Nathaly Murillo Kevin Brew 04/20/08
Experiment 2: Partial Molal Volume Abstract
Densities of a small range of concentrations of aqueous potassium chloride and aqueous sodium chloride were recorded with a density meter so that the partial molal volumes, and ultimately, the partial molal volumes at infinite dilution, could be calculated. For potassium chloride and sodium chloride, the partial molal volumes at infinite dilution of the salts were calculated to be 25.18 mL/mol and 15.19 mL/mol, respectively. These differ from literature values by 6.23% and 26.85%, respectively. Error sources include inadequate mixing of the solutions, evaporation and the small range of the solutions.
Introduction:
Amagat’s law states that volumes are approximately additive. However, this does not apply to solutions whose concentrations are to be known to a high degree of accuracy. Preparation of a solution with accurate molality is generally done by adding an amount of water to a measured amount of salt and obtaining the weight of water by difference. In 1770 Millero reported that volume decreases when salts are added to a specific volume of water. This effect was explained as electrostriction: the volume contracts due to
interaction of the polar solvent around the ions. However, this phenomenon occurs in non-ionic solutions well, reflecting differences in intermolecular forces. Thermodynamics explains this deviation from ideal behavior through partial molal quantities. The most important partial molal quantity is chemical potential:
(1)
For this experiment, partial molal volume will be measured:
(2)
In high pressure systems, partial molal volume is related thermodynamically to chemical potential by the following:
(3)
The partial molal volume considers the change in molal volume with the increase in moles of material:
Since partial molal volumes are functions of concentration but not the total number of moles, equation 4 can be expressed as:
where V is total volume. Taking component 1 to be water and component 2 to be the salt, the volume of solution can be determined with static amounts of solvent (water) and varying amounts of salt. Since molality is the concentration of solute per kg of solvent, it is intuitive to take the amount of water fixed at 1000 g. With the molality of the solution and the molecular weight of the salt used and the measured density of the solution, the volume can be calculated:
The graph of experimental data for volume as a function of molality can be fit with a power series, yielding a fit equation whose derivative with respect to molality yields the partial molal volume as a function of molality or amount of salt added:
Replacing equation 7 into equation 5, taking n1 = 55.508 mol of water
(1000g/18.015g/mol), n2 = m, and rearranging, the partial molal volume of solvent can be
expressed as:
Since both partial molal volumes are functions of concentration, they can be expressed at infinite dilution for a single value. At infinite dilution for the partial molal volume of water, the effects of solvated ions on the solvent are null. The partial molal volume of salt at infinite dilution reflects the effects of electrostriction on water due to the solvated ions. The values of partial molal volumes at infinite dilution depend on the equation used to fit the data and how well is extrapolates to m = 0. Thus, it is imperative that density be measured accurately because slight deviations can result in poor results.
Procedure:
Five solutions of KCl with varying molalities between 0.05 m and 2.00 m were prepared by weighing salt by difference in a jar with lid. 20 mL of distilled water was added to the jar and the mass was recorded. This was used to calculate the molality of the
solution. The DMA 4500 was turned on and its temperature was adjusted to 25.00°. Distilled Water was injected and then the air line was reconnected and the pump was turned on. The density was then taken. Once the density read that of air (between 0.0011-0.0014 g/mL), a syringe of distilled water was put into the injection port and distilled water was injected. The density for water was recorded at least 3 times for different portions until consistency (within 0.0001g/mL). Then the syringe was rinsed twice with small portions of the KCl solution and was then filled with the solution. The solution was injected partially and density was recorded. This was repeated until 3 consistent values of density were reported for the solution, again using different portions. The syringe was rinsed with another solution of KCl and the density was measured as before. This was repeated for the remaining KCl solutions. Then the entire procedure was repeated using NaCl instead of KCl.
