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I. About Graphs
What is a graph?In mathematics, a graph is an abstract representation of a set of objects where some
pairs of the objects are connected by links. The interconnected objects are represented by
mathematical abstractions called vertices, and the links that connect some pairs of
vertices are called edges. Typically, a graph is depicted in diagrammatic form as a set of
dots for the vertices, joined by lines or curves for the edges.[1]
What are the types of graphs?There are many types of graphs and charts that are commonly used for showing
different reports. These are listed as follows.
Bar Graphs Line Graphs Scatter Plot Graphs Pie Charts
Bar GraphsThese are one of the most common types of graph used to display data.
Sometimes known as "column charts", bar graphs are most often used to show amounts
or the number of times a value occurs. The amounts are displayed using a vertical bar
or rectangle. The taller the bar, the greater number of times the value occurs. Bar
graphs make it easy to see the differences in the data being compared.[2]
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Figure 1.1 A Bar Graph [3]
Line GraphsThese are often used to plot changes in data over time, such as monthly
temperature changes. They can also be used to plot data recorded from scientific
experiments, such as how a chemical reacts to changing temperature or atmospheric
pressure.
Similar to most other graphs, line graphs have a vertical axis and a horizontal axis.
If you are plotting changes in data over time, time is plotted along the horizontal or x-
axis and your other data. When the individual data points are connected by lines, they
clearly show changes in your data and you can use these changes to predict future
results.[2]
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Pie Charts or Circle GraphsThese graphs are a little different from the other three types of graphs discussed. For
one, pie charts do not use horizontal and vertical axes to plot points like the others. They
also differ in that they are used to chart only one variable at a time. As a result, it can
only be used to show percentages.
The circle of pie charts represents 100%. The circle is subdivided into slices
representing data values. The size of each slice shows what part of the 100% it
represents.[2]
Figure 1.4 A Circle Graph[4]
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II. Line Graphs Importance of constructing and interpreting line graphs
Line Graphs play an important role in the modeling and understanding data in
analytical chemistry. Graphs are visual representations of numerical systems and
equations. Graphs help us to visualize the relationship of one bit of data to another. The
relationship can also be translated into a mathematically meaningful equation. The
equation for a line (y = mx + b) is one such equation.
In constructing a line graph, plotting known data can help us to visualize the
behavior of systems in situations that have not been measured. Thus, we can predict
whether the data gathered are decreasing or increasing with respect to its variables.
Graph Variables Independent and Dependent variables
DefinitionsIndependent Variables these variables are ones that are more or less controlled.
Scientists manipulate these variables as they see fit. They still vary, but the variation isrelatively known or taken into account.
Dependent Variables these not controlled or manipulated in any way, but
instead are simply measured or registered. These vary in relation to the independent
variables, and while results can be predicted, the data is always measured. There can be
any number of dependent variables, but usually there is one to isolate reason for variation.
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y vs. x Convention in plotting variablesBy convention, the independent variable is plotted along the horizontal (x) axis
and the dependent variable is plotted up the vertical (y) axis. To plot a point, you simply
move along thex axis to the point that corresponds to the value of your independent
variable, then find the point up the y that corresponds to your dependent variable's score,
and mark a spot on the graph where lines from these points meet.
When choosing a title for a chart, say y plotted againstx. That is, 'dependent
variable' plotted against 'independent variable'.
Equation of a Line Mathematical equation
The equation of a line with a given slope m and the y-intercept b is
y = mx + b
This is obtained from the point-slope equation by setting a = 0. It must be
understood that the point-slope equation can be written for any point on the line, meaning
that the equation in this form is not unique. The slope-intercept equation is unique
because if the uniqueness for the line of the two parameters: slope and y-intercept.[5]
Parameters: slope and y-interceptThe slope of a line is a number that measures its "steepness", usually denoted by
the letterm. It is the change in y for a unit change in x along the line.[6]
The y-intercept of a line is the point at which is crosses either the y axis. If we do
not specify which one, the y-axis is assumed. It is usually designated by the letterb.
