Calibration and Standards

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Lecture 3 Calibration and Standards
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Least-squares curve fitting
Carl Friedrich Gauss in 1795
The points (1,2) and (6,5) do not fall exactly on
the solid line, but they are too close to the
line to show their deviations. The Gaussian curve
drawn over the point (3,3) is a schematic
indication of the fact that each value of y is
normally distributed about the straight line.
That is, the most probable value of y will fall
on the line, but there is a finite probability of
measuring y some distance from the line.
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Least squares
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ykxb straight line equation
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ykxb straight line equation
k Slope ?y / ?x
b - blank!
Let us subtract blank y-b Y kx
Y1kx1 Y2kx2
One standard
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• Procedure
• Measure blank.
• Measure standard.
• Measure unknown.
• Subtract blank from standard and from unknown.
• Calculate concentration of unknown

If you have several (N) standards, do it several
(N) times
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Standard addition Why and when?
Matrix (interfering components) can affect the
slope
In equation Ykx you do not know k any more!
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Use your sample as a new blank Add a known
amount to your sample
Ixstandard
Ix
X
X standard
Increase in intensity because of this addition
Ixstandard Ix
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• Procedure
• Measure unknown.
• Add a known amount to the unknown and measure
  this sample.
• 3. Subtract (2) from (1).
• 4. Calculate concentration of unknown

If you have several (N) standard additions, do it
several (N) times
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Internal Standard
Why and when?
All intensities vary from sample to sample
Sample 1 x s Sample 2 0.6x
0.6s Sample 3 1.2x 1.2s
No reproducibility!
Let us divide intensity in the first column by
the intensity in the second
Sample 1 x s x/s Sample 2
0.6x 0.6s x/s Sample 3 1.2x 1.2s
x/s
Now they are the same!
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Restriction you need to measure 2 values
simultaneously
You may prefer to have the same amount of
internal standard in all your samples
Procedure 1. Add equal amounts of the internal
standard to all your standards and analytes. 2.
Measure intensities of your target compound
(atom) and your internal standard in your
solutions. 3. For each pair of measurements,
divide the intensity coming from your target
compound by the intensity of the internal
standard. 4. Process these new normalized
intensities like you did before.
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Least squares does not work?
Good Plot!
Y 1.5 x 1
One VERY BAD point
Bad plot!
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A possible solution
k 2.33, 1.5 1.50
b 0.3 1 1
Median Y 1.5 x 1
robust
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Weighed least squares
Straight line
Any function