A Two-Dimensional Model of a Liquid: The Pair-Correlation Function Marlin Conheras and Jorge V a l e m e l a Universidad de Chile, Casilla 653, Santiago, Chile In the undergraduate chemistry major curriculum, little attention is given to the structure of liquids as compared to that of solids and gases. This may be due to the fact that basic elements of statistical mechanics are not taught in introductory physical chemistry courses. Only a t the graduate level, the students get some knowledge on liquids when they take a course on statistical thermodynamics. Some basic aspects of liquids, in elementary physical chemistry, can be didactically introduced through the fundamental concept of the modern theory of liquids called pair-correlation function-also known as pair-distribution function or radial distribution function. This function, which can be obtained from X-ray and fast-neutron scattering experiments, provides direct structural information concerning the molecular organization in liquids ( 1 4 ) . The pair-correlation function, g(R), may be defined as (6) g(R) = p(R)lpO
(1)
where po is the hulk density of the liquid expressed as the number of molecules per unit volume. p(R) is the average number of molecules in an element of volume a t a distance R from an arhitrarv molecule--called the central molecule -which is locatedat the origin of coordinates. p(R) provides a descri~tionof the local densitv of molecules. averaged over a long period of time, a t any histance R from thecentral molecule. Thus, the dimensionless function g(R) is a measure of the fluctuations which the local density undergoes with R (6). The building of a model for liquids and the determination of the pair-correlation function are very useful for understanding the meaning of this function. In this sense, a twodimensional model consisting of simple geometrical molecules is the most appropriate for this purpose. Here we have chosen circular and triangular molecules. Two-Dlmenslonal Llquid Model We first consider the model consisting of circular molecules shown in Figure 1. A radius unit (ru) was assigned to each circular molecule. Hereafter. all distances are " eiven in ru. Figure 1 shows rhat 66.4 molecules are contained in area of 400 ru2, resultinc! in a bulk densirv. - . .or.. -. of 0.166 molecules/ru2. A; a ~ e r a ~ e ~ n t e r m o l e c udistance lar do = 2.4 ru was obtained from do = (l/po)"2. The hulk density and the average intermolecular distance only provide approximate molecular information, because they are evaluated by considering the liquid as a "molecular continuum". Thus, local details are not taken into account. T o estimate them, a molecule, labeled by 0, was chosen as the coordinate origin (Fig. 1). Then the number of molecules surrounding the molecule 0 was counted within determined radii R. The density, p(R) was estimated a t different values of R by a simple and practical procedure as follows. Figure 1 was drawn on a square piece of cardboard 0.24 mm thick and 36 cm on each side. Each circle of 1.8 cm radius represented a molecule (1 ru = 1.8 em). Circles of radii 1.25,1.50,1.75,. . ., 10.00 ru were drawn from the center. Then. the cardboard was cut out following the circumferences of each one of theae circle'i. Molecules, or portions of molecules, contained in the
resulting rings were, in turn, cut out. Then, the "fractions of the molecules" beloneine to each ring were weighed out in an analytical balance.~henumherof m'hlecules n i o he found in each rina was obtained bv dkidine the total weieht of molecules in each ring by theaeight or one molecule^ The whole procedure was repeated three times, and the results were averaged. The same procedure was applied to the model consisting of triangular molecules as shown in Figure 2. In this case, each molecule has 2.66 A per side, and the bulk density, po, is 0.258 molecules/ru2. Table 1shows some illustrative results from both models. The function g(R) was determined by direct evaluation of density as follows,
where n is the number of molecules contained in each ring; Rj and Rj are the external and internal radii of a ring, respectively. g(R) was calculated by using expressions 1 and 2. Values of p(R) and g(R) are listed in Table 1.Then g(R) was plotted against R, as shown in Figure 3a. As R increases, g(R) fluctuates around unity and becomes definitely equal to 1 approximately over 9 ru. This means
0
2
i
6
8
R(ru) Flgure 1. A twodimensional model of a liquid. formed by circular molecules of unit radius. Only one ring, whose shaded area corresponds lo the fraction of the moiecules in the ring, has been drawn. Dashed lines show the limits of the lirsl, second, and third coordination shells.
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that the greater R, the greater the number of molecules considered, and therefore, for highR valuss the local density p(R) approaches the bulk density po of the liquid. For R < 2 ru, g(R) decays rapidly to zero, since the central molecule does not allow other molecules to approach closer than 1ru. Therefore, the probability of finding a second molecule within this region is zero. In the region 1ru < R < 3 ru, p(R) clearly differs from po, because of the presence of the first coordination shell. A more informative form of the radial distribution function is 2aRp(R). This form allows better insight into the local density variation with R. The quantity 2r7Rp(R)dR provides the average number of molecules to he found in a ring of dR width a t a distance R from the central molecule. Values of 2aRp(R) and 2rRPo were plotted against R as shown in Figure 3h. Coordination Shells The position and the shape of the maxima of g(R) or 2rRpog(R) functions-the latter being slightly shifted to greater values of R-are important features in a study of liquids. First, we will refer to the position of the maxima. The presence of a maximum a t a distance R indicates an increase of local density around R, because of the formation d a kind of molecular arrangement called coordiuation shell. Here we use the term "coordination" only in the sense of a geometric arrangement without specifying any kind of association. In the model with circular .nolecules. the first. the second, and rhe third coordinalion shrllc are found at the dizrance 2.3 i 0.2.4.8 zIJ.2, and 7.2 f 0.2 ru from thr rentral molecule, respectively. A fourth shell is not clearly observed in Figure 1. The coordination numbers, n,, for each shell were calculated using the following expression:
where RI and Rp are the corresponding distances for two
Table 1.
Pair-Correlation Function of a Two-Dimensional Liquid
Ri
RI
n
1.75
2.00
0.625
0.212
9.00
9.25
2.37
0.165
1.50 1.75 3.00 4.00 5.50 6.75 7.75
1.75 2.00 3.25 4.25 5.75 7.00 8.00
Triangular model 0.984 0.384 1.20 0.407 1.64 0.335 1.30 0.201 2.55 0.289 2.79 0.258 2.99 0.242
P(R)
2rRo(R1
g(R1
2.50
1.28
Circular model
Table 2.
Coordination Numbers tor the Two-Dlmensional Llqulds Shown in Figures 1 and 2
nca
Shell
RI
R,
1 2 3
1.0 3.5 6.0
Circular model 3.5 6.0 6.3
5.3 11.5 16.2
5 13 16
1 2 3
1.0 2.5 4.5
Triangular model 2.5 4.5 6.2
4.6 9.9 12.7
5 11 13
nca
sObtained by procedure desorlbed in text.
boMalnedby direct muntlng.
0 ' 2 . i 6 8 '
Rlrul Figure 2 . Model of triangular molecules. Dashed lines show the limits of the lirst, second. and third cwrdination shells.
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Journal of Chemical Education
minima around a maximum. The integral represents the area under the curve between R, and R2. The coordination numbers of the first three coordination shells were determined as follows: 1)By drawing Figure 3h on cardboard, 2) cutting out the area under the curve, 3) then making vertical cuts a t 3.5, 6.0, and 8.2 ru, 4) weighing each one of those sections and dividing them by the weight of a circle made of the same cardboard of radius 1ru. Table 2 shows the coordination numbers, n,, obtained by this procedure. I t is also ~ossibleto determine the coordination numbers bv direct counting of molecules whose centers are situated within the dashed lines in the Figure 1.As expected, hoth procedures practically yield the same coordination numbers. Since the model allowed us only to distinguish up to the third coordination shell, the correlation disappears beyond 8.2 ru. Thus, the central molecule should he correlated only with a maximum of 33 molecules. This number was estimated by summing then, values. The coordination numbers for triangular molecules, obtained by the same procedure, are given in Table 2. With regard to the shape of the maximaof g(R), the higher and sharper the maximum a t distance R, the more probable the structural arrangement a t that point. The first shell is more likely to be structured around the central molecule, which is a property shown by all liquids. The second and third shells are less likely t o be structured, and so on. If the first maximum of g(R) is very sharp, i t means that
those in the circular model. This fact is due to the more compact structure of the triangular model. This method can be extended to "molecules" with other geometrical shapes, such as squares, ellipses, or rods. Three-Dlmenslonal Liquld In a real liquid, g(R) and 4rrR2pog(R) functions exhibit approximately the same shape as shown in Figure 3. Note that in a three-dimensional liquid 4?rR2has been substituted for 2rR. However, 4aR2p, has a parabolic shape when plotted against R. In this case, the coordination number may be determined by
If the g(R) function is determined by a diffraction experiment, it is extremely difficult to interpret it in terms of a polyhedric arrangement of molecules surrounding a central molecule. A maximum of the function g(R) is frequently interpreted using a model which proposes a compact structure with more than one coordination sphere. When a liauid is investieated bv a diffraction ex~eriment. an annular diffuse pattern of scattering results. Our analysis of circular cuttine and weiehine vields onlv an averaee density in each ring 2 radius i,wiich represents an analogous situation to the diffuse rings. The problem is still more complicated in a solution, where g(R) may be expressed by g(R),~~ci~. = g(R),t.,t
+ ~(R),I~*,,.*
+ gR(w,,,) (5)
Figure 3. Representation 01: (a) g ( 0 , and (b) Z r R p ( 0 as a function d R. Circular model (0); triangular madel (A).
the molecules belonging to the shell are found a t almost the same distance from the center. This is indicative of a compact structural arrangement. If the molecules in a shell are randomly arranged, the corresponding maximum is lowered and the distribution becomes broader. Due to the simplicity of our model, remarkable broadening differences in the first three maxima are not observed. As observed in Figure 3a, the triangular model exhibits higher maxima, and a t closer distances from the center than
Further details on g(R) functions can he found in the Literature Cited. Acknowledgment We thank F. Urihe and Mrs. Ch. Wampler for reading the manuscript and for comments. We also thank Mrs. P. Moreno and M. Gonzalez for their assistance during the preparation of this paper. Literature Clted
( 5 ) Cantzeras, M.; Va1enzuela.J. Re". Chil. E d w . Quim. LSSS.8.3. (6) ~ircnberg,~ . ; ~ a u z m a nW. n , "~hestruavreand~ m p e n i e of s wator";oxford. 1969; p 156.
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