Unless that line is exactly vertical, it will always cross the y-axis somewhere, even if it is
way off the top or bottom of the chart.[7]
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Plotting data of a linear curve ProcedureA standard linear curve is a graph relating a measured quantity (radioactivity,
fluorescence, or optical density, for example) to concentration of the substance of interest
in "known" samples. You prepare and assay "known" samples containing the substance in
amounts chosen to span the range of concentrations that you expect to find in the
"unknown" samples. You then draw the standard curve by plotting assayed quantity (on
the Y axis) vs. concentration (on the X axis). Such a curve can be used to determine
concentrations of the substance in "unknown" samples.
Line of best fitA straight line drawn through the center of a group of data points plotted on a
scatter plot. Scatter plots depict the results of gathering data on two variables; the line of
best fit shows whether these two variables appear to be correlated.
A more precise method for determining the line of best fit is a mathematical
calculation called the least squares method. The line of best fit is used in regression
analysis, and is a key input in statistical calculations such as the sum of squares.[19]
Method ofLeast Squares (Linear Regression)The method of least squares is a standard approach to the approximate solution
of overdetermined systems, i.e., sets of equations in which there are more equations than
unknowns. "Least squares" means that the overall solution minimizes the sum of the
squares of the errors made in solving every single equation.
[8]
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Using Linear Regression (LR) function of a scientific calculatoro Procedure of encoding/key-in of data
1. Clear all the data firstPress SHIFT > Press CLR > Press 3(ALL) > Display: Reset All > Press
= > Press AC.
2. Set calculator to Linear Regression modePress MODE twice > Press 2 (REG) > Press 1 (Lin)
3. Entry of dataDo data entry one by one with x comma y format
Time (min) Distance (m)
1 3.5
3 5.6
5 7.4
Then do entry of data:
Press 1 > Press , > Press 3.5 > Press M+ > Display n=1
Press 3 > Press , > Press 5.6 > Press M+ > Display n=2
Press 5 > Press , > Press 7.4 > Press M+ > Display n=3
o Determination of linearity of data (linearity check)Press SHIFT > Press "2" > Press >" (next button) twice >>> Press "3" (r) >
Press "=" > Display (for the above example: r = 0.998
). If you need R square, just use thesquare button.
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o Determination of slope and y-interceptFor slope (B)
Press "S" > Press "2" > Press ">" (next button) twice > Press "2" (B) >>> Press "=" >Display (for the above example: B =0.975)
Fory- intercept (A)
Press "SHIFT" > Press "2" > Press ">" (next button) twice > Press "1" (A) >>> Press "="
> Display (for the above example: A=2.575)
Now, you get the equation: y = Bx+A
y = 0.975x +2.575 ( r = 0.988)
o Calculating value of x for a given value of yGiven that y = 5.0, x can be calculated using the equation y = 0.975x +2.575 that was
made earlier. Substitute y from the equation and find x;
y = 0.975x +2.575
x = 5.0 2.575 / 0.975
x = 2.49
o Calculating value of y given the value of xGiven that x = 4.0, y can be calculated using the equation y = 0.975x +2.575 that was
made earlier. Substitute x from the equation and find y;
y = 0.975x +2.575
y = 0.975(4.0)+2.575 = 6.475
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III. Calibration Curve Definition (what is a calibration curve?)
In analytical chemistry, a calibration curve is a general method for determining
the concentration of a substance in an unknown sample by comparing the unknown to a
set of standard samples of known concentration. A calibration curve is one approach to
the problem of instrument calibration; other approaches may mix the standard into the
unknown, giving an internal standard.
The calibration curve is a plot of how the instrumental response, the so-
called analytical signal, changes with the concentration of the analyte (the substance to
be measured).[9]
Procedure (how to prepare a calibration curve?)A calibration curve is prepared by plotting the data or by fitting them to a suitable
mathematical equation, such as the linear relationship used in the method of least squares.
The operator prepares a series of standards across a range of concentrations near the
expected concentration of analyte in the unknown. Analyzing each of these standards
using the chosen technique will produce a series of measurements. For most analyses a
plot of instrument response vs. analyte concentration will show a linear relationship. The
operator can measure the response of the unknown and, using the calibration curve,
can interpolate to find the concentration of analyte.[9]
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Determination of value of unknown concentration of an analyte from a calibrationcurve
One of the most common applications of spectrophotometry is to determine the
concentration of an analyte in a solution. The experimental approach exploits Beer's Law,
which predicts a linear relationship between the absorbance of the solution and the
concentration of the analyte (assuming all other experimental parameters do not vary).
In practice, a series of standard solutions are prepared. A standard solution is a
solution in which the analyte concentration is accurately known. The absorbances of the
standard solutions are measured and used to prepare a calibration curve, which is a graph
showing how the experimental observable varies with the concentration. Thus, the points
on the calibration curve should yield a straight line (Beer's Law). The slope and intercept
of that line provide a relationship between absorbance and concentration:
A = slope c + intercept
The unknown solution is then analyzed. The absorbance of the unknown
solution,Au, is then used with the slope and intercept from the calibration curve to
calculate the concentration of the unknown solution, cu.[10]
cu =
Au- intercept
slope
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Illustration (graph)
Figure 3.1 A Calibration Curve
Sample problems Applications to Spectroscopy
Standard solutions of iron complex were prepared and the absorbances of
these solutions were measured by spectronic 21. The results are presented in table
3.1
Table 3.1 Absorbance reading of iron complex
Solutions Concentration (ppm) Absorbance
Blank 0.00 0.00
1 1.00 0.21
2 3.00 0.433 5.00 0.59
4 7.00 0.70
5 9.00 0.89
Unknown ? 0.46
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Applications to ElectrochemistryStandard solutions of AR grade sodium chloride were prepared and the conductance
of each solution was measured. The gathered data were presented in table 3.2.
Table 3.2 Measured conductances of various solutions with respect to its concentration
Concentration (ppm) Conductance (mS/cm)
0.00 0.00
10.0 356
20.0 663
30.0 955
40.0 1265
Unknown 781
a. Construct a calibration curve of conductance versus concentration.b. From the calibration curve, calculate the concentration of the unknown solution.
Solution:
Figure 3.3 Calibration curve of AR grade NaCl standard solutions
y = 31.29x + 22
R = 0.9987
0
200
400
600
800
1000
1200
1400
0 10 20 30 40 50
Conductance(mS/cm)
Concentration (ppm)
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The equation of the line was found to be y = 31.29x + 22, let x = unknown
concentration. To solve for x:
y = 31.29x + 22
x= 781 22/31.29
x = 24.26ppm
Applications to ChromatographyStandard solutions of AR graded CCl4 of known concentration were prepared
(0.00, 20.00, 40.00, 60.00 and 80.0) and each concentration gave a peak area of (0.00,
325, 635, 910 and 1230). Construct a calibration curve of peak versus concentration and
calculate for the concentration of the unknown if its peak area is 721.
Solution:
Figure 3.4 Calibration curve of AR graded cyclohexane
y = 15.225x + 11
R = 0.9993
0
200
400
600
800
1000
1200
1400
0 20 40 60 80 100
Peakarea
Concentration
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Standard Addition Curve Procedure (how to prepare a standard addition curve)
The instrument response is then measured for all of the diluted solutions and the
data is plotted with volume standard added in the x-axis and instrument response in the y-
axis. Linear regression is performed and the slope (m) and y-intercept (b) of the
calibration curve are used to calculate the concentration of analyte in the sample.[11]
Interpretation of data (how to determine the unknown concentration of ananalyte from a standard addition curve)
If the response is linear, then when the total analyte concentration is 0, the
response is 0. We can extrapolate the line back to a point at which Response = 0 (as
presented in figure 2.2). The Concentration value at which Response = 0 is the x-intercept
of the regression line. Here it happens to be at 0.0015 units of added analyte. What does
this mean in the context of the analysis? In order to have a total of 0 analyte causing 0
response, we would have to add 0.0015 units analyte, or remove 0.0015 units. Why?
Because the sample of unknown that was placed in every container has 0.0015
concentration units. Remember, [Analyte]total = [added] + [unknown]. So, 0 = -0.0015added
+ [unknown], and thus 1.7 = [unknown].[13]
Another approach is possible using the linear regression to determine the
unknown concentration of the analyte,
S = mVS+ b [Equation 1]
Where: S = instrument response (signal)
VS = volume of standard
Conceptually, if the curve started where the instrument response is zero, the volume of
standard [(Vs)0] from that point to the point of the first solution on the curve (x = 0)
contains the same amount of analyte as the sample. So:
Vxcx = |(VS)0|cS [Equation 2]
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Where: Vx = volume of the sample aliquot
cx = concentration of the sample
cs = concentration of the standard
Combining Equation 1 and Equation 2 and solving for cx results in:
Cx = bCs/mVx [Equation 3]
And one can then calculate the concentration of analyte in the sample from the slope and
intercept of the standard addition calibration curve.[12]
Illustration (graph)
Figure 2.1 Standard Addition Method Plot (absorbance vs. concentration of standard)
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Multiple Addition Method (multiple data)
o Identify variables in equationWhere,
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Sample problems Application to spectroscopy
The single-point standard addition method was used in the determination of
phosphate by the molybdenum blue method. A 2.00-mL urine sample was treated with
molybdenum blue reagents to produce a species absorbing at 820 nm, after which the
sample was diluted to 100-mL. A 25.00-mL aliquot of this solution gave an absorbance
of 0.428 (solution 1). Addition of 1.00-mL of a solution containing 0.0500 mg of
phosphate to a second 25.0-mL aliquot gave an absorbance of 0.517 (solution 2). Use
these data to calculate the concentration of phosphate in milligrams per millilitre of the
specimen.
Solution:
C = A1c1Vs / A2Vt A1Vx [eq. 4.2]
Here we substitute into eq. 4.2, and obtain
= 0.00780 mg PO43-
mL
This is the concentration of the diluted sample. To obtain the concentration of the
original urine sample, we need to multiply by 100.00/2.00. Thus,
[PO43-]= 0.390 mg/mL
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Application to electrochemistryA cell containing of a saturated calomel electrode and a silver ion developed a
potential of -0.4706 V when immersed in 50.00-mL of a sample. A 5.00-mL addition of
standard 0.02000 M silver solution caused the potential to shift to -0.4490 V. Calculate
the molar concentration of lead in the sample.
Solution:
[eq. 4.1]We shall assume that the activity of Pb
2+is approximately equal to [Pb
2+] and
apply eq. 4.1. Thus,
whereEcell is the initial measured potential (-0.4706 V).
After the standard addition is added, the potential becomes Ecell(-0.4490 V), and
Subtracting this equation from the first leads to
= 0.7297
[Ag+]= 4.08 x 10
-3
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Figure 5.1 A Continuous Variation Plot (Absorbance vs. Mole fraction)
Sample ProblemA series of Ni2+ and dimethylglyoxime (DMG) solutions of equal volume was
prepared to determine the stoichiometry of the complex.The absorbance of each solution
was measured.
Table 5.1 Absorbance reading of nickel complex solution
Sample 0.0050M
Ni2+
(mL)
0.0050M
DMG
(mL)
X Ni2+
X DMG Absorbance
1 1.0 9.0 0.1 0.9 0.65
2 2.0 8.0 0.2 0.8 1.25
3 3.0 7.0 0.3 0.7 1.39
4 4.5 5.5 0.45 0.55 1.19
5 5.5 4.5 0.55 0.45 1.07
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Solution:
Figure 5.2 Plot Continuous Variation
Beer-Lambert's law states that the absorbance is proportional to the concentration
of the absorbing species. Since the sum, nM+ nL, is kept constant, the maximum amount
of complex is formed when the ratio nM/nL corresponds to the stoichiometry of the
complex.
Maximum absorbance is observed when the x/y =XNi2+
/XDMG. The maximum
occurs at aroundXNi2+ = 0.3. Thus x/y = 0.3/0.70 = 2/4.7 = 2/5.
Mole-Ratio Method Description ofMethod
In the mole-ratio method, a series of solutions is prepared in which the analytical
concentrations of one reactant (usually the cation) is held constant while that of the other
is varied. A plot of absorbance versus mole-ratio of the reactants is then made. If the
formation constant is reasonably favourable, we obtain two straight lines of different
slopes that intersect at a mole ratio corresponding to the combining ratio in the
complex.[14]
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GraphTypical mole-ratio plot is shown in figure 5.2. Formation constants can be
evaluated from the data in the curved portion of mole-ratio plot where the reaction is least
complete.[14]
Figure 5.2 Mole-ratio Method Plot (1:1)
Sample ProblemThe accompanying data were obtained on a slope ratio investigation of the
complex formed between NH4 and Quinol. The measurements were made at 420nm
wavelength.
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Slope-Ratio Method Description of the Method
This approach is particularly is useful for weak complexes but is applicable only
to systems in which a single complex is formed. The method assumes (1) that the
complex-formation reaction can be forced to completion by a large excess of either
reactant, (2) that Beers Law is followed under these circumstances, and (3) that only the
complex absorbs at the wavelength chosen for the experiment.[14]
Consider the reaction in which the complex MxLy is formed by the reaction ofx
moles of the cation M withy moles the ligand.
xM+yL = MxLy
Mass-balance expressions for this system are
cM= [M] +x[MxLy]
cL = [L] +y[MxLy]
where cM and cL are the molar analytical concentrations of the two reactants. We now
assume that at very high analytical concentration ofL, the equilibrium is shifted far to the
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right and [M]
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Graph
Figure 5.3 A Plot of Slope-ratio Method
Sample Problem
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VI. SpectrophotometricTitration Description
In analytical chemistry, Spectrophotometric titration is an analytical method in
which the radiant-energy absorption of a solution is measured spectrophotometrically
after each increment of titrant is added.[15]
Procedure Techniques forEnd-point determination
The spectrophotometer is an optical device that responds only to radiation within
a selected very narrow band of wavelengths in the visual, ultraviolet, or infrared regions
of the spectrum. The response can be made both quantitatively and linearly related to the
concentration of a species that absorbs radiation within this band. Titrations at
wavelengths within the visual region are by far the most common.
An example is the titration of iron(II) in dilute sulfuric acid solution with a
standard solution of potassium permanganate. The spectrophotometer is adjusted to
measure the absorbance of the highly colored titrant. Very little absorbance is observed
until the end point is reached; iron(II) and the reaction products are essentially colorless
and are not seen by the spectrophotometer. However, the absorbance rises as titration is
continued beyond the end point because the titrant is no longer being destroyed.[16]
The
titration curve is shown in figure 6.1.
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Graph
Figure 6.1 Spectrophotometric titration curve of iron(II) in dilute sulfuric acid solution
with potassium permanganate as the titrant. The curve shows the response when the
titrant absorbs beyond the end point.
Sample Problem
The concentrations of Fe3+ and Cu2+ in a mixture can be determined following their
reaction with hexacyanoruthenate (II), Ru(CN)64, which forms a purple-blue complex
with Fe3+ (max = 550 nm), and a pale green complex with Cu2+ (max = 396 nm) . The
mo l a r absorptivities (M 1
cm 1) for the metal complexes at the two wavelengths are
summarized in the following table.
Table 6.1 Molar absorptivities of Copper andIron
550 396
Fe3+
9970 84
Cu2+
34 856
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When a sample containing Fe3+ and Cu2+ is analyzed in a cell with a pathlength
of 1.00 cm, the absorbance at 550 nm is 0.183, and the absorbance at 396 nm is 0.109.
What are the molar concentrations of Fe3+ and Cu2+ in the sample?
Solution:
(Am)1 = (X)1bCX+ (Y)1bCY [eq. 6.1]
(Am)2 = (X)2bCX+ (Y)2bCY [eq. 6.2]
Substituting known values into equations 6.1 and 6.2 gives
A550 = 0.183 = 9970CFe + 34CCu
A396 = 0.109 = 84CFe + 856CCu
To determine the CFe and CCu we solve the first equation for CCu
and substitute the result into the second equation.
Solving for [Fe] gives the concentration of Fe3+
as 1.80 105
M. Substituting this
concentration back into the equation for the mixtures absorbance at a wavelength of 396
nm gives the concentration of[Cu] as 1.26 104
M.
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VII. PotentiometricTitration Description
In analytical chemistry, potentiometric titration is a titration in which an electrode
reversible to one of the ionic components of the analyte or titrant is immersed in the
analyte solution and the electric potential of the electrode (relative to that of an inert
reference electrode) is measured during addition of titrant. Because the potential of an
electrode is a logarithmic function of the activity of the ion to which the electrode is
reversible, an abrupt change in its potential is observable as an equivalence point is
approached.[17]
Procedure Techniques forEnd-point determination
If a pH meter is used, acid-base titration curves can be plotted or recorded. The
meter and its associated electrodes are first standardized by use of a buffer solution of
known pH. The electrodes are then immersed in the well-stirred solution to be titrated,
and the titration is begun. For routine purposes, interest is in rapid and reasonably close
end-point location. Titration is first carried out quite quickly until the meter shows signs
of rapid response, and then is slowed so that it can be stopped at the pH jump or fall that
marks the end point. Obviously, potentiometric titration can be used for a highly colored
titrand solution, in which the response of a color-change indicator could not be seen.
There are two approaches to higher precision. In the first the pH at which the
desired end point occurs is determined, and the titration of the actual sample is then
arrested exactly at, or very close to, this pH value. The other approach is to stop at the
point of steepest slope of the titration curve. This can be done by plotting (or, with
suitable instrumentation, sensing) the first derivative (pH/V) or the second derivative
(2pH/V
2) against the volume Vof titrant added. In theory this method is applicable to
any potentiometric titration, without the need to predetermine the end-point conditions.
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By suitable choice of electrodes, these potentiometric methods can also be applied
to combination titrations and to oxidation-reduction titrations. The advent of modern ion-
selective electrodes has greatly extended the scope of potentiometric titration and other
branches of titrimetry.[16]
Graph
Figure 7.1 Measurements, first and second derivative in a potentiometric titration.
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Sample ProblemCalculate the titration curve for the titration of 50.0 mL of 0.0500 MSn2+ with
0.100 M Tl3+
Both the titrand and the titrant are 1.0 M in .HCl. The titration reaction is
Sn2+
(aq)+ Tl3+
(aq) Sn4+
(aq) + Tl+(aq)
Solution:
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VIII. Molecular Weight Determination using HPLC Description ofLiquid Chromatographic Methods employed in molecular weight
determination
High-performance liquid chromatography, HPLC, is a chromatographic
technique that can separate a mixture of compounds and is used in biochemistry and
analytical chemistry to identify, quantify and purify the individual components of the
mixture. HPLC typically utilizes different types of stationary phases, a pump that moves
the mobile phase(s) and analyte through the column, and a detector to provide a
characteristic retention time for the analyte. Analyte retention time varies depending on
the strength of its interactions with the stationary phase, the ratio/composition of
solvent(s) used, and the flow rate of the mobile phase.
The widely used technique for the molecular weight determination (usually
polysaccharides) is the Size-exclusion chromatography (SEC), also known as gel
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permeation chromatography or gel filtration chromatography. It separates particles on
the basis of size. It is generally a low resolution chromatography and thus it is often
reserved for the final, "polishing" step of purification. SEC is used primarily for the
analysis of large molecules such as proteins or polymers. SEC works by trapping these
smaller molecules in the pores of a particle. The larger molecules simply pass by the
pores as they are too large to enter the pores. Larger molecules therefore flow through the
column quicker than smaller molecules, that is, the smaller the molecule, the longer the
retention time.[19]
ProcedureThe sample to be analyzed is introduced, in small volumes, into the stream of
mobile phase. The solution moved through the column is slowed by specific chemical or
physical interactions with the stationary phase present within the column. The velocity of
the solution depends on the nature of the sample and on the compositions of the
stationary (column) phase. The time at which a specific sample elutes (comes out of the
end of the column) is called the retention time; the retention time under particular
conditions is considered an identifying characteristic of a given sample. The use of
smaller particle size column packing (which creates higher backpressure) increases the
linear velocity giving the components less time to diffuse within the column, improving
the chromatogram resolution. Common solvents used include any miscible combination
of water or various organic liquids (the most common are methanol and acetonitrile).
Water may contain buffers or salts to assist in the separation of the sample components,
or compounds such as trifluoroacetic acid which acts as an ion pairing agent.
A further refinement ofHPLC is to vary the mobile phase composition during theanalysis; gradient elution. A normal gradient for reversed phase chromatography might
start at 5% methanol and progress linearly to 50% methanol over 25 minutes; the gradient
depends on how hydrophobic the sample is. The gradient separates the sample mixtures
as a function of the affinity. This partitioning process is similar to that which occurs
during a liquid-liquid extraction but is continuous, not step-wise. In this example, using a
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water/methanol gradient, more hydrophobic components will elute (come off the column)
when the mobile phase consists mostly of methanol (giving a relatively hydrophobic
mobile phase).
