Research: Science and Education
Regression Methods To Extract Partial Molar Volume Values in the Method of Intercepts Leon F. Loucks* Chemistry Department, University of Prince Edward Island, Charlottetown, PE C1A 4P3, Canada
Partial molar quantities, while fundamental to the thermodynamics of solutions, are conceptually difficult for many students. In the laboratory, only volume is studied in most physical chemistry course offerings. Volume has the advantage of concrete visualization, which assists students and may permit them to extrapolate to other thermodynamic properties, once partial molar volume has been mastered. For the evaluation of the partial molar volumes of the components of binary solutions, either “the slope method” (1–4) or “the method of intercepts” (3, 5–8) is used. The method of intercepts is a very satisfying approach in mathematical and graphical theory; however, in practice it is imprecise because the graphical assignment of tangents is too arbitrary for many binary systems that show only minor nonideality in the plots of molar volume against mole fraction. This paper presents the use of a regression technique to generate an equation that accommodates the curvature in the plot of molar volume vs mole fraction for systems exhibiting a regular pattern of nonideality, in which the deviation is zero at the mole fractions of zero and unity and increases monotonically to a maximum at 0.5 mole fraction. This equation is then used to generate the first derivative so that the slope of the tangent line can be evaluated at all points along the curve; in turn, the intercepts of the tangent line are evaluated. This method provides precision far superior to manual graphical methods. In the following presentation, the notation follows that of Alberty and Silbey (5). The total volume of an ideal bi– – – nary solution is given by V = V *1n1 + V *2n2, where V *1 and – V *2 are the molar volumes of the pure components 1 and 2 and n1 and n2 are the number of moles of 1 and 2. In practice, however, the total volume of a binary solution is usually less than predicted by ideal mixing (although in some cases it is more). Such volume changes arise from the relative strengths of the intermolecular forces between the like and unlike molecules of the binary solution and from differences in the packing of unlike molecules, due to size and shape. In cases where the volume of a solution is less than the sum of the volumes mixed, the presence of the second substance provides for more efficient packing of the molecules or generates stronger interactions between unlike molecules than between like molecules. In some cases, dislocations in the hydrogenbonded structure by the insertion of foreign molecules cause collapse of structure and volume decreases. Similarly, the solvation of ions to create a tightly packed solvation shell may lead to a net volume decrease. In cases where the solution volume exceeds the sum of volumes mixed, the second substance disrupts a rather strong intermolecular interaction of a pure material and substitutes a weaker interaction between the unlike molecules; the result is a larger volume per molecule, on average. *Email:
[email protected] The differential of binary solution volume change caused by component additions to a solution being kept at constant temperature and pressure is:
dV = ∂V ∂n 1
dn 1 + ∂V ∂n 2
n2
dn 2
(1)
n1
The partial derivative coefficients in this equation are–called the – partial molar volumes and are given the symbols V1 and V2. Thus: – – dV = V1dn1 + V2dn2 (2) The partial molar volumes depend on the solution composition. Their values can be appreciably different from the molar volumes of the pure components. For a solution of particular composition, eq 2 may be integrated to yield a total volume given by – – V = V1n 1 + V2n2 (3) Division by the total number of moles (n1 + n2) results in – – – V = x 1V1 + x 2V2 (4) – where V is the molar volume of any binary solution; that is, the volume of one mole of solution. Upon insertion of x1 = (1 – x2), this latter equation rearranges to – – – – V = V1 + ( V2 – V1)x2 (5) The tangent line at a particular concentration is given by – this equation. The tangent line has an intercept of V1 at x2 = 0; – similarly, the right-side intercept at x2 = 1 must be V2, as the – – the slope V2 – V1 can only thus be accommodated. At any – – different composition, the tangent line differs and V1 and V2 take on different values. Figure 1 shows an example of one – tangent drawn to a plot of V vs x 2. The molar volumes of binary solutions are readily calculated from density measurements. In our studies we have found that densities measured at 25 °C with a 25-mL Weldstyle density bottle for about 10 different mole fractions provide a sufficient data base for the ensuing analysis. It is helpful to select compounds of similar molar volumes in choosing the binary set, so the plot of molar volume vs mole fraction can easily reveal the curvature indicative of deviation from ideality. We have found the ethanol–ethylene glycol and t-amyl alcohol–toluene systems to be excellent selections for the negative and positive deviations from ideality. Table 1 shows typical student data for the two systems. The filled circles of Figure 1 show the dependence of the molar volume on the mole fraction of ethanol in the ethanol– ethylene glycol system. The regular pattern shows a clear downward curvature between the intercepts, which are the molar volumes of the pure compounds. If there existed no deviation from ideality, the intercepts would be joined by a straight line.
