In the Laboratory
Partial Molar Volumes from Refractive Index Measurements Anthony F. Fucaloro Joint Science Department, Claremont McKenna, Pitzer, and Scripps Colleges, 925 N. Mills Avenue, Claremont, CA 91711;
[email protected] Background The use of measurements of index of refraction as a quick, convenient, and accurate way to estimate densities of liquid mixtures has been reported (1–3). The method is based – upon the assumption that the molar refraction (R) of a mixture is a linear function of mole fraction. Thus, for a binary mixture: –
–
–
–
–
–
(1)
–
where V is the molar volume of the mixture, f is equal to (n2 – 1)/(n2 + 2) and is a function of the mixture’s index of refraction (n), xi is the mole fraction of the i th component, – and R i is the molar refraction of pure component i. Equation 1 is based upon the Lorentz–Lorenz equation, which relates molecular polarizability α to density ρ and index of refraction n (1, 4 ). Thus, for a pure substance i,
R i = V i f i =
M i n i 2 – 1 ∝ αi ρi n 2 + 2 i
(2)
where
V i =
Mi ρi
(3)
and
f i =
n i 2 – 1
(4)
n i 2 + 2
Mi is the molar mass and ni is the index of refraction. Assuming that the molecular polarizability of each component is relatively invariant as a function of composition implies eq 1. Thus, by knowing the densities and refractive indexes of the pure components, one may estimate the density of a mixture of known composition by measuring its index of refraction and using eq 1. Molar refraction is not strictly linear with respect to mole fraction, but is considerably more linear than molar volume and therefore may be used to yield better estimates of the density of mixtures than estimates based upon the assumption that molar volume is linear with respect to mole fraction. The latter is true only for ideal solutions, as indicated in eq 5. –
–
–
–
–
–
Videal = x1V1 + x2V2 = (V1 – V2) x1 + V2
(5) –
E
Figure 1 shows the ratio of excess molar refraction (R ) and – excess molar volume (V E) to their respective ideal quantities versus– mole fraction for mixtures of methanol and water. R E and V E are defined as follows: –
–
–
–
R E = R – (x1R1 + x2R 2) –
E
–
V =V –
– (x1V1
+
– x2V2)
(6) (7)
The data used here as well as for the remainder of this report came from the CRC Handbook of Chemistry and Physics (5).
0.00
Excess Molar Quantity Ideal Molar Quantity
–
-0.01
-0.02
-0.03
-0.04 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mole Fraction of Methanol Figure 1. The ratios of (ⵧ) excess molar volume to ideal molar volume – – E/ V (V ) and (䉬 ) excess molar refraction to ideal molar refraction ideal – – (R E/Rideal) for water-plus-methanol mixtures as a function of mole fraction of methanol.
1.0
Density / (g/mL)
–
R = R ideal = V f = x1 R1 + x2R 2 = (R1 – R 2)x1 + R 2
The Handbook reports all measurements at 20 °C. The measurements of index of refraction were made using Na yellow light and the densities were reported relative to the density of water at 4 °C. All points reported in all the graphs are at the mole fractions reported in the Handbook. Figure 1 is fairly – – typical with regard to the relative linearity of R and V for many (perhaps all) aqueous alcohol and polyalcohol mixtures, as well as for other binary mixtures. Figure 2 shows the actual and calculated densities for the aqueous methanol system as a function of mole fraction of methanol. Experience suggests that deviations of measured densities from those estimated using molar refractions are some five to ten times smaller than those estimated assuming linear dependence of molar volume on mole fraction. Are these calculated values for molar volume sufficiently good to allow for reliable estimates of partial molar volumes
0.9
0.8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mole Fraction of Methanol Figure 2. Densities for water-plus-methanol mixtures as a function of mole –fraction of methanol. (䉬 ) Actual– density; (䉭) density calculated from R; (ⵧ) density calculated from Videal.
JChemEd.chem.wisc.edu • Vol. 79 No. 7 July 2002 • Journal of Chemical Education
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In the Laboratory
1. Systems previously investigated may be used in teaching laboratories to introduce students to the use and significance of partial molar volumes without having them engage in tedious, time-consuming density measurements. While it is true that measuring densities using a densimeter is not time-consuming, densimeters are not generally available in undergraduate laboratories or in the research laboratories of nonspecialists. While the development of the equations used here to compute partial molar volumes is not treated in undergraduate physical chemistry texts (see Methods) and requires some additional preparation by students, this is more than offset by the considerable body of data obtained in one or two laboratory sessions. This allows students to consider more fully molecular packing and intermolecular interactions.
