Homework Assignment 02 '''is from Chapter 3' Problems assigned are: 5,11,12,15,21,22 This assignment

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Homework Assignment 02

• Homework Assignment 02 ...is from Chapter 3.
  Problems assigned are 5,11,12,15,21,22This
  assignment is due at class time
• Friday, Sept 17
• Prepare on regularly sized paper, one side only
  with multiple pages stapled.

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Chapter 3 Math Toolkit Logarithms
Consider the common logarithm of the
number 4.265 x 10 ?2, that is log10 (4.265 x 10
?2). This number has 4 significant figures, so
its logarithm has 4 digits in its mantissa. The
characteristic is related to the power of ten.
The common log of 4.265 x 10 ?2. is ?1.3701
which appears to have 5 significant figures, but
the characteristic (?1) is the power of ten. The
pre?exponential part of the number (4.265) has 4
significant figures so the mantissa Should also
have 4 digits.
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Chapter 3 Math Toolkit
For large numbers the characteristic of the
logarithm is positive. Consider the common
logarithm (base 10) of the number 78,436 or
7.8436 x 104. log10 7.8436 x 104 4.89452.
For numbers greater than 10, the characteristic
is positive. (101 10). The characteristic is
the same as if the number is written in
scientific notation. There needs to be the
same number of significant figures in the
mantissa as present in the number itself.
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Chapter 3 Math Toolkit Types of Errors

• Systematic or determinate errors are repeatable.
• Often systematic errors may be identified and
  then
• corrected.
• For example you might measure a length with a
• faulty ruler, say one that had a cm cut off, or
• measured the pH of a solution having standardized
• with a buffer that you thought was 7.00, but
  whose
• value was actually 8.00

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Chapter 3 Math Toolkit Types of Errors

The tolerance for a 50-mL class A buret is ?0.05
mL. If you were pushing the limits of use of
this device, you might calibrate within
individual ranges as was discussed last time.
Typical results of this calibration are shown
above.
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Chapter 3 Math Toolkit Types of Errors

2. Random or indeterminate errors are those
that arise from our limitations to make physical
measurements. Random errors have equal chances
to be positive or negative. They should occur at
the level of the uncertainty of the measuring.
Random errors cannot be eliminated, but may be
reduced by better performed experiments.
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Chapter 3 Math Toolkit Standard Reference
Materials

Standard Reference Materials are prepared
and certified by the U.S. National Institute of
Standards and Technology (NIST, formerly National
Bureau of Standards, NBS). Standard reference
materials are very important in establishing new
methods of analysis and checking the results
obtained by various laboratories and
testing agencies.
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Chapter 3 Math Toolkit Standard Reference
Materials

Errors in the analysis of anticonvulsant drugs
before and after reference to standard materials.
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Chapter 3 Math Toolkit Precision vs. Accuracy

Precision is a measure of the reproducibility of
the individual data within a set of results.
Accuracy is a measure of how close the measured
value is to the true value.
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Chapter 3 Math Toolkit Precision vs. Accuracy


Cheryl is neither accurate nor precise Cynthia is
accurate but not precise Carmen is both accurate
and precise Chastity is precise but not accurate
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Chapter 3 Math Toolkit Absolute vs. Relative
Error
Absolute Error is the measurement of the
uncertainty associated with the measurement. It
is always given in terms of the same units as the
measurement itself, such as for a balance, 0.1
mg. Relative Error is the measurement of the
uncertainty expressed in terms of the magnitude
of the measurement. Relative errors have no
units, though they are often expressed as or
parts per thousand. relative error absolute
error / magnitude of measurement

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Chapter 3 Math Toolkit Absolute vs. Relative
Error
What is the relative error whose absolute error
is 0.1 mg if the measurement is 105 mg?
relative error 0.1 mg / 105 mg 0.00095 or
0.095 or 0.95 ppt What is the relative
error whose absolute error is 0.1 mg if the
measurement is 40 mg? relative error 0.1 mg
/ 40 mg 0.25 or 2.5 ppt (Just like means
parts per 100 per centum, Latin, ppt means
parts per thousand or ppt X 10.)

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Chapter 3 Math Toolkit Propagation of
Uncertainties in calculations

Whenever data is processed by doing the
mathematical operations of addition, subtraction,
multiplication or division, the uncertainties of
each measurement is incorporated in the
uncertainty we can associate with the final
answer. Like the rules for significant figures,
there are ways to account for the cumulative
uncertainty 1) for addition or subtraction,
and then 2) for multiplication or division.
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Chapter 3 Math Toolkit Propagation of
Uncertainties Addition/Subtraction


The uncertainties whenever data is added or
subtracted is given by the expression
_________________ en ?
e12 e22 e32 ..
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Chapter 3 Math Toolkit Propagation of
Uncertainties Addition/Subtraction


Consider the following measurements and their
uncertainties. The desired result is X1 X2
X3. What is the value and the uncertainty in the
final value? X1 1.76 (?0.03) X2 1.89
(?0.02) X3 0.59 (?0.02) 1.76 1.89
0.59 3.06 _____________________
______ e ? (0.03)2 (0.02)2 (0.02)2 ?
0.0017 0.0412
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Chapter 3 Math Toolkit Propagation of
Uncertainties Addition/ subtraction


The resulting value could then be stated as 3.06
? 0.041 (The authors somewhat unique use of
subscripts in his way to indicate what the next
digit is, although it is beyond what can be
reported in chain calculations where additional
mathematical processing happens, carry
one additional value, and then round off the
final result.)
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Chapter 3 Math Toolkit Propagation of
Uncertainties Addition/subtraction


The error of ? 0.041 is in absolute measurements
and in the same units as the 3.06 value. The
error may also be expressed as relative
uncertainty, commonly referred to as percent
relative uncertainty, which is defined as RU
e X 100 / value, or in this case RU
(0.041)(100) / 3.06 1.34 1.3 the same
expressed in ppt is 13 ppt
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Chapter 3 Math Toolkit Propagation of
Uncertainties


Multiplication and Division First convert all
errors to relative errors as described in the
proceeding slide. _____________________
e ? ( e1)2 ( e2)2 ( e3)2 Note that
e is the error in absolute measurement units.
The relative error is e X 100 / value. (See
example bottom of page 62)
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Chapter 3 Math Toolkit Propagation of
Uncertainties


Mixed operations - whenever the calculation
involves both addition or subtraction and
multiplication or division, work through the
addition/subtration first, then the
multiplication/division. Note the example,
middle of page 63.
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Chapter 3 Math Toolkit Summary Regarding
Significant Numbers


The first uncertain figure of the answer is
the last significant figure.
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Chapter 3 Math Toolkit Excel Spreadsheets


A spreadsheet consists of a collection of columns
(vertical arrangement) and rows (horizontal
arrangement). The intersection of a given
column and row defines a cell. Within a cell you
may enter text, numerical values, or formulas and
functions. Click to Link to Spreadsheet