K.A. Cruz, D.L.C. Fernando, R.A. Soriano, Partial Molar Volume of a Substance, 2014
Partial Molar Volume of a Substance K.A. Cruz, D.L.C. Fernando, R.A. Soriano Department of Chemical Engineering, College of Engineering, University of the Philippines Diliman 28 January 2014
_________________________________________________________ ABSTRACT _________________________________________________________________________________ Usage of simple mixing rules for non-ideal solution results to a high error, hence, an understanding of partial molar properties is of utmost importance to be able to predict what changes occur upon changing the composition of a solution. The experiment aims to establish a method to determine the partial molar volume of ethanol and water in a solution with each other by measuring the volume of mixtures with different composition. An expression of volume as a function of composition is established to be able to determine its derivative which will be used in partial molar volume calculations. Experimental values of partial molar volume of ethanol had a maximum deviation of 1.295% while that of water had 3.536%. Measured molar volume of the ethanol-water solution reached a deviation of 0.982%. Keywords: partial properties, solution, composition, molecular interaction, summability relation, GibbsDuhem equation
_________________________________________________________
1. Introduction Some properties of a pure chemical species change when it is in a solution with other species. These properties that are affected by solution composition are called partial molar properties. The definition of a partial molar property of species i in a solution is: ̅
*
(
)
pressure will result to a total property change represented by ΔnM and the corresponding partial molar volume of species i is ̅ . The molar property of the solution in terms of the partial molar properties of all the species in it are expressed in the summability relations: ∑
+
The symbol ̅ stands for any property dependent on composition such as enthalpy and volume. It is a response function; it represents the change of total property nM because of the addition of a differential number of moles of species i, ni, to a solution at constant pressure and temperature. As such, partial molar properties are intensive variables. Conventions state that Mi is the molar property of pure species i and M is the molar property of the solution[1][2]. If we apply this to volume, addition of Δni of species i to a solution with total property nM at constant temperature and
∑
̅
̅
Partial molar properties are important because they can denote the degree of non-ideality of a system. In a solution, the constituents are intimately mixed. Due to molecular interactions between the species in the solution, their individual properties are modified to some degree. This implies that substances in a solution cannot have private properties, or ones that remain truly unaffected despite being in the presence of another material[1].
1
K.A. Cruz, D.L.C. Fernando, R.A. Soriano, Partial Molar Volume of a Substance, 2014 For a binary system, it is found that the partial molar volumes and solution molar volume are: ̅
( ̅
)
[ ]
[ ] [D1]
̅
̅
[ ]
This provides us with a method to determine the partial molar properties of 2 chemical species when mixed with each other. This is done by preparing mixtures with different composition and measuring the resulting mass of a known volume. An expression for V as a function of xi will be obtained here and its derivative will provide the second term of the right hand side of the equation. This experiment aims to determine the partial molar volumes of ethanol and water in solutions of varying concentrations. This will be done through the use of a pycnometer, which consists of a small glass flask and a glass stopper with a capillary hole running through the center. Excess liquid is ejected from the pycnometer through this capillary hole in order to obtain the specified solution volume with a very high accuracy.The experimentally obtained values for the volume of the solutions will also be plotted against the ethanol mole fractions. The generated curve will be compared to the theoretical one. 2. Material & Methods The materials for this experiment were a pycnometer for accurately measuring 10 mL of the sample solutions, 6 50-mL volumetric flasks to hold the ethanol solutions, a 10-mL pipette for transferring the ethanol into the volumetric flasks, a 10mL graduated cylinder and a 1-mL pipette for transferring the solutions into the pycnometer, 3 1000-mL beakers for the water baths of the ethanol solutions and the pycnometer, a 2000-mL beaker for temporarily holding the waste solutions, a thermocouple for measuring the
