CHE 322
CHAPTER5_CAL
CHAPTER 5
CALIBRATIONS, STANDARDIZATIONS, AND BLANK
CORRECTIONS
A. CALIBRATING SIGNALS
B. STANDARDIZING METHODS
C. LINEAR REGRESSION
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CHE 322
A.
CHAPTER5_CAL
CALIBRATING SIGNALS
Goal of calibration
To detect and correct for systematic errors.
y  b0  b1 x
Periodic calibration of equipment and instruments is highly recommended,
because the response of most instruments changes with time as a result of
wear, operators’ abuse, change in room conditions (temperature, moisture,
dust, etc…).
Operator/ personal error must and should be minimized by care and selfdiscipline in checking instrument readings, notebook entries, calculations,
etc…
Balance calibration
***Correction of mass for the buoyancy of air
  1
1
WV  Wa  1  

  Do DW
WV :
Wa :
Do :
DW :


  0.0012 


object's true weight in vacuo
object's weight in air
object's density
density of calibrating weights
0.0012 g cm 3 :
density of air under laboratory conditions
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CHAPTER5_CAL
Spectrophotometer calibration
Problem 4.18
60.06 ppm K2Cr2O7 in 0.0050 M H2SO4 has a known absorbance of 0.640 at 350.0 nm.
A   l C

extinction coefficient/ molar absorptivity

S. Ebel, Fresenius J. Anal. Chem. 1992, 324, 769
Wavelength
(nm)
233
Absorbance of K2Cr2O7 (60.06
mg/L) in 5.0 mM H2SO4 in 1-cm
cell
0.748  0.010
257
0.865  0.010
313
0.292  0.010
350
0.640  0.010
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CHE 322
CHAPTER5_CAL
STANDARDIZING METHODS
Determine k experimentally  Determine the relationship between the measured signal/
response and the amount of analyte.
Measure the signals of standard materials
B.1
Primary and Secondary Standards
A primary standard: reagent of known purity and composition, so that the
mass or volume determined is an accurate measure of the number of moles
of the reagent.
Important requirements for a primary standard
1) High purity
Established methods for confirming purity should be available.
Have been carefully analyzed by supplier and the assay is printed on the
container label.
2) Atmospheric stability: stable in solid or solution form during long-term
storage
3) Not hygroscopic
A secondary standard is a reagent of composition determined by use of a
primary standard.
Several secondary standards are hygroscopic or hydrated reagents.
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CHAPTER5_CAL
Some Common Primary standards
(Appendix 2, P729) (National Institute of Standards and Technology/ NIST)
Zinc
Zn metal
Strong reducing agent for
(Cd, Cu)
Magnesium
Mg metal
Potassium
bromide
Calcium
carbonate
KBr
119.01
CaCO3
100.09
Acid-Base and Redox Standards
Reagent
Formula
KHC8H4O4
FW
(g/mol)
204.23
Potassium
Hydrogen
Phthalate (KHP)
Sodium
carbonate
Sodium Oxalate
Standard acid
Na2CO3
106.00
Standard base
Na2C2O4
134.00
Potassium
Dichromate
Potassium
Bromide
Potassium iodide
K2Cr2O7
294.19
Reducing agent (used to
standardize permanganate
solutions)
Oxidizing agent
KBr
119.01
KI
166.00
Mild reducing agent
Potassium iodate
KIO3
214.00
Oxidizing agent
Reagent grade chemicals

Reagent conforming to standards set by the American Chemical Society.
 Must be used to prepare primary and secondary standards.
Preparation of Standard Solutions
By weighing and dissolving solid standard material or by serial dilutions of a stock
solution.
Sources of errors: mass determination, volumetric glassware calibration, dilutions.
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B.2
CHAPTER5_CAL
Single Point Standardization
A single standard solution used to determine k
S
k  s tan d
Cs
Assumptions
a) proper 'blank' has been used
b) k is constant throughout the full range of concentration of interest.
B.3
Multiple-Point Standardization
A least three standard solutions are used, and the signal measured is plotted against the
concentration to generate a calibration curve.
B.3.1 External Standards
Standards are analyzed in the absence of the matrix of the sample to determine k. The
signal generated by a sample containing the analyte is measured to obtain the
concentration.
Signal versus [analyte] in standards can be:
1. Linear (normal calibration curve) with 0 intercept and with intercept  0
Sstd= kCs
Sstd= kCs + Sreag
2. Nonlinear
Limitation: differences between the standards matrix and sample matrix may introduce
systematic errors.
Solution:
Use matrix matching or use standard addition method
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CHE 322
CHAPTER5_CAL
B.3.2 Standards Additions
(A) Single point standard addition
V
S samp  kC A o
Vf
V
V
S spike  k (C A o  C std std )
Vf
Vf
CA 
S samp C std Vstd
Vo ( S spike  S samp )
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CHE 322
CHAPTER5_CAL
(B) Multiple-point standard addition
Spike a series of sample with increasing amounts of standard. The measured signal
( S spike ) is given by equation
kC AVo kC sVs

