In the Laboratory edited by
The Microscale Laboratory
R. David Crouch Dickinson College Carlisle, PA 17013-2896
Rapid Microdetermination of Partial Molar and Excess Molar Volumes
W
Bruno Lunelli* and Francesco Scagnolari Università di Bologna, Dipartimento di Chimica “G.Ciamician”, 2 via F. Selmi, I-40126 Bologna, Italy; *
[email protected] Partial Molar Quantities and Volumes Partial molar quantities play a fundamental role in the thermodynamics of homogeneous phases. The most intuitive appear to be the partial molar volumes, which can be thought of as generalizations of the molar volumes. The indirect assignment of their values, however, leads to a certain sense of arbitrariness, which could be removed by a “hands-on” determination in the laboratory. The partial molar volume VB of component B of a mixture can be calculated by means of the expression V B = V m – Σ xD ( ∂ V m / ∂ xD )
(1)
D≠B
where Vm = V/Σ B nB = ΣxBVB is the molar volume of the mixture, V its total volume, nB the amounts (1, p 41) of the single components, and the xB’s are their mole fractions (2). The key quantity is Vm, whose experimental determination requires mass and volume measurements, which traditionally involve quantities of the order of several grams and milliliters (3). Turning to a Microscale Method Present concern with operator safety, environmental pollution, and cost favors the use of microtechniques, but this would complicate the procedure because it would no longer be possible to neglect surface effects. In particular, when emptying a container of wetting liquids, films whose thickness
depends on the nature of the solid surface and the composition of the mixture remain on all former solid–liquid interfaces. This surface effect can be neglected in macro measurements (4 ) because the volumes are proportional to the cube of the characteristic size of the system, whereas the areas of the interfaces are proportional to its square; the relative influence thus becomes important when the size of the sample decreases. The use of a smart micropychnometer (5) would then be impractical because it would be necessary to remove all traces of liquid by washing with a solvent and then drying after each measurement. It follows that a rapid microdetermination of partial molar volumes could be feasible only if it were readily possible to avoid this effect of interfacial tension. The use of gas-tight microsyringes offers such a possibility. Their specially designed piston tip, made of low-friction, hydrophobic, flexible Teflon, adheres to the smooth precision-made surface of the bore up to molecular sizes, and prevents gas-phase molecules from leaking through the sliding seal up to pressure differences of the order of 10 bar (1 bar = 14.50 psi). Such a feature also makes the syringe liquid-tight, and emptying by means of the piston removes the liquid films from all liquid–glass interfaces. Thus it becomes possible to fix the volume of a liquid and weigh it, avoiding surface effects by weighing the filled and empty syringe. Furthermore, all parts in contact with the sample consist of chemically resistant materials (borosilicate glass, AISI 304 stainless steel, and Teflon),1 allowing the method
Table 1. Features of the Water–Acetone Mixtures at 22 °C and Atmospheric Pressure Components
Run or Sample No.
Vexptl /µL vw
va
Mixtures
M calcd /mg a Mw
Mole Fractiona
Ma
xw
xa
ρ(i)/(g mL1) Exptl
Calcdb
Vm(i)/ V Em(i)/ (mL (mL mol1) mol1)
1
0
200
0.00
156.10
0.0000
1.0000
0.7809
0.7805
74.41
0.00
2
20
180
19.86
140.49
0.3131
0.6869
0.8206
0.8231
55.32
1.47
3
40
160
39.72
124.88
0.5063
0.4937
0.8551
0.8526
44.33
–1.59
4
60
140
59.58
109.27
0.6374
0.3626
0.8813
0.8784
37.05
–1.49
5
80
120
79.45
93.66
0.7322
0.2677
0.8991
0.9013
31.89
–1.31
6
100
100
99.31
78.05
0.8040
0.1960
0.9196
0.9216
28.07
–1.10
7
120
80
119.17
62.44
0.8602
0.1398
0.9398
0.9395
25.14
–0.87
8
140
60
139.03
46.83
0.9054
0.0946
0.9560
0.9554
22.82
–0.64
9
160
40
158.90
31.22
0.9426
0.0574
0.9677
0.9695
20.96
–0.41
10
180
20
178.76
