Chem. 31 – 2/18 Lecture

Announcements• Turn in AP1.2• Quiz today• Exam 1 coming up (1 week from next

Monday)• Today’s Lecture

– Chapter 4 Material• Calibration and Least Square’s Analysis

– Chapter 6 Material• Equilibrium Expressions from Reactions

Calibration• For many classical methods direct

measurements are used (mass or volume delivered)

• Balances and Burets need calibration, but then reading is correct (or corrected)

• For many instruments, signal is only empirically related to concentration

• Example Atomic Absorption Spectroscopy– Measure is light absorbed by “free”

metal atoms in flame– Conc. of atoms depends on flame

conditions, nebulization rate, many parameters

– It is not possible to measure light absorbance and directly determine conc. of metal in solution

– Instead, standards (known conc.) are used and response is measured

Light beam

To light Detector

Method of Least Squares• Purpose of least squares method:

– determine the best fit curve through the data– for linear model, y = mx + b, least squares determines

best m and b values to fit the x, y data set– note: y = measurement or response, x = concentration,

mass or moles• How method works:

– not required to know math to determine m and b– the principle is to select m and b values that minimize

the sum of the square of the deviations from the line (minimize Σ[yi – (mxi + b)]2)

– in lab we will use Excel to perform linear least squares method

Example of Calibration Plot

Mannosan Calibration

y = 541.09x + 6.9673

R2 = 0.9799

0

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5 0.6

Conc. (ppm)

Pe

ak

Are

a

Best Fit Line Equation

Best Fit Line

Deviations from line

Assumptions for Linear Least Squares Analysis to Work Well

• Actual relationship is linear• All uncertainty is associated with the

y-axis• The uncertainty in the y-axis is

constant

Calibration and Least Squares- number of calibration standards (N)

N Conditions1 Must assume 0 response for 0 conc.; standard must

be perfect; linearity must be perfect2 Gives m and b but no information on uncertainty

from calibrationMethods 1 and 2 result in lower accuracy, undefined precision

3 Minimum number of standards to get information on validity of line fit

4 Good number of standards for linear equation (if standards made o.k.)

More standards may be needed for non-linear curves, or samples with large ranges of concentrations

Use of Calibration Curve

Mg Example:An unknown solution

gives an absorbance of 0.621

Use equation to predict unknown conc.

y = mx + bx = (y – b)/mx = (0.621 + 0.0131)/2.03x = 0.312 ppmCan check value graphically

y = 2.0343x - 0.0131

R2 = 0.9966

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.10 0.20 0.30 0.40 0.50

Mg Conc. (ppm)

Ab

sorb

ance

Calibration “Curve”

Use of Calibration Curve- Uncertainty in Unknown Concentration

2

2

)(

)(11

xxm

yy

nkm

SS

i

iyx

Uncertainty given by Sx (see below):

Notes on equation: m = slope, Sy = standard error in yn = #calibration stds k = # analyses of unknown, xi = indiv std conc., yi = unknown responseThe biggest factors are Sy and mTwo other parameters that often indicate calibration quality are R2 and b. R2 should be close to 1 (good is generally >0.999); b should be small relative to y of lowest standard.

Use of Calibration Curve- Quality of Results

• Quality of Results Depends on:– Calibration Results

• R2 value (measure of variability of response due to conc.)

• Reasonable fit– Range of Unknown

Concentrations• Extrapolation outside

of range of standards should be avoided

• Best concentration range (see next slide)

Good Calibration

y = 0.3634x - 0.1009

R20.9998 =

0.0000

2.0000

4.0000

6.0000

8.0000

10.0000

12.0000

0 5 10 15 20 25 30

Conc. (ppm)

Re

lati

ve

Pe

ak

Are

a

Line fit through Curve

y = 262.44x + 37.034R2 = 0.9772

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5

LG Conc. (ppm)

Pea

k A

rea

MN

Linear (MN)

Poor R^2 Value

y = 0.0041x + 0.0107

R2 = 0.9622

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60

Galactose Standard (ug)

Ab

sorb

ance

(49

0 n

m)

Better fit by curve

Use of Calibration Curve- Quality of Results

• Quality of Results Depends on:– Calibration Results

• on last slide– Range of Unknown

Concentrations• Extrapolation

outside of range of standards should be avoided

• Best concentration range

y = 2.0343x - 0.0131

R2 = 0.9966

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.10 0.20 0.30 0.40 0.50

Mg Conc. (ppm)

Ab

sorb

ance

Range of Standards (0.02 to 0.4 ppm)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.00 0.10 0.20 0.30 0.40 0.50

Mg Conc. (ppm)

Un

cert

ain

ty i

n C

on

c. (

pp

m)

Absolute Uncertainty

0

10

20

30

40

50

60

0.00 0.10 0.20 0.30 0.40 0.50

Mg Conc. (ppm)

% U

nce

rtai

nty

Relative Uncertainty

Best Range: upper 2/3rds of standard range

Calibration Question

• A student is measuring the concentrations of caffeine in drinks using an instrument. She calibrates the instruments using standards ranging from 25 to 500 mg/L. The calibration line is:Response = 7.21*(Conc.) – 47The response for caffeine in tea and in

espresso are 1288 and 9841, respectively. What are the caffeine concentrations? Are these values reliable? If not reliable, how could the measurement be improved?

Ch. 3 and 4 – What you need to know

• Equations you need to know:– Average calculation– t and Z based confidence intervals– line equation

• Equations I will provide:– Propagation of uncertainty for +/-, *//,

and exponent– Standard deviation– Case 2 and 3 t-test, F-test and Grubbs

test (if needed)

Equilibrium Equations

Equilibrium Equations from Chemical Equations (Reactions)

Generic Example:aA + bB ↔ cC + dD (Reaction)

ba

dc

BA

DCK

Equilibrium Equation

Compounds are in equation if in solution (not present as solid, or solvent). Concentrations are in M but K is unitless

Similar equation for gases (except with PAa replacing

[A]a)

Equilibrium Equations

Example problem:Write equation for reaction:

AgCl(s) + 2NH3(aq) ↔ Ag(NH3)2+(aq) + Cl-(aq)

23

23)(

NH

ClNHAgK

AgCl not included because it is a solid

Equilibrium Equations- manipulating reactions

a) Flipping Directions- If for A ↔ B, K = K1, then for B ↔ A, K =

1/K1

b) Adding Reactions1) NH4

+ ↔ NH3(aq) + H +

2) H+ + OH- ↔ H2O(l)

3) NH4+ + OH- ↔ NH3(aq) + H2O(l)

Reaction 3) = rxn1) + rxn2)So K3 = K1K2