Experiment 2: Partial Molar Volume

Yipei Wang and Stephen Tereniak

CHEM445-024L

Due Date: 10/17/07

Submitted: 10/17/07

Abstract

Densities of a range of concentrations of aqueous potassium chloride and aqueous sodium

chloride were recorded with a density meter so that the partial molar volumes, and ultimately, the

partial molar volumes at infinite dilution, could be calculated. For potassium chloride and

sodium chloride, the partial molar volumes at infinite dilution of the salts were calculated to be

0.0261 (± 0.0001) L/mol and 0.0176 (± 0.0001) L/mol, respectively. These differ from literature

values by 1.14% and 6.02%, respectively. Theories explaining the percent error include

inadequate mixing of the solutions and leaving the caps off the jars containing the sodium

chloride solutions as their densities were being measured

Introduction

When considering the volume of a solution with two components, a useful approximation

is that the volume of the solution is equal to the sum of the volumes of the two components.

However, due to attractions between the two components, this approximation is not correct; the

total volume of the solution is less than the sum of the volumes of its two components. If the

molality of an aqueous salt solution is known, then the volume of the solution is

(1)

In order to determine the density of a solution, an aliquot of the solution is injected into an

instrument that records the density. The density of a solution can then be fit to a power series

with molality:

(2)

where m is molality and a0, a1, a2, … are constants acquired from a curve fitting program.

Likewise, the volume of a solution can be fit to a power series with molality in the following

manner:

(3)

where b0, b1, b2, … are constants acquired from a curve fitting program. The partial molar

volume of the salt is the partial derivative of the total volume of the solution with respect to

molality (Equation 3) at constant temperature and pressure:

(4)

Then, the partial molar volume of the solvent (water) can be calculated:

(5)

where n1 is the number of moles of water in the solution. By extrapolating a curve of against

molality to a molality of 0, the partial molar volume of the salt at infinite dilution can be

obtained ( ) and compared to literature values. The partial molar volume of a salt at infinite

dilution provides a quantitative measure of the properties of the solvated ions and how they

affect the structure of water.

Experimental

Sodium chloride and potassium chloride solutions were prepared by weighing the salt in

a capped jar, adding approximately 20 milliliters of distilled water, and weighing the solution.

The solutions were stirred to ensure that all of the salt dissolved. The density meter was cleaned

with distilled water multiple times until the density of air stopped changing at 0.00134 g/mL.

Once the density meter was cleaned, ~4 mL of the lowest concentration solution was injected

into the instrument so that all of the water was removed, and only the solution remained. The

density of the solution was recorded once the temperature stabilized to 25.00°C. Then, a ≥2 mL

aliquot of the solution was injected into the instrument, and the density was recorded again. A

total of at least four density measurements was recorded for each solution concentration;

typically, the first aliquot’s density was lower than the following aliquots’ densities for a given

concentration, indicating that some mixing with the previous concentration solution analyzed

occurred during the measurement of the first aliquot. The syringe was rinsed twice before

analyzing a new solution concentration. A total of five concentrations were measured for both

potassium chloride and sodium chloride. The density meter was cleaned with distilled water and

pumped with air twice before analyzing a different salt.

Results and Discussion

The density data for pure water, potassium chloride, and sodium chloride are given in

Table 1. One clear trend evident from observing the data is that the density of the first trial for a

given concentration was usually much lower (>0.0001 g/mL) than the subsequent trials. One

theory is that mixing occurred between the previous trial (of a solution of lower concentration)

and the first trial of a new concentration, resulting in a density intermediate between the two

solutions. This problem was resolved (in almost all cases) with the second trial of a new

concentration. In addition, the 0.4973 molal sodium chloride trials were excluded from the data

analysis since they were far off the best-fit curve. Possible reasons for this include a misread

analytical balance (for either the salt itself or the total aqueous solution) or an incompletely-

mixed aqueous solution.

