u311 topic 1 - people · 2006-09-05 · chemistry 311: topic 1: figures of merit and calibration...

Chemistry 311: Topic 1: Figures of Merit and Calibration Techniques

Instructor: Rob O’Brien (Sci 163) 807-9569 email: [email protected] Course Description: Overview of instrumental methods of chemical analysis including spectroscopic methods, mass spectrometry, surface analysis, and chromatography. OUC equivalent: CHEM 325. [3-3-0] Required Prerequisites: CHEM 211. STAT 230 or BIOL 304 is recommended. Lectures: 7:00 pm - 8:30pm; Tuesday, Wednesday. Sci 337 Laboratory: 2:00 pm - 5:00 pm, Thursday: Sci 221


Required Text / Material: 1. Principles of Instrumental Analysis: Forth Edition, Skoog, D.A., Holler, J.F., & Nieman, T.A. (QD 79 .I5 S58 1992) Note: Harris is also acceptable. 2. Chemistry 311 Laboratory Manual, Fall 2006 version Helpful Reference Material: 1. My Chem 311 Web site (http://people.ok.ubc.ca/orcac/chem311.html)

2. Quantitative Chemical Analysis, Sixth Edition, Daniel C. Harris 3. Harris Textbook Web site (http://bcs.whfreeman.com/qca) 4. The ACS Style Guide: A Manual for Authors & Editors, Editor: Janet S. Dodd, American Chemical Society, 1998 (ISBN: 0-841-23462-0) 5. Statistics and Chemometrics for Analytical Chemistry 4th edition, Miller, J.N. & Miller, J.C., Prentice Hall, Toronto, 2000 ISBN 0-130-22888-5 (QD 75.4.S8 M54) Evaluation: Midterms (x2) 30% (October 4, November 6) Laboratory 30% Final Exam 40% Note: The laboratory and the lecture must be passed independently. Students are expected to use a word processor as well as a spreadsheet program.


Course Outline: Topic 1: Basic Statistics and Calibration Techniques: Figures of merit and calibration techniques. Skoog Ch. 1 Topic 2: Atomic Spectroscopy: A brief introduction to spectroscopic instruments followed by a review of atomic spectroscopic techniques; Atomic Absorption, Graphite Furnace, ICP and XRF. Skoog Ch. 6-10, 12 Topic 3: Molecular Spectroscopy: A brief survey of major molecular spectroscopic techniques, FTIR, UV/VIS, and Fluorescence. Skoog Ch. 13-18 Topic 4: Mass Spectrometry: Introduction to Mass Spectrometry. Quadrupoles, Ion Traps, Magnetic and Electrostatic sectors, Time-of-flight. Ionization techniques, ESI, EI, CI, FAB, and MALDI. Skoog Ch. 20, 11 Topic 5: Basic Chromatography: Introductory Theory, Basic Components, Qualitative and Quantitative applications. HPLC, GC, Ion Chromatography. Skoog Ch. 26-29 Topic 6: Electrophoresis and Ion Mobility: Brief Introduction to Separations based on ion flux and applications. Skoog Ch. 30


Statistic terms and techniques you need to know; • The Standard Deviation (s)

• Degrees of Freedom

• Variance - Square of the standard deviation (s2)

• Relative Standard Deviation or the coefficient of variation.

• Gaussian distribution

• Confidence Limits & Confidence Interval

• Student T-test

• F-Test (see also stats Excel file - web site: http://people.ok.ubc.ca/orcac/stats.xls)


µ = X ± t s

√N

Summary of Confidence Limit and Data Reporting:

• For data sets where n>30, report confidence limit using standard deviation and a k factor (typically 2) i.e., 30.1 ± ks where k= 2 for 95% CL

• For data sets where n<30, report confidence limit using standard t value intervals. i.e., using the formula

• When reporting results indicate confidence level, confidence

interval, standard deviation and number of measurements, ie., o The measured value at the 95% confidence level is;

30.1 ± 0.5 where s= 0.41 and n = 5 • The confidence limit value only has one significant figure. The magnitude of

the confidence limit “sets” the number of significant digits in the measured value. For example, if the standard deviation above was 0.9 rather than 0.41, the reported 95% confidence level would be

30 ± 1 where s= 0.9 and n = 5 • It is acceptable to carry an extra, non-significant digit when reporting values.