15.61
0.9736
0.0264
0.9849
0.9820
19.42
–0.20
11
200
0
198.62
0.00
1.0000
0.0000
0.9925
0.9931
18.14
0.00
the densities ρw = 0.9931 and ρa = 0.7805 obtained from the density function. bFrom the density function ρ(x ) = 0.9931 – 0.4333x + 0.3799x2 – 0.1593x3. a aUsing
626
Journal of Chemical Education • Vol. 79 No. 5 May 2002 • JChemEd.chem.wisc.edu
In the Laboratory
Putting It into Practice The actual experiment involved the binary mixture water–acetone because it complies with the demands of safety, operative simplicity, and speed of completion. Operating at a constant laboratory temperature of 22 °C and using a 100-µL syringe with the needle tip cut at 90° (Hamilton 1710 with cemented needle #3) we prepared 11 samples of running index i (i = 1–11; see Table 1) of different composition by mixing 0 to 200 µL of distilled water (CAS registry number 7732-185, component w, molecular weight Mw, molar volume Vm,w, density ρw, volume vw(i), corresponding amount of substance nw(i)) and 99.5% commercial acetone (CAS registry number 67-64-1, component a, molecular weight Ma, molar volume Vm,a, density ρa, volume va(i), corresponding amount of substance na(i)) in steps of 20 µL to a total of about 200 µL in 1-mL containers. Volumetric measurements were preferred to gravimetric ones because they greatly reduce evaporation and atmospheric moisture pickup by the samples; the minimum volume measured was 20 µL to avoid large relative errors. The densities ρ(i) of the mixtures were determined as 1/100 of the difference of weight M(i) in milligrams of the syringe filled to 100 µL and then empty: ρ(i) = M(i)/100 µL. To obtain the partial molar volumes we need the molar volume of the mixture Vm against the molar fraction of one component, say acetone: Vm = Vm (xa). From this, one can directly calculate the partial molar volumes of the components and also the excess (molar) volume of the mixture (1, p 73) V Em. We first determined the density ρ of the mixtures as a function ρ = ρ(xa), obtained as a least-squares polynomial based on all measurements. We consider the function more reliable than any single measurement, being based on 11 different
0.0
VmE / (mL/mol)
to be used with aggressive chemicals. Taking into account safety, time, and ease of measurement as well as cost, we found that the most convenient microsyringe size for determining the density of mixtures is about 100 µL. The 100-µL syringe used had a stated volume precision of ±1% and there are 100 divisions, so that aligning the piston tip with a division, we may expect a volume error of ± 1⁄4 of a division2 or 0.25 mL in 100 mL (i.e., 0.25%). The weight of the mixtures in the filled syringe is of the order of 100 mg; and since (6, p 144) the maximum change of density on mixing with respect to the linear combination of the densities of the pure components is of the order of 1%, the maximum change of weight will be about 1 mg. Thus, to be able to evaluate weight changes with about the same errors as in the volumes, we must be able to measure weights with a sensitivity not lower than one hundredth of a milligram. Utilizing such a responsive balance entails many precautions (7); but use of the same syringe both to prepare the mixtures and to weigh 100 µL of each of them to determine the density ρ introduces a factor of relativity, allowing the assumption of some canceling of errors. To achieve this reproducibility requires the use of cots or gloves. Higher quality results can be obtained by preparing the mixtures either volumetrically, using screw-plunger syringes, or gravimetrically, by weighing the components with the highsensitivity balance used to find the density of the mixtures, and by using more sophisticated data handling (8) and measurement procedures (9).
-0.5
-1.0
-1.5
0.0
0.2
0.4
0.6
0.8
1.0
Xa Figure 1. Excess volume of the mixture water–acetone as a function of the molar fraction of acetone. Filled squares are data points. Thick line—4th degree, thin line—3rd degree, dashed line—2nd degree polynomial fit.
