Abstract



BE309 Project:

Potentiometric Titration

Dr. Litt

T4

December 9, 1997

George Bell

Satyam Sarma

Greg Saybolt

Keng Shi Mahesh Swaminathan

Abstract

The concentrations of ferrous sulfate solutions were determined using potentiometric titration referenced to a silver/silver chloride electrode. Five solutions of FeSO4 (titrate), ranging in molarity from 0.015 M to 0.1 M, were prepared and titrated with potassium dichromate (titrant). An Ag/AgCl electrode and a platinum electrode were used to monitor potential changes of the solution during titration. Potential as a function of dichromate volume was plotted for each solution. The end point of the titration was found by taking the derivative of the voltage-volume plot and finding the largest increase in slope. The experimental molarity of the iron sulfate solutions was calculated two methods: the volumetric method, based on the volume of dichromate needed to reach the equivalence point and the potential method, based on the potential of the solution at the equivalence point. Using the volumetric method, the average percent error associated with the experimental iron molarities was 14.5% ( 5.1%. Using the potential method, the average percent error of the iron solution molarities was 1.13E22 %. The systematic error for the volumetric and potential methods was 1.6% and 2.0E6 % respectively. The systematic error for both methods could not entirely account for the high random error. The sensitivity of both methods was analyzed and the major source of error for the volumetric method was the volume of the ferrous sulfate solutions. The major source of error for the potential method was the measured voltage potentials.

Background

Biological cells contain large concentrations of electrolytes and behave as complex electrochemical systems. Understanding simple electrochemical systems, such as an electrochemical cell, is an important first step to understanding the electrochemical nature of biological cells. Potentiometric titration is admirably suited for this instructional purpose. It calculates the concentration of ions in a solution using the measured electric potential of the sample. A variety of instruments and equipment used in biological engineering applications are based on this technique of measurement.

The experimental setup in this experiment is equivalent to an electrochemical cell (figure 1).

Figure 1

[pic]

Figure 1: Experimental Apparatus

The species of interest in this experiment, FeSO4, ionizes when placed in water.

FeSO4 ( Fe+2 + SO4-2 (1)

When the Ag/AgCl electrode is placed in the ferrous sulfate solution, two cell reactions take place:

Fe+3 (aq) + e- (aq) ( Fe+2 (aq) EoR1 = .771 V (2)

AgCl (s) + e- ( Ag (s) + Cl- (aq) EoL1 = .222 V (3)

These reactions are equivalent to:

Fe3+ + Ag (s) + Cl- (aq) ( Fe2+ (aq) + AgCl (s) Eo1 = .549 V (4)

The cell reaction gives a measurable potential.

The measured potential is a function of only the activity of Fe+2 and Fe+3 ions in the solution since the reactants in the reference electrode stay at constant concentration through the experiment. Activity is a function of concentration. When the concentration of either iron species is changed, a change in potential reading can be measured. This change in potential can be used to measure the concentration of the test solution.

When K2Cr2O7 titrates with Fe+2 ion in the test solution two additional half-cell-reactions take place.

Cr2O7-2(aq) + 14H+(aq) + 6e-(aq) ( 2Cr3+(aq) + 7H2O(l) EoR2 = 1.232 volt (5)

Fe3+(aq) + e-(aq)( Fe2+(aq) EoL2 = 0.771 volt (6)

These two half-reactions are combined to form the following overall reaction.

6 Fe2+(aq) + Cr2O7-2(aq) + 14H+(aq) ( 2Cr3+(aq) + 6Fe3+(aq) + 7 H2O(l)

Eo2 = 0.461 volt (7)

As K2Cr2O7 is added to the solution, a proportional amount of Fe+2 is oxidized to Fe+3. If sufficient K2Cr2O7 titrates with the Fe+2 ions, all Fe+2 ions will convert to Fe+3. The point, where the amount of K2Cr2O7 added is sufficient to completely convert Fe+2 to Fe+3 is the equivalence point of the titration. By finding the equivalence point, the amount of K2Cr2O7 added to the solution at the equivalence point can then be used to calculate the original amount of FeSO4 in the solution.

In a more traditional titration, an indicator is added to the system that changes color when the equivalence point is reached. In this experiment, the measured solution potential is used to determine equivalent point. This is the potentiometric titration.

When K2Cr2O7 is added incrementally to the solution and the potential of the solution measured after each addition, a titration curve of potential vs. volume of K2Cr2O7 added can be plotted (Figure 2).