Ideally that should be subscripted but in practice it is not always, for example;

30.1 ± 0.51 where s= 0.41 and n = 5 30.1 ± 1.1 where s= 0.9 and n = 5


Performance Characteristics of Instruments - Figures of Merit

A figure of merit is a number, derived from measurements, that is used to evaluate an instrument or an analytical technique.

The number corresponds to a criterion that can be used to evaluate the performance of the instrumental method.

To decide whether a given instrumental method is suitable for solving an analytical problem.

Some figures of merit are:

Accuracy Precision Bias Sensitivity Detection limit Dynamic range Selectivity


Accuracy: Indicates how close a measured value is to the true value. Normally expressed as the relative percent error. (can be absolute error)

o 1% error indicates that the measured concentration is within 1% of the true analyte concentration

Precision

A measure of the reproducibility of the method. o How close are the results obtained in the same way?

Standard deviation is an effective monitor of random error Usually expressed as a percent relative standard deviation.

Accurate & Precise Precise not Accurate Accurate not Precise


Error Types:

Two types of error in an analytical measurement o Random error indicated by Standard Deviation o Systematic or determinate error indicated by?

Bias

Measure of the systematic (determinate) error of a measurement. Expressed as an absolute error

o bias = Xave – µ where; Xave = mean measured value, µ = True value Bias has both a definite magnitude and sign. Typical sources are Instrumental errors, Personal errors, and Method errors.

Shots very precise but offset by a fixed amount.

Clear Systematic error


Sensitivity Indicates the response of the instrument to changes in analyte concentration or a measure of a method’s ability to distinguish between small differences in concentration in different samples. Effected by slope of calibration curve & precision.

Indicated by the slope of the calibration curve or in other words, a change in analytical signal per unit change in [analyte]

For two methods with equal precision, the one with steeper calibration

curve is more sensitive. Calibration sensitivity If two methods have calibration curves with equal slopes, the one with

higher precision is more sensitive. Analytical sensitivity


Calibration sensitivity (S):

The slope of the calibration curve evaluated in the [analyte] range of interest.

o S = mc + Sbl (m = slope; c = conc; Sbl = Signal of Blank)

Advantage: With linear calibration curve, sensitivity independent of [analyte]

Disadvantage: does not account for precision of individual measurements

Analytical Sensitivity (γ)

Defined by Mandel and Stiehler to include precision in sensitivity definition

o γ = m/Ss (m = slope; Ss is the standard deviation of measurement)

Advantage: o Insensitive to amplification factors i.e. increasing gain also increases m

but Ss also increases by same factor hence γ stays constant.

o Independent of measurement units for S

Disadvantage: concentration dependent as Ss usually varies with [analyte]


cm = Sm—Sbl

m

Detection Limit The smallest [analyte] that can be determined with statistical confidence.

Analyte must produce an analytical signal that is statistically greater than the

random noise of blank. (i.e. analytical signal = 2 or 3 times std. dev. of blank

measurement (approx. equal to the peak-peak noise level).

Calculation of detection limit

o The minimum detectable analytical signal (Sm) is given by: Sm = Sbl + k(stdbl); for detection use k =3 To Experimentally Determine

o Perform 20 – 30 blank measurements over an extended period of time.

o Treat the resulting data statistically to obtain Sbl (mean blank signal) and

stdbl (std. dev. of blank signals). Use these to obtain Sm value.

o Using slope (m) from calibration curve. Detection limit (Cm) is calculated

by: (Rearranged from Sm = mc + Sbl)


Detection Limit Example 1: An analysis of the calibration data for the determination of lead based upon its flame emission spectrum yielded an equation: S = 1.12C + 0.312 where C is the Pb concn in ppm and S is a measure of the relative emission intensity. The following replicate data were obtained:

Concn (C), ppm

No. of replicate

s

Mean value of S

Std. dev.