measurements. We thus obtained3 ρ(xa) = 0.9931 – 0.4333x + 0.3799x2 – 0.1593x3
(2)
The degree of the polynomial was taken as three (6, p 146) because it realized the best compromise between a high value of R 2, the coefficient of multiple determination, and the standard error of the polynomial coefficients. Actually, we have errors in both ρ(i) and xa(i): in the former from measurement of the desired 100 µL of the mixture and its weight M(i), and in the latter from the volumes vw(i) and va(i) of the pure components of the mixture. But we did not consider the details of processing data with errors in both abscissa and ordinate (8) suitable for an undergraduate laboratory class. From eq 2 we recalculated the mole fractions xw(i) and xa(i) (columns 6 and 7 of Table 1) using ρw = 0.9931 and ρa = 0.7805, values also obtained from this equation rather than from the direct measurements or literature values (at 22 °C ρw = 0.9977, ρa = 0.7876) (10). This procedure should minimize the consequences of systematic errors in the volume measurements. The values of the molar volume Vm(i) Vm i =
v w i ρw + v a i ρa V i = n w i + n a i ρ i v w i ρw Mw1 + v a i ρaM a1
(3)
and of the excess molar volume V Em(i) V Em(i) = Vm(i) – [xw(i)Vm,w + xa(i)Vm,a ]
(4)
are listed in Table 1. By fitting a polynomial equation to the calculated excess volumes of the mixture V Em(xa), we found that R 2 is .99771 for the second degree, .99813 for the third degree, and .99997 for the fourth degree. The first equation would imply that the mixture of water and acetone is a regular mixture (11)— an implausible occurrence, because both species have an appreciable dipole moment and are capable of hydrogen bonding. The modest increment of R 2 by adding a third degree and its jump on addition of the fourth degree term might indicate
JChemEd.chem.wisc.edu • Vol. 79 No. 5 May 2002 • Journal of Chemical Education
627
In the Laboratory
that the suitable Margules expression for G Em requires B0, B1, and B 2. This result, however, may be an artifact due to the limited accuracy of our experiment. The result shown in Figure 1 compares reasonably well with that of the literature (12), but shows the need of some additional points on the high value side of the xa axis: the large differences of the xa there are due to the large difference in molecular weight of water and acetone. Students’ measurements could be readily accomplished in a single 4-hour laboratory session. Their results, not shown here, closely resembled ours and in a few cases appeared even better. Hazards This procedure is intrinsically safe, but it should be remembered that acetone is a powerful solvent, which should not contact eyes, mucous membranes, skin, or plastics, including contact lenses. The syringe should be discharged slowly, to avoid the loss of contents or dangerous spurts. If components different from those indicated above are chosen, the instructor should be alert to toxic, aggressive, or otherwise hazardous effects, including strongly exothermic mixing (which would occur, for instance, in the case of water–sulfuric acid mixtures). W
Supplemental Material
Instructions for students are available in this issue of JCE Online. Notes 1. We refer to the Hamilton specifications. 2. Such error can be further reduced by using a viewing lens or a cathetometer, but the larger relative error in weighing would make the effort useless.
628
3. We used Costat by Cohort Software, Minneapolis, MN 55419, USA.
Literature Cited 1. Mills, I.; Cvitas, T.; Homann, K.; Kallay, N.; Kuchitsu, K. Quantities, Units, and Symbols in Physical Chemistry, 2nd ed.; Blackwell: Oxford 1993. 2. van Ness, H. C. Classical Thermodynamics of Non-Electrolyte Solutions; Pergamon: Oxford, 1964; p 87. 3. Coch, J. A.; Lopez, V. J. Chem. Educ. 1970, 47, 270. 4. McGlashan, M. L. Chemical Thermodynamics; Academic: London, 1979; p 35. 5. McCullough, T. J. Chem. Educ. 1973, 50, 546. 6. Bauer, N.; Lewin, S. Z. In Physical Methods of Organic Chemistry, 3rd ed.; Weissberger, A., Ed.; Interscience: New York, 1959; Part 1. 7. Battino, R. J. Chem. Educ. 1984, 61, 51. Goodwin, B. L. Chem. Br. 1996, 32 (10), 22. Kowalski, T. Chem. Br. 1996, 32 (10), 22. 8. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge, 1988; p 539. Reed, B. C. Am. J. Phys. 1992, 60, 59. Macdonald, J. R. Am. J. Phys. 1992, 60, 66. Bechhoefer, J. Am. J. Phys. 2000, 68, 424. 9. Loucks, L. F. J. Chem. Educ.1999, 76, 425. 10. CRC Handbook of Chemistry and Physics, 78th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1997–1998; p 6-5. 11. Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions; van Nostrand-Reinhold: New York, 1970. 12. Fort, R. J.; Moore, W. R. Trans. Faraday Soc. 1965, 61, 2102.
Journal of Chemical Education • Vol. 79 No. 5 May 2002 • JChemEd.chem.wisc.edu
