Diffusion Coefficient of 2m NACL in Water

This experiment aims to determine the diffusion coefficient of a liquid mixture, sodium chloride solution, in de-ionized water. This is done through the use of a honeycomb diffusion cell which consists of 121 vertical capillaries which are 5mm long and 1mm in diameter. Fick’s Law of Diffusion is used in this experiment to calculate the diffusion coefficient of sodium chloride solution in de-ionized water. The change in concentration of the sodium chloride in the de-ionized water is measured by using the conductivity meter, since sodium chloride dissociates in water. The calculated value of diffusion coefficient was compared with the literature value derived from Eletrochemistry textbook, and the difference in values will be discussed.

Some assumptions are being made to make the calculations much simpler. The concentration at the lower end of the diffusion cell is assumed to be equal to 2M, which is the molarity of the sodium chloride solution. The concentration at the upper end of the diffusion cell is taken to be effectively zero.

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Certain factors that affect diffusion and the calculation of the diffusion coefficient are also discussed to understand more about their effects on diffusion. However, this experiment did not place emphasis on these factors, and these factors are assumed to be negligible in the experiment as steps have been taken to minimise their presence in this experiment.

Objective

The objective of this experiment is to determine the diffusion coefficient of 2M Sodium Chloride solution in de-ionised water.

Principles

1. Diffusion

Molecular diffusion, or simply known as diffusion, is a spontaneous process that results in the net transport of molecules from a region of higher concentration to another region of lower concentration. This net transport is a result of the intrinsic thermal energy present in all molecules at temperatures above absolute zero.

The phenomenon of diffusion can be mathematically described using Fick’s Law (or Fick’s 1st Law), which states that for a two component system consisting of “A” and “B”, the molar flux, Ni , of the individual component (i = A, B) is defined as:

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(1)

where D is the diffusion coefficient , Ci is the concentration of component i, and is the concentration gradient in the z-direction. The negative sign indicates the flow of transport is from region of higher concentration to region of lower concentration. In the context of this experiment, Fick’s law in one-dimension is considered, and there is no mean flow of mixture in the experimental set-up. Therefore, Equation (1) for the molar flux is valid for this experiment.

2. Experimental Set-up

A diffusion cell is an instrument that is used to study the diffusion of liquid or gaseous mixtures. The diffusion cell used in this experiment consists of a honeycomb of 121 accurately dimensioned vertical capillaries that are 5mm long and 1mm wide in diameter. This is to restrict the diffusion to a one -dimensional diffusion.

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To prepare the diffusion cell for use, a small volume of concentrated solution is first placed on one side of the honeycomb, whilst the other side consists of a large volume of pure solvent (de-ionized water). The concentration within the larger volume will increase gradually as diffusion of the solute occurs. This change in concentration is monitored closely with a conductivity sensor and meter. To ensure a uniform concentration within the larger volume, the mixture is continuously stirred with a magnetic stirrer. (Figure 1)

The concentration at the lower end of the tube is in fact constant and is equal to the molarity of the solution. The concentration at the upper end of the tube is effectively zero (Figure 2).

Honeycomb structure with capillaries

Concentration of solution outside tube is zero

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Concentration of solution inside tube is 2M NaCl

Therefore it can be said that:

(2)

where M is the molarity of the solution and L is the length of the tube. The amount of sodium chloride diffusing per unit time from the capillary is approximately:

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(3)

where d is the diameter of the capillary, n is the number of capillaries and L is the length of the capillary.

Outside the tube, since sodium chloride is able to dissociate in water into sodium and chloride ions, the accumulation of the sodium chloride salt in the de-ionized water can be easily calculated by the measurement of change in the conductivity k of the solution with respect to time t. The amount of sodium chloride accumulating per unit time outside the vessel is approximately:

(4)

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where V is the volume of water in outer vessel and CM is the conductivity change for unit molarity change and has the value of 0.41Ω-1M-1 for the solution used in this experiment.

Mass balance requires that the amount of sodium chloride going out of the capillaries must be equals to the amount of sodium chloride accumulated in the de-ionized water.

(5)

Therefore,

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(6)

By rearranging Equation (6),

(7)

All the terms in equation (7) are constants and known, except for . However, to determine, we can plot a graph of conductivity as a function of time and find the gradient, which is equals to. Thus, by plotting conductivity as a function of time, the value of the diffusion coefficient D for the sodium chloride solution can be determined.

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Equipment and Materials

One – litre vessel with cell holder (Figure 3)

Magnetic stirrer with variable speed (Figure 3)

Magnetic stir bar (Figure 3)

Diffusion cell (Figure 4)

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Conductivity meter (Figure 5)

Conductivity leads

Stopwatch (Figure 6)

A bottle of de-ionized water (Figure 7)

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50 ml beaker (Figure 8)

2M sodium chloride solution in a 100ml glass bottle (Figure 9)

Experimental Procedure

The one litre vessel was filled with de-ionized water up to 1cm below the graduation (black) mark.

The magnetic stirrer was set to the lowest setting.

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The conductivity meter was connected to the electrodes and then switched on.

The reading on the conductivity meter was noted to be less than 10-4Ω-1 (1 Ω-1 = 1 Siemens).

The diffusion cell was completely filled with 2M (molarity) sodium chloride solution (116.9g NaCl/litre). Excess solution on top of the capillaries and on the sides was wiped off gently with a soft tissue.

The diffusion cell was clamped in its position with the top of the capillaries lying parallel with the graduation mark on the vessel.

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The vessel was carefully filled to the graduaton mark with the provided de-ionized water bottle. The nozzle of the bottle was aimed away from the capillaries and towards the wall of the vessel, so as to minimise the turbulence caused.

The timer was started immediately when the vessel was filled to the graduation mark. The reading on the conductivity was noted down at the same time.

The conductivity readings were noted down every 200 seconds till 3600s.

After the end of Run 1, the solution in the vessel and diffusion cell was disposed. Both the vessel and the diffusion were washed thoroughly with de-ionized water.

Steps (1) to (10) were repeated to get another set of readings for Run 2.

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Results

The results are tabulated in Table 1.

Questions

Plot conductivity (k) versus time (t) and determine the slope of the best fit straight line through the points.

From the plot, the gradient of slope of the best fit straight line for Run 1 and Run 2 are

4 x 10-8 and 3 x 10-8 respectively.

Determine the diffusion coefficient of 2M NaCl using equation (6).

To use equation (6), we need the value for . The value for is determined from the graph. We will take the average of the gradient of the 2 slopes we get from the graph.

= = 3.5 x 10 -8

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Where,

V = 1 Litre = 0.001 m3

L = 5mm = 0.005 m

CM = 0.41Ω-1M-1

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n = 121

M = 2M

d = 1mm = 0.001 m

(gradient of slope) = 3.5 x 10^-8 (Ωs)-1

For = 3.5 x 10^-8 (Ωs)-1,

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How does your value(s) compare with literature values? Comment on sources of error.

Table 2 is obtained from the literature of an Electrochemistry textbook. The data in the table are obtained by various experimental techniques, such as tracer diffusion determination.

The diffusion coefficient of Na+ and Cl- ions in water, extracted from relevant data in Table 2, are found to be 1.33 x 10-5 cm2/s and 2.03 x 10-5 cm2/s respectively. These ionic diffusion coefficients do not take into consideration the effect of solvation of the water molecules on the ionic species.

From Table 2, both the sodium ion and chloride ion have different diffusion coefficient from each other. However sodium chloride diffuses with only one coefficient, therefore we need to compute the average diffusion coefficient, DNaCl.

From the equation:

(8)

j1 = j2 = – Dc1 = – [] c1

The average diffusion coefficient of the electrolyte NaCl ,

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DNaCl = 2 (9)

(1/DNa+ + 1/DCl- )

= 2

(1/1.33 + 1/2.03) x 105

= 1.607 x 10-5 cm2/s

= 1.607 x 10-9 m2/s

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Percentage difference between experimental and literature values of diffusion coefficient

= (2.246 – 1.607)/(1.607) x 100%

= + 39.8%

The experimental diffusion coefficient values of 2.246 x 10-9 m2/s is slightly higher than the literature value of 1.607 x 10-9 m2/s, and this could be due to some possible sources of error (Will be further discussed in the next section).

The calculated value (DNaCl) based on Table 2, equation (8) and (9) does not take into consideration the effect of hydration of the water molecules on the ionic species. The phenomenon of solute-solvent interaction between sodium chloride and water is called hydration. The solute and the solvent both interact to form a new species, which is the actual species diffusing in the solvent

Hence, it should be expected that our experiment value will be different from the literature value (DNaCl), as hydration do play a part in the experiment. In addition, the high concentration of the sodium chloride solution (2M) may result in convection, which will also affect the rate of diffusion. (Will be further under Discussions)

Possible Sources of Error

S/N

Possible Sources of Error

Solution

1.

