Chem 232: Quantitative Analysis Lecture Noteshuffman/include/lecture3.pdf · Scott Hu man Chem 232: Quantitative Analysis Lecture Notes chap 3: Basic tools of Analytical Chemistry

chap 3: Basic tools of Analytical Chemistry

Chem 232: Quantitative Analysis Lecture Notes

Scott Hu�man

September 8, 2014

Scott Hu�man Chem 232: Quantitative Analysis Lecture Notes


Topic

1 chap 3: Basic tools of Analytical Chemistry



De�nition: SF

Signi�cant Figures:

minimum number of digits required to express a value inscienti�c notation without loss of accuracy.

harris p39

SF means Signi�cant Figures

used in measurements and calculated numbers. NOT in counts

where the numbers are certain or integers.

How many people in the room?



De�nition: SF



harris p39







De�nition: SF



harris p39







Example: SF

If you have a number written:

1.2356 g

the 6 is the least signi�cant digit. This means that you are the

least con�dent in its accuracy. Another way of saying this is that it

has (the most ) some uncertainty.



Example: SF

If you have a number written:

1.2356 g

the 6 is the least signi�cant digit. This means that you are the

least con�dent in its accuracy. Another way of saying this is that it

has (the most ) some uncertainty.



Example: Learn By Example

Balance reports a mass of 1.2345 g. How many sf? -> 5


Volume is measured at 5.001 mL. How many sf?

4

Volume is measured at 5.001× 10−3 mL. How many sf?

4

NOTE: in regular numbers the decimal place is not important

for SFs?







4


4


for SFs?







4


4


for SFs?







4


4


for SFs?







4


4


for SFs?







4


4


for SFs?



Example: Scienti�c notation and Signi�cant Figures:

Scienti�c notation is useful, because the number of SFs is always

correct.

Ex. Take the following example: 5001 mL.

How many SF? 4

What about this example: 5000 mL.

How many SF? 4

This answer is ambiguous.

It could be 4 or 1 depending on the rules that you learned. So we

will used scienti�c notation whenever there is ambiguity. Convert

the previous example to scienti�c notation with 4 SFs

5.000× 103





correct.


How many SF? 4


How many SF? 4





5.000× 103





correct.


How many SF? 4


How many SF? 4





5.000× 103





correct.


How many SF? 4


How many SF? 4





5.000× 103





correct.


How many SF? 4


How many SF? 4





5.000× 103



Rules of Addition and Subtraction and SF:

When you add numbers together, how many signi�cant �gures do

you report in your answer?



Learn by Example: Easy example

1.0453× 10−1

+1.0000× 10−1

2.0453× 10−1




1.0453× 10−1

+1.0000× 10−1

2.0453× 10−1



Learn by Example: Harder Example

1.0453× 10−1

+1.00× 10−1

2.0453× 10−1

Is this correct?

NO!!!!! The second number contains the limiting number of SFs: 3

So, always use the limiting number of signi�cant �gures



Learn by Example: Harder Example

1.0453× 10−1

+1.00× 10−1

2.0453× 10−1

Is this correct?





Rules of Multiplication and Division and SF:

Rules of Multiplication and Division and SF

When you multiply numbers together, how many signi�cant �gures

do you report in your answer?




1.0463× 10−1

×1.0000× 10−12

1.0463× 10−13

Is this correct?

yes

Both numbers contain the same number of SF




1.0463× 10−1

×1.0000× 10−12

1.0463× 10−13

Is this correct?

yes

Both numbers contain the same number of SF



Learn by Example: Harder example

1.0463× 10−1

×1.00× 10−12

1.0463× 10−13

Is this correct?



1.05 Ö 10-13




1.0463× 10−1

×1.00× 10−12

1.0463× 10−13

Is this correct?



1.05 Ö 10-13




1.0463× 10−1

×1.00× 10−12

1.0463× 10−13

Is this correct?



1.05 Ö 10-13



Rules of logarithms and antilogarithms and SF:

what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017

The logarithm of a number contains two parts the

the characteristic is the integer count of how far from the

decimal the leading digit is

The mantissa is not an integer and therefore subject to sf rules.

log10( 16.1︸︷︷︸number

) = 1.207︸︷︷︸logarithm




what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017










what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017










what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017










what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017










what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017










what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017










what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017










what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017










what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017










what is the:

log10(1) = 0

log10(10) = 1

log10(100) = 2

log10(1000) = 3

log10(10.4) = 1.017









Example: example log

log10(0.001234) = -2.90868484

How many sf in the original number? 4

How many sf in the log? 4

so the result should be -2.9087




log10(0.001234) = -2.90868484







log10(0.001234) = -2.90868484







log10(0.001234) = -2.90868484







log10(0.001234) = -2.90868484






Example: antilog

antilog10(4.37) = 2.344228× 104

how many sf in the original numer? 2

how many in the result? 2

so the answer should be 2.3 Ö 104



Example: antilog

antilog10(4.37) = 2.344228× 104






Example: antilog

antilog10(4.37) = 2.344228× 104






Example: antilog

antilog10(4.37) = 2.344228× 104






Example: antilog

antilog10(4.37) = 2.344228× 104






Rounding:

how do you round

1.2345 to the nearest

ones place 1.

tenths place 1.2

hundredths place 1.23

thousandths place 1.234 or 1.235. which is it?