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To fit the data well, an appropriate regression should allow for the curvature as an effect superimposed on what would otherwise be a straight line. A parabolic term centered on the midpoint of the mole fraction axis provides a way to introduce the correction symmetrically on each side of x 2 = 0.5; thus – we assume that V is given by a + bx2 as a first approximation, which can be improved by adding a parabolic term. The use of such a parabolic term has been reported in the literature (9) as the simplest term of a general method. Thus the regression needs to be conducted to satisfy – V = a + bx2 + c(x 2 – 0.5)2 (6) The coefficients a, b, and –c are evaluated in a spreadsheet multiple regression of the V values as a function of both x2 and (x2 – 0.5) 2. In the case of ethanol–ethylene glycol, the equation becomes – V = 55.287 + 2.793x2 + 2.08(x 2 – 0.5)2 (7) The data for t-amyl alcohol–toluene give – V = 108.16 + 2.416x2 – 2.20(x 2 – 0.5)2
(8)
Notice the change of sign for the coefficient c as the nonideality changes character between the two systems. The R 2 values are .9995 and .9982 in these two respective regressions. – The solid curve of Figure 1 has been constructed from the V values calculated from eq 7; the fit is obviously very good. Once equipped with an equation for the dependence of – V on x2, one can extract the equation for the slope of the tangent line as the first derivative. Thus – dV /dx 2 = b – c + 2cx2 (9)
Molar Volume / mL
59
58
57
56
55 0.0
0.2
0.4
0.6
0.8
1.0
Mole Fraction Ethanol Figure 1. A plot of the molar volumes of ethanol–ethylene glycol solutions as a function of mole fraction ethanol. Circles are experimental data; the curve is calculated from the regression equation; and the straight line is the tangent calculated from the regression coefficients at a mole fraction of 0.5.
Table 1. Molar Volumes of Binary Solutions at 25 8C
t - Amyl Alcohol –Toluene X amyl alcohol V¯ /mL
Ethanol – Ethylene Glycol
X ethanol
V¯ /mL
0
55.828
0
107.554
0.1092
55.902
0.1130
108.155
0.2244
56.060
0.2232
108.549
As expected, the slope of the tangent varies with the mole fraction. The slope of the tangent at a particular value of V¯ can be expressed as – – ( V – V1)/x 2 = slope (10)
0.3321
56.245
0.3307
108.915
0.4393
56.513
0.4407
109.240
0.5499
56.866
0.5505
109.452
0.6529
57.178
0.6621
109.650
0.7818
57.621
0.7702
109.859
or as:
0.8686
58.004
0.8741
109.976
1
58.591
1
110.041
– – ( V2 – V )/(1 – x 2) = slope (11) – The expressions for V given by eq 6 and for the slope by eq 9 can then be used to obtain: – V1 = a + 0.25c – cx22 (12) – 2 V2 = a + b – 0.75c + 2cx 2 – cx 2 (13) – – These relationships permit us to calculate V1 and V2 values at every mole fraction value that we choose. Table 2 shows a number of values calculated–for the ethanol–ethylene glycol system. The data reveal how V2 approaches the molar volume of pure ethanol (58.60 mL) as x2 approaches unity; in contrast, as x2 approaches zero the molar volume of ethanol has reached a quite different limit of 56.52 mL, representing the volume of one mole of ethanol dissolved in an infinite amount of ethylene glycol. Conversely, as x2 approaches zero, the partial molar volume of ethylene glycol becomes the molar volume of that pure substance (55.81 mL). Meanwhile, at x 2 equal to unity, the partial molar volume of ethylene glycol has taken a limiting value of 53.73 mL for the volume of one mole dissolved in an infinite amount of –ethanol.–Figure 2 shows a plot of the calculated values of V1 and V2 as a function of the mole 426
