Physical Chemistry Lab 2:
Section 0101
Quantitative Analysis of a Mixture
Experimenters:
Mr. Zimmermann
Mr. Baniszewski
Experiment Dates: March 2 and March 9, 2004
Report Written: March 15, 2004
Report Submission Date: March 16, 2004
Title (5)
Abstract (10):
In this experiment we used absorption spectroscopy to quantitatively measure the concentrations of three similar compounds, Ortho-cresol, Meta-cresol, and Para-cresol, which were presented to us in an unknown mixture. We calculated that the concentrations of the unknowns in solution O3A were co = 0.019 +/- 0.025 g/L, cm = 0.0066 +/- 0.036 g/L, and cp = 0.018 +/- 0.126 g/L by using the Beer’s Law equation and simultaneously solving for the three unknown concentrations using a series of absroptivity constants which we calculated by doing linear regressions on known concentrations of pure samples of each compound. On the way, we measured the absorptivities of three known mixtures as well to confirm Beer’s Law behavior of the compounds. The mixture data ranged from -40% to +27% off of the actual concentrations.
Introduction (10):
The purpose of this experiment was to determine the concentrations of three cresol isomers, ortho, meta, and para, in an unknown solution, O3A. To do this, we resorted to an spectrographic analysis of the unknown. Beer’s Law (1), A = ebc, where A is the absorption, e is the absorptivity, b is the path length (1 cm in our case), and c is the concentration of the sample, was used in conjunction with the fact that absorptions should be additive (2) in cases where the compounds do not interact (an assumption which we made before conducting the experiment).
Since the absorption in Beer’s Law (1) is directly proportional to the concentration, we could take an absorbance spectrum measurement of each different isomer at several different concentrations and use a linear regression to find the absorptivity constant. Once we are armed with these constants, we can then find the concentrations of each component in a three component mixture by solving a system of linear equations. Our lab manual was kind enough to work through the lengthy algebra for us, and provided us with the following three equations (3,4,5) which yield the concentrations of the ortho, meta, and para forms of the compounds in solution. In (3,4,5) the eXY are the epsilon values from the Beer’s Law equation for isomer X at wavelength Y.
Equations for Introduction:
(1) A = ebc
(2) Atot = A1 + A2 + A3 + ... + An
(3) co = (1/b)((Ax(emy*epz-epy*emz)-Ay(emx*epz-epx*emz)+Az(emx*epy-epx*emy))/(eox(emy*epz-epy*emz)-emx(eoy*epz-epy*eoz)+epx(eoy*emz-emy*eoz))
(4) cm = (1/b)((Ax-eoxbco)epy-epx(Ay-eoybco))/(emx*epy-epx*emy)
(5) cp = (1/b)((Ax-eoxbco-emxbcm)/epx)
Experimental Set-up (5):
We followed the experimental directions provided to us in Dr. Rebbert’s lab handout (1).
We were armed with a U-3010 dual-beam scanning UV-Vis spectrophotometer from the Hitachi Company (serial number 1381-003) and computer software on an attached workstation when we conducted this experiment. We measured volumes of solution with three uncalibrated burettes which were provided in the lab room.
During part one of the experiment, we decided on using the absorpances at 272.25 nm, 280 nm, and 286 nm. Our measurements of the unknown and known samples were taken at these concentrations.
Data (10):
Wavelengths Chosen:
Lambda x = 272.25 nm,
Lambda y = 280.0 nm,
Lambda z = 286.0 nm.
The measured absorbances of each of the standards and the mixtures are attached on the following page.
We used 1 cm quartz cells to hold the samples in the Hitachi spectrophotometer. Thus, b = 1 cm in all cases.
The concentration of the Para Cresol was 0.072 g/L
The concentration of the Meta Cresol was 0.022 g/L
The concentration of the Ortho Cresol was 0.072 g/L
However, to make the calculations easier, we used a scaled concentration system where the units were divided by the numeric value of the concentrations listed above. It was necessary to convert back to normal concentrations by multiplying by the concentrations of the original solutions before reporting the final results (See the calculations section for an example).
Calculations (30):
The attached spreadsheet of calculations was prepared by Merle Zimmermann. Handwritten samples of each type of calculation are attached following the spreadsheet pages. Note that several of the same calculation appeared in different places; for example, the error propagation in the calculation of the unknown sample concentrations repeats a lot. In these cases I only wrote ONE handwritten sample of the calculation, since the guidelines only ask for one representative sample of each calculation type. If there are ANY calculations you wish me to duplicate then please let me know and I will gladly provide you with them.