[pic]

Figure 2: Sample Titration Curve

At the equivalence point, Fe+2 has been entirely converted to Fe+3. Equation (7) is not longer the dominant titration reaction with further addition of dichromate solution. The addition of another aliquot of titrant will cause a dramatic jump in the measured potential of the system due to the switch of dominant cell reaction. By computing the slope of the titration curve at each point on the titration curve, the equivalence point will take place with the greatest slope. In more precise mathematical terms, the equivalence point occurs at the point where the derivative of the titration curve approaches infinity. Upon identifying the equivalence point, the original concentration of ferrous sulfate can be calculated using the stoichomatric coefficient of equation (7), the concentration of dichromate titrant, and the volume of titrant used.

Another application of potentiometric titration is that it allows the computation of the concentration of each species at various points in the titration using the Nernst equation and the measured potential. Referring to Figure 3, a titration curve consists of three regions important to this discussion: the buffer region, equivalence point, and over titration region.

The concentration of the iron species in the buffer region can be computed using the Nernst equation for Fe+2/Fe+3. (All the solutions in the Nernst equations are assumed ideal. Hardy, 1997)

[pic] (8)

At the beginning of titration, no significant levels of Cr2O7-2 exist because it is assumed the entire aliquot of Cr2O7-2 reacts with Fe+2 to form Fe+3. The equation took only into account the effects of iron on the electrode. Dichromate can also affect the electrode. Reaction (15) gives the reaction between dichromate and the Ag/AgCl electrode. This reaction is not taken into account in order to build a simpler model of the electrochemical reaction that occurs in the buffering region. The dominant reaction is the conversion of Fe+3 into Fe+2. Any dichromate added is assumed immediately converted to Cr+3 by reacting with Fe+2. Thus, the conversion of dichromate into Cr+3 does not leave any dichromate to react with the electrode. This assumption breaks down as the concentration of Fe+2 decreases near the equivalence point. There is less Fe+2 for the dichromate to react with and instead, the dichromate reacts with the electrode. As a result, the potential of the solution quickly rises. The expected voltages calculated for the solution are accurate at low concentrations of dichromate added, where Fe+2 is in relatively high concentrations. The error in the calculated voltage increases as the volume reaches the equivalence point.

Equation 8 permits the computation of the concentration of either iron species in the buffer region given the measured potential.

The potential at the equivalence point can be calculated using the Nernst equation. The Nernst equations for reaction (5) and (6) respectively at the equivalence point are:

[pic] (9)

[pic][pic] (10)

By adding equation (9) to (10) gets an expression that relates the equivalence point to all the reactants in the system.

[pic] (11)

The equivalence point that the following relationships between Fe+2, Fe+3, Cr2O7-2, and Cr+3 is the following (assume reaction (7) is complete):

[pic]

With these relationships applied into equation (10), the result is an expression that can be used to compute the electric potential at equivalence point.

[pic] (12)[1]

Equation (12) shows that the potential at the equivalence point is a function of the concentration of Cr+3 and H+ only. The derivation of equation 12 assumed all solutions to be ideal and the reaction goes to completion. These assumptions apply best when the solutions of interest are very dilute solutions. Further more, equation 12 required the concentration of H+ ion to stay constant at 1 M. This assumption only applies when H+ ion presented in the solution is in great excess compared with other reaction species.

In the over titration period, the concentration of Cr+3, Cr2O7-2, and H+ become the important factors that determine the potential of the solution. The dominant helf-cell reactions becomes:

Cr2O7-2(aq) + 14H+(aq) + 6e-(aq) ( 2Cr3+(aq) + 7H2O(l) EoR2 = 1.232 volt (13)

AgCl (s) + e- ( Ag (s) + Cl- (aq) EoL1 = .222 V (14)

which is equivalent to:

Cr2O7-2(aq) + 14H+(aq) + 6Ag (s) + 6Cl- (aq) ( 2Cr3+(aq) + 6AgCl (s) +7H2O(l)

E = 1.009 V (15)

All of the Fe+2 species in the sample have converted to Fe+3 ions. The Fe+3 species no longer causes a change in the measured potential because the ion stays at a constant concentration. Equation (10) can be used to compute the potential of points in the over titration region.

Methods and Materials

Reaction (7) gives the stoichiometric relation of K2Cr2O7, FeSO4 and HCl. 400 ml of .083 M K2Cr2O7 was made from solid K2Cr2O7 and de-ionized water. The concentrations of the FeSO4 solutions were made so they would require between 3 ml and 20 ml of K2Cr2O7 to reach an equivalence point. Excess HCl was added to the ferrous sulfate solution. The FeSO4 were titrated immediately to limit the side oxidation reaction of Fe+2 ion by oxygen.

