Chapter 3Math Toolkit

3-1 Significant Figures

• The number of significant figures is the minimum number of digits needed to write a given value in scientific notation without loss of accuracy.

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Figure 3-1

Figure 3-1 Scale of a Bausch and Lomb Spectronic 20 spectrophotometer.

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Significant Figures

• Measurement: number + unit• Uncertainty • Ex:

0.92067 five0.092067 five9.3660105 five936600 four7.270 four

3-2 Significant Figures in Arithmetic

Addition and Subtraction• If the numbers to be added or subtracted hav

e equal numbers of digits, the answer is given to the same decimal place.

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• The number of significant figures in the answer may exceed or be less than that in the original data.

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Multiplication and Division• In multiplication and division

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Example ： Significant Figures in Molecular Mass

• Find the molecular mass of C14H10 with the correct number of significant digits.

SOLUTION:

• 14×12.010 7=168.1498←6 significant figures because 12.010 7 has 6 digits

• 10×1.007 94=10.079 4←6 significant figures because 1.007 94 has 6 digits

178.2292

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Logarithms and Antilogarithms

• The base 10 logarithm of n is the number a, whose value is such that n=10a:

• The number n is said to the antilogarithm of a.

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• A logarithm is composed of a characteristic and a mantissa.

• The number 339 can be written 3.39×102. The number of digits in the mantissa of log 339 should equal the number of significant figures in 339.

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• In converting a logarithm to its antilogarithm, the number of significant figures in the antilