Homework Chapter 31. Consider a laboratory experiment in which your goal is to determine the amount of

chloride (Cl-) in an unknown solution. You place 25 mL of unknown solution into a

large beaker and add 5% aqueous silver nitrate (AgNO3) until all of the Cl- precipitates as

silver chloride (AgCl). Then, you filter the resulting mixture to collect the solid. You

dry the AgCl (s), weigh it, and use its mass to calculate the molarity of Cl- in the

unknown. List at least 2 sources of each type of error (systematic, random, and

blunder) that may occur in the experiment. Briefly explain your reasoning for each.

2. Calculate the absolute uncertainty and express each answer with the appropriate number

of significant figures.

a. 9.43 (±0.05) x 0.016 (±0.001)

b. [6.2 (±0.2) ‐ 4.1(±0.1)] ÷ 9.43 (±0.05)

3. You prepared a solution by diluting 8.45 (±0.04) mL of 28.0 (±0.7) wt % ammonia (NH3)

up to 500.0 (±0.2) mL. The density of the 28.0 (±0.7) wt % NH3 is 0.899 (±0.003) g/mL,

and the molar mass of NH3 is 17.0305 g/mol (negligible uncertainty). Calculate the

molarity of the resulting solution and its absolute uncertainty with an appropriate

number of significant figures.

8/26/2021

Chapter 3: Experimental Error

Familiar Topics:

• Significant Figures

• Types of Error

• Precision vs Accuracy

Significant Figures

“Minimum number of digits needed to write a given

value in scientific notation without loss of precision”

How many significant figures?

1.35102

Newer Topics:

• Absolute and Relative Uncertainty

• Propagation of Uncertainty

3501

0.00023

Rounding Note:

ONLY round at the FINAL ANSWER to

avoid accumulating round-off errors

62.300

602

1

2

Significant Figures

Significant Figures

Logarithms and anti-logs

Addition and Subtraction:

• Final value has the same number of decimal points as the original

value with the least number of digits after decimal point

A logarithm has 2 parts: Characteristic (integer part)

Mantissa (decimal part)

152.32

+ 18.5

170.82 =170.8

• # of sig figs in mantissa = # of sig figs in n

• # of sig figs in antilog = # of digits in mantissa

Multiplication and Division:

• The final answer has the same number of significant figures as the

original value that had the least

log 339 = 2.530

2.431.2 = 2.916 = 2.9

(8.2610 )/(6.489210 ) = 1.272788… 10

−2

4

6

log 3.39 x 10-5 = -4.470

antilog (-3.42) = 10-3.42 = 3.8 x 10-4

= 1.27 106

3

4

Significant Figures

Types of Error

Systematic Error (also called determinate error)

Consistent error that can be detected and corrected

A logarithm has 2 parts: Characteristic (integer part)

Mantissa (decimal part)

•

•

•

•

• # of sig figs in mantissa = # of sig figs in n

• # of sig figs in antilog = # of digits in mantissa

pH = – log (3.39 x 10-5) = 4.470

Reproducible

Can be discovered and corrected

Always + OR always Due to flaw in experimental design or equipment

[H3O+] = 10-3.42 = 3.8 x 10-4 M

Sig Figs for pH: only numbers AFTER decimal place are significant

5

6

1

8/26/2021

Types of Error

Types of Error

Systematic Error (also called determinate error)

Consistent error that can be detected and corrected

Random Error (also called indeterminate error)

Can NOT be eliminated, but may be reduced

•

•

•

•

How do you detect systematic error?

• Analyze known certified reference material

• Use a different method and compare results

• Different labs analyze same sample and compare

• Analyze blank sample

ALWAYS present

Can NOT be corrected

Maybe + or – , and can change

Due to uncontrolled variables in measurements

How do you correct systematic error?

• Calibrate glassware

• Calibration curves, standard addition, etc.

7

8

Types of Error

Blunders (also called gross error)

Unrecoverable errors due to procedural, instrumental,

or clerical mistakes

• Extreme random or systematic error

• Due to accidental procedures

Accuracy and Precision

Accuracy: nearness to “true” value

Precision: reproducibility of the result

Analytical Chemistry requires accuracy AND precision

9

10

Uncertainty and Error

Uncertainty and Error

Error: difference between measured and “true” value

Accuracy: nearness to “true” value

Error: difference between measured and “true” value

Uncertainty: variability within measurements

Precision: reproducibility of the result

Uncertainty: variability within measurements

Low error

Low uncertainty

Ideally, you want to MINIMIZE error and uncertainty

11

High error

Low uncertainty

Low error

High uncertainty

High error

High uncertainty

Ideally, you want to MINIMIZE error and uncertainty

12

2

8/26/2021

Absolute and Relative Uncertainty

Absolute Uncertainty: margin of uncertainty associated with

a measurement

Ex: buret reading is ± 0.02 mL

Relative Uncertainty: compares size of absolute uncertainty

with size of associated measurement

Ex: relative uncertainty of buret reading 12.35 ± 0.02 mL is:

Propagation of Uncertainty

Each measurement has an uncertainty associated with it

• But what happens when there is > 1 measurement?

• How do we calculate the combined uncertainty?

Propagation of Uncertainty

Rules are different for systematic and random error

Percent Relative Uncertainty: (%e)

13

14

Propagation of Uncertainty: Systematic Error

When dealing with systematic error:

ADD the uncertainties

Propagation of Uncertainty: Systematic Error

When dealing with systematic error:

ADD the uncertainties

A pipette delivers 25.00 ± 0.03 mL.