temperature of the tap water baths, ice for lowering the temperature of the water baths, an alcohol thermometer for measuring the wet bulb temperature, a piece of cloth for drying the pycnometer, a paper tong for handling the pycnometer, a stopwatch for measuring time intervals, and masking tape for making flask labels. The reagents that were used in this experiment were distilled water and ethanol. The apparatus that were used for this experiment were the analytical balance and the hot plate. First, the room temperature and pressure were recorded. The wet bulb temperature was also determined. This was done by wrapping a small piece of cotton around the bulb of an alcohol thermometer and securing it in place with a rubber band. The cotton was dipped in water afterwards and the thermometer was rapidly, but cautiously, swung in a circular manner for 30 seconds. The alcohol thermometer reading was recorded and 2 more trials were done. The empty volumetric flasks were labeled using pieces of masking tape and were covered. They were weighed using the analytical balance and their masses were recorded. The ethanol solutions were then prepared according to the following table: Table 1 Ethanol Mole Fractions Solution Mole Fraction Ethanol 0 A 0.2 B 0.4 C 0.6 D 0.8 E 1 F
The corresponding ethanol volume for each solution was calculated through the following equation:
(
) 2
K.A. Cruz, D.L.C. Fernando, R.A. Soriano, Partial Molar Volume of a Substance, 2014
Isolating Vethanol on one side of the equation through algebraic manipulation will yield an expression on the other side in terms of the desired ethanol mole fraction, the densities of water and ethanol, and the molar masses of water and ethanol. The ethanol volumes for each solution are then obtained: Table 2 Correct Ethanol Volume for Each Solution Solution Ethanol Volume A 0.0000 B 22.4176 C 34.2139 D 41.4916 E 46.4296 F 50.0000
The solutions were all prepared in their corresponding volumetric flasks. Ethanol was transferred into the flasks via the 10mL pipette and after the correct amount had been transferred, the flasks were filled to the mark with distilled water. The flasks were then covered afterwards and were weighed once again. The masses were recorded. The flasks were placed in a 2 tap water baths afterwards. Using a thermocouple, the temperature of the baths was monitored and was kept at room temperature by the use of a hot plate, when the temperature had dropped below 26.5OC, and ice, when the temperature had risen above 26.5OC. The mole fraction of ethanol in each flask was recalculated using the respective masses of ethanol and water.
weighed thrice with the analytical balance and the mass for each trial was recorded. The volumetric flask with solution A was then removed from the tap water bath and 10 mL of the solution was transferred to the pycnometer using a graduated cylinder. The capillary stopper was then slowly inserted into the pycnometer opening, making sure that there were no spaces or trapped bubbles within the capillary space. For the times when the space wasn’t completely filled with the solution, the stopper was removed and a small amount of the solution was placed into the pycnometer via the 1-mL pipette. The capillary stopper was then cautiously replaced back onto the pycnometer. On the other hand, for times when the solution overflowed from the top of the capillary space, a piece of cloth was used to dry the pycnometer and the stopper, and remove some of the solution that stayed on the top of the stopper but outside of the capillary space as a bead of liquid. The pycnometer was then placed in the water bath for five minutes in order for the solution inside to reach the temperature of the bath. Afterwards, the pycnometer was carefully removed from the bath by handling it by the neck. The pycnometer was dried with a piece of cloth and was transferred to the analytical balance through the use of a paper tong. The mass of the pycnometer with the solution was recorded. The solution in the pycnometer was then placed in the 2000-mL waste beaker. Two more trials with the pycnometer and solution A were done to ensure the consistency of the experimental data. The pycnometer, graduated cylinder, and the 1mL pipette were then cleaned afterwards.
Figure 1 The 6 Sample Solutions in the Water Baths
Following this, the empty pycnometer was washed and dried. It was 3
K.A. Cruz, D.L.C. Fernando, R.A. Soriano, Partial Molar Volume of a Substance, 2014 D E F
0.7108 0.7938 0.8505
0.5135 0.2153 0.0000
0.5806 0.7866 1.0000
Similarly, the masses of the solution that were placed within the pycnometer were determined by same procedure that was employed for table 2: Table 5 Masses of the Pycnometer Solutions
Figure 2 Pycnometer in the Water Bath
The same procedure was repeated for solutions B to F. Upon finishing, all the used glass wares were washed, and the contents of the waster beaker and the leftover solutions were disposed in the designated waste jar.