A 50-ml buret was used to add the titrant, K2Cr2O7, to the titrate, FeSO4. Two AgCl/Ag reference electrodes and a platinum electrode attached to a multimeter were placed inside the ferrous solution. Incremental aliquots of titrant were added to the test solution. The multimeter recorded the electric potential of the solution after the addition of each aliquot and plotted against the volume of dichromate used in the titration.

The platinum electrode was constructed with a three-foot strand of platinum wire with small diameter purchased LRSM and a glass stir. The wire was folded and then twisted to form a thicker 1-ft wire. This increased the surface area of the wire while providing sufficient length to make the electrode. This process was cheaper than buying a wire of comparable length and diameter.

Six inches of the wire were tightly wrapped around one end of the six-inch, standard, glass stir. This maximized the surface area of the platinum exposed to the solution. The remaining six inches of the wire extended to the other end of the stirring rod. 12-inch piece of plastic-covered copper wire was soldered to the free end of the platinum wire to the rod. The other end of the insulated wire was attached to a metal lead used to plug the electrode into the multimeter.

Results

Color Observations during Titration

As the titrant, potassium dichromate (orange color) is added to the titrate, ferrous sulfate, a color change is observed in the ferrous sulfate solution. The color starts at pale green, and gets darker as more iron is oxidized and more chromium III is produced. The color becomes a mix of green and brown at the equivalence point. The color of the solution becomes increasingly brown as the titration is continued past the equivalence point.

Determination of Equivalence Point

Two methods were used to find the concentration of the iron solutions: the volume method and the potential method. The volume method uses the titrated volume of potassium dichromate to determine the concentration of ferrous sulfate. The potential method uses the voltage measured by the multimeter to compute the unknown concentration of ferrous sulfate.

Volumetric Method

The volumetric method uses the volume of dichromate added at the equivalence point to calculate the iron concentration of the test samples. The equivalence point of titration is determined by examining the rate of potential change at each point on the titration curve. Table 1 gives information for a titration of a 0.1M concentration of ferrous sulfate.

Table 1

|Concentration of potassium dichromate solution |0.0832 M |+/-.00013 |

|Molarity of iron sulfate | |0.1M +/-.0012 |

|Molarity of HCl | |1 M+/-.005 |

|Grams of potassium dichromate |9.795 g+/-.005 |

|Water added (ml) | |400 ml+/-.0004 |

|Grams of iron sulfate | |2.778 g+/-.005 |

|Water added (ml) | |100 ml+/-.001 |

|Volume of HCl added (ml) |50 ml+/-.0005 |

Table 1: Molarities and solution volume for ferrous sulfate titration. The molarity of ferrous

sulfate and dichromate solutions are given.

The titration curve for the data in Table 1 is shown in Figure 3. Dichromate was added at 4-ml intervals for the early points of the titration. As the titration approached the equivalence point, the titrant was added in smaller increments.

Figure 3

[pic]

Figure 3: The titration curve for a 0.1M ferrous sulfate, titrated by a 0.0832M-dichromate solution. The early points were added in increments of 4 ml until the titration reached closer to the equivalence point, where smaller increments were used. Electrode 1 and 2 refer to the two reference electrodes immersed in the solution. (Electrode 1 lies behind Electrode 2 and is barely visible.)

Figure 3 gives the titration curve for two Ag/AgCl electrodes. Both reference electrodes were immersed in the solution. The purpose of using two electrodes is to determine the consistency of the electrodes. Both electrodes gave similar potentials, differing by no more than 4 mV at any given point. The equivalence point was found by taking the derivative of the curve and finding the derivative’s maximum value. Table 2 shows the data around the equivalence point of the plot.

Table 2

|Volume of | |Delta V1 |Delta V2 |Delta ml |Slope 1 |Slope 2 |

|dichromate | | | | |mV/ml |mV/ml |

|22.5+/-.05 |-9+/-.74 |-9+/-.74 |-1+/-.1 |9+/-.30 |9+/-.31 |

|23+/-.05 |-17+/-.74 |-17+/-.76 |-0.5+/-.1 |34+/-1.2 |34+/-1.1 |

|23.5+/-.05 |-52+/-.76 |-53+/-.76 |-0.5+/-.1 |104+/-3.1 |106+/-3.12 |

|24+/-.05 |-137+/-.77 |-138+/-.77 |-0.5+/-.1 |274+/-4.6 |276+/-4.34 |

|24.5+/-.05 |-8+/-.77 |-8+/-.77 |-0.5+/-.1 |16+/-.40 |16+/-.42 |

|25+/-.05 |-4+/-.78 |-3+/-.78 |-1+/-.1 |4+/-.32 |3+/-.29 |

Table 2: The dichromate volume changes cause a voltage change in the solution. The equivalence point corresponds to the point where the greatest change in potential occurs with the addition of a small amount of dichromate. The largest slope is found by dividing the change in measured potential between two points and the change in dichromate volume. Delta V1 and V2 are the voltage changes for electrode 1 and 2 respectively. Delta ml is the volume change of dichromate. The largest change in slope occurs between 23.5 ml and 24 ml.