You use the pipette 4 times at 25mL each.

Why? The systematic error is always the same – it’s

reproducible error, and we can correct for it.

How much volume did you deliver?

25.00 mL

+ 25.00 mL

+ 25.00 mL

+ 25.00 mL

100.00 mL

When would we do this:

pipetting with same pipette

titrating with same buret

calculating molecular mass

15

16

Propagation of Uncertainty: Random Error

When dealing with random error:

calculations are based on arithmetic you are doing …

Propagation of Uncertainty: Random Error

When dealing with random error:

calculations are based on arithmetic you are doing …

Why?

• Random error means some error is (+) and some (-)

• This results in some cancelation of error.

Table 3-1 Summary of rules for propagation of uncertainty

Funtion

Uncertainty

Functiona

Uncertainty b

y = x1 + x2

ey = ex21 + ex22

y = xa

%ey = a ( %ex )

y = x1 − x2

ey = e + e

y = log x

y = x1 x2

%ey = %ex21 + %ex22

y = ln x

x

y= 1

x2

%ey = %e + %e

y = 10 x

2

x1

2

x1

2

x2

2

x2

y = ex

17

What is the uncertainty?

± 0.03 mL

+ ± 0.03 mL

+ ± 0.03 mL

+ ± 0.03 mL

± 0.12 mL

ey =

ey

y

e

1 ex

0.43429 x

ln10 x

x

e

ey = x

x

= (ln10 ) ex 2.3026ex

ey

y

= ex

18

3

8/26/2021

Propagation of Uncertainty: Random Error

Addition and subtraction

use absolute uncertainty

efinal = ei2 = e12 + e22 + e32 +

of the individual terms

i

(include units)

1.76 m (0.03)

+ 1.89 m (0.02)

– 0.59 m (0.02)

3.06 m (0.041)

efinal =

Propagation of Uncertainty: Random Error

Multiplication and Division

use percent relative

2

2

2

2

%efinal = % ei = %e1 + %e2 + %e3 +

uncertainty of the

i

individual terms

0.494 M(0.004 M) 5.00 mL (0.01mL) 0.494 M ( 0.81% ) 5.00 mL ( 0.20% )

=

100.00 mL (0.08 mL)

100.00 mL ( 0.080% )

%efinal = (0.81)2 + (0.20)2 + (0.080)2 = 0. 84 %

To convert relative uncertainty to absolute uncertainty:

0.008 4 0.024 70 M = 0.000 21 M

(0.03)2 + (0.02)2 + (0.02)2 = 0.041 m

0.024 7 M 0.000 21 M

0.024 7 M 0.84%

3.06 m 0.041 m (absolute uncertainty)

19

(absolute uncertainty)

(relative uncertainty)

20

Propagation of Uncertainty: Random Error

Multiplication and Division

use percent relative

2

2

2

2

%efinal = % ei = %e1 + %e2 + %e3 +

uncertainty of the

i

individual terms

** Remember, we need to convert to percent relative uncertainties! **

** OR – instead of converting everything to a % (by×100),

you could also just use a relative uncertainty **

Propagation of Uncertainty: Random Error

Fun table full of summary equations … we’ll mostly use the +/- or x/÷

Table 3-1 Summary of rules for propagation of uncertainty

Funtion

Uncertainty

Functiona

Uncertainty b

y = x1 + x2

ey = ex21 + ex22

y = xa

y = x1 − x2

ey = ex21 + ex22

y = log x

y = x1 x2

%ey = %ex21 + %ex22

y = ln x

%ey = %ex21 + %ex22

y = 10 x

y=

x1

x2

%ey = a ( %ex )

ey =

ey

y

e

1 ex

0.43429 x

ln10 x

x

e

ey = x

x

= (ln10 ) ex 2.3026ex

y = ex

21

ey

y

= ex

22

The REAL Rule for Sig Figs

The first digit of the absolute uncertainty is the

last significant digit in the answer!

Propagation of Uncertainty: Random Error

Example:

[1.76 (± 0.03) – 0.59 (± 0.02)]

1.89 (± 0.02)

Example:

4 sig figs

0.002364 (± 0.000003)

0.02500 (± 0.00005)

4 sig figs

Try this on your own … solve for answer with absolute uncertainty.

= 0.0946 (±0.0002)

Hints:

Remember your order of operations!

3 sig figs

Addition or

e

= ei2 = e12 + e22 + e32 +

subtraction: final

i

Multiplication

%efinal =

or Division:

23

2

2

2

2

% ei = %e1 + %e2 + %e3 +

i

24

4

8/26/2021

Propagation of Uncertainty: Random Error

Example:

[1.76 (± 0.03) – 0.59 (± 0.02)]

1.89 (± 0.02)

Addition or

e

= ei2 = e12 + e22 + e32 +

subtraction: final

i

25

Propagation of Uncertainty

Example:

[1.76 (± 0.03) – 0.59 (± 0.02)] = 1.17 (± 0.036)

1.89 (± 0.02)

1.89 (± 0.02)

Multiplication

%efinal =

or Division:

2

2

2

2

% ei = %e1 + %e2 + %e3 +

i

26

Propagation of Uncertainty

Example:

[1.76 (± 0.03) – 0.59 (± 0.02)]

1.89 (± 0.02)

27

5

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