Rounding:

how do you round


ones place 1.

tenths place 1.2





Rounding:

how do you round


ones place 1.

tenths place 1.2





Rounding:

how do you round


ones place 1.

tenths place 1.2





Rounding:

how do you round


ones place 1.

tenths place 1.2





Rounding:

how do you round


ones place 1.

tenths place 1.2





Rounding:

how do you round


ones place 1.

tenths place 1.2





Rounding:

how do you round


ones place 1.

tenths place 1.2





Precollege Rounding:

digit past rounding digit is a 5 round up

problem: this provides a bias toward higher numbers



Precollege Rounding:

digit past rounding digit is a 5 round up

problem: this provides a bias toward higher numbers



Rounding for the rest of your life

digit past rounding digit is > 5 round up

1.23456 round to thousandths place 1.235

digit past rounding digit is < 5 round down


digit past rounding digit is exactly 5 round up and down.

how: many methods:

simplest is round to the nearest even signi�cant digit.

so our answer for 1.2345 rounded to thousandths place is 1.234










how: many methods:












how: many methods:












how: many methods:












how: many methods:












how: many methods:












how: many methods:












how: many methods:






Example Problem: Types of Error

2 types

systematic or deterministic

asystematic or random



De�nition

Systematic Error:

error that is deterministic or consistently o�set from the

correct value



Detection of Systematic Error:

Analyze a known sample, such as a standard reference material

your method of measurement should reproduce the knownvalues

Analyze a blank sample,if your method gives a non-zero value, your method respondsto more than you expected. (interference)

Use di�erent analytical methods to measure the same quantity.

If the two methods don't agree, then there is error in one orboth of the methods.This is usually what is done in forensic labs. Agreementbetween methods is required.

Round robin experiment: di�erent people in severallaboratories analysis identical samples by the same or di�erentmethods.

Disagreement beyond estimated random error is systematicerror.



Example: Volumetric Glassware

What if a 100.0 mL volumetric �ask actually, consistently contains

a 100.12 mL.

This is corrected by performing all volume calculations with the

actual volume. Such as concentration: 0.0010000.1000 = 1.000× 10−2 M

actual concentration → 0.0010000.10012 = 0.9988× 10−2 M

Correction of Systematic Error

Calibration



Example: Random Uncertainty or Error

These are due to uncontrollable �uctuations in measurement.

These errors �uctuate above and below the �true� value.

Correction of Random Error

This type of error is corrected by many measurements or replication.



Example: True Value is 10.390

10.079

10.579

10.871

10.469

10.419

mean = 10.483



Example: True value is 10.390

10.356

10.850

10.858

10.303

10.462

10.261

10.245

10.022

10.281

10.341

10.054

10.980

10.617

10.124

10.145

mean = 10.393Scott Hu�man Chem 232: Quantitative Analysis Lecture Notes


De�nition: Accuracy

Accuracy:

nearness to the truth



De�nition: Precision

Precision:

reproducibility



Special Notation: recorded vs. reported

NOTE:

A special notation in this book is to write in your notes one extra

digit after the least signi�cant digit. But this digit is subscripted to

indicate that it is only to prevent rounding errors.

example:

1.2314 M

the 4 is not signi�cant, but if

We have to use this value in any additional calculations

We have kept and extra digit to help midigate rounding errors

We will call this notation the recorded value

We will call 1.231 M the reported value



Special Notation: gaps and commas

number convention Comment

1000000.017 di�cult to read

1,000,000.017 US easy for us to read

1.000.000,017 European strange for us

1 000 000.017 more universal book's convention

found in reference material





























































































De�nition: uncertainty

uncertainty:

expressed as a range or standard deviation

example: using a buret reading 12.54 ± 0.02 mL means that

the volume reading is between 12.52 and 12.56 mL

example: written on glassware



De�nition: absolute uncertainty

absolute uncertainty:

same units as measurments

Example: 12.01 ± 0.01 mL



De�nition: relative uncertainty

relative uncertainty:

a ratio of absolute uncertainty to magnitude of measurement

relative uncertainty =

absolute uncertainty

magnitude of measurement

dimensionless quantity



De�nition: Percent Relative Unvertainty

Percent Relative Uncertainty:

relative uncertainty times 100 %

Example 12.01 mL (± 0.08%)



De�nition: Propagation of Random Error

Propogation of Random Error :

a method for estimating the uncertainty in a determined value, that

was determined using multiple measurements.



Example: Addition and subtraction

+ 1.76 ( ± 0.03) e1 mL

+ 1.89 ( ± 0.03) e2 mL

- 0.59 ( ± 0.02) e3 mL

+ 3.06 ( ± e4) mL

what is e4?

You cannot just add the uncertainties together, why?

because of ±. There should be some cancellation.