Analysis and Results
Weights, molalities, and densities for water, sodium chloride and potassium chloride were recorded in Table 1. It must be noted that instead of using 0.5 to 2.0 molal solutions as the procedure indicated, 0.01 to 0.5 molal solutions for sodium chloride and 0.06 to 0.5 molal solutions for potassium chloride were used. With the data obtained, Figure 1, which shows the relationship between density and molality for each salt, was produced. The graphs indicate a quadratic relationship between density and molality; as molality increases, density increases as well. R-squared values of 0.99872 for sodium chloride and 0.99346 for potassium chloride indicate that the data obtained is precise. Table 2 contains the calculated volume as a function of molality, V{m}, partial molal volume of water, V1, the partial molal volume of the salts, V2, and the apparent
partial molal volume, φ. The volume as a function of molality was calculated using equation 6, the partial molal volume of water using equation 8, the partial molal volume of the salts using equation 7 and the apparent molal volume using equation 11. It is to be noted that the partial molal volume of water is somewhat constant across different molalities but the partial molal volume of the salts decreases greatly with increasing molality.
Figure 2 represents the relationship between the partial molal volume of the salt and molality; both graphs show a quadratic relationship. As molality increases, partial molal volume of the salt increases as well. R-squared values for figure 2 are not as high as those for figure 1 but still show about 90% reliability.
Table 3 is a summary of the values for an infinite dilution using three different methods of calculation. By taking the derivative of the fit equation for volume versus molality in the form V = A + B*m+C*m2. An expression for the partial molal volume is
obtained. This is V2 = B + 2*C*m. The infinite dilution can be found as a limit of
molality approaching 0. This results in the infinite dilution of V2 being equal to the fit
parameter B. A second method to find V2 at infinite dilution is to take the limit of m 0
again, but use the fit equation obtained in figure 3. A third method is to do the same but use figure 4. This data shows that method 2 is the most reliable with only a 8.6% deviation from the literature value for NaCl and a 6.2% deviation for KCl.
In figure 3, φ is plotted against m1/2 for both salts. It was found that there is a
linear relationship between φ and m1/2 for NaCl but a quadratic relationship for KCl. This
could be due to the small range of molalities used. The values for R-squared are not as desirable as those in previous graphs, values of 0.5847 for NaCl and 0.96381 for KCl
were acquired. The quadratic relationship for KCl, although more accurate, does not fit the mason equation (12) which is clearly linear.
φ =φº + am1/2 + bm (12)
If the salt solutions followed the Debye-Huckel theory, the equation for φ{m} would provide a single slope of 1.868 for all 1,1-electrolites at 25ºC. This slope changes depending on charge and temperature. The relationship between φ-1.86m1/2 and m1/2 is
shown in figure 4. φº is the intercept at m=0. The value φº for NaCl was found to be 14.24 and 25.18 for KCl. This means a deviation from the literature value of 14.3% and 6.2% respectively. The graph for NaCl is linear whereas KCl is quadratic. Once again, KCl does not fit the equation (13) provided.
φ =φº + 1.868m1/2 + bm (13)
Table 4 presents information on the differences between the partial molal values of KCl and NaCl, and between KBr and NaBr at an infinite solution. It is noted that the
difference between the partial molal volumes and the apparent molal volumes of KCl and NaCl decreases with decreasing molality. We determined that since both KCl and KBr, and NaCl and KBr are 1,1 electrolytes the difference between them would be equal. The literature indicates a difference of 6.9 between the partial molal volumes of ions of Cl and Br. The reason for the disparity between the literature value and the experimental values may be due to the low molality solutions used.