Delta V1 and V2 are the voltage changes between two points for electrode 1 and 2 respectively. The voltage change divided by the volume change between the two points gives the derivative of the titration curve. The greatest slope of voltage vs. volume graph is at the equivalence point. This occurs between the points 23.5 ml and 24 ml of dichromate added. The dichromate volume needed to reach the equivalence point was taken as the average of the two ends and was calculated to be 23.75ml. Table 3 gives the dichromate volumes needed to reach the equivalence point for other titration curves. Comparisons are made between the experimental volume of dichromate used and the expected volume. The molarity of iron sulfate solutions was also calculated from the dichromate volumes and compared to the original prepared iron sulfate molarity.

Table 3

|Expected |Expected |Experimental |Experimental Iron |Percent Error of Iron Molarity |

|Iron Molarity |Dichromate |Dichromate Volume |Molarity | |

| |Volume | | | |

|(M) |Ml |ml |(M) |% |

|0.20+/-.0090 |20.0+/-.1664 |22.75+/-.05 |0.2247+/-.0068 |12.4 |

|0.10+/-.0035 |20.0+/-.167 |23.75+/-.05 |0.119+/-0.0020 |19.0 |

|0.05+/-.0032 |10.0+/-.084 |12.25+/-.05 |0.061+/-.0034 |22.0 |

|0.03+9.6E-05 |25.0+0.036 |29+0.05 |0.034+9.082E-05 |13.3 |

|0.03+/-.0023 |6.1+/-.075 |6.75+/-.05 |0.0337+/-5.023E-4 |12.3 |

|0.015+/-.003 |3.02+/-.028 |3.25+/-.05 |0.0162+/-5.67E-4 |8.0 |

| | | |Average |14.5% + 5.1 |

Table 3: Comparisons between expected iron sulfate molarities and experimental molarity values. The expected molarities were obtained from the amount of iron sulfate weighed on a Mettler balance divided by the solution volume. The expected dichromate volume is the amount of dichromate needed to reach the equivalence point calculated from the amount of moles of iron in the solution. The experimental volume of dichromate used was higher in all trials compared to expected volumes. The molarity of the iron solution was calculated from the experimental volume of dichromate.

Table 3 compares the expected values for iron sulfate molarity to the experimental values. The expected dichromate volume is the amount of dichromate that is needed to reach equivalence point based on the prepared iron sulfate molarity. The experimental volume of dichromate to reach equivalence in all trials was higher than the calculated amount. The iron sulfate molarities calculated based on the experimental values of dichromate were also higher than the expected iron sulfate molarity. The average percent error of the iron sulfate molarities was 14.5% + 5.1. The percent deviation from expected molarity ranges from 8.0% to 22.0%.

Potential Method

Voltage offers another parameter that can be used to compare the experimental titration curve to the expected titration curve. The expected curve can be derived piecewise by dividing the titration curve into four regions and applying different potential equations for each region. Four regions of the titration curve were analyzed: initial conditions, buffering region, equivalence point, and over-titration. The expected potential readings were compared to actual experimental data. Figure 4 plots the expected titration curve and the experimental curve for a 0.03 M ferrous sulfate solution.

Figure 4

[pic]

Figure 4 labels the four regions of the titration curve. A different equation was used to calculate the expected voltage for each region. Each region and their respective equations will be analyzed separately.

Initial Point

The solution, containing FeSO4 and HCl, had a measurable potential before the addition of K2Cr2O7. The active species in the solution were Fe+2 and Fe+3. Equation (8) was used to calculate the expected and experimental concentrations for each ion. The initial voltage readings of the iron sulfate solutions and the computed concentrations of Fe+3 ions appear in Table 5.

