# Wilkes University Chemistry Questionnaire

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?
1.35102
• 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
• 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
2.431.2 = 2.916 = 2.9
(8.2610 )/(6.489210 ) = 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:
Propagation of Uncertainty: Systematic Error
When dealing with systematic error:
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
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:
3 sig figs
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)
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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