Data and Figures
Table 1: Salt Solutions Molalities and Densities
Salt wt.(g)Salt wt(g)H2O Molality(m) m2 Density
Water 0.99808 0.99708 0.99708 NaCl 0.0151 19.8885 0.01299122 0.00016877 0.99765 0.99765 0.99765 0.1568 19.5541 0.13720918 0.01882636 1.00278 1.00267 1.00267 1.00269 1.00271 0.2995 19.7263 0.25979221 0.06749199 1.00758 1.00757 1.00758 1.00757 0.4416 19.6673 0.38420167 0.14761093 1.01260 1.01259 1.01258 1.01259 0.5849 19.20197 0.52120763 0.27165739 1.01692 1.01693 1.01694 1.01694 Water 0.99609 0.99589 0.99668 0.99709 0.9971 0.9971 KCl 0.0919 19.8768 0.0620102 0.00384527 0.99766 0.99769 0.99764 0.2013 19.9749 0.13516158 0.01826865 1.00342 1.00345 1.00343 0.3813 19.9664 0.25613041 0.06560279 1.00891 1.00894 1.00895 1.00895 0.5641 19.6936 0.38417145 0.1475877 1.01475 1.01471 1.01474 0.7457 19.5996 0.51028292 0.26038866 1.0202 1.02022 1.02023 1.02022
Figure 1: Density vs Molality for NaCl and KCl Solutions 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.995 1.000 1.005 1.010 1.015 1.020 NaCl Data Polynomial Fit Data: Data1_B Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting Chi^2/DoF = 7.3015E-8 R^2 = 0.99872 A 0.99719 ±0.00011 B 0.04246 ±0.00107 C -0.00829 ±0.00202 D en sity ( g /mL)
Molality (mol NaCl/kg H2O) Density versus Molality for NaCl Solutions
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.995 1.000 1.005 1.010 1.015 1.020
Density versus Molality for KCl Solutions
KCl Data Polynomial Fit Data: Data1_D Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting Chi^2/DoF = 5.9627E-7 R^2 = 0.99346 A 0.99619 ±0.00028 B 0.05073 ±0.00314 C -0.00683 ±0.00612 D en sity (g /m L ) Molality (mol KCl/kg H2O)
Table 2:Volumes as a function of molality, V{m}, partial molal volumes of water, V1, the partial molal volumes of the salts, V2, and the apparent partial molal volumes, φ for KCl and NaCl Salt m1/2 m m2 d(g/ml) V{m} V1 V2 (ml/mol)φ Water 0 0 0 0.99808 998.08 17.98083 34.13482 0 0 0 0.99708 997.08 17.96282 34.13482 0 0 0 0.99708 997.08 17.96282 34.13482 NaCl 0.113979 0.012991 0.000169 0.99765 1003.117 18.06369 33.64091 11.66608 0.113979 0.012991 0.000169 0.99765 1003.117 18.06369 33.64091 11.66608 0.113979 0.012991 0.000169 0.99765 1003.117 18.06369 33.64091 11.66608 0.370418 0.137209 0.018826 1.00278 1005.224 18.03806 28.91832 16.46577 0.370418 0.137209 0.018826 1.00267 1005.335 18.04004 28.91832 17.26951 0.370418 0.137209 0.018826 1.00267 1005.335 18.04004 28.91832 17.26951 0.370418 0.137209 0.018826 1.00269 1005.314 18.03968 28.91832 17.12336 0.370418 0.137209 0.018826 1.00271 1005.294 18.03932 28.91832 16.97722 0.509698 0.259792 0.067492 1.00758 1007.546 18.03782 24.25789 17.63171 0.509698 0.259792 0.067492 1.00757 1007.556 18.038 24.25789 17.6702 0.509698 0.259792 0.067492 1.00758 1007.546 18.03782 24.25789 17.63171 0.509698 0.259792 0.067492 1.00757 1007.556 18.038 24.25789 17.6702 0.61984 0.384202 0.147611 1.0126 1009.731 18.05556 19.52803 17.61029 0.61984 0.384202 0.147611 1.01259 1009.741 18.05574 19.52803 17.63625 0.61984 0.384202 0.147611 1.01258 1009.751 18.05592 19.52803 17.6622 0.61984 0.384202 0.147611 1.01259 1009.741 18.05574 19.52803 17.63625 0.721947 0.521208 0.271657 1.01692 1013.315 18.12084 14.31926 19.85797 0.721947 0.521208 0.271657 1.01693 1013.305 18.12066 14.31926 19.83885 0.721947 0.521208 0.271657 1.01694 1013.295 18.12049 14.31926 19.81973 0.721947 0.521208 0.271657 1.01694 1013.295 18.12049 14.31926 19.81973 Water 0 0 0 0.99609 996.09 17.94498 64.29841 0 0 0 0.99589 995.89 17.94138 64.29841 0 0 0 0.99668 996.68 17.95561 64.29841 0 0 0 0.99709 997.09 17.963 64.29841 0 0 0 0.9971 997.1 17.96318 64.29841 0 0 0 0.9971 997.1 17.96318 64.29841 KCl 0.249018 0.06201 0.003845 0.99766 1006.98 18.07704 57.4027 64.7444 0.249018 0.06201 0.003845 0.99769 1006.95 18.07649 57.4027 64.25611 0.249018 0.06201 0.003845 0.99764 1007 18.0774 57.4027 65.06995 0.367643 0.135162 0.018269 1.00342 1006.635 18.01499 49.26805 27.15236 0.367643 0.135162 0.018269 1.00345 1006.605 18.01444 49.26805 26.9297 0.367643 0.135162 0.018269 1.00343 1006.625 18.0148 49.26805 27.07814 0.506093 0.25613 0.065603 1.00891 1010.097 18.03206 35.81597 27.84565 0.506093 0.25613 0.065603 1.00894 1010.067 18.03152 35.81597 27.72839 0.506093 0.25613 0.065603 1.00895 1010.057 18.03134 35.81597 27.6893 0.506093 0.25613 0.065603 1.00895 1010.057 18.03134 35.81597 27.6893 0.619816 0.384171 0.147588 1.01475 1013.692 18.11275 21.57744 27.92209 0.619816 0.384171 0.147588 1.01471 1013.732 18.11347 21.57744 28.0261 0.619816 0.384171 0.147588 1.01474 1013.702 18.11293 21.57744 27.94809 0.714341 0.510283 0.260389 1.0202 1017.493 18.26113 7.553479 28.47113 0.714341 0.510283 0.260389 1.02022 1017.473 18.26077 7.553479 28.43204 0.714341 0.510283 0.260389 1.02023 1017.463 18.26059 7.553479 28.41249 0.714341 0.510283 0.260389 1.02022 1017.473 18.26077 7.553479 28.43204
Figure 2: Volume vs Molality for NaCl and KCl Solutions 0.0 0.1 0.2 0.3 0.4 0.5 0.6 996 998 1000 1002 1004 1006 1008 1010 1012 1014 Data: Data1_B Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting Chi^2/DoF = 2.44104 R^2 = 0.90637 A 1000.2363 ±0.63684 B 34.13482 ±6.20282 C -19.00948 ±11.6541 NaCl Data Polynomial Fit Vol ume (mL)
Molality (mol NaCl/kg H2O) Volume versus Molality for NaCl
0.0 0.1 0.2 0.3 0.4 0.5 0.6 995 1000 1005 1010 1015 1020
Volume versus Molality for KCl
Data: Data1_D Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting Chi^2/DoF = 5.07679 R^2 = 0.91877 A 998.41765 ±0.82639 B 64.29841 ±9.15573 C -55.60144 ±17.87097 KCl Data Polynomial Fit Vol ume (mL)
Table 3: Values for Infinite Dilutions via 3 different methods
V2{Method 1} V2{Method 2} V2{Method 3} Literature
NaCl 34.13 15.19 14.24 16.63
KCl 64.30 25.18 25.18 26.85
% Deviation from literature for
NaCl 105.26049 8.64396 14.37589
% Deviation from literature for
Figure 3: φ vs m1/2 for NaCl and KCl Solutions 0.3 0.4 0.5 0.6 0.7 0.8 16 18 20 Data: Data1_B Model: Line Equation: y = A + B*x Weighting: y No weighting Chi^2/DoF = 0.25021 R^2 = 0.58474 A 15.19251 ±0.59586 B 4.71179 ±1.14624 Data Linear Fit φ ( ml/mol ) Molality1/2 (mol/kg)1/2
φ versus Molality1/2 for NaCl
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 26.8 27.0 27.2 27.4 27.6 27.8 28.0 28.2 28.4 28.6
φ versus Molality1/2 for KCl
Data: Data1_D Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting Chi^2/DoF = 0.01125 R^2 = 0.96381 A 25.17714 ±0.60707 B 5.88164 ±2.31566 C -1.9165 ±2.11332 Data Quadratic Fit φ ( m l/m o l) Molality1/2 (mol/kg)1/2
Figure 4: φ-1.86m1/2 vs m1/2 for NaCl and KCl Solutions 0.3 0.4 0.5 0.6 0.7 0.8 16 17 18 19 Data: Data1_B Model: Line Equation: y = A + B*x Weighting: y No weighting Chi^2/DoF = 0.3561 R^2 = 0.58821 A 14.23929 ±0.60596 B 5.00133 ±1.08048 Data Linear Fit φ – 1 .86m 1/2 m1/2(mol/kg H2O) φ – 1.86m1/2 vs. m1/2 for NaCl 0.3 0.4 0.5 0.6 0.7 26.2 26.4 26.6 26.8 27.0 27.2 Data: Data1_D Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting Chi^2/DoF = 0.01126 R^2 = 0.87445 A 25.17698 ±0.60723 B 4.02248 ±2.31626 C -1.91707 ±2.11387 φ – 1.86m1/2 vs. m1/2 for KCl Data Quadratic Fit φ – 1.86m 1 /2 m1/2(mol/kg H2O)
Table 4: Differences between partial and infinite molal volumes for KCl-NaCl and KBr-NaBr
m {KCl} m{NaCl} V2{KCl} V2{NaCl} V2{KCl}-V2{NaCl} V2{KBr}-V2{NaBr} φ{KCl} φ{NaCl}
φ{KCl}-φ{NaCl} 0.06201 0.012991 57.40269698 33.64091262 23.76178435 23.76178435 64.69015192 11.66608 53.02406807 0.135162 0.137209 49.26805297 28.91832471 20.34972827 20.34972827 54.2057591 17.02108 37.18468228 0.25613 0.259792 35.81597116 24.2578943 11.55807685 11.55807685 27.73816096 17.65096 10.08720568 0.384171 0.384202 21.57743825 19.52802562 2.049412634 2.049412634 27.96542569 17.63625 10.32917607 0.510283 0.521208 7.553479211 14.31925645 -6.765777243 -6.765777243 28.43692338 19.83407 8.602852776 Theoretical:
m V2{NaCl} V2{KCl} V2{KCl}-V2{NaCl} φ{NaCl} φ{KCl} φ{KCl}-φ{NaCl}
0 34.13482 64.29841 30.16359 9.37781 142.98282 133.60501 0.1 30.33296 53.178122 22.845158 15.58444855 45.90652811 30.32207956 0.5 15.12554 8.69697 -6.42857 19.09436622 30.30297781 11.20861159 1 -3.88374 -46.90447 -43.02073 18.70873 94.2506 75.54187 1.5 -22.893 -102.50591 -79.61289 16.66103396 187.2577965 170.5967626 2 -41.9023 -158.10735 -116.20505 13.75299244 295.3073356 281.5543432
Conclusion
The sodium chloride partial molar volume at infinite dilution, 15.19 mL/mol, is significantly different than a literature value of 16.63 mL/mol by 8.64%. Running more determinations at greater range of molalities might have lead to the better results than those obtained. The potassium chloride partial molar volume at infinite dilution, 25.18 mL/mol, is significantly different than a literature value of 26.85mL/mol. The percent error between the literature and experimental values for the partial molar volume of sodium chloride at infinite dilution is 6.23%. This error is smaller than the error in the potassium chloride measurements. Reasons for these errors include evaporation of water from the salt chloride solutions during density measurements, not mixing the solutions thoroughly could have lead to errors, and also the small range of molalities may not reflect the behaviors at a larger range of molalities.
References:
A. Poisson and J. Chanu, Limnology and Oceanography, Vol. 21, No. 6. (Nov., 1976), pp. 853-861.
Coulture, A.M., Laidler, K.J.. "Partial Molal Volume of Ions in Aqueous Solutions." Canadian Journal of Chemistry 34(1956): 1209-16.