10.0 10 11.62 0.15 1.00 10 1.12 0.025

0.000 24 0.0296 0.0082 Calculate (a) the calibration sensitivity, (b) the analytical sensitivity at 1 and 10 ppm of Pb, and (c) the detection limit.


Detection Limit Example Solution: a) calibration sensitivity is the slope of the calibration curve = 1.12 b) Using γ= m/Ss : at 10 ppm γ= 1.12 / 0.15 = 7.5 at 1 ppm γ= 1.12 / 0.025 = 45

c) Using: Sm = Sbl + k(stdbl) Sm = 0.0296 + 3(0.0082) = 0.054 Therefore from:

cm = Sm—Sbl

m

cm 0.054 - 0.0296

1.12 = 0.022 ppm Pb


Dynamic Range [Range] over which measurements can be made. Extends from LOQ to LOL.

LOQ (limit of quantitation): [lowest] at which quantitative measurements can

reliably be made. Equal to 10 x Average Signal for blank i.e. 10Sbl.

LOL (limit of linearity): point where signal is no longer proportional to

concentration.


Selectivity Degree to which a measurement is free from interferences by other species contained in the matrix. Analytical Signal Detected is a sum of the Analyte signal plus interference signals

S = maCa + mbCb + mcCc + Sblank

Selectivity is a measure of how easy it is to distinguish between the analyte signal and the interference signal.

Selectivity of an analytical method can be described using a figure of merit called selectivity coefficient.

kb,a = mb/ma : kc,a = mc/ma

S = ma (Ca + kb,aCb + kc,aCc) + Sblank

Selectivity coefficients range from 0 to >> 1. Can be negative if interference reduces the observed signal.


Instrument Calibration Methods Limited number of techniques that we can use to find the relationship between analyte concentration and instrument response or signal. The most common approaches are;

• External Calibration Method (Calibration curve)

• Standard Addition Method

• Internal Standard Method


Calibration of Instrumental Methods Calibration curve (also analytical curve or working curve) Basis of quantitative analysis is that the magnitude of the measured property is proportional to concentration of analyte.

It is technically possible to determine the proportionality constant (k) between the [analyte] and resulting signal using a single standard;

k = (SStandard)/(CStandard) some times called k factor.

However, both the systematic and indeterminate errors would also be included. Instead, a plot of the analytical signal (of the measured property) versus the analyte concentration is used to determine a linear calibration relationship. This is typically obtained by measuring the analytical signals for a series of standards i.e. analyte solutions of known concentration.


Method of Least Squares: Ideally the analyte content and the measured signal should exhibit a perfectly linear relationship of formula: Y = mX + b or Signal = m (Conc.) + Sblank In reality such perfectly linear relationships don’t exist when real samples and standards are used.

Therefore the “best” line relationship among the experimental points is used finding the best-fit line of the data. Visual estimation of the “best” line through a set of points is subject to error, both in plotting the points and in fitting the line. A preferable approach for finding such a line is the method of least squares or sometimes referred to as linear regression.


Method of Least Squares - Assumptions That the values plotted on x-axis are known with much higher precision and accuracy than those on the y-axis. The uncertainties in the y values are greater than those in the x values. The line representing the data should be drawn so that deviations of the y values are minimized.

Thus the best fit line or the least squares line is the straight line drawn that minimizes the vertical deviations between the points and the line.

These vertical deviations are called residuals Since some of the deviations are positive and some are negative we preferably minimize the sum of the squares of the deviations. Hence its name the method of least squares. That is we draw the straight line that has the least value for the sum of squares of the deviations In practice, for a set of data points, it is possible to calculate the values of the slope (m) of line and the intercept (b) which minimizes the sum of squares.


Method of Least Squares - Reliability Notice: the least-squares line need not pass right through the data points. This is the consequence of indeterminate errors affecting the signal. The reliability of the least squares method can be estimated by calculating the uncertainties in the slope (m) and the intercept (b) of the line.