0
Excess Partial Molar Volume / (mL/mol)
for the components as a function of composition—good enough, that is, for the purposes listed below?
-1 -2 -3 -4 -5 -6 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mole Fraction of Ethanol Figure 3. Excess partial molar volumes of water-plus-ethanol mixtures. Partial molar volumes for (䉬 ) water and (ⵧ) ethanol determined directly from the densities and for (䉭) water and () ethanol estimated from refractive index measurements .
Methods Molar volumes of binary mixtures may be estimated from the refractive indexes and densities of the pure components and the measured refractive indexes of the mixtures as described – above. The computed values of V E (see eq 7) may then be fit to a general polynomial in mole fraction or to the Redlich– Kister equation (6, 7 ) k
V E = x 1x 2 Σ A j 1 – 2x 1
Excess Partial Molar Volume / (mL/mol)
2. Workers may use this approach as a convenient way to survey systems to determine whether extensive density measurements are warranted. 0.05
0.00
-0.05
-0.10 0.00
j
0.02
Experience in this laboratory indicates that there is usually a negligible difference between these two related methods for – expressing V E as a function of mole fraction. Thus, for the former method: –
V E = a0 + a1xi + a 2 x i 2 + … + a n x i n
–
V = m1V1 + m2V2 where m1 and m2 are the number of moles, and
866
0.06
0.08
0.10
0.12
0.14
(9) – V1
Figure 4. Excess partial molar volumes of water for water-plus-ethanol mixtures over a limited range of compositions. Partial molar volumes (䉬 ) determined directly from densities and (ⵧ) estimated from the refractive index measurements.
(8)
where xi is the mole fraction of the ith component and {a i} is the set of coefficients determined using standard polynomial fitting algorithms. The regression analysis routine of Mathcad 8 was used where standard statistical tests may be applied for error analyses. The Redlich–Kister equation ensures that the excess function goes to zero for the pure components and is compatible with the Gibbs–Duhem equation. For the more general polynomial to yield zero values for the pure components, a0 must equal zero and the sum of the coefficients, a i , must also equal zero. This happens in all cases studied. Moreover, the general polynomial is also Gibbs–Duhem compatible. What follows is a brief development of a convenient mathematical procedure that may be used to compute partial molar volumes from molar volumes, either determined directly from measurements of density or estimated from measurements of refractive index. It is particularly well suited for spreadsheets. It can be shown that the total volume, V, of a binary mixture may be represented as –
0.04
Mole Fraction of Ethanol
j=0
and
– V2
the partial molar volumes for components 1 and 2, respectively (8). The partial molar volume for component 1 is defined as
V 1 = ∂V ∂m 1
T,P, m2
= ∂V ∂m 1
(10) m2
Partial molar quantities may be calculated from the molar quantities as a function of mole fraction (9), as shown here for molar volumes.
V 1 = V – x 2 ∂V ∂x 2
(11)
It then follows that
V 1E = V 1 – V 1 = V E – x 2
V 1E = V E + x 2
∂V E ∂x 1
are
Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu
∂V E ∂x 2
(12)
(13)
In the Laboratory 1
Excess Molar and Partial Molar Volume / (mL/mol)
Excess Partial Molar Volume / (mL/mol)
0 -1 -2 -3 -4 -5 -6 -7 -8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 -1 -2 -3 -4 -5
1.0
0.0
Mole Fraction of Acetic Acid
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mole Fraction of Methanol
Figure 5. Excess partial molar volumes of water-plus-acetic acid mixtures. Partial molar volumes for ( 䉬) water and (ⵧ) acetic acid determined directly from the densities and for (䉭) water and () acetic acid estimated from refractive index measurements .
Figure 7. Excess molar volumes and partial molar volumes for water plus methanol. The (䉬) molar volume and partial molar volumes for (ⵧ) water and (䉭) methanol were determined directly from the densities.
1
Excess Molar and Partial Molar Volume / (mL/mol)
Excess Partial Molar Volume / (mL/mol)
0 -2 -4 -6 -8 -10 -12 0.00
0.01
0.02
0.03
0.07
0.08
0.09
0.07
Mole Fraction of Silver Nitrate
0 -1 -2 -3 -4 -5 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mole Fraction of Methanol
Figure 6. Excess partial molar volumes of water + silver nitrate mixtures. Partial molar volumes for (䉬) water and (ⵧ) silver nitrate determined directly from the densities and for (䉭) water and () silver nitrate estimated from refractive index measurements .
