Lab 2

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 0.00110.0014 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

Salt wt.(g)

H2O wt(g)

Molality(m)

m2

Water 0.0151

19.8885

0.01299122

0.00016877

0.1568

19.5541

0.13720918

0.01882636

0.2995

19.7263

0.25979221

0.06749199

0.4416

19.6673

0.38420167

0.14761093

0.5849

19.20197

0.52120763

0.27165739

0.0919

19.8768

0.0620102

0.00384527

0.2013

19.9749

0.13516158

0.01826865

0.3813

19.9664

0.25613041

0.06560279

0.5641

19.6936

0.38417145

0.1475877

0.7457

19.5996

0.51028292

0.26038866

NaCl

Water

KCl

Density 0.99808 0.99708 0.99708 0.99765 0.99765 0.99765 1.00278 1.00267 1.00267 1.00269 1.00271 1.00758 1.00757 1.00758 1.00757 1.01260 1.01259 1.01258 1.01259 1.01692 1.01693 1.01694 1.01694 0.99609 0.99589 0.99668 0.99709 0.9971 0.9971 0.99766 0.99769 0.99764 1.00342 1.00345 1.00343 1.00891 1.00894 1.00895 1.00895 1.01475 1.01471 1.01474 1.0202 1.02022 1.02023 1.02022

Figure 1: Density vs Molality for NaCl and KCl Solutions

Density versus Molality for NaCl Solutions 1.020

Density (g/mL)

1.015

NaCl Data Polynomial Fit

1.010 Data: Data1_B Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting

1.005

Chi^2/DoF = 7.3015E-8 R^2 = 0.99872

1.000

A B C

0.99719 0.04246 -0.00829

±0.00011 ±0.00107 ±0.00202

0.995 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Molality (mol NaCl/kg H2O)

Density versus Molality for KCl Solutions 1.020

1.015

Density (g/mL)

KCl Data Polynomial Fit

1.010 Data: Data1_D Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting

1.005

Chi^2/DoF = 5.9627E-7 R^2 = 0.99346

1.000

A B C

0.99619 0.05073 -0.00683

±0.00028 ±0.00314 ±0.00612

0.995 0.0

0.1

0.2

0.3

0.4

Molality (mol KCl/kg H2O)

0.5

0.6

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 Water

NaCl

Water

KCl

m1/2

m

m2

d(g/ml)

V{m}

V1

V2

0 0 0 0.113979 0.113979 0.113979 0.370418 0.370418 0.370418 0.370418 0.370418 0.509698 0.509698 0.509698 0.509698 0.61984 0.61984 0.61984 0.61984 0.721947 0.721947 0.721947 0.721947 0 0 0 0 0 0 0.249018 0.249018 0.249018 0.367643 0.367643 0.367643 0.506093 0.506093 0.506093 0.506093 0.619816 0.619816 0.619816 0.714341 0.714341 0.714341 0.714341

0 0 0 0.012991 0.012991 0.012991 0.137209 0.137209 0.137209 0.137209 0.137209 0.259792 0.259792 0.259792 0.259792 0.384202 0.384202 0.384202 0.384202 0.521208 0.521208 0.521208 0.521208 0 0 0 0 0 0 0.06201 0.06201 0.06201 0.135162 0.135162 0.135162 0.25613 0.25613 0.25613 0.25613 0.384171 0.384171 0.384171 0.510283 0.510283 0.510283 0.510283

0 0 0 0.000169 0.000169 0.000169 0.018826 0.018826 0.018826 0.018826 0.018826 0.067492 0.067492 0.067492 0.067492 0.147611 0.147611 0.147611 0.147611 0.271657 0.271657 0.271657 0.271657 0 0 0 0 0 0 0.003845 0.003845 0.003845 0.018269 0.018269 0.018269 0.065603 0.065603 0.065603 0.065603 0.147588 0.147588 0.147588 0.260389 0.260389 0.260389 0.260389

0.99808 0.99708 0.99708 0.99765 0.99765 0.99765 1.00278 1.00267 1.00267 1.00269 1.00271 1.00758 1.00757 1.00758 1.00757 1.0126 1.01259 1.01258 1.01259 1.01692 1.01693 1.01694 1.01694 0.99609 0.99589 0.99668 0.99709 0.9971 0.9971 0.99766 0.99769 0.99764 1.00342 1.00345 1.00343 1.00891 1.00894 1.00895 1.00895 1.01475 1.01471 1.01474 1.0202 1.02022 1.02023 1.02022

998.08 997.08 997.08 1003.117 1003.117 1003.117 1005.224 1005.335 1005.335 1005.314 1005.294 1007.546 1007.556 1007.546 1007.556 1009.731 1009.741 1009.751 1009.741 1013.315 1013.305 1013.295 1013.295 996.09 995.89 996.68 997.09 997.1 997.1 1006.98 1006.95 1007 1006.635 1006.605 1006.625 1010.097 1010.067 1010.057 1010.057 1013.692 1013.732 1013.702 1017.493 1017.473 1017.463 1017.473