Table 1: Salt Solution Molalities and DensitiesType of Salt Wt. salt (g) Wt. H2O (g) Molality Density (g/mL)

(mol salt/kg H2O)

- - - 0.0000

0.996640.996570.99657

0.99659 ± 0.00004

KCl

0.1574 20.3483 0.1038

0.999531.000041.000041.00004

0.99991 ± 0.00026

0.7732 20.3797 0.5089

1.020211.020301.020301.02031

1.02028 ± 0.00005

1.5265 19.8383 1.0321

1.042521.042771.042771.04276

1.04027 ± 0.00012

2.2287 20.1984 1.4801

1.061101.061161.061141.061171.06116

1.06115 ± 0.00003

2.9716 21.1648 1.8833

1.076811.076831.076821.076851.07684

1.07683 ± 0.00002

NaCl

0.1243 20.4087 0.1042

1.001701.001381.001391.001401.00140

1.00145 ± 0.00014

0.1309 21.2776 0.1053

1.001331.001291.001361.001341.001351.00134

1.00134 ± 0.00002

0.6004 20.6574 0.4973

1.011301.011281.011311.01130

1.01130 ± 0.00001

0.8206 20.5879 0.6820 1.024001.02428

1.024131.024121.024071.02406

1.02411 ± 0.00010

1.2569 21.9746 0.9787

1.035111.035381.035411.035421.03542

1.03535 ± 0.00013

2.0849 22.1803 1.6084

1.058341.058411.058421.058441.05847

1.05842 ± 0.00005

2.8600 19.2407 2.5434

1.090531.090651.090661.09066

1.09063 ± 0.00006*Temperature at 25.00ºC; red indicates data excluded from the calculations. The density data excluded from the subsequent data analysis were included in calculating the average and standard deviation of the densities at each concentration.

In Figure 1, density is plotted as a function of molality for both potassium chloride and

sodium chloride. Each curve is a quadratic function that increases with increasing molality and

has a decreasing rate of increase with increasing molality. The R-squared values for each curve

are rather good, indicating that the data used in the data analysis is precise. Also of note is that

the two curves cross around 0.15 molal; the potassium chloride curve lies above the sodium

chloride curve except for below ~0.15 molal.

Tabulated in Table 2 are the data for the volume of the salt solutions as a function of

molality as well as the partial molar volumes of the salt and water at a given molality; the mass

of water is assumed to be 1.000 kg in these calculations. The volume of solution was calculated

from Equation 1; the partial molar volume of salt was calculated with Equation 4, and the partial

Figure 1: Density of Salt Solutions vs. Molality

d = -0.0029(±0.0003)m2 + 0.0484(±0.0006)m + 0.9960(±0.0002)

R2 = 0.9998

d = -0.0016(±0.0001)m2 + 0.0408(±0.0001)m + 0.9970(±0.0001)

R2 = 1

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

0 0.5 1 1.5 2 2.5 3

Molality (mol/kg)

Den

sity

(g/

mL

)

KCl

NaCl

Poly. (NaCl)

molar volume of water was calculated with Equation 5. Note that the partial molar volume of

water barely changes across a wide range of molalities (the change is a very slight decrease). On

the other hand, the partial molar volumes of the salts do increase by a fairly large amount.

Table 2: Volumes of Solution and Partial Molar Volumes as a Function of MolalityType of Salt Molality

(mol/kg)V{solution}

(L)

KCl

0.00001.0034 0.0261 0.01811.0034 0.0261 0.01811.0034 0.0261 0.0181

0.10381.0077 0.0265 0.01811.0077 0.0265 0.01811.0077 0.0265 0.0181

0.50891.0173 0.0278 0.01811.0173 0.0278 0.01811.0173 0.0278 0.0181

1.03211.0328 0.0296 0.01811.0328 0.0296 0.01811.0328 0.0296 0.0181

1.4801

1.0463 0.0311 0.01801.0464 0.0311 0.01801.0463 0.0311 0.01801.0463 0.0311 0.0180

1.8833

1.0591 0.0325 0.01801.0590 0.0325 0.01801.0590 0.0325 0.01801.0590 0.0325 0.01801.0590 0.0325 0.0180

NaCl0.0000

1.0034 0.0176 0.01811.0034 0.0176 0.01811.0034 0.0176 0.0181

0.1042

1.0047 0.0178 0.01811.0047 0.0178 0.01811.0047 0.0178 0.01811.0047 0.0178 0.0181

0.1053

1.0048 0.0178 0.01811.0049 0.0178 0.01811.0048 0.0178 0.01811.0048 0.0178 0.01811.0048 0.0178 0.01811.0048 0.0178 0.0181