Table 2. Partial Molar Volumes in Ethanol–Ethylene Glycol System X ethanol V¯ ethanol /mL V¯ethylene glycol /mL 0
56.519
55.807
0.05
56.722
55.802
0.15
57.097
55.760
0.25
57.430
55.677
0.35
57.721
55.552
0.45
57.971
55.386
0.55
58.179
55.178
0.65
58.345
54.928
0.75
58.470
54.637
0.85
58.553
54.304
0.95
58.595
53.929
1
58.600
53.726
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Research: Science and Education
0.0
58
-0.1 Excess Molar Volume / mL
Partial Molar Volume / mL
59
57
56
55
54
53 0.0
-0.2 -0.3 -0.4 -0.5 -0.6
0.2
0.4
0.6
0.8
1.0
0.0
0.2
Mole Fraction Ethanol
0.4 0.6 Mole Fraction Ethanol
0.8
1.0
Figure 2. A plot of the partial molar volumes of ethanol and ethylene glycol as a function of the mole fraction of ethanol in ethanol– ethylene glycol solutions. All values are intercepts of tangents of Fig. 1, calculated from the regression coefficients and the mole fraction. The upper curve is for ethanol; the lower, for ethylene glycol.
Figure 3. A plot of the excess molar volumes of ethanol–ethylene glycol solutions as a function of the mole fraction of ethanol. The curve shows the values calculated from a regression fit of the experimental data to a parabola centered on a mole fraction of 0.5.
fraction of ethanol and we see that the partial molar volumes are continuously variable with the composition of the solution. Many thermodynamicists prefer to monitor the molar volume behavior of binary solutions in terms of excess molar – volume. This quantity, VE, is the volume deviation per mole from the expected volume if no contraction or expansion occurs upon mixing the two components. Thus – – – – VE = V – x1V *1 – x2V *2 (14) – For systems–that exhibit – contraction upon mixing, V will be less than x1V *1 + x2V *2; hence the excess molar volumes are negative values. Such is the case for the ethanol–ethylene glycol system, whereas the t-amyl alcohol–toluene system yields positive excess molar volumes. Figure 3 shows a plot of the excess molar volumes of the ethanol–ethylene glycol system with the maximum in the negative value occurring at about x2 = 0.5. The functional behavior is approximately parabolic centered on x2 = 0.5. The data can be fit to – VE = m + c(x 2 – 0.5) 2 (15) – Regression of the VE experimental values as a function of the square of (x2 – 0.5) gives the constants: m = {0.5255 and c–= 2.079. These constants can be used to calculate values of VE
for any x 2; the solid curve shown on Figure 3 displays the values calculated from the regression equation. Once again a regression method using a parabolic term centered on x2 = 0.5 has been found very useful in the data analysis of a wellbehaved binary system. Literature Cited 1. Levine, I. N. Physical Chemistry, 3rd ed.; McGraw-Hill: New York, 1988; p 234. 2. Moore, W. J. Basic Physical Chemistry; Prentice Hall: Englewood Cliffs, NJ, 1982; p 179. 3. Barrow, G. M. Physical Chemistry, 6th ed.; McGraw-Hill: New York, 1996; p 282. 4. Reid, C. E. Chemical Thermodynamics; McGraw-Hill: New York, 1990; p 150. 5. Alberty, R. A.; Silbey, R. J. Physical Chemistry, 1st ed.; Wiley: New York, 1992, p 126. 6. Noggle, J. H. Physical Chemistry, 2nd ed.; Scott-Forseman: Glenview, IL, 1989; p 334. 7. Atkins, P. W. Physical Chemistry, 4th ed.; Freeman: New York, 1990; pp 182–183. 8. Salzberg, H. W.; Morrow, J. I.; Cohen, S. R.; Green, M. E. Physical Chemistry; Academic: New York, 1969; p 375. 9. Redlich, O.; Kister, A. T. Ind. Eng. Chem. 1948, 40, 345–348.
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