Transcript Slide 1

Calibration Methods
Introduction
1.) Graphs are critical to understanding quantitative relationships

One parameter or observable varies in a predictable manner in
relationship to changes in a second parameter
2.) Calibration curve: graph showing the analytical response as a
function of the known quantity of analyte

Necessary to interpret response for unknown quantities
Time-dependent measurements of
drugs and metabolites in urine
samples
Generally desirable to graph data to generate a straight line
Calibration Methods
Finding the “Best” Straight Line
1.) Many analytical methods generate calibration curves that are linear or
near linear in nature
(i)
Equation of Line:
y  mx  b
where:
x = independent variable
y = dependent variable
m = slope
b = y-intercept
y
slope 
m
x
Calibration Methods
Finding the “Best” Straight Line
2.) Determining the Best fit to the Experimental Data
(i)
Method of Linear Least Squares is used to determine the best values for
“m” (slope) and “b” (y-intercept) given a set of x and y values

Minimize vertical deviation between points and line
d i  ( y i  y )  ( y i  m( x i )  b )

Use square of the deviations  deviation irrespective of sign
d i2  ( yi  y )2  ( yi  m( xi )  b )2
Calibration Methods
Finding the “Best” Straight Line
4.) Goodness of the Fit
(i)
R2: compares the sums of the variations for the y-values to the best-fit line
relative to the variations
to a horizontal line.
2 based
R
on these relative differences

R2 x 100: percent of the variation of the y-variable that is explained by
Summed
for each point
the variation of the
x-variable.

A perfect fit has an R2 = 1; no relationship for R2 ≈ 0
R2=0.9952
Very weak to
no relationship
R2=0.5298
Strong
direct
relationship
99.5% of the y-variation is due to the x-variation
53.0% of the y-variation is due to the x-variation
What is the other 47% caused by?
Calibration Methods
Calibration Curve
1.) Calibration curve: shows a response of an analytical method to known
quantities of analyte
Procedure:
a)
Prepare known samples of analyte covering
convenient range of concentrations.
b)
Measure the response of the analytical
procedure.
c)
Subtract average response of blank (no analyte).
d) Make graph of corrected response versus
concentration.
e)
Determine best straight line.
Calibration Methods
Calibration Curve
2.) Using a Calibration Curve

Prefer calibration with a linear response
- analytical signal proportional to the quantity of analyte

Linear range
- analyte concentration range over which
the response is proportional to
concentration

Dynamic range
- concentration range over which there
is a measurable response to analyte
Additional analyte does not
result in an increase in response
Calibration Methods
Calibration Curve
3.) Impact of “Bad” Data Points


Identification of erroneous data point.
- compare points to the best-fit line
- compare value to duplicate measures
Omit “bad” points if much larger than average ranges and not reproducible.
- “bad” data points can skew the best-fit line and distort the accurate
interpretation of data.
y=0.091x + 0.11 R2=0.99518
y=0.16x + 0.12 R2=0.53261
Remove “bad” point
Improve fit and
accuracy of m and b
Calibration Methods
Calibration Curve
4.)
Determining Unknown Values from Calibration Curves
(i)
(ii)
Knowing the values of “m” and “b” allow the value of x to be determined
once the experimentally y value is known.
Know the standard deviation of m & b, the uncertainty of the determined xvalue can also be calculated
Calibration Methods
Calibration Curve
4.)
Determining Unknown Values from Calibration Curves
(iii) Example:
The amount of protein in a sample is measured by the samples absorbance of light at a given
wavelength. Using standards, a best fit line of absorbance vs. mg protein gave the following
parameters:
m = 0.01630
sm = 0.00022
b = 0.1040
sb = 0.0026
An unknown sample has an absorbance of 0.246 ± 0.0059. What is the amount of protein in the
sample?
Calibration Methods
Calibration Curve
5.)
Limitations in a Calibration Curve
(iv) Limited application of calibration curve to determine an unknown.
- Limited to linear range of curve
- Limited to range of experimentally determined response for known
analyte concentrations
Uncertainty increases further
from experimental points
Unreliable determination
of analyte concentration
Calibration Methods
Calibration Curve
6.)
Limitations in a Calibration Curve
(v)
Detection limit
- smallest quantity of an analyte that is significantly different from the blank
Signal detection limit:

ydl  yblank  3 s
where s is standard deviation
- need to correct for blank signal
Corrected signal:
ycs  y sample  yblank
- minimum detectable concentration
Detection limit:



3s
c
m
Where c is concentration
s – standard deviation
m – slope of calibration curve
Calibration Methods
Calibration Curve
6.)
Limitations in a Calibration Curve
(vi) Example:
Low concentrations of Ni-EDTA near the detection limit gave the following counts in a
mass spectral measurement: 175, 104, 164, 193, 131, 189, 155, 133, 151, 176. Ten
measurements of a blank had a mean of 45 counts. A sample containing 1.00 mM Ni-EDTA
gave 1,797 counts. Estimate the detection limit for Ni-EDTA
Calibration Methods
Standard Addition
1.)
Protocol to Determine the Quantity of an Unknown
(i)
Known quantities of an analyte are added to the unknown
- known and unknown are the same analyte
- increase in analytical signal is related to the total quantity of the analyte
- requires a linear response to analyte
(ii)
Very useful for complex mixtures
- compensates for matrix effect  change in analytical signal caused by
anything else than the analyte of interest.
(iii) Procedure:
(a) place known volume of unknown sample in multiple flasks
Calibration Methods
Standard Addition
1.)
Protocol to Determine the Quantity of an Unknown
(iii) Procedure:
(b) add different (increasing) volume of known standard to each unknown
sample
(c) fill each flask to a constant, known volume
Calibration Methods
Standard Addition
1.)
Protocol to Determine the Quantity of an Unknown
(iii) Procedure:
(d) Measure an analytical response for each sample
- signal is directly proportional to analyte concentration
Concentration of analyte in initial solution
signal from initial solution

Concentration of analyte plus s tandard in final solution signal from final solution
Standard addition equation:
X i
S  f  X  f

IX
IS X
Total volume (V):
V  Vo  VS , Vo  unknown initial volume, VS  addedvolume of s tandard
X  f  X i  Vo 
V 
S  f  S i  VS 
V 
Calibration Methods
Standard Addition
1.)
Protocol to Determine the Quantity of an Unknown
(iii) Procedure:
(f) Plot signals as a function of the added known analyte concentration and
determine the best-fit line.
X-intercept (y=0) yields X  f
which is used to calculate X i from:



 Vo 
X i  X  f  V
Calibration Methods
Standard Addition
1.)
Protocol to Determine the Quantity of an Unknown
(iii) Example:
Tooth enamel consists mainly of the mineral calcium hydroxyapatite, Ca10(PO4)6(OH)2. Trace
elements in teeth of archaeological specimens provide anthropologists with clues about
diet and disease of ancient people. Students at Hamline University measured strontium in
enamel from extracted wisdom teeth by atomic absorption spectroscopy. Solutions with a
constant total volume of 10.0 mL contained 0.750 mg of dissolved tooth enamel plus
variable concentrations of added Sr. Find the concentration of Sr.
Added Sr (ng/mL = ppb)
Signal (arbitrary units)
0
28.0
2.50
34.3
5.00
42.8
7.50
51.5
10.00
58.6
Calibration Methods
Internal Standards
1.)
Known amount of a compound, different from analyte, added to the
unknown.
(i)
Signal from unknown analyte is compared against signal from internal standard

Relative signal intensity is proportional to concentration of unknown
- Valuable for samples/instruments where response varies between runs
- Calibration curves only accurate under conditions curve obtained
- relative response between unknown and standard are constant


Widely used in chromatography
Useful if sample is lost prior to analysis
Area under curve proportional
to concentration of unknown (x)
and standard (s)
 area of s tan dard signal 
Area of analyte signal

 F 
Concentration of analyte
 Concentration of s tan dard 
Ax
A 
 F S 
 X   S  
Calibration Methods
Internal Standards
1.)
Example:
A solution containing 3.47 mM X (analyte) and 1.72 mM S (standard) gave peak areas of
3,473 and 10,222, respectively, in a chromatographic analysis. Then 1.00 mL of 8.47 mM S
was added to 5.00 mL of unknown X, and the mixture was diluted to 10.0 mL. The solution
gave peak areas of 5,428 and 4,431 for X and S, respectively
(a)
(b)
(c)
(d)
Calculate the response factor for the analyte
Find the concentration of S (mM) in the 10.0 mL of mixed solution.
Find the concentration of X (mM) in the 10.0 mL of mixed solution.
Find the concnetration of X in the original unknown.