0.6820 1.0154 0.0187 0.01811.0154 0.0187 0.0181

1.0154 0.0187 0.01811.0154 0.0187 0.0181

0.9787

1.0211 0.0192 0.01811.0210 0.0192 0.01811.0210 0.0192 0.01811.0210 0.0192 0.0181

1.6084

1.0336 0.0202 0.01801.0336 0.0202 0.01801.0336 0.0202 0.01801.0336 0.0202 0.0180

2.54341.0532 0.0217 0.01801.0532 0.0217 0.01801.0532 0.0217 0.0180

In Figure 2, the volume of aqueous salt solution is plotted against molality. The curves

for both sodium chloride and potassium chloride in Figure 2 are quadratic functions that increase

with increasing molality. The potassium chloride curve remains greater than the sodium chloride

curve for all molalities from 0 to 3 molal. Both R-squared values are above 0.999, further

reinforcing the notion that the data are accurate.

For Figure 3, the partial molar volume of each salt is plotted as a function of molality.

For both potassium chloride and sodium chloride, these curves are linear functions that increase

with increasing molality. Both R-squared values are excellent (each is 0.99995, carried out to

five decimal places). This information means that the partial molar volume of each salt at

infinite dilution (at a molality of 0 on the curve of Figure 3 and in Table 2) can be considered

precise calculations.

Tabulated in Table 3 are the partial molar volumes for potassium chloride and sodium

chloride at “round” molality values. Note that the difference between the two solutions’ partial

molar volumes increases by 0.0009 L/mol between two successive fixed values of molality; the

difference between the two solutions’ partial molar volumes, if plotted as a function of molality,

would itself be a linear function.

Figure 2: Volumes of Salt Solutions vs. Molality

V = 0.0008(±0.0001)m2 + 0.0176(±0.0001)m + 1.0030(±0.0001)

R2 = 0.9999

V = 0.0017(±0.0003)m2 + 0.0261(±0.0006)m + 1.0040(±0.0002)

R2 = 0.9995

1.0000

1.0200

1.0400

1.0600

1.0800

1.1000

0 0.5 1 1.5 2 2.5 3

Molality (mol/kg)

V (

L)

KCl

NaCl

Table 3: Calculated Partial Molar Volume Differences Between KCl and NaCl

m (mol/kg)(L/

mol) (L/mol) -

(L/mol)0.0 0.0261 0.0176 0.00850.5 0.0278 0.0184 0.00941.0 0.0295 0.0192 0.01031.5 0.0312 0.0200 0.01122.0 0.0329 0.0208 0.01212.5 0.0346 0.0216 0.01303.0 0.0363 0.0224 0.0139

From the - data at infinite dilution, an estimate for -

at infinite dilution can be calculated. Several assumptions must be made: at infinite

dilution, it must be assumed that the interionic distances are infinite; i.e., no attractions between

ions in solution exist (so that ion pairing is neglected). Also, one assumes that the ions are not

solvated by water, for the extent of solvation varies between ions (and the increase in ionic size

is not easy to quantify). Using these assumptions, the difference in partial molar volumes of the

bromides at infinite dilution can be calculated from Pauling ionic radii (1):

- =

L/mol

Examining partial molar volume data for copper(II) sulfate demonstrates that partial

molar volumes are not directly proportional to ionic size. In Table 4, density, volume of

solution, and partial molar volume of copper sulfate data is presented as a function of molality.

The densities steadily increase, but the volumes of solution first decrease, reach their minimum

Figure 3: Partial Molar Volume {Salt} vs. Molality

PMV = 0.0016(±0.0001)m + 0.0176(±0.0001)

R2 = 1

PMV = 0.0034(±0.0001)m + 0.0261(±0.0001)

R2 = 1

0.0175

0.0195

0.0215

0.0235

0.0255

0.0275

0.0295

0.0315

0.0335

0.0355

0.0375

0 0.5 1 1.5 2 2.5 3

Molality (mol/kg)

Par

tial

Mol

ar V

olum

e {s

alt}

(L

/mol

)

KCl

NaCl

at 0.194 molal (using more significant figures than are displayed in the table), and then steadily

increase with increasing molality. The partial molar volume of copper sulfate actually is a

negative value until 0.128 molal; this requires that partial molar volumes not be proportional to

ionic size, for then negative partial molar volumes would not be possible since ionic radii cannot

have negative values. A negative partial molar volume merely indicates that the volume of

solution decreases with increasing copper sulfate concentration in solution.

