CHEM 213Instrumental Analysis

Lab Lecture – Copper by AA& Least Squares Analysis

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Flame Atomic Absorption

In the gas phase, atomic species will absorb light.

General steps:1. M(ABC) M(XYZ) (aq) M (g)2. Perform spectroscopy on M (g)

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Flame AA

Flame atomizes most molecular speciesCu(NO3)2 Cu + NO + NO2 +…

Cu in gas phase will absorb light according to Beer’s LawA = aλbc; b = length of flame, c = concentration of vapour in

flame

Wavelength of absorption depends on the electronic structure of the atomic species (here Cu vapour)

Use light source of appropriate wavelength for species being measured.

Generate calibration curve and off you go, but…

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But…

In many methods, the matrix can have significant effects on instrument response.

This sample matrix has…Cu, Ni, excess acid, other unknown elementsDifficult to reproduce and duplicate

Problem:How to account for matrix we cannot reproduce?!?

Inst

Res

p

Conc of analyte

Normal response

Reduced responsedue to matrix

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Calibration by Standard Addition

THE method of choice when matrix effects are present/expected

U S

…S0 S1 S2 S3 S4

1. Add SAME known volume of unknown (U) to each vial

2. Add increasing SMALL volumes of standard (S) to each vial

Volume of standard must be << Volume of sample.

Std must be >> concentration of sample

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Plotting Std Addition Data

+

+

+

+

+

•

Asolution

cstd added0

Intercept

Plot CORRECTED Absorbance on Y-Axis

Plot VOLUME of STD Addedon X-Axis

Get a straight line with X-intercept of

– Vol std. added

[U] = - intercept × [S] / Vol U

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General notes:

• Prepare stock standard (1.2 to 1.4 mg/mL) and unknown solutions (both in 100 mL).

• Pipet 10 mL of unknown into 5×50 mL flasks.• Pipet different amounts of standard to each flask (0, 50,

100, 150, 200 µL). Use 50 µL micropipet

• Calibrate the micropipet ahead of time per the instructions on page 46. (The time consuming step)

• Record Pipet number and save the pipet tip!

• No data printout, so you need to write down all the numbers

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Micropipettes

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Calibration Curves

• In 211 used replicate standards of the same concentration to standardize the titrant.

• With instruments, response is

measured for several concentrations and a calibration curve is developed.

• The concentration of an unknown is determined from the curve.

0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)A

bsor

banc

e0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)

0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)A

bsor

banc

e0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)A

bsor

banc

e0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)

0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)A

bsor

banc

e

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Rules for Calibration curves

0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)

Abs

orba

nce

0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)

0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)

Abs

orba

nce

0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)

Abs

orba

nce

0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)

0.20

0.40

0.60

0.80

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10

Fe Conc (g/mL)

Abs

orba

nce

Unknown must fall within range of standards

Res

po

nse

Var

iab

le

Concentration

X-Axis assumed to be error free

Y-Axis assumed to contain all errorError independent of magnitude

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Graphical Method (plot by hand)

• Advantage: By visually inspecting the data, it becomes obvious if the data fall on a line.

• Disadvantages: – Line is drawn by “eye” subjective process, imprecise – Difficult to read graph to less than a few parts per thousand.

0.0 2.0 4.0 6.0 8.0 100.20

0.40

0.60

0.80

1.0

1.2

Discard point

Use individual measurements

Obtain individual concentrations

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Method of Least Squares

• Advantage: Method is objective and without systematic bias. – Results are usually chosen from this over graphical

• Disadvantage: Method is accurate only if the data truly fall on a straight line. – The method itself does not discard points. – Always compare the least squares results with those

from the graphical method.

• Note: Do NOT simply use the “add trendline” function in Excel. It includes all data points, does not allow for error calculation.

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x

ymslope

Least Squares Fit

Least squares – process of fitting a mathematical function to a set of measured points by minimizing the sum of the squares of the distances from the points to the curve

Fig 4-9

residuals

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10

15

20

25

30

35

40

0 20 40 60

10

15

20

25

30

35

40

0 20 40 60

Look at your DATA!!!

BIG Difference in your result!!!

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How does it work?

Minimize the sum of the squares of the residualsy = mx + b; yi is measured value, y is predicted value

(from equation of line)

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15

20

25

30

35

40

0 20 40 60

di = yi - ydi

yi

y

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• Data points: xi and yi

xi = concentration of standard for point i

yi = observed response for concentration i

• Linear Equation estimate of y:

y = mxi + b

y residual = yi - (mxi + b), that is yi - y

Least Squares Statement:

Q = ∑ [yi - (mxi + b)]2

- find the values of m and b that minimize Q- data that deviates significantly from the line has a large effect on

least squares fit

Least Squares Fit

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• Lab. Manual, page 75-76, Text p 66-67.Least Squares Fit

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Lab. Manual, page 77

Least Squares Fit

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1. Manually on graph paper.

2. Mathematically using the least squares method.• Is the calibration curve linear?

- can I use y = mx + b ?• What is the best straight line?

- what are m and b?• What are the errors in m and b?

- what are sm and sb?

• What is the error in a determined concentration?

- what is sx?

Drawing a Calibration Curve

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• Deviation Table Lab. Manual, page 77Least Squares Fit

y = mx + b deviation = yi - y

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Least Squares Fit

Errors (-) in m, b and y (signal)Lab Manual, pg 78

2

2

n

ds iy

22

222

)(

)(

ii

yib

xxn

sxs

22

22

)( ii

ym

xxn

sns

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xunk = (yunk - b)/m• Set up tables to calculate input values for the least-

squares equations.• Evaluate b and m.• Evaluate xunk from individual measurements (yi

unk) and calculate average.

• Evaluate uncertainty in the average value.• Evaluate uncertainty in reported value.

-specific to each experiment!!!!

Least Squares Fit

Calculation of unknown concentrationLab Manual, pg 78

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Least Squares Fit

Calculation of error in avg unknown concentrationLab Manual, pg 79

k = number of measurements of unknownn = number of points in calibration curvexi = individual x values in calibration curvex = average of all x values in calibration curvey = average of all the y values in the calibration curvey = average of all the values of the y unknown

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2

)(

)(11

xxm

yy

nkm

ss

i

yx

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Could also use (easier when doing by hand):

Least Squares Fit

22

222

)(

21

ii

iiyx

xxn

xxxnx

km

ss

k = number of measurements of unknownn = number of points in calibration curvexi = individual x values in calibration curvex = average of all x values FOR UNKNOWN

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Chem 213 Least Squares Fit Excel Program

Download New_LSQ.xls from www.chem.ualberta.ca/courses/Chem 213/also available on course website in the Laboratory section

En

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En

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Y v

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En

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Y v

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Results appear here

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Chem 213 Least Squares Fit Excel Program

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• The final answer for the unknown value to be reported most often involves further calculations. – Such calculations will normally require propagation of error

calculations to arrive at a final uncertainty.

• In particular, note the explicit examples presented in Appendix A of the Laboratory Manual.

• Note again the rules for Propagation of Uncertainty and Sig. Fig.

Review Chapter 3 of Text!

Final Answer Reported