17.98083 17.96282 17.96282 18.06369 18.06369 18.06369 18.03806 18.04004 18.04004 18.03968 18.03932 18.03782 18.038 18.03782 18.038 18.05556 18.05574 18.05592 18.05574 18.12084 18.12066 18.12049 18.12049 17.94498 17.94138 17.95561 17.963 17.96318 17.96318 18.07704 18.07649 18.0774 18.01499 18.01444 18.0148 18.03206 18.03152 18.03134 18.03134 18.11275 18.11347 18.11293 18.26113 18.26077 18.26059 18.26077

34.13482 34.13482 34.13482 33.64091 33.64091 33.64091 28.91832 28.91832 28.91832 28.91832 28.91832 24.25789 24.25789 24.25789 24.25789 19.52803 19.52803 19.52803 19.52803 14.31926 14.31926 14.31926 14.31926 64.29841 64.29841 64.29841 64.29841 64.29841 64.29841 57.4027 57.4027 57.4027 49.26805 49.26805 49.26805 35.81597 35.81597 35.81597 35.81597 21.57744 21.57744 21.57744 7.553479 7.553479 7.553479 7.553479

φ (ml/mol)

11.66608 11.66608 11.66608 16.46577 17.26951 17.26951 17.12336 16.97722 17.63171 17.6702 17.63171 17.6702 17.61029 17.63625 17.6622 17.63625 19.85797 19.83885 19.81973 19.81973

64.7444 64.25611 65.06995 27.15236 26.9297 27.07814 27.84565 27.72839 27.6893 27.6893 27.92209 28.0261 27.94809 28.47113 28.43204 28.41249 28.43204

Figure 2: Volume vs Molality for NaCl and KCl Solutions Volume versus Molality for NaCl 1014 1012

NaCl Data Polynomial Fit

1010

Volume (mL)

1008 Data: Data1_B Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting

1006 1004 1002

Chi^2/DoF = 2.44104 R^2 = 0.90637

1000

A B C

998

1000.2363 34.13482 -19.00948

±0.63684 ±6.20282 ±11.6541

996 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Molality (mol NaCl/kg H2O)

Volume versus Molality for KCl 1020

KCl Data Polynomial Fit

1015

Volume (mL)

1010 Data: Data1_D Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting

1005

Chi^2/DoF = 5.07679 R^2 = 0.91877

1000

A B C

995

0.0

0.1

0.2

0.3

998.41765 64.29841 -55.60144

0.4

Molality (mol NaCl/kg H2O)

±0.82639 ±9.15573 ±17.87097

0.5

0.6

Table 3: Values for Infinite Dilutions via 3 different methods NaCl KCl % Deviation from literature for NaCl % Deviation from literature for KCl

V2{Method 1} 34.13 64.30

V2{Method 2} 15.19 25.18

V2{Method 3} 14.24 25.18

105.26049

8.64396

14.37589

139.47266

6.23039

6.23099

Literature 16.63 26.85

Figure 3: φ vs m1/2 for NaCl and KCl Solutions 1/2

φ versus Molality

for NaCl

Data: Data1_B Model: Line Equation: y = A + B*x Weighting: y No weighting

20

φ (ml/mol)

Chi^2/DoF = 0.25021 R^2 = 0.58474 A B

15.19251 4.71179

±0.59586 ±1.14624

18

Data Linear Fit 16 0.3

0.4

0.5

0.6 1/2

Molality

0.7

(mol/kg)

1/2

φ versus Molality

0.8

1/2

for KCl

28.6 28.4

Data Quadratic Fit

28.2

φ ( ml/mol)

28.0 27.8

Data: Data1_D Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting

27.6 27.4

Chi^2/DoF = 0.01125 R^2 = 0.96381

27.2

A B C

27.0

25.17714 ±0.60707 5.88164 ±2.31566 -1.9165 ±2.11332

26.8 0.35

0.40

0.45

0.50

0.55 1/2

Molality

0.60 1/2

(mol/kg)

0.65

0.70

0.75

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

19

Data Linear Fit

φ – 1.86m

1/2

18

Data: Data1_B Model: Line Equation: y = A + B*x Weighting: y No weighting

17

Chi^2/DoF = 0.3561 R^2 = 0.58821 A B

16

0.3

0.4

0.5

0.6

14.23929 5.00133

0.7

±0.60596 ±1.08048

0.8

1/2

m (mol/kg H2O)

1/2

φ – 1.86m1/2 vs. m

for KCl

27.2

Data Quadratic Fit

φ – 1.86m

1/2

27.0

26.8 Data: Data1_D Model: Parabola Equation: y = A + B*x + C*x^2 Weighting: y No weighting

26.6

Chi^2/DoF = 0.01126 R^2 = 0.87445

26.4

A B C

25.17698 4.02248 -1.91707

26.2 0.3

0.4

0.5 1/2

m (mol/kg H2O)

0.6

0.7

±0.60723 ±2.31626 ±2.11387

Table 4: Differences between partial and infinite molal volumes for KCl-NaCl and KBr-NaBr m {KCl}

V2{NaCl}

V2{KCl}-V2{NaCl}

V2{KBr}-V2{NaBr}

φ{KCl}

V2{KCl}

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

V2{NaCl}

V2{KCl}

Theoretical: m

V2{KCl}V2{NaCl}

φ{NaCl}

φ{KCl}φ{NaCl}

φ{KCl}

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

φ{NaCl}

φ{KCl}φ{NaCl}

m{NaCl}

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.