The choice of solvents, additives and gradient depend on the nature of the column
and sample. Often a series of tests are performed on the sample together with a number of
trial runs in order to find the HPLC method which gives the best peak separation.[19]
Graph
Figure 8.1 An example of a Plot of log mol. wt. versus Ve/Vo of a series of molecular
weight standards
Sample ProblemStandard solutions of polystyrene of known molecular weight were prepared and
the retention time of each solution was measured using size exclusion chromatography.
Prepare a plot of log MW versus retention time. Find the molecular weight of the
unknown with a retention time of 14.55.
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Table 8.1 Retention time of Polystyrene
MW Log MW Retention time (min)
6.75x106
6.829 12.78
5.99 x106 6.777 13.28
5.04 x106
6.702 13.77
4.27 x106
6.630 14.20
3.55 x106
6.550 14.76
Solution:
Figure 8.2
The equation of the line was found to be y = -0.1443x + 8.6836, let y = unknown
molecular weight. To solve for y:
y = -0.1443x + 8.6836
y = -0.1443(14.55)+ 8.6836
y = 6.5840
y = 106.5840
y = 3.84x106
y = -0.1443x + 8.6836
R = 0.9944
6.5
6.55
6.6
6.65
6.7
6.75
6.8
6.856.9
12.5 13 13.5 14 14.5 15
logMW
Retention time (tR)
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LIST OF REFERENCES
[1] http://en.wikipedia.org/wiki/Graph_%28mathematics%29
[2] http://spreadsheets.about.com/od/spreadsheetlessons/ss/excel_graph_use.htm
[3] http://www.ncsu.edu/scivis/chemistry.html
[4] http://www.ground-water-
models.com/products/aquachem_details/aquachem_details.html
[5] http://www.cut-the-knot.org/Curriculum/Calculus/StraightLine.shtml
[6] http://www.mathopenref.com/coordslope.html
[7] http://www.mathopenref.com/coordintercept.html
[8] http://en.wikipedia.org/wiki/Least_squares
[9] http://en.wikipedia.org/wiki/Calibration_curve
[10] http://www.chm.davidson.edu/vce/spectrophotometry/unknownsolution.html
[11] http://docs.google.com/viewer?a=v&q=cache:scrY_M8zXxkJ:www.asi-
sensors.com/ASI/learning/standard_addition.pdf+Standard+Addition+Method&hl=tl&gl
=ph&pid=bl&srcid=ADGEESjAQVV1Lot9tOocsmrK3wHb1t8_tfz8WJq0dS3eBYGS0X
o_xrOg8DvKxHpc-F9KB-wFlyKWo4cowgtHTLmgF02P6CaoHt6vGHuSvpXopPjRB9qT9-
oagcQAs4LMWlAIZ6hLW7aX&sig=AHIEtbS1_9diVy7uWdJBWyyGn0kIHaAATA
[12] http://en.wikipedia.org/wiki/Standard_addition
[13]http://www.google.com.ph/url?sa=t&rct=j&q=how%2Bto%2Bprepare%2Ba%2Bstan
dard%2BADDITION%2Bcurve&source=web&cd=4&ved=0CDMQFjAD&url=http%3A
%2F%2Fwww.asdlib.org%2FonlineArticles%2Fecourseware%2FAnalytical%2520Chem
istry%25202.0%2FText_Files_files%2FChapter5.pdf&ei=roStTuroDof9iQLZprWICw&
usg=AFQjCNHhMgB5ZFVPsIcNIkPWZiEk-_6FlQ
[14] Fundamentals of Analytical Chemistry, 8th
ed., D. A. Skoog, D. M. West, F. J.
Holler, and S. R. Crouch. Brooks/cole, a division of Thomson Learning, 2004.
[15] http://www.answers.com/topic/spectrophotometric-titration
[16] http://accessscience.com/content/Titration/699100
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[17] http://www.answers.com/topic/potentiometric-titration-1
[18] http://en.wikipedia.org/wiki/High-performance_liquid_chromatography#Size-
exclusion_chromatography
[19] http://www.investopedia.com/terms/l/line-of-best-fit.asp#ixzz1cLBcCTQD