Table 5

|Solution |Initial |Expected Fe+3 (nM) |Experimental Fe+3 (μM) |Experimental Fe+2/Fe+3 |

| |Potential (mV) | | |Ratio |

|1 |357 |1.052 |113.6 |1759 |

|2 |354 |0.0526 |49.58 |2015 |

|3 |364 |0.0263 |38.02 |1314 |

|4 |379 |0.0158 |39.32 |762 |

|5 |403 |0.00789 |50.00 |299 |

Table 5: The potential readings of the ferrous sulfate solutions prior to titration with dichromate. The initial potential represents the experimentally measured potential of the solution. The experimental concentration of Fe+3 was higher than expected. This may be the result of oxidation caused by the presence of oxygen in the solution.

The expected Fe+3 concentrations at the initial point were calculated using equation (8). It was assumed that the initial ferrous sulfate solution would be at equilibrium, i.e. E = 0 and Fe+2/Fe+3 ratio = 1.9E9. Since the initial number of moles of iron was known, the expected distribution of Fe+3 to Fe+2 could be calculated.

All experimental ferrous sulfate solutions show a higher concentration of Fe+3 compared to expected concentrations. The experimental concentration of Fe+3 is three orders of magnitude of higher than expected concentration. The experimental Fe+2/Fe+3 ratio is lower than the expected ratio of 1.9E9. The relatively high amount of Fe+3 at the initial point may be a result of oxidation by oxygen present in the test solutions.

Buffer Region

The buffer region is the potential that is dominated by the iron species (Fe+2, Fe+3) interacting with the Ag/AgCl electrode. Using equation (8), the expected potential for different Fe+2/Fe+3 concentrations can be calculated. The expected concentration of Fe+2 ion at each point of the titration is calculated based on knowing the initial Fe2+ ion present and the amount of dichromate ions added. As dichromate is added, it is assumed that the reaction between dichromate and Fe+2 goes to completion, producing Fe+3 and Cr+3 ions. The ratio of Fe+2/Fe+3 can be used to calculate the expected voltage (equation (8)). Figure 4 gives the expected voltage and the experimental potential as functions of titrant volume added in the buffer region. The experimental potential for the buffer region deviated from the calculated expected potential. Table 6 gives the average error between expected and experimental potentials in the buffer region for five trials.

Table 6

|Solution |Fe Molarity |% Difference from buffer region|

|1 |0.1 M |11.8% + 1.8 |

|2 |0.05 M |9.0% + 1.0 |

|3 |0.03 M |11.0% + 1.0 |

|4 |0.2 M |11.0% + 1.8 |

|5 |0.015 M |8.0% + 0.4 |

All five solutions show a similar trend of 8% to 11% deviation from expected potential readings.

Equivalence point

The equivalence point potential was determined from the titration curve as the voltage corresponding to the halfway point of the steepest slope. The equivalence point found experimentally was compared to the expected equivalence point, calculated using equation (12). Each solution had a different [H+] and [Cr+3] concentration and so the equivalence, causing the potential of at the equivalence point for each solution to vary. Table 4 compares the experimental equivalence points.

Table 7

|Molarity (M) |Expected Equivalence Potential (mV)|Equivalence Potential (mV) |% Error in potential (%) |

|0.200 |851 |700.5 |17.7% |

|0.100 |848 |728 |14.2% |

|0.050 |878 |656.5 |25.2% |

|0.030 |829 |737 |11.1% |

|0.015 |899 |615 |31.6% |

| | |Average |20% |

Table 7: % error between expected equivalence potential and experimental equivalence potential. The expected equivalence potential is calculated based on equation (12) in background section. The average % error between expected and obtained potential is 20%.

Table 7 shows the percent difference between the expected equivalence potential experimental potential. The expected equivalence potential was calculated using equation (12). The average percent error between the expected and experimental potential is 20%.

Theoretically, the potential equations can be used to solve for concentration of ions if the voltage is known. Each experimental point on the titration curve can supply the necessary data for potential to calculate iron concentration. The equivalence point potential was used to calculate the original concentration of the iron molarity solution. When applying equation (12) to calculate the original concentration of ferrous sulfate, the concentration of H+ ion is assumed constant in order to overcome the limitation of computation software. The assumption is not quite valid because the initial H+ concentration is not large enough so its change during the titration is not negligible (see discussion section). The calculated concentration of ferrous sulfate using equation (12) is compared with the expected concentration. The result is tabulated in Table 8.

Table 8

|Expected Molarity (M) |Calculated Molarity (M) |% Error in concentration (%) |

|0.200 |6.58E+10 |3.29E+11 |

|0.100 |3.69E+07 |3.69E+08 |

|0.050 |1.05E+16 |2.10E+17 |

|0.030 |3.18E+06 |1.06E+08 |

|0.015 |8.48E+20 |5.65E+22 |

| |Average |1.13E+22 |

Table 8: % error between expected molarity of ferrous sulfate solution and calculated molarity using equation (12). The concentration of ferrous sulfate can be calculated from equation (12). Comparing the expected molarity with the calculated molarity, the average % error is 1.13E+22 %.

Table 8 shows the % error between expected molarity of ferrous sulfate solution and the calculated molarity from equation (12). Comparing the expected molarity with the calculated molarity, the average % error is 1.13E+22 %. The high % error is believed to come from the exponential and power terms in the equation. A small error in the concentration of H+ ion has a great impact on the calculated molarity [H+] is raised to the 14th power in equation (12). Small errors in voltage readings also lead to gigantic error in calculating the ferrous solution’s concentration because of the logarithmic term in equation (12).

Over-Titration

Figure 4 compared the expected voltage and experimental voltage in the over-titration region. The over-titration region is where dichromate molecules dominate the reaction with the Ag/AgCl electrode. The solution is primarily Fe+2 and the dichromate reacts directly with the electrode. Table 9 gives the percent difference between expected and experimental potentials in the over-titration region.

Table 9

|Solution |Fe Molarity |% Difference from buffer region|

|1 |0.1 M |21.0% + 0.8 |

|2 |0.05 M |28.0% + 2.6 |

|3 |0.03 M |26.0% + 0.9 |

|4 |0.2 M |21.0% + 0.8 |

|5 |0.015 M |33.0% + 2.4 |

The percent difference in the over-titration region was higher than the percent difference in the buffer region. This can also be seen in Figure 4 where the expected curve is much higher than the experimental curve in the over-titration region than in the buffer region. The percent difference varies between 21% to 33%.

Discussion

Color Observations during Titration

As the titrant, potassium dichromate (orange color) was added to the ferrous sulfate, a color change was observed in the ferrous sulfate solution. A change from clear to green indicated the presence of chromium (Cr+3). The color started at pale green, and became darker as more iron was oxidized and more chromium III produced. As more dichromate was added, the color became a mix between green and brown when all of the iron was oxidized (at the equivalence point). The brown color was a result of the mixing of the orange color of the dichromate and the green of the chromium. As the titration continued past the equivalence point, the color became increasingly brown as the orange dichromate mixed with the green chromium.

Volumetric Method

Table 3 shows the iron molarities calculated using the volume method. It was found that the average percent deviation of the experimental concentration from the expected concentration of each solution was 14.5% ( 5.1%. The percent deviation was in the range from 8.0% to 22.0%. The experimental concentration for each sample was higher than the expected concentration. The experimental molarities predicted a higher amount of iron in the solutions. The systematic error associated with this method ranged between 0.41% and 3.63%. The systematic error of the volumetric method cannot fully account for the high percent error leaving a significant random error.

The high random error is believed to come from several possible sources. The concentration of solutions made depend on the random error in measuring volume of water added and chemical amount added. To better analyze how the effects of random chemical impurities affect the percent error, a sensitivity analysis was carried out. A sensitivity analysis would determine the affect a parameter had on the experimental and expected values. A certain parameter was changed and all other parameters were held constant. This was done for each parameter determine the relative effects of each on the calculated values. Table 10 shows the change in calculated iron molarity when individual parameters are changed. The variables K2Cr2O7, FeSO4, volume of water to dilute both the dichromate and ferrous sulfate were all changed by an amount equal to their systematic error.

Table 10: Percent Deviation

|Species Changed |Expected K2Cr2O7 M |Expected FeSO4 M |Expected K2Cr2O7 Vol. |Experimental FeSO4 M |

|0.005g K2Cr207 |0.051% |0% |0.051% |0.051% |

|0.005g FeSO4 |0% |0.36% |0.34% |0% |

|0.4 ml H20 with K2Cr207 | | | | |

| |0.10% |0% |0.10% |0.10% |

|1.0 ml H2O with FeSO4 | | | | |

| |0% |1.00% |0% |1.00% |

Table 10: The percent deviation from the original value after the change of the indicated measurement by the systematic error is shown. Reaction (7) was used to calculate the original values and values after incremental change. The difference between the values before and after the incremental change was made gave the percent deviation.

The result of sensitivity analysis shows that the biggest potential source of random error in this method comes from the molarity of the initial FeSO4 solution. If the volume of water added to the FeSO4 is off by 1.0 ml, the expected molarity of the solution will be off by 1%. A more accurate method of measuring the volume of titrate would greatly improve the accuracy of the lab. An error in the measured volume of titrant would have 10% the effect that equivalent errors in titrate would on the experimental concentration.

