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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
• Addition and subtraction
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
answer with the correct
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
• Addition and subtraction:
• 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    0.2
2 2
2
0.002 x 0.00946 = 0.00019
0.00005
 100  0.2
0.025
 0.2
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: