are the partial molal quantities are difficult for many students to follow and one finds, in fact, that many such proofs do not stand up to critical examination. The main difficultv arises in rieorouslv relatine
(g)m
values to
-
values. The mkthods,
0 Figure 7.
Partial Miscibility and Limited Solubility
Figure 6. Model for illustrating a mold diagram. The source of light the wigin is o 6 v , 6-w outmobile bulb; 0 one-mole-total diagram is drown in black ink on 0 shed d gla~s. The insert [above right) is a selfilluminated photograph of the model. at
Strictly these two terms are synonymous, but usage tends to associate the former with mixtures of two liquids and with solid solutions and the latter with mixtures whose pure components are of different physicad states. The only difference these phenomena make to a molal diagram is to divide it into regions according to the number of phases present. (The
usually adopted are unsatisfactory for the mathematically-minded and bewildering to the novice, principally due to a failure to specify precisely the conditions of partial differentiation. The use of the molal diagram to prove the method of intercepts is particularly straightforward. Referring to 14gure 7, and bearing in mind the straight line generators through the origin, it is clear that the three tangents BE, LIE, and B'D' lie in a plane. Moreover ODXB is a parallelogram and hence its opposite sides make equal slopes with the basal plane. Therefore the intercept relation can be derived in one line:
rz), an,
=
Similarly
slope of line BE
=
slope of line OD =
GD/m = (intercept D'G')/l
r$)-, =
(intercept B C H ' ) / 1
A further important equation can be equally easily derived. From the one-mole total diagram (Fig. 8), using similar triangles:
giving
X = z,K, + 2,K2 This is the relation usually obtained by the application of Euler's theorem of homogeneous functions to the dependence of X upon n, and a2. The geometric approach is obviously much sin~pler.
number of moles of each component present are those for the total system irrespective of the number and nature of the phases.) For each such region all the geometric features are preserved which result from the generation of the surface by straight lines through the origin. But there are discontinuities in the surface at the boundaries common to two adjacent regions. These boundaries lie on vertical planes through the origin which are fixed for a given system (at the same temperature and total pressure) whatever the extensive property of interest. Figure 9 shows the molal surface giving the heat of mixing (excess enthalpy, see next section) for the system KOH.ZH,O and water (4). This molal surface has two distinct regions, that to the left representing one phase, namely KOH solution, and that to the right two phases, namely solid KOH.2He0 and saturated solution. Note that the right-hand surface is a plane; this is always the case for binary mixtures when two phases are present whatever the extensive property, because the phases change merely in extent Volume
38, Number 3, March 196 1
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it depends only upon nl/nl. where f, is any function. Thus
X = OA.j,(n,/nx)
But
0.4 = (nlZ n2)Ii2
+
=
Therefore X or
Figure 9. Molal diogrom for KOH.ZHa0, woter system showing two regions.
and not, in nature when the total composition is changed. Reference States
Volume and heat capacity have unambiguous ahsolute values, unlike the thermodynamic functions E, H, F, G, etc., whose values eaimot be fixed in the absence of an absolute datum. I t is simpler t,o introduce the student to the former type first because the notion of excess extensive properties follo\vs naturally. To plot excess volume or excess heat capacity is simply to choose the reference datum in a special way, t,hat is, so that the pure components themselves have zero values of t,he property. Excess functions are a useful means of amplifying small deviations of the system from simple additivity such as is usually shown by volume for example. I t is now but a simple step to apply the not,ion of reference states to those extensive properties of t,he second type which have no absolute datum. In the case of enthalpy. for example, one can const,ruct the molal diagram so that each point on the surface represents the enthalpy relative to the enthalpies of t,he elements (in t,heir standard states) from which the component,^ are made. Alternatively one can estimate enthalpies relative to the pure components themselves, in which case the surface is that for the excess enthalpy, or, as it is more usually called, the heat of mixing. In the case of free energy there are several useful reference states which can be used in order to construct the molal diagram. Ilowevec in all cases in which excess propert,ies are depicted, the surface terminat,es in t,he nl and n9 axes themselves (see Fig. 9). This can be conveniently demonstrated on the projection apparat,us Figure 6.
X
It is in fact fi(nl/n2)
+ 1lU1
na((n1ln2
=
Mnl/nd
=
n2f2(nl/nl),fl(nl/n2)
=
n2j(njh2).
Other equally valid forms are n,f(nl/nz), nrf(?~r/nl), et,c. The results of measurements on extensive properties are often summarized in the form of equat,ions in order t,o dispense with tabulated data and to facilitate interpolation. As long as this is all that is demanded of t,he equation the only criterion of suitabilit,~is its goodness-of-fit. For example an equation giving t,he heat of mixing which results when n2 moles of HC1 (gas) are dissolved in 1000 g water a t about 18°C is given hy Klotz (6) as
and, on integration ( H r ) n , _ j j .=s -17,300 n9
+ 215 nz2+ constant
As we have seen, any excess extensive property is zero for either pure component,. Putting n2 = 0, we obtain an integration constant of zero. The resultant. equat,ion is that applying to a constant-nl diagram (n,, the number of moles of water, being 1000/18 = 55.5), and it closely represent,^ the experimental data over the range to which it can be applied, namely n2 between 0 and 20.0 (saturated HCI)* except for small errors in the value of H E for n2 very close t,o zero. IYom this equation, we can generate one for the mold surface between the nl axis (pure water) and the discontinuity which occurs a t saturation. We can put the equat,ion in the form n2f(n2/nl) as follows: HE = n&li,300
(ndn,)"
221 nl(n21n,)l
Equations to Molal Surfaces
Although not essential to the empirical study of extensive properties, equat,ions giving the relation between X and composition are essential for any theoretical understanding of mixtures. From the geometric properties of the molal surface we can dedure the most general form the equation can take. The value of X a t any point on the surface is given by the product of the two factors (Fig. 10) :
Taking tan 0 first we see that for a given mixture 126
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Figure 10.
In this equation n, = 55.5, hence the equation to the surface is: for the range between nl/nl = 0 and 0.36.
* Data are frequently quoted for HCI molalitiea far greater than 20 but these are not experimental. They originate from speeulat,ions of Thomsen @a).
This equation to the molal surface carries no theoretical implications. A theoretical model for the mixture will restrict the type of equation used to represent the molal surface. Usually, the more sophisticated the model, the more elaborate the equation. Thus Moelwyn-Hughes (6) cites an equation for the excess enthalpy of mixtures which is of the form
where the a's are constants appropriate to the mixture. Guggenheim (7) uses a similar equation which has one higher power both in the denominator and numerator. I n all cases where equat,ions are used t o represent molal surfaces or vertical cut,s t,hrough them, it is essential to confine applications within the experimental limits. Extrapolation outside these limits is a dangerous exercise. Consider again equation (1) for the HCl-water mixture. We can obtain the equation to the one-mole-total cut by puttsing n, = X I and n2 = 1 - x l , whence H B = -17,300 (1 - 1,)+ ll,(iW ( 1 - Z ~ ) ~ / Z ~ but this will apply only to the experimental range. Outside t,his range HE would apparently be infinitely large a t XI = 0. No finite system could have an infinite enthalpy; moreover, even if the value of H" were finite a t XI = 0 and were theoretically intelligible, it would refer not t o pure KC1 (gas) but t o HC1 in a hypothetical liquid state under the same temperature and pressure conditions. Accordingly we recommend
that students using such equations should be warned against applying them outside the experimental limits. Conclusions
The molal diagram is a useful device both for interpreting import,ant properties of mixtures and for deriving important algebraic relation8hips in a simple way. It is true that the one-mole-total diagram summarizes all the properties of the system in a convenient way but as an introduction t o the study of extensive properties we believe the molal diagram t o be invaluable. Literature Cited
(1) GIBBS, J. W I I . I . . ~ D"C~llueted , Works," Yale University Presn, N e w Haven, Reprinted 1948. (2) R o o z ~ ~ o B om ~ ,~ a r s ,"Die Heterogenen Gleichgenirhtr," 2 vols., Braunschweig, 1904-1919. (3) Data from: CRAIG,1). N., A N D VINAI., G. W., .I. IZewmch Sat. BUT.Standards, 24, 475 (1940); KUNZLER, J. E., A N D