The second significant potential source of random error comes from a change in mass and purity of the iron sulfate. A change of 0.05 g of iron sulfate effects the calculated concentration by 0.35%. The change in mass (in term, purity) had twice the effect that a corresponding change in mass of the potassium dichromate had. Overall, the concentration of iron sulfate affects experimental accuracy more than the concentration of potassium dichromate does. Using more accurate measurement tools for the mass and volume of iron sulfate or increasing titrate mass and volume to damp the fixed systematic errors of the measuring tools would improve the accuracy of the lab.

Another possible source of error that could account for the 14.5% random error would be the amount of [H+] in the solution. [H+] ions were added to the solution in excess, but not in large enough excess. In all of the trials, at the end point of the titration, over 20% of the hydrogen ions in the solution initially were consumed. The hydrogen ions could have been a rate-limiting step and therefore could have accounted for the excess amount of dichromate needed to reach the endpoint.

Potential Method

This method was used to calculate the expected voltage readings at each point along the titration curve. This gave another method of comparing experimental to expected data. To analyze the titration curve, it is broken into four regions: initial condition, buffering, equivalence point, and over titration regions. In this experiment, the focus will be just on the initial condition, buffering region, and the equivalence point.

The initial condition refers to the ferrous sulfate solution before dichromate is added. The solution is composed of FeSO4 and the Ag/AgCl electrode only. The buffering region is the lower region of the titration curve. The addition of dichromate produces minimal increases in potential due to buffering effects of the ferrous sulfate. The equivalence point corresponds to the volume of dichromate that produces the largest change in potential. The over-titration region occurs after the equivalence point and represents the Ag/AgCl electrode interaction with excess dichromate.

Initial Condition

Table 7 shows the amount of Fe+3 present exceeds the expected value by more than 1000 times. The solutions with lower concentrations of ferrous sulfate have a lower Fe+2/Fe+3 ratio (Table 7). This trend implies that the amount of dichromate required to reach an equivalence point should be lower than the expected volume because of the smaller amount of Fe+2 in solution. This was not the case. Table 3 shows that the experimental volume of dichromate at the equivalence points for all trials were greater than the expected value for all trials.

Buffer Region (volume of dichromate added less than equivalence point)

As seen in Figure 4, the percent error between experimental and expected potential is small in the buffer region. The average percent error is around 11.0 % + 0.6%. This percent error is in the same magnitude as the percent error from the volumetric method. The expected voltages calculated for the solution are accurate at low concentrations of dichromate added, where Fe+2 is in relatively high concentrations. The error in the calculated voltage increases as the volume reaches the equivalence point, where Fe+2 is in low concentration.

A source of the observed percent error in the buffer region is from the assumption that reaction goes to completion. In the buffer region, no dichromate is assumed to be in the solution, only Cr+3. However, dichromate can also affect the electrode. Reaction (15) gives the reaction between dichromate and the Ag/AgCl electrode. This reaction is not taken into account in order to build a simpler model of the electrochemical reaction that occurs in the buffering region. The assumption breaks down as the concentration of Fe+2 decreases. At the equivalence point, there is less Fe+2 for the dichromate to react with, and instead, the dichromate reacts with the electrode leaving higher error to the model.

Equivalence Point

Equation (12) applies to the system at the equivalence point. By using equation (12) at the equivalence point, the concentration of the initial Fe+2 ion can be calculated. This method yields an average of 1.13*1022 % error between calculated concentration and the known concentration. The outrageous % error is believed to come from the nature of the equation. In equation (12), the concentration of the H+ ion is raised to the power of 14. Thus a slight alteration in the concentration can greatly affect the calculated Fe+2 concentration. The equation assumes H+ ion is in great excess and that its concentration does not change throughout the titration. This assumption does not hold in this experiment because the amount of H+ ion present is in excess, but not in sufficient excess. The H+ ion concentration serves as the rate limiting step in the oxidation of Fe+2. Overall, the potential method to calculate the expected voltages does not work if the [H+] concentration is low.

Another major source of error that leads to the gigantic % error value is the error associated with the potential reading. In calculating the Fe+2 concentration using equation (12), the electric potential term becomes the power term of base 10. Again, a small error in the potential reading can greatly affect the accuracy of calculated Fe+2 concentration. The electrode used in this experiment is incapable of reading electric potential when the ion concentrations are too low.

Since the % error of concentration is a biased indicator of the model’s performance (due to the nature of the equation), further analysis is done by calculating expected potential readings and comparing them with the experimental potential readings.