If thewalls of the capillary tubes arewet, or if the diffusion tube is filled up too slowly or wrongly, air bubbles may form underthe capillariesin the honeycomb structure. The bubbles can significantly impede diffusion through the capillaries, affecting the rate of diffusion.

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Ensure that no air bubbles are formed in or under the honeycomb structure. This can be achieved by filling the diffusion cell slowly.

2.

Evaporation of water from the 1 litre vessel during the experiment could affect thevolume present in the vessel and hence diffusion coefficient would be affected, since the calculation of the diffusion coefficient involves the volume of water present in the vessel.

The vessel could be covered with a cover or lid to minimise evaporation of water.

3.

The stopwatch should be startedas close as possible to the instant when the water covers the surface of the capillaries. This is because significant changes in concentration can occur at this pointintime due to the steep concentration gradient between the 2 solutions. However, human reaction time will result in some delay.

Have one person to fill up the vessel until the honeycomb structure is fully submerged in water, while another person looks from the side and starts the stopwatch once the honeycomb structure is fully submerged. This will minimise the human reaction time.

4.

The solution may not be well-stirred enough, thus resulting in a higher concentration at the diffusion surface than the rest of the solution.

Have more than one conductivity sensor at various locations or depth along the vessel enables us to get more accurate conductivity readings by averaging the values obtained.

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5.

The needle was fluctuating at some of the instant when readings were supposed to be taken, making it hard to determine the value to be recorded. By the time the needle stabilised, it has already exceeded the stipulated time (e.g. 1000s), and the reading obtained from the meter would not reflect the conductivity of the solution at that particular time but instead for another timing.

A data logging system should be used where signals from the conductivity meter used for the diffusion cell can be recorded into a computer at fixed regular time interval. This helps to eliminate human error from poor judgment in the reading of results as well as allows for a more accurate reading of the conductivity.

6.

The vessel may not be thoroughly cleaned from previous experiment by other students, therefore leaving sodium chloride residue at the wall of the vessel. This may add to the conductivity of the solution during the diffusion experiment when the vessel is filled with de-ionised water and therefore lead to an error when calculating the diffusion coefficient of sodium chloride.

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Clean the equipments thoroughly once more with de-ionised water to ensure that no residues are present on the equipment.

7.

Small air bubbles could have been trapped at the opening of the capillaries of the diffusion cell before the start of the experiment. This would affect the diffusion rate of the sodium chloride solution.

A porous plate can be used. It enables the solute and solvent to pass through freely, minimizing the chance where the molecules are ‘stuck’ at the pores or in the capillaries.

8.

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Convection occurred in the experiment is a major factor that will affect the accuracy of our experiment. Effort has been made to minimized convection. However, the extent in which convection has affected our experiment values is unknown. The rotation of the magnetic stirrer could introduce convection currents in the water.

The addition of dye to the vessel containing watercan be used to determine whether the magnetic stirrer is causing too much convection. This should be done before the commencement of the actual experiment to ensure minimal or no convection are present. The dye should never be used during the commencement of the experiment as it might interfere with the diffusion rate of sodium chloride solution.

9.

The changes in the conductivity may be so small such that the analog meter is not able to detect the changes. The use of an analog meter also depends on how the person judges the values of the readings when the needle is in between the markings on the meter. This might result in random errors.

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A digital conductivity meter or data logger should be used to minimise the random errors that might be introduced if an analog meter is used. A digital conductivity meter or data logger would be more able to reflect the small changes in conductivity of the solution, giving us readings that are more accurate and more precise.

10.

The temperature of the sodium chloride solution in the diffusion cell and the water in the vessel is assumed to be the same and constant throughout the experiment. However, this may not be true due to the fact that the experimental setup is exposed to the surroundings, thus it is very difficult to ensure constant temperature within the system, unlike the case of a water bath. Temperature, however, does affect the diffusion coefficient.

Conduct the experiment in an enclosed environment to minimise contact with the surroundings. Measure the temperature of the water and sodium chloride solution constantly to ensure constant and identical temperature throughout the experiment.

Discussions

Solute-Solvent interaction (hydration)

In the experiment, the species that are diffusing are not just pure sodium and chloride ions. In fact, sodium chloride and water interact to give a new species which is the actual species diffusing in the solvent.

The effect of hydration can be studied by using the following flux equation:

(10)

j1 = – D0 (1 + ) c1

(11)

j1 = – (1 + ) c1

where D0 is the new diffusion coefficient, μ is the solvent viscosity, R0 is the solute radius, and Ɣ1 is the activity coefficient.