Figure 8. Excess molar volumes and partial molar volumes for water plus methanol. The (䉬) molar volume and partial molar volumes for (ⵧ) water and (䉭) methanol were estimated from the refractive index measurements.
since
Results and Discussion dx1 = ᎑ dx2 –
E
Thus, by fittingV to a polynomial (see eqs 7, 8) and taking the analytic derivative of the polynomial, one may compute the partial molar volume as a function of mole fraction. This procedure is well suited for use on standard spreadsheets. Students are asked to make binary mixtures and measure their refractive indexes using a thermostatted refractometer. The spreadsheet includes, in part, a column each for mass fractions, mole fractions, computed ideal molar volumes (eq 5), estimated molar volumes (eq 1), and estimated excess – – molar volumes ( V – Videal). The columns containing mole fractions and excess molar volumes are transported to a mathematics program such as Mathcad, where the data are fit to a polynomial (see eq 8). The coefficients of the polynomial are used to create another spreadsheet column for the fitted excess molar volumes. The students take the derivative of the polynomial in order to create a new column – (∂V E/∂x1), from which the excess partial molar volumes may be calculated for each mole fraction (eq 13).
Figure 3 shows the excess partial molar volumes for water–ethanol mixtures determined directly from measurements of density and estimated from measurements of refractive index. Figure 4 shows the results for the excess partial molar volumes of water for these mixtures over a limited range of mole fractions. From these, one may get a sense of the extent to which this method estimates partial molar volumes, including its capacity to capture the structural profile of the partial molar volumes as a function of composition. Results for water plus methanol, 1-propanol, or 2-propanol are quite similar. Figures 5 and 6 show the same results as Figure 3 for aqueous mixtures of acetic acid and silver nitrate, respectively. The density and refractive index of solid silver nitrate were used to calculate the molar refraction of pure silver nitrate. The excess molar and partial molar volumes for aqueous methanol mixtures determined directly from the densities and estimated from the measurements of refractive index are shown in Figures 7 and 8, respectively. In each case, it may
JChemEd.chem.wisc.edu • Vol. 79 No. 7 July 2002 • Journal of Chemical Education
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In the Laboratory –
–
–
clearly be seen that V1E and V2E intersect V E at its– minimum, a necessary mathematical condition. Setting (∂V E/∂x1)m2 to zero (an extremum) in eq 13 yields –
–
–
–
V 1E = V E
interactions. Moreover, this method may be expanded to analyze systems more complex than binary systems. It could prove invaluable for surveys of such systems that would avoid time-consuming density measurements until and unless the results of a study reveal information that warrants them.
In the same way, V 2E = V E –
at an extremum of V E. This requirement is seen for all systems studied with the exception of aqueous silver nitrate where no extremum is observed because of the limited range of available data. Conclusion Clearly, one may exploit this technique to introduce advanced undergraduates to the concept of partial molar quantities by allowing them to accumulate data quickly and to treat those data systematically to estimate partial molar volumes as a function of composition. Systems selected for study may be those for which partial molar volumes are known and those for which they are not. For the former, students may be asked to analyze the effectiveness of the method; for the latter, they may be asked to rationalize their results on the basis of their knowledge of molecular structure and intermolecular
868
Literature Cited 1. Fajans, K. In Physical Methods of Organic Chemistry, 2nd ed., Vol. I Part II; Weisburger, A., Ed.; Interscience: New York, 1949; pp 1159, 1169. 2. Zamvil, S.; Pludow, R.; Fucaloro, A. F. J. Appl. Crystallogr. 1978, 11, 163. 3. Fucaloro, A. F.; Billington, C.; Varnavas, M. J. Chem. Educ. 1978, 55, 793. 4. Lorentz, H. A. The Theory of Electrons, 2nd ed.; Dover: New York, 1952; p 45. 5. CRC Handbook of Chemistry and Physics, 55th ed.; Weast, R. C., Ed.; CRC: Cleveland, OH, 1978; pp D194–D236. 6. Redlich, O.; Kister, A. T. Ind. Eng. Chem. 1948, 40, 345. 7. Loucks, L. F. J. Chem. Educ. 1999, 76, 425. 8. See for example, Atkins, P. Physical Chemistry, 6th ed.; Freeman: New York, 1997; pp 164–165. 9. Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworths Scientific: London, 1982; pp 88–90.
Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu
