Nathaly MurilloKevin 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

SaltSalt wt.

(g)H2O wt(g)

Molality(m) m2 Density

Water0.998080.997080.99708

NaCl

0.0151 19.8885 0.01299122 0.00016877 0.997650.997650.99765

0.1568 19.5541 0.13720918 0.01882636 1.002781.002671.002671.002691.00271

0.2995 19.7263 0.25979221 0.06749199 1.007581.007571.007581.00757

0.4416 19.6673 0.38420167 0.14761093 1.012601.012591.012581.01259

0.5849 19.20197 0.52120763 0.27165739 1.016921.016931.016941.01694

Water

0.996090.995890.996680.997090.99710.9971

KCl

0.0919 19.8768 0.0620102 0.00384527 0.997660.997690.99764

0.2013 19.9749 0.13516158 0.01826865 1.003421.003451.00343

0.3813 19.9664 0.25613041 0.06560279 1.008911.00894

1.008951.00895

0.5641 19.6936 0.38417145 0.1475877 1.014751.014711.01474

0.7457 19.5996 0.51028292 0.26038866 1.02021.020221.020231.02022

Figure 1: Density vs Molality for NaCl and KCl Solutions

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 V2φ

(ml/mol)

Water0 0 0 0.99808 998.08 17.98083 34.134820 0 0 0.99708 997.08 17.96282 34.134820 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 11.666080.113979 0.012991 0.000169 0.99765 1003.117 18.06369 33.64091 11.666080.113979 0.012991 0.000169 0.99765 1003.117 18.06369 33.64091 11.666080.370418 0.137209 0.018826 1.00278 1005.224 18.03806 28.91832 16.465770.370418 0.137209 0.018826 1.00267 1005.335 18.04004 28.91832 17.269510.370418 0.137209 0.018826 1.00267 1005.335 18.04004 28.91832 17.269510.370418 0.137209 0.018826 1.00269 1005.314 18.03968 28.91832 17.123360.370418 0.137209 0.018826 1.00271 1005.294 18.03932 28.91832 16.977220.509698 0.259792 0.067492 1.00758 1007.546 18.03782 24.25789 17.631710.509698 0.259792 0.067492 1.00757 1007.556 18.038 24.25789 17.67020.509698 0.259792 0.067492 1.00758 1007.546 18.03782 24.25789 17.631710.509698 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 17.610290.61984 0.384202 0.147611 1.01259 1009.741 18.05574 19.52803 17.636250.61984 0.384202 0.147611 1.01258 1009.751 18.05592 19.52803 17.66220.61984 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 19.838850.721947 0.521208 0.271657 1.01694 1013.295 18.12049 14.31926 19.819730.721947 0.521208 0.271657 1.01694 1013.295 18.12049 14.31926 19.81973

Water

0 0 0 0.99609 996.09 17.94498 64.298410 0 0 0.99589 995.89 17.94138 64.298410 0 0 0.99668 996.68 17.95561 64.298410 0 0 0.99709 997.09 17.963 64.298410 0 0 0.9971 997.1 17.96318 64.298410 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 64.74440.249018 0.06201 0.003845 0.99769 1006.95 18.07649 57.4027 64.256110.249018 0.06201 0.003845 0.99764 1007 18.0774 57.4027 65.069950.367643 0.135162 0.018269 1.00342 1006.635 18.01499 49.26805 27.152360.367643 0.135162 0.018269 1.00345 1006.605 18.01444 49.26805 26.92970.367643 0.135162 0.018269 1.00343 1006.625 18.0148 49.26805 27.078140.506093 0.25613 0.065603 1.00891 1010.097 18.03206 35.81597 27.845650.506093 0.25613 0.065603 1.00894 1010.067 18.03152 35.81597 27.728390.506093 0.25613 0.065603 1.00895 1010.057 18.03134 35.81597 27.68930.506093 0.25613 0.065603 1.00895 1010.057 18.03134 35.81597 27.68930.619816 0.384171 0.147588 1.01475 1013.692 18.11275 21.57744 27.922090.619816 0.384171 0.147588 1.01471 1013.732 18.11347 21.57744 28.02610.619816 0.384171 0.147588 1.01474 1013.702 18.11293 21.57744 27.948090.714341 0.510283 0.260389 1.0202 1017.493 18.26113 7.553479 28.471130.714341 0.510283 0.260389 1.02022 1017.473 18.26077 7.553479 28.432040.714341 0.510283 0.260389 1.02023 1017.463 18.26059 7.553479 28.412490.714341 0.510283 0.260389 1.02022 1017.473 18.26077 7.553479 28.43204

Figure 2: Volume vs Molality for NaCl and KCl Solutions

Table 3: Values for Infinite Dilutions via 3 different methodsV2{Method 1} V2{Method 2} V2{Method 3} Literature

NaCl 34.13 15.19 14.24 16.63KCl 64.30 25.18 25.18 26.85% Deviation from literature for NaCl 105.26049 8.64396 14.37589% Deviation from literature for KCl 139.47266 6.23039 6.23099

Figure 3: φ vs m1/2 for NaCl and KCl Solutions

Figure 4: φ-1.86m1/2 vs m1/2 for NaCl and KCl Solutions

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.