Table 4: CuSO4 Solution Volumes and Partial Molar Volumes as a Function of Molality2

Molality (mol/kg) Density (g/mL) V{m} (L) (L/mol)0.000 0.99823 1.0018 -0.00150.031 1.00332 1.0016 -0.00110.095 1.01370 1.0015 -0.00020.128 1.01899 1.0014 0.00020.194 1.02957 1.0014 0.00110.261 1.04026 1.0014 0.00200.329 1.05104 1.0015 0.00280.400 1.06202 1.0017 0.00380.472 1.07310 1.0021 0.00470.544 1.08428 1.0024 0.00570.620 1.09556 1.0031 0.00670.697 1.10704 1.0037 0.00770.854 1.13040 1.0053 0.00981.020 1.15455 1.0072 0.01201.106 1.16693 1.0082 0.01311.193 1.17961 1.0091 0.01421.375 1.20586 1.0113 0.0167

Note: Measurements taken at 20.0°C.

Using the Table 4 data, density against molality is plotted in Figure 4. The curve has a

positive slope; the change in the slope decreases with increasing molality. With a R-squared

value of 1, the data are precise. Depicted in Figure 5 is the volume of solution versus molality.

Note the minimum in the parabola around 0.100 molal; this occurs at a slightly lesser

concentration than the minimum solution volume in the data at 0.194 molal. The R-squared

value for this chart is not terribly good. In Figure 6, the partial molar volume of copper sulfate is

plotted against molality. The crossover from negative partial molar volumes to positive partial

molar volumes can clearly be seen in Figure 6; it occurs at a molality a little above 0.100 molal.

Excluding the y-intercept of the data (the partial molar volume at infinite dilution), the linear

curve in Figure 6 is not notably different from the linear curves in Figure 3.

Figure 4: Copper Sulfate Density vs. Molality

d = -0.0082(±0.0004)m2 + 0.1617(±0.0005)m + 0.9984(±0.0002)

R2 = 1

0.95

1

1.05

1.1

1.15

1.2

1.25

0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400

Molality (mol/kg)

Den

sity

(g/

mL

)

Figure 5: Volume of Copper Sulfate Solution vs. Molality

V = 0.0066(±0.0004)m2 - 0.0015(±0.0006)m + 1.0015(±0.0001)

R2 = 0.9939

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400

Molality (mol/kg)

Vol

ume

(L)

Figure 6: Partial Molar Volume of CuSO4 vs. Molality

PMV = 0.0132(±0.0001)m - 0.0015(±0.0001)

R2 = 1

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400

Molality (mol/kg)

Par

tial

Mol

ar V

olum

e C

uSO

4 (L

/mol

)

Conclusions

The potassium chloride partial molar volume at infinite dilution, 0.0261 (± 0.0001)

L/mol, is significantly different than a literature value, 0.0264 L/mol, at the 95% confidence

interval since tcalc is greater than ttable for 20 degrees of freedom (3). Running more

determinations might have lead to the conclusion that the literature and the experimental values

are not significantly different at the 95% confidence interval; the two values differ by only

0.0003 L/mol (or 1.14%). The sodium chloride partial molar volume at infinite dilution, 0.0176

(± 0.0001) L/mol, is significantly different than a literature value, 0.0166 (± 0.0001) L/mol, at

the 95% confidence interval since tcalc is greater than ttable for 25 degrees of freedom (4). The

percent error between the literature and experimental values for the partial molar volume of

sodium chloride at infinite dilution is 6.02%. This error is much larger than the error in the

potassium chloride measurements. Reasons for these errors include water’s evaporation from the

sodium chloride solutions during density measurements since the lid was sometimes left off of

the jar of the solution whose density was being measured. Also, not mixing the solutions enough

could have been a culprit.

References

1. Winter, M. “Ionic Radius (Pauling) for M(I) Ion.” WebElements. 2007. WebElements, Ltd. 17 Oct 2007 <http://www.webelements.com/webelements/properties/text/definitions/radius-ionic-pauling-1.html>.

2. Munson, B. “Partial Molar Volume,” pg. 6. 8/23/07.3. Gucker, F. T.: Chem. Rev. 13, 126 (1933).4. Redlich, O., and Bigeleisen, J.: Chem. Rev. 30, 175 (1942).