The DATA collected in the experiment appears in each section where it is most relevant.
Each different stage of the experiment is placed in its own box.
The calculations for the unknown sample concentration and errors are reproduced on the second page.
Pages 3 – 11 are linear regression for the charts which appear on pages 12 – 20 of this calculations section.
Note that the final end-calculation results from this section are in scaled concentration units which have no dimensions. Sample conversions are shown in the handwritten pages which demonstrate converting back to g/L dimensions.
Results (10):
A table of Results is attached on the page immediately following this one.
(I included the mixture data from the first week of the experiment because I wasn’t sure if we were supposed to put it in this report or not)
Discussion and Conclusions (20):
Although we found the concentrations to be co = 0.019 +/- 0.025 g/L, cm = 0.0066 +/- 0.036 g/L, and cp = 0.018 +/- 0.126 g/L, the associated errors were comparatively large. This was caused by the arithmetical difference in the numerator of the first co equation ending up pretty small, which amplified the uncertainty in each individual measurement.
However, the percentage errors in the mixtures of which we knew the concentrations in advance were pretty poor too, ranging from -40% to +27% wrong in the cases of co and cm for the first mixture. I would probably finger the linear regression calculations of the absorptivity calculations for these errors.
When we based the calculations solely on the absorptivities of the pure compounds, the results for the mixtures were a lot closer to the actual concentrations. This data is not reproduced here, since the directions for the lab specified using linear regressions to solve the problem, but the results were only off by 7% or 8% at the worst, an improvement of nearly eight times in quality. This might suggest that the dilutions which we prepared did not have the concentrations which we believed them to have. I discuss this possibility in detail in the next subsection.
Possible Sources of Errors:
It is likely that the main source of error in these cases was that the system was not actually behaving in a Beer’s Law manner. If we closely examine the Meta Cresol graph of absorpances at 272.25nm, it appears that the points would be better fitted by a parabolic curve rather than a linear regression. If we examine the errors in the calculated absorptivity constant we can see that the meta cresol is also gifted with errors which are proportionally twice as large as the errors in the other measurements. I believe that this is the reason that the errors in the final results are so large (the unitless concentrations, for example, were co = 0.27 +/- 0.35, cm = 0.30 +/- 1.64, cp = 0.25 +/- 1.75).
Another possible source of error was the set of three uncalibrated burettes which we used to measure the Cresol solutions for dilution. It is quite likely that any differences in measured volumes which we made with these tools were all distorted by the same amount, which would result in a systematic error which could grow or shrink as the concentration increased. The results for the Meta Cresol solution, for example, could also be accounted for if the burette dispensed more than the marked amount of solution. This would cause the amount of Meta Cresol in the more dilute solutions to be somewhat larger than we believed, which would result in the concave upwards appearance of the Meta Cresol graphs.
Equations of the Plots:
The line equations for each plot are printed on the associated linear regression pages. In the case of the Meta Cresol solution at 280 nm, for example, our linear regression gave us an equation of
A = ebc = f(c) = -(0.0936 +/- 0.0195) + (1.058 +/- 0.025)c.
The y-intercept is pretty close to zero, and the slope is not too far away from the simple e = A(1) calculations which we used in the prelab.
Evaluation of Experiment:
Overall, the experiment could be improved in several ways, all of which would require more experimental lab time to implement.
First: we could take a few hours off at the beginning of the experiment by spending some time calibrating our glassware. This would defend against the burette systematic error possibility.
Second: we could take many more concentration measurements, which would allow us to be more sure of the linear regressions. This would also allow us to confirm that any deviations observed would be actually coming from a Beer’s Law violation and not random instrumental error.
Third: we could take measurements at more than three wavelengths, which would allow us to select the three wavelengths for doing the regressions after we see the data rather than before. This would allow us to avoid some non-obvious problems which we might run into if the compounds (such as Meta Cresol) violate Beer’s Law behavior at some wavelengths and not at others.
Fourth: we could dilute the unknown by a known factor. This would allow us to check to see if the results are stable when the concentration is reduced. This would reduce the amount of cross-component interaction which might be occuring in the unknown and mixture solutions.
Fifth: if we had access to more customized linear regression formulae, we might be able to produce a better regression. It is known for certain that the y-intercept in each case should be 0, but none of our linear regressions hit zero exactly. An upgraded linear regression formula would solve this problem.
References:
(1) Dr. Rebbert, et al., Laboratory Handout 2, University of Maryland, College Park MD, 2004; pp 1-6.
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