Estimation of uncertainty in a result from a calibration curve

Frequently in instrumental analysis a calibration curve is prepared, and values for unknown samples are read from it. If the calibration is a straight line, the standard deviation of the unknown, Sc, can be estimated from the measurements from which the straight line was obtained and the measurements made on the sample.

Note that Sc is smallest when yc = y.

o Measurements of the unknown should be near the center of the calibration plot for minimum error.

o Since the uncertainty increases as yc moves away from y, results obtained by extrapolation are prone to increased error.


Calibration of Instrumental Methods - Regression


Method of Least Squares - Calculations using Excel Manual Calculations of Least Squares can be done by manually using calculation techniques outlined in Appendix 1 In practice, spreadsheets such as Excel provide the most practical method for calculating Least Squares. Excel contains three main functions that can be used to fit Least Squares

Trendline o Operative only in Graph mode o Very limited stats available

Regression

o Available in Tools/Data Analysis/Regression o Requires to be added on o Numerous Parameters Reported. Many not needed.

LINEST

o Most useful set of data o Difficult to access and unlabelled data


Calibration of Instrumental Methods

[Analyte] # Replicates Mean Signal Std Dev Predicted Residuals Functions0.00 25 0.031 0.0079 0.030 0.001 Intercept 0.03002.00 5 0.173 0.0094 0.164 0.009 Slope 0.06706.00 5 0.422 0.0084 0.432 -0.010

10.00 5 0.702 0.0084 0.700 0.00214.00 5 0.956 0.0085 0.969 -0.01318.00 5 1.248 0.0110 1.237 0.011

y = 0.0670x + 0.0300R2 = 0.9996

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.00 5.00 10.00 15.00 20.00

Series1

Linear(TRENDLINE)


[Analyte] # Replicates Mean Signal Std Dev

0.00 25 0.031 0.00792.00 5 0.173 0.00946.00 5 0.422 0.0084

10.00 5 0.702 0.008414.00 5 0.956 0.008518.00 5 1.248 0.0110

Results Produced Using Tools/Data Analysis/RegressionSUMMARY OUTPUT

Regression StatisticsMultiple R 0.999783818R Square 0.999567683Adjusted R Square 0.999459604Standard Error 0.010873998Observations 6

ANOVAdf SS MS F Significance F

Regression 1 1.093574358 1.09357436 9248.468238 7.00969E-08Residual 4 0.000472975 0.00011824Total 5 1.094047333

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Lower 95.0%Upper 95.0%Intercept 0.030013699 0.007311135 4.10520381 0.014789982 0.009714692 0.050313 0.009715 0.050313X Variable 1 0.067038356 0.000697089 96.1689567 7.00969E-08 0.065102922 0.068974 0.065103 0.068974

RESIDUAL OUTPUTObservation Predicted Y Residuals

1 0.030013699 0.0009863012 0.164090411 0.0089095893 0.432243836 -0.0102438364 0.70039726 0.001602745 0.968550685 -0.0125506856 1.23670411 0.01129589


Calibration of Instrumental Methods - Regression

Plot Generated by Regression

00.20.40.60.8

11.21.4

0.00 5.00 10.00 15.00 20.00

X Variable 1

Y YPredicted Y


X Variable 1 Residual Plot

-0.015-0.01

-0.0050

0.0050.01

0.015

0.00 5.00 10.00 15.00 20.00

X Variable 1

Res

idua

lsCalibration of Instrumental Methods - Regression


Calibration of Instrumental Methods - LINEST X Y[Standard] Mean Signal Predicted Y Residual Y

0.00 0 0.209 0.2090.10 12.36 12.279 -0.0810.20 24.83 24.350 -0.4800.30 35.91 36.420 0.5100.40 48.79 48.491 -0.2990.50 60.42 60.561 0.141

@LINEST(Y-Values, X-Values,Intercept true or False, Stats true or False)Select 2 x 5 Region, Insert Function LINEST, ctrl-shift-return