Equation (10) and (11) can be affected by hydration in two ways. These two factors are the hydrated species radius R0, and the concentration dependence of diffusion.

From equation (11), the solute radius R0 is that of the hydrated species. This can be related to the true solute radius R0′ by the following equation:

Ï€ R03 = Ï€ (R0′)3 + n() (12)

Where VH2O is the molar volume of water and n is the hydration number.

The third column of Table 3 shows the values obtained from equation (12).

If we were to take the values of Na+ (0.5) and Cl- (-0.7) and compute the average radius R0, the radius obtained will be a smaller value as compared to the case when no hydration is considered. Therefore, we will obtain a larger D0 if we substitute a smaller R0 into equation (11). This explains why we get a larger D0 from experimental results as compared to literature values.

Convection

In our experiment, the sodium chloride solution has a molarity of 2M, which is a concentrated solution. Such concentrated solution may cause convection in our experiment, thus affecting the rate of diffusion.

Total mass transported = Mass transported by diffusion + Mass transported by convection

(13)

The total mass flux n1, (mass transported per unit area per unit time):

n1 = j1a + c1va

where j1a is the diffusion flux and c1va represents the convection.

From equation (7):

(7)

D =

If we add in the convection term, it becomes

(7.1)

D = + (convection)

From equation (7.1), we can see that the presence of convection in the experiment will mean a bigger value for diffusion coefficient since the total mass transported is now dependent not solely on diffusion only. Convection does play a part in the transporting of mass.

In order to examine whether the effect of convection is significant in our experiment, we can carry the experiment with sodium chloride solution at lower concentrations. The diffusion coefficient obtained will then be compared against literature values. If the degree of difference between experimental results and literature values is similar to that of using 2M sodium chloride solution, then we can say that the effect of convection is negligible in our experiment, and vice versa.

Temperature – dependence of diffusion coefficient

In our experiment, the temperature of the water in the vessel and the sodium chloride solution is assumed to be the same and constant throughout the experiment.

The dependence of diffusion coefficient on temperature in liquids can be determined using the Stokes- Einstein equation:

(14)

where T1 and T2 denote temperature 1 and 2 respectively

D is the diffusion coefficient

is the dynamic viscosity of the solvent

From the equation, we can see that temperature does affect the diffusivity of the sodium chloride ions. In order to ensure that this problem does not affect our experiment, we should continuously measure the temperature of the water in the vessel and the sodium chloride solution to ensure identical and constant temperature throughout the experiment.

Conclusions

The aim of the experiment was to determine the diffusion coefficient of 2M Sodium Chloride solution in de-ionised water. The purpose of the experiment is achieved through the use of diffusion cell and the conductivity meter to study the change in concentration along the vessel at regular intervals.

The value of our experimental result is of the order of 10-9 m2/s, which is common for liquid mixtures. This implies that the method of using honeycomb diffusion cell and measuring the change in conductivity to calculate the accumulation of the sodium chloride in the de-ionized water is appropriate for the determination of diffusion in a liquid mixture in this experiment. However, one should be aware that the conductivity method is only applicable for solute which fully dissociates in water. Solutes which do not dissociate in water will not give any changes in conductivity of the de-ionized water.

It was found that the diffusion coefficient determined in the experiment is 39.8% larger than the literature values. This is most likely due to the fact that the literature values did not consider the effect of hydration on the sodium and chloride ions. The effect of hydration tends to increase the rate of diffusion, and this was the case for sodium chloride solution. Therefore, if we were to take into consideration the effect of hydration on the ions in our calculation of the literature values, the diffusion coefficient calculated will be close to our experimental value.

We have discussed that convection is an important factor that might affect the reliability and accuracy of the experimental results. Due to the limitations of the experiment procedure, we were unable to determine the extent in which convection affects the experiment. Dyes could have been used to detect the presence of any significant convection currents due to the rotation of the magnetic stirrer.

This experiment has showed that it is very difficult to calculate the actual diffusion coefficient very accurately using mathematical models and scientific concepts that we currently have. Due to the fact that the values of diffusion coefficients for liquid mixtures are very small, of the order of 10-9 m2/s, any slight error during the experiment will result in a big difference from the actual value. This difference might be small in magnitude, but the application of this value may have large implications in actual situations, such as in chemical plants. Therefore, much effort should be utilised for future research works to accurately determine the actual diffusion coefficients under different situations and conditions.

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