The average % difference between the expected equivalence potential and the experimental potential is 20% (Table 7). The 20% difference is compared with the systematic error of 4.43mV. The average % difference is considerably high when compared to the systematic error of 4.43 mV, suggesting that the experimental outcome deviates significantly from the model. The problem may lie in the derivation of the mathematical model. The equations derived using the Nernst equation assume that all solutions are ideal. This approximation is more appropriate for dilute solutions. The solutions used in this experiment may be too concentrated for this assumption to be rationally accurate.

Another source of error is the experiment procedure. Due to time constraints, potential readings were not given enough time to stabilize; each data point was taken with a time interval of 2 to 3 min. This source of error is less important when using the volumetric method to compute the concentration of the test solution because the volume method relies upon changes in potential rather than absolute potential readings. The change in potential at the equivalence point is usually very rapid and most of the potential change occurs in 2 minutes (the time given for stabilization of potential readings). The unstable nature of the potential readings becomes more important when the potential method is applied to compute the concentration (in this case the analysis is replaced by computation of electric potential) of the test solution because it uses absolute potential readings (see equation (12)). With the given equipment and time constraints, the volume method is the better method

Lastly, from Table 10, it is obvious the high average % difference is due to the high % difference from the very diluted solutions. This phenomenon is expected since at low concentration, the system is most sensitive to volume of dichromate added to ferrous solution. The incremental dichromate added to ferrous solution is set to be 0.5 ml near the equivalence point. Because the incremental volume added is quite high for such diluted ferrous solution, the equivalence point volume calculated by the average of the point before and after the equivalence point can greatly underestimate the real electric potential at the equivalence point. To justify this argument, a sensitivity analysis was performed. By shifting the equivalence points in the two diluted solutions by 0.15 ml (for 0.015M solution) of average dichromate solution added, the average % difference drops by 1%. Also in calculating the average equivalence potential for 0.05M ferrous solution, it is realized that the average experimental potential reading is extremely sensitive to volume of dichromate added; the tangential slope of the titration curve, delta potential/ delta dichromate added, is 255 mV / 1 ml dichromate volume. If the % difference associate with 0.05M ferrous solution is also taken into account by shifting the average volume of dichromate added by 0.2 ml, the overall average % difference is reduced by 1.98% and becomes 18.0%.

The three sources of error contribute most to the observed % error. It is critical to be aware that due to insufficient data in this experiment the validity of this model can neither be rejected nor supported. The experimental outcomes, however, do show a high possibility that the model is too crude to be applied without using highly sensitive instruments.

The potential method should not be used to calculate the initial concentration of Fe+2 due to the high systematic and random error associated with this method. Instead, the volumetric method provides a more accurate means to determine Fe+2 concentration. The potential method should not be completely discarded because it can be used to calculate expected voltages along the titration curve.

Future recommendations

The molarity of the ferrous sulfate changes while the molarity of the potassium dichromate remains constant. High concentrations of ferrous sulfate gives the possibility of better results, since the equivalence volumes are closer to each other relative to the total volume used. The lower concentrations use a smaller amount of dichromate, so each addition of titrant will have more of an effect on the reaction. The electrode also has a lower limit to the concentration of the ferrous sulfate.

A better apparatus to determine iron concentrations would be an Iron ion specific electrode (ISE). The electrode is similar to a pH meter in that it detects one type of ion. The ISE would be more accurate in monitoring changes in iron potentials in solution, and would be unaffected by the presence of potentially active impurities. The ISE would also be easier to use, since it would not require construction of a platinum electrode.

References

A. Castellan, GW, 1983. Physical Chemistry, 3rd Ed. Reading, Massachusetts, Addison-Wesley Pub. Co.

B. BE 309 Bioengineering Laboratory III: Laboratory Manual.

C. CRC Handbook of Chemistry and Physics

D. “Titration”, “Physical and Chemical Analysis and Measurement Inert-Indicator-Electrode Potentiometry”, and “Physical and Chemical Analysis and Measurement: Ion-Selective Electrodes” from the Encyclopedia Britannica Online ().

E. Schenk, George H., Jr., 1966. Quantitative Analytical Chemistry, Massachusetts, Allyn and Bacon, Inc.

F. Kolthoff, IM & Sandell, EB, 1952. Textbook of Quantitative Inorganic Analysis, 3rd Edition, New York, The Macmillan Company.

G. Web Page for Analytical Chemistry, University of Akron



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[1] Dr. Hardy of the University of Akron

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