Solution
Trial 1
Trial 2
Trial 3
A B C D E F
9.9874 9.1903 8.8044 8.4235 8.1118 7.8738
10.0005 9.2158 8.8122 8.4368 8.1305 7.8642
10.0326 9.2297 8.8363 8.4612 8.1289 7.8823
The average was taken and by dividing these by the volume of the solution inside the pycnometer, the densities of the 6 sample solutions were obtained. Table 6
3. Results & Discussion
Densities of the 6 Sample Solutions
The mole fractions of the solutions were recalculated by solving for the actual masses of ethanol and water inside the flasks. The ethanol masses were obtained by subtracting the mass of the empty flasks from the recorded masses with the ethanol. The water masses were obtained similarly: Table 3 Ethanol and Water Masses in each Solution Solution
Ethanol mass, g
Water mass, g
A B C D E F
0.0000 20.3565 26.9526 32.7548 36.5771 39.1921
49.7933 25.2183 16.9129 9.2531 3.8803 0.0000
These values were converted to moles by dividing the masses in the previous table with the appropriate molar mass, 46.08 g/mol for ethanol and 18.02 g/mol for water: Table 4 Moles of Ethanol and Water in each Solution Solution
Ethanol moles
Water moles
Ethanol mole fraction
A B C
0.0000 0.4418 0.5849
2.7632 1.3995 0.9386
0.0000 0.2399 0.3839
Solution
Average mass, g
Density, g/mL
A B C D E F
10.0068 9.2119 8.8176 8.4405 8.1237 7.8734
1.00068 0.92119 0.88176 0.84405 0.81237 0.78734
Since the mass of the sample solutions are known, as outlined in table 2, their molar volumes can be calculated as well. This was done by taking the quotient of the solution masses and their respective density, which was then subsequently divided by the total number of moles in the parent solution: Table 7 Molar Volume of the 6 Sample Solutions Solution
Molar volume, mL/mol
A B C D E F
18.0077 26.8700 32.6540 40.6508 49.3520 58.5259
Plotting the molar volumes, as the ordinate, against the mole fraction of
4
K.A. Cruz, D.L.C. Fernando, R.A. Soriano, Partial Molar Volume of a Substance, 2014 abscissa,
yields
the
Molar Volume of Solution (mL/mol)
70 y = 3.7972x2 + 36.84x + 17.944 R² = 1
60
59 Partial Molar Volume of Ethanol (mL/mol)
ethanol, as the following graph:
50 40 30 20
58 57 56 55
Theoretical
54 Experimental
53 52 0.0
10
0.5
1.0
Ethanol mole fraction 0 0
0.5
1
Figure 4 Ethanol Partial Molar Volume vs xEthanol
Ethanol Mole Fraction
Figure 3
Using the equation for best fit curve, the derivative at the solution mole fractions can be determined and through equations 1 and 2, the partial molar volumes at those points can be determined: Table 8 Partial Molar Volume of Ethanol and Water in the 6 Sample Solutions ̅ ̅ Solution A 54.7841 17.9442 B 56.3876 17.7256 C 57.1401 17.3844 D 57.9133 16.6642 E 58.4084 15.5946 F 58.5813 14.1470
19 Partial Molar Volume of Water (mL/mol)
Solution Molar Volume versus Ethanol Mole Fraction
18 17 16 Theoretical 15
Experimental
14 0
0.5 Ethanol mole fraction
1
Figure 5 Water Partial Molar Volume vs xEthanol
Theoretical values were obtained by digitizing the points in the following figure:
Plotting the partial molar volumes of the two species against the ethanol mole fraction separately:
Figure 6 Theoretical Partial Molar Volume versus Ethanol [3] Mole Fraction
5
K.A. Cruz, D.L.C. Fernando, R.A. Soriano, Partial Molar Volume of a Substance, 2014
It should be noted that as the mole fraction of ethanol in the solution increases, the partial molar volume of ethanol approaches the molar volume for the pure liquid, which is approximately 58.700 mL/mol. The same can be said for water; as the ethanol mole fraction approaches zero, the molar volume of the solution approaches the molar volume of water, which is 18.056 mL/mol at room temperature. Outside of these two extremes, the molar volumes of the two species possess different values. This is a result of the molecular interactions between the ethanol and water. Their volumes are no longer private properties; their volumes are modified as they exist in a solution[7]. Another quantity that can be observed from the plots above is the infinite dilution molar volume of water and ethanol. These are the values of their partial molar volume at a very minute concentration. For water, this is approximately 13.93 mL/mol. For ethanol, this is approximately 54.31 mL/mol[6]. The theoretical values for the molar volume of the solution were obtained from this plot as well by using equation 3. Superimposing this with the experimental data:
Molar Volume of Solution (mL/mol)
70 60
y = 3.7972x2 + 36.84x + 17.944 R² = 1
50 40
It can be seen from the plots that the experimental data is generally in good agreement with the theoretical values. The partial molar volume of ethanol consistently displays positive errors with respect to the theoretical curve. On the other hand, the partial molar volume for water was coherent with the theoretical values. Most of the experimental data points coincided with the curve produced from literature values. Deviations noticeably increased with the mole fractions and the error peaked at the 5th data point. Finally, minute deviations were only observed with the experimental and theoretical molar volume of the solutions. Looking at figure 7, the 6 data points all fall on the curve that was generated using equation 3. Quantitatively, the ethanol and water partial volume, and solution volume percent errors for each data point are as follows: Table 9 Percent Error for each Data Point ̅ ̅ Solution A 0.879 0.812 B 1.295 0.049 C 1.053 0.353 D 1.055 1.637 E 1.000 3.536 F 1.002 1.532
0.461 0.119 0.477 0.734 0.982 0.907
Reasons for these deviations are mainly due to equipment limitations. The loose, damaged or unfit rubber stoppers of the volumetr c flasks could’ve allowed some of the alcohol to escape via volatilization. Some of the stoppers were too loose and left small gaps along the brim of the flasks. Damaged ones had small holes in certain areas.
30 20 Experimental
10
Theoretical
0 0
0.5 Ethanol Mole Fraction
1
Figure 7 Solution Molar Volume versus Ethanol Mole Fraction with Theoretical Data
Difference in the water bath temperatures and room temperature could’ve also caused the errors for the density measurement via the pycnometer. The volume of the solutions were very sensitive to temperature, such that 10 mL measured at 26.5OC would become significantly less at a lower temperature and greater at a higher one. As such, it is possible to obtain erroneous density values if the temperature of the room and the tap water baths were not equal. This could have been remedied by using a single 6
K.A. Cruz, D.L.C. Fernando, R.A. Soriano, Partial Molar Volume of a Substance, 2014 water bath for all of the flasks and pycnometer, with a built-in heat source in order to keep the temperature of the system constant and close to the room temperature[4].
Figure 8
volumes of the 6 parent solutions incorrect to some degree. 4. Conclusions & Recommendations The procedure was able to generate experimental data for the partial molar volumes of ethanol and water and the molar volume of ethanol-water solution at different ethanol mole fractions that is consistent with the theoretical values obtained from literature. Minimum error was observed with the solution molar volumes, while the ethanol partial molar volumes exhibited constant deviations, ranging from 0.879% to 1.295%. The partial molar volumes of water were in good agreement with the theoretical values, except for a few data points; the error range was 0.049% to 3.536%. The gathered data deviated from the expected values for a number of reasons.
[8]
Recommended Experimental Set-up
Furthermore, adhesion of material to the external and internal surface of the pycnometer would alter the measured masses and, as a result, the solution densities as well. This could take place at several instances in the procedure, such as during the removal of the pycnometer from the water bath for weighing due to improper handling. Failure to sufficiently clean and dry the pycnometer before being placed in the analytical balance would cause positive errors for the mass of the sample and the density of the solution, and negative errors for the volume of the parent solutions. Also, it is possible for water vapor to condense on the surface of the pycnometer if the solutions had a sufficiently low temperature and if the room had a high humidity, as indicated by the wet bulb temperature. After weighing, failure to completely dry the pycnometer after solution disposal and cleaning would also consequently cause the calculated volumes of the parent solutions to increase, since water will make the 10-mL sample solutions less dense. Finally, in line with the previous situation, this could also take place during the measurement of the empty pycnometer mass. This would render the calculated
