# Slide 1

```CHEMISTRY 59-320
ANALYTICAL CHEMISTRY
Fall - 2010
Lecture 4
Chapter 3 Experimental error
3.1 Significant Figures
The minimum number of digits needed to write
a given value in scientific notation without loss of
accuracy
A Review of Significant Figures
How many significant figures in the following examples?
•
0.216
90.7 800.0 0.0670 500
•
88.5470578%
•
88.55%
•
0.4911
The needle in the figure appears to be at an absorbance value of 0.234. We
say that this number has three significant figures because the numbers 2 and
3 are completely certain and the number 4 is an estimate. The value might be
read 0.233 or 0.235 by other people.
The percent transmittance is near 58.3. A reasonable estimate of
uncertainty might be 58.3 ± 0.2. There are three significant figures in the
number 58.3.
3.2 Significant figures in arithmetic
The number of significant figures in the
answer may exceed or be less than that in
the original data. It is limited by the leastcertain one.
• Rounding: When the number is exactly
halfway, round it to the nearest EVEN digit.
• Multiplication and division: is limited to the
number of digits contained in the number with
the fewest significant figures:
• Logarithms and antilogarithms
A logarithm is composed of a characteristic and a mantissa.
The characteristic is the integer part and the mantissa is the
decimal part. The number of digits in the mantissa should equal the
number of significant figures.
• Problem 3-5. Write each
number of digits.
•
•
•
•
(a) 1.021 + 2.69 = 3.711
(b) 12.3 − 1.63 = 10.67
(c) 4.34 × 9.2 = 39.928
(d) 0.060 2 ÷ (2.113 ×
104) = 2.84903 × 10−6
• (e) log(4.218 × 1012) = ?
• (f) antilog(−3.22) = ?
• (g) 102.384 = ?
•
•
•
•
•
•
•
(a) 3.71
(b) 10.7
(c) 4.0 × 101
(d) 2.85 × 10−6
(e) 12.6251
(f) 6.0 × 10−4
(g) 242
3-3 Types of errors
• Every measurement has some uncertainty, which is
called experimental error
• Random error, also called
indeterminate error, arises
from the effects of uncontrolled
(and maybe uncontrollable)
variables in the measurement.
• Systematic error, also called
determinate error, arises from
a flaw in equipment or the
design of an experiment. It is
always positive in some region
and always negative in others.
•
• A key feature of systematic
error is that it is reproducible.
Random error has an equal
chance of being positive or
negative.
• It is always present and cannot
be corrected. It might be
reduced by a better experiment.
• In principle, systematic error
can be discovered and
corrected, although this may
not be easy.
Accuracy and Precision:
Is There a Difference?
• Accuracy: degree of agreement between
measured value and the true value.
• Absolute true value is seldom known
• Realistic Definition: degree of agreement
between measured value and accepted
true value.
Precision
• Precision: degree of agreement between
replicate measurements of same quantity.
• Repeatability of a result
• Standard Deviation
• Coefficient of Variation
• Range of Data
• Confidence Interval about Mean Value
You can’t have accuracy without good precision.
But a precise result can have a determinate or systematic error.
Illustration of Accuracy and precision.
Absolute and relative uncertainty:
• Absolute uncertainty expresses the margin of uncertainty
associated with a measurement. If the estimated uncertainty
in reading a calibrated buret is ±0.02 mL, we say that ±0.02
mL is the absolute uncertainty associated with the reading.
3-4 Propagation of Uncertainty from Random
Error
• Multiplication and Division: first convert all uncertainties
into percent relative uncertainties, then calculate the
error of the product or quotient as follows:
The rule for significant figures: The first digit of the
absolute uncertainty is the last significant digit in the
answer. For example, in the quotient
0.000003
 100  1.61045  102
0.002364
1.61045 10

2 2
0.00005
 100  0.2
0.025
  0.2   0.2
2
0.002 x 0.00946 = 0.00019
100
0.002
3-5 Propagation of uncertainty:
Systematic error
• It is calculated as the sum of the
uncertainty of each term
• For example: the calculation of oxygen
molecular mass.
3-C. We have a 37.0 (±0.5) wt% HCl solution with a density of 1.18 (±0.01)
g/mL.
To deliver 0.050 0 mol of HCl requires 4.18 mL of solution. If the uncertainty
that can be tolerated in 0.050 0 mol is ±2%, how big can the absolute
uncertainty in 4.18 mL be? (Caution: In this problem, you have to work
backward).
You would normally compute the uncertainty in mol HCl from the uncertainty
in volume:
But, in this case, we know the uncertainty in mol HCl (2%) and we need to
find what uncertainty in mL solution leads to that 2% uncertainty.
The arithmetic has the form a = b × c × d,
for which %e2a = %e2b+%e2c+%e2d.
If we know %ea, %ec, and %ed,
we can find %eb by subtraction: %e2b = %e2a – %e2c – %e2d )
0.050 0 (±2%) mol =
Error analysis:
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