In the Laboratory
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The Measurement of Activity Coefficients in Concentrated Electrolyte Solutions An Experiment for the Physical Chemistry Laboratory Judith M. Bonicamp,* Ashley Loflin, and Roy W. Clark Department of Chemistry, Middle Tennessee State University, Murfreesboro, TN 37132; *
[email protected] 1.0
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γ±
In the first year of college chemistry students learn about ideal behavior: the ideal gas laws, Henry’s law, Raoult’s law, and the colligative property and equilibrium relationships. In quantitative analysis students begin to appreciate the approximate nature of these relationships and consider nonideal behavior as they are introduced to the concept of activities. If students progress to using the simple or advanced Debye–Hückel (D–H) theory, they eventually become aware of the limitations of the theory, which uses electrical interaction models to predict activity coefficients. Each electrolyte is said to have a mean ionic activity coefficient, γ±, which is a function of concentration and which when multiplied by the concentration gives an “activity”. This activity can be used in the ideal solution relationships to make realistic predictions under circumstances where the assumption of ideal behavior would fail. In the physical chemistry sequence, students should wonder where tables of γ± come from, particularly those having concentrations exceeding the D–H limits. What are the limits of the D–H theory? Ira Levine once characterized these limits as zero to “slightly contaminated distilled water” (1). Yet there are various tables of γ± for very high concentrations (2). There must be experimental techniques that yield activity coefficients for electrolytes. Indeed there are. One can measure activity coefficients by measuring deviations from the colligative property relationships (osmotic coefficients), by measuring deviations from the electrical potential laws (junction potentials, emf of cells without transference), or by applying any equilibrium technique that involves only one electrolyte in solution and its departures from ideal theory. Many tables of γ± were published in the years 1930–1950. In order that the reader might appreciate the work that went into these measurements (temperature measurements to ±0.0007 degrees), we suggest reading Electrolyte Solutions, R. A. Robinson and R. H. Stokes’s classic work (3), in which the authors summarize the experiments of Scatchard and coworkers in the early 1930s on freezingpoint depression (4). We present here an experiment for a 3-hour session in a computer-equipped laboratory in which there is available a vapor pressure osmometer.1 We used the Vapro Vapor Pressure Osmometer (Wescor, Inc., Logan, UT 84321), which is actually a dew-point osmometer. Its basis is a measurement of vapor pressure depression made possible by thermocouple hygrometry. A thermocouple hygrometer is incorporated within the sample chamber. This sensitive temperature sensor operates by a thermal energy balancing principle to measure the dew-point temperature depression within the chamber. In the experiment students learn (i) the effect of nonideality on the colligative property relationship; (ii) the formulation of this as the practical osmotic coefficient, φ; and (iii) the method by which γ± for a salt is calculated from measure-
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m Figure 1. Student plot of the mean ionic activity coefficient, γ± vs molality for KNO3 solutions. The solid line represents smoothed student experimental data; circles represent data from ref 2.
ments of the practical osmotic coefficient. Spreadsheets are used to curve fit the data using Excel’s Solver (5, 6 ) or similar software, and for computations, which convert the experimentally determined practical osmotic coefficient to the mean ionic activity coefficient. These computations include integration (trapezoidal method) and a choice of plotting functions to avoid indeterminate intercepts. By no means will this experiment reproduce the painstaking work of Scatchard and coworkers previously cited, but it will provide experimental data that, when properly treated, yield mean activity coefficients for simple 1:1 electrolytes. A plot of student-determined γ± versus molality of KNO3 is in remarkably good agreement with literature values despite the simplicity of the data collection method for the osmotic coefficients (Fig. 1). Although the students collect osmotic coefficient data only in the region of 0.1 to 1 m salt solutions, the results in Figure 1 include a complete graph of experimental γ± calculated from 0 to 1 m. Theory The practical osmotic coefficient φ is defined by eq 1 for a 1:1 electrolyte in water where mOsm is the osmolality in mmol/kg and m is the molality of the solution.
φ = mOsm 2000 m
(1)
Osmolality is an expression of the concentration that gives the total number of particles (ions and molecules) in millimoles per kilogram of solvent. Then φ is the fraction of the “ideal” osmolality, which, for 1:1 electrolytes, would
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m
ln γ± = φ – 1 + 0
φ–1 dm m
m
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(2)
This equation indicates that φ values are not sufficient and one must have φ values from the desired m down to m equals zero to do the integration. For convenience in doing the integration, we used the form
ln γ± = φ – 1 –
25
1–φ m
presumably be 2 times m. If m is expressed in the usual mol/kg, 1000 is necessary as a conversion factor for mmol to mol. The students may jump to the conclusion that φ is the activity coefficient. It is not, at least not the activity coefficient for the solute. If φ is plotted versus m, the shape of the plot is reminiscent of similar γ± plots, but the values are wrong. According to thermodynamic arguments (7, 8) the relationship between the two is
1–φ dm m
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1–φ m
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m Figure 3. As suggested by Randall and White (10), a plot of (1 – φ)/√m vs √m has a definite intercept. The sample graph results from student data for KNO3 solutions.
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1–φ d m m
This function has a finite y intercept of 0.39 (Fig. 3). Because of the finite intercept, this integral is easily evaluated for each value of m. Twice this integral is the desired integral term in eq 2 (see derivation onlineW).