Vf
Vf
Plot S spike versus Vs
S spike 
Slope =
k 
y-intercept =
kC AVo
Vf
x-intercept =
 C AVo
Cs
Slope  V f
 CA 
kCs
Vf
Cs

y int ercept  V f
C AVo
y int ercept  C s
Slope  Vo
Limitation: a standard addition calibration curve can not be used for other samples.
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CHAPTER5_CAL
B.3.3 Internal Standard
Internal standard is a known amount of a compound different from the analyte added to
the unknown. The signal from the unknown is compared to the signal from the internal
standard to find out how much analyte is present.
Useful when the sample analyzed or the instrument response varies slightly from run to
run for reasons that are difficult to control.
e.g.
-sample concentration changes during the analysis
-Non-controlled Instrumental drifts
Requirements
1) must not react with analyte or matrix
2) its signal must not overlap with the analyte signal
3) but have properties very similar to those of the analyte
Example: Problem 5.10
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CHE 322
B.
CHAPTER5_CAL
LINEAR REGRESSION AND CALIBRATION CURVES
What is the best straight line through calibration data?
Least square method: minimize the sum of the squares of the residuals
C.1
UNWEIGHTED LINEAR REGRESSION WITH ERROR IN Y
Errors in y are not dependent on value of x.
All standards contribute equally to the error in the regression analysis
d i  yi  (b0  b1 xi )
residual error
d i2   yi  b0  b1 xi 2
n
 d i2
i 1
Assumptions
1) error about regression is due to indeterminate errors affecting the values of the
signal measured (y).
2) inderteminate errors on the y values are similar and normally distributed
3) errors on y values do not depend on the value of x
4) errors on standards are similar and much smaller than error on y.
Results






S xx   xi  x
S yy   yi  y
2
 xi 
  xi2 
n
2
2
2  yi 
  yi 

n
S xy   xi  x yi  y   xi yi 
b1 
2
 xi  y i
n
S xy
S xx
b0  y  b1 x
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CHAPTER5_CAL
Standard deviation about regression
sr 
S yy  b02 S xx

n2
  yi  b0  b1 xi 2
n2
Standard deviation of the slope
s r2
S xx
sb 
1
Standard deviation of the intercept
sb  s r
0
 xi2
n  xi2   xi 2
Confidence Intervals
1  b1  tsb
1
 0  b0  tsb
0
Standard deviation on a calculated x value
y  b0
x x
b1
s
sx  r
b1
m

2
yx  y
1 1
 
2
m n b2
x

x

i
1


number of replicate measurements
Confidence interval for the analyte's concentration
 x  x  ts x
t for (n-2) degrees of freedom
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CHE 322
C.2
CHAPTER5_CAL
WEIGHTED LINEAR REGRESSION WITH ERRORS IN Y
C.2.1 Checking the validity of the assumptions used for unweighed linear
regression analysis
Plot residuals versus x.
C.2.2 When error in y depends on x
b1 
b0 
wi 
n wi xi yi   wi xi  wi yi
n wi xi2   wi xi 2
 wi yi  b1  wi xi
n
nsi 2
 si 2
si : standard deviation of y i
Read example 5.13
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CHE 322
C.3
CHAPTER5_CAL
CURVILINEAR AND MULTIVARIATE REGRESSION
Curvilinear
1) linearize relationship between the two variables
take log, reciprocal, square roots, exponentials
2) Use nonlinear regression
Multivariate
S meas  k AC A  k I C I  S reag
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D
CHAPTER5_CAL
BLANK CORRECTIONS
Blank solutions: solutions containing reagents and solvents used in analysis but 'no
deliberatly added analyte.
Typical blanks measure the response of the analytical method to impurities and
interfering species in reagents.
Blanks correct for constant sources of errors
An optimal Blank must correct for
1) Solvent effects
2) Reagent effects
3) Effects of interactions between analyte and sample matrix
Types of blank corrections
1) Calibration blank (CB)
Missing matrix
2) Reagent blank (RB)
Missing analyte
3) CB and RB
Missing analyte and matrix interactions
4) Total Youden Blank/ true blank correction
Signal versus samples of different sizes
Suggested problems
5.6, 5.7, 5.8, 5.13
Homework (40 points)
5.12
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