Procedural mistakes are deduced to be the primary reason for these deviations. Improper handling of the pycnometer prior to weighing, and failure to completely dry the said glass ware after sample solution disposals and even during the initial weighing could all lead to errors in the calculated volumes of the 6 ethanol-water solutions. Difficulties were also introduced by problems with the equipment, such as the unsuitable covers for the volumetric flasks. Uncertainties were brought about by the precision of the instruments, such as the 10-mL pipette and the thermocouple. Nevertheless, the procedure proved to be a decent way of obtaining the partial molar volumes of water and ethanol at a specified ethanol mole fraction. Several modifications can be made to the procedure in order to obtain more accurate results. For future endeavors, it is recommended that a larger container be made available for the water bath of the volumetric flasks and the pycnometer. This will keep the temperature of the flasks and the pycnometer at a constant value throughout the experiment and will minimize any volume contractions and/or expansions due to differences in the temperature of the bath and the room. Furthermore, this negates the necessity of repeatedly 7
K.A. Cruz, D.L.C. Fernando, R.A. Soriano, Partial Molar Volume of a Substance, 2014 removing the flasks and the pycnometer from the bath for the transferring of the 10mL aliquots. Doing so will also decrease the waiting time between weighing trials since the temperature of the solutions are kept nearly constant all throughout. Furthermore, an additional facet can be added to the experiment by performing trials involving electrolytes, such as salt. This will require modifications to the equations that were used due to the increase in solution activity. Pertinent equations for this is the Debye-Huckle equation and the like[5]. The results will also be compared with values derived from theoretical models and literature. 5. References [1]
Smith, J., et. al. (2004). Introduction to Chemical Engineering Thermodynamics, (7th ed.). United States: McGraw-Hill.
6. Appendix [D1] Derivation of partial molar volumes of a binary system The summability relation for volume: ∑ ̅ For results to:
binary
̅ ̅
̅
[4]
Petek, A., Pecar, D. & Dolecek, V. (2001). Volumetric Properties of Ethanol-Water Mixtures Under High Pressure. Acta Chim. Slov., 48, 317-325.
[5]
oučka, F. & Nezbeda, I. (2009). Partial Molar Volume of Methanol in Water: Effect of Polarizability. ResearchGate. doi:10.1135/cccc2008202.
[6]
Armitage, D., et al. (1978). Partial Molar Volumes and Maximum Density Effects in Alcohol–Water Mixtures. Nature, 219, 718-720. doi:10.1038/219718a0.
̅
̅
̅ ̅
̅
̅
̅
̅ ̅
̅
̅
̅
̅
Substituting summability relation,
these
)̅
(̅
(
to
the
)
̅ ̅
( (̅
)
) )̅
(
̅ ̅
[7]
Sakurai, M. (1988). Partial Molar Volumes in Aqueous Mixtures of Nonelectrolytes. II. Isopropyl Alcohol. Journal of Solution Chemistry, 17(3), 267-275.
̅
̅
Chang, R. (2007). Physical Chemistry for the Biosciences. United States: McGraw-Hill. Atkins, P. & De Paula, J. (2006). Atkins’ Physical Chemistry, (8th ed.). Great Britain: Oxford University Press.
expansion
If ̅ is a function of xi at constant T and P, from Gibbs/Duhem,
[2]
[3]
systems,
Sample Calculations 1.) m are in terms of g, ef: empty flask
[8]
PHYWE. (2014). Pycnometer Water-Bath Set-up [Image]. 8
K.A. Cruz, D.L.C. Fernando, R.A. Soriano, Partial Molar Volume of a Substance, 2014 E F
36.7283 36.938
73.3054 76.1301
77.1857 76.1301
Pcynometer Data Solution Trial 1, g Empty 15.8457 A 25.8331 B 25.036 C 24.6501 D 24.2692 E 23.9575 F 23.7195
Trial 2, g 15.8458 25.8463 25.0616 24.658 24.2826 23.9763 23.71
Trial 3, g 15.8456 25.8782 25.0753 24.6819 24.3068 23.9745 23.7279
Table 11 ̃
̃
̃ Δ Δ
Δ
̃ Δ ̃
Δ
2.) n are in terms of mol
̃
̃
̃ ̃
Δ
Δ ( ̃ Δ ̃
̃ Δ ̃
)
̃ Δ
̃
( )
̃
Δ
Raw Data Tables Table 10 Mole Recalculation Data Mass (g)
Solution
A B C D
36.1657 39.5045 36.2671 37.5238
36.1657 59.861 63.2197 70.2786
85.959 85.0793 80.1326 79.5317
9