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φ
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Figure 2. Integration of (1 – φ)/m from student data for KNO3 solutions. Below m = 0.002 the area of the trapezoid must be estimated because of the indefinite intercept.
This makes the integration values positive. One method of calculating results is to use Excel’s Solver to do curve fitting on the students’ data using the equation φ = 1 – 0.39 m – C 2m (3) 1 + C1 m where C1 and C2 are adjustable constants. This is similar to the equation Scatchard et al. used, which they found fit φ data well (4 ). Actually, Scatchard et al. used 0.3738 as the first constant, since the temperature was below 0 °C. We use 0.39 because the experiment is carried out near 25 °C (9). This constant is the D–H theory constant divided by 3 (see derivation onlineW). The points of φ versus m to be fitted to the curve range from 0.1 to 1 m, and the equation is forced through the point φ = 1 at 0.0 m. Once φ versus m has been plotted and an equation has been fit to the data, this equation can be used to calculate φ values for example from 0.002 m to 1 m, in steps of 0.002. Numerical integration of the function (1 – φ)/m from m = 0.002 to each m value is easily accomplished in Excel, but the integration from 0.0 m to 0.002 m is not possible because of the indefinite intercept (Fig. 2). One can estimate this segment from D–H theory and add it in, but we believe it is better to use Randall and White’s ingenious method (10) of changing the variable in eq 2 to m1/2. Then one evaluates the integral
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Experimental Procedure The student prepares six or more solutions from 0.1 to 1.0 m of the chosen 1:1 electrolyte. These must be made as molal, not molar, solutions. Solutions of KCl, KNO3, NaCl,
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m Figure 4. Student φ data for KNO3 solutions fitted to eq 3.
Journal of Chemical Education • Vol. 78 No. 11 November 2001 • JChemEd.chem.wisc.edu
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and NaNO3 give good results. The details are online.W Very small samples of these solutions, about 10 µL depending on the instrument, are introduced into the osmometer, and the osmolality is measured and recorded in mmol/kg. Three readings are sufficient if the demands on osmometer time require less precision. Once the osmolality figures are obtained, the remainder of the experiment involves using three spreadsheet steps: curve fitting the φ versus m data, integrating the proper function, and calculating γ± using eq 2. Excel spreadsheets serve as examples in the supplemental material, but almost any spreadsheet or graphing package will do as well provided it has a regression add-in tool such as Solver, which is provided with Excel.2 Hazards The experiment approximates a safety level acceptable even in a freshman laboratory. Preparing solutions of KCl, KNO3, NaCl, and NaNO3 poses little risk. Results and Discussion Figure 4 presents a set of student data, which illustrates what can be expected of the experiment. These data were obtained using a vapor pressure osmometer and studentprepared KNO3 solutions. The data for φ are plotted versus m and the curve fit performed. Notice that the fit is forced through the point φ = 1 at 0.0 m, even though that is not an observed datum. After the mathematical manipulations, the results of the student’s determination of γ± are as shown in the plot in Figure 1 and compared with values from ref 2. Solutions of KCl, KNO3, NaCl, and NaNO3 give γ± results that are within about 5% of literature values. Acknowledgments We thank the Faculty Research and Creative Achievement Committee for funds to purchase the osmometer and the Undergraduate Research Council for Ashley Loflin’s grant.
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Supplemental Material
Supplemental material for this article is available in this issue of JCE Online. It includes instructor notes, a discussion of instruments, an example spreadsheet for the instructor and a skeleton spreadsheet for the students, student materials and procedures, and other information needed to implement the experiment. Notes 1. Vapro Model 5520 Vapor Pressure Osmometer, Wescor, Inc., Logan, UT 84321, or similar osmometer. 2. Quattro Pro comes with a regression add-in tool called “Optimizer”. A regression add-in tool is available for Lotus 123 at additional cost. Nonlinear regression can also be done with other programs independently and the results imported into your spreadsheet.
Literature Cited 1. Levine, I. N. Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1995; p 277. 2. CRC Handbook of Chemistry and Physics, 71st ed.; Lide, D. R., Ed.; CRC Press: Boston, 1990; Section 5, pp 99–100. 3. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd rev. ed.; Butterworths: London, 1968; pp 183–189. 4. Scatchard, G.; Jones, P. T.; Prentiss, S. S. J. Am. Chem. Soc. 1932, 54, 2676–2705. 5. Harris, D. C. J. Chem. Educ. 1998, 75, 119–121. 6. Clark, R. W. Chem. Educator 1999, 4 (3), S 1430–4171 (99) 03296-7; http://link.springer.de/link/service/journals/00897/ tocs.htm (accessed Aug 2001). 7. Lewis, G. N.; Randall, M. Thermodynamics; revised by Pitzer, K. S.; Brewer, L.; McGraw-Hill: New York, 1948; pp 322–323. 8. Rysselberghe, P. V. R.; Hunt, G. J. J. Chem. Educ. 1948, 25, 87–89. 9. Condon, F. E. Study Projects in Physical Chemistry; Academic: New York, 1963; pp 173–177. 10. Randall, M.; White, A. M. J. Am. Chem. Soc. 1926, 48, 2514– 2517.
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