+ 1.76 ( ± 0.03) e1 mL

+ 1.89 ( ± 0.03) e2 mL

- 0.59 ( ± 0.02) e3 mL

+ 3.06 ( ± e4) mL

what is e4?






+ 1.76 ( ± 0.03) e1 mL

+ 1.89 ( ± 0.03) e2 mL

- 0.59 ( ± 0.02) e3 mL

+ 3.06 ( ± e4) mL

what is e4?





Propagating Random Uncertainty: Addition and Subtraction

derived in Appendix C using Calculus.

You will not be held responsible for coming up with these

equations, but you will have to remember them

e4 =√

e21+ e2

2+ e2

3

e4 =√

(0.03)2 + (0.03)2 + (0.02)2 = 0.046904 = 0.0470.047 ← recorded uncertainty

so ± 0.05 mL is the absolute uncertainty.

We you report the value

3.06 ± 0.05 mL







e4 =√

e21+ e2

2+ e2

3

e4 =√(0.03)2 + (0.03)2 + (0.02)2 = 0.046904 = 0.047

0.047 ← recorded uncertainty



3.06 ± 0.05 mL







e4 =√

e21+ e2

2+ e2

3

e4 =√(0.03)2 + (0.03)2 + (0.02)2 = 0.046904 = 0.047




3.06 ± 0.05 mL







e4 =√

e21+ e2

2+ e2

3

e4 =√(0.03)2 + (0.03)2 + (0.02)2 = 0.046904 = 0.047




3.06 ± 0.05 mL



Example: Using a Buret

Uncertainty in every reading is ± 0.02

Initial volume reading: 0.05 mL

Final volume reading: 17.88 mL

The volume delivered is the di�erence between these two numbers

17.88 (± 0.02) mL

- 0.05 (± 0.02) mL

17.83 (± e)

where e =√0.022 + 0.022 = 0.028 ≈ 0.03

So the volume delivered with the buret was 17.83 ± 0.03 mL



De�nition:

Propagating Random Uncertainty Multiplication and Division:

For multiplication �rst convert all uncertainties to percent relative

uncertainties then calculate the error of the product or quotient

using %e4 =√

(%e1)2 + (%e2)2 + (%e3)2



Example: Propagate the Uncertainty

1.76(±0.03)×1.89(±0.02)0.59(±0.02) = 5.64± e4

First convert ( ± ei) to ( ± %e1)

1.76(±1.7%)×1.89(±1.1%)0.59(±3.4%) = 5.64± e4

Next calc %e4%e4 =

√(1.7)2 + (1.1)2 + (3.4)2 = 4.0%

So the recorded value is 5.64 (± 4.0 %)

(NOTE: don't report the 4 just keep it for future calculations.)

Sometimes you want to covert the uncertainty back to absolute

uncertainty.



Example: Propagate the Uncertainty (continued)

Sometimes you want to covert the uncertainty back to absolute

uncertainty. 4.0%× 5.64 = 0.040 × 5.64 = 0.23

This gives the recorded value 5.64 (± 0.23).

And when you report this value you would report

5.6 (± 0.2) absolute uncertainty

or

5.6 (± 4%) relative uncertainty



Example: Mixed operations

What do you do for mixed operations?[1.76(±0.03)−0.59(±0.02)]

01.89(±0.02) = 0.6190 ± e4

First the numerator: [1.76(±0.03)− 0.59(±0.02)] = 1.17(±0.036)where 0.036 =

√(0.03)2 + (0.02)2

Then covert the two absolute uncertainties to % relative

uncertainties1.17(±0.036)1.89(±0.02) = 1.17(±3.1%)

1.89(±1.1%) = 0.6190(±3.3%)



Example: mixed operations continued

and for the absolute uncertainty

0.033 × 0.6190 = 0.020


0.619 (± 3.3 %) % relative uncertainty

To report your values: �nal absolute uncertainty is 0.02 so the

hundredths place in the number 0.619 is the least certain number

NOT the thousandths place.

Reporting Values:


0.62 (± 3. %) % relative uncertainty



SF for real

REAL rule for Signi�cant Figures:

The �rst digit of the absolute uncertainty is the last digit in the

reported value.

So why did we have all those rules earlier about + - Ö ö

For test taking



SF for real



reported value.


For test taking



SF for real



reported value.


For test taking



Example: Why this is the real way

0.002 364(±0.000 003)0.025 00(±0.000 05) = 0.094 6(±0.000 2)and0.821(±0.002)0.803(±0.002) = 1.022(±0.004)



error propagation systematic

Also a problem

How do you �x this?



error propagation systematic

Also a problem

How do you �x this?



Example Problem: Exercise 3-B



Problem 3-11

How do you �x this problem?



Problem 3-20



Example Problem: Exercise 3-C



Problem 3-18



Problem 3-19



Problem 3-4


Documents

Chem 232: Quantitative Analysis Lecture Noteshuffman/include/lecture3.pdf · Scott Hu man Chem 232: Quantitative Analysis Lecture Notes chap 3: Basic tools of Analytical Chemistry