Values Generated using @LINEST FunctionCol. 1-Linest ArraCol 2-Linest Array

Slope => 120.7057143 0.208571429 <= InterceptStd Dev m=> 0.964064525 0.29188503 <= Std Dev bR^2=> 0.999744903 0.403297125 <= Variance = Std Dev RegressionF function=> 15676.29604 4 <= # degree of freedomSum Res.^2 2549.727156 0.650594286 <=Sum Res.Y^2


Calibration of Instrumental Methods - LINEST

X Y[Standard] Mean Signal Predicted Y Residual Y

0.00 0 0.209 0.2090.10 12.36 12.279 -0.0810.20 24.83 24.350 -0.4800.30 35.91 36.420 0.5100.40 48.79 48.491 -0.2990.50 60.42 60.561 0.141

Values Generated using @LINEST FunctionSlope => 120.7057143 0.208571429 <= InterceptStd Dev m=> 0.964064525 0.29188503 <= Std Dev bR^2=> 0.999744903 0.403297125 <= Std Dev regF function=> 15676.29604 4 <= # DoFSum Res.^2 = 2549.727156 0.650594286 <=Sum Res.Y^2

y = 120.71x + 0.2086R2 = 0.9997

0

10

20

30

40

50

60

70

0.00 0.10 0.20 0.30 0.40 0.50 0.60


Example Problem for Practice: Do linear regression analysis on each of the following data sets. Indicate which is the “best fit”. What parameter was most indicative of your conclusion? Data Set 1 Data Set 2 Data Set 3 Data Set 4X Y X Y X Y X Y

4 4.26 4 3.1 4 5.39 8 6.585 5.68 5 4.74 5 5.73 8 5.766 7.24 6 6.13 6 6.08 8 7.717 4.82 7 7.26 7 6.42 8 8.848 6.95 8 8.14 8 6.77 8 8.479 8.81 9 8.77 9 7.11 8 7.04

10 8.04 10 9.14 10 7.64 8 5.2511 8.33 11 9.26 11 7.81 8 5.5612 10.84 12 9.13 12 8.15 8 7.9113 7.58 13 8.74 13 12.74 8 6.8914 9.66 14 8.1 14 8.84 19 12.5


Standard Addition: If the sample matrix is both complex and problematic an external calibration curve may not be acceptable. In standard addition, the sample is used as the matrix for the calibration. If the volume of standard added is large enough to dilute the solution, then the dilution factor must also be considered. For Example: A volume, V0 sample is diluted to a final volume, Vf, and the signal, Ssamp is measured. A second identical aliquot of sample is spiked with a volume, Vs, of a standard solution for which the analyte's concentration, Cs, is known. The spiked sample is diluted to the same final volume and its signal, Sspike, is recorded.

Ssamp = kCA V0

Vf ( + ) Sspike = k CAV0

Vf

CsVs

Vf

=[Initial Analyte]

[Analyte + Standard]

Initial Signal from Analyte alone

Signal from Analyte & Standard


In other words; A one step standard addition is effectively equilivent to a one point calibration curve

Ssamp = kCA V0

Vf

( + ) Sspike = k CA V0

Vf Cs

Vs

Vf


A multi-point Standard addition calibration can also be used and this has similar advantages to the normal external calibration curve. The y-intercept will correspond to the signal of the “unspiked” sample


Internal Standard Method: An internal standard is a compound other than the analyte that is added to the unknown. The signal response of the internal standard is compared to the signal response of the analyte to determine the amount of analyte present. In order for this method to work the relative response factor of the analyte to the internal standard must be known. Response Factor (RF) for Analyte (A) and Internal Standard (IS):

Relative Response Factor (RRF):

RFA= SignalA

[A] RFIS=

SignalIS

[IS]

SignalA[A]

SignalIS

[IS]

RRF = RFA

RFIS =

u311 topic 1 - people · 2006-09-05 · chemistry 311: topic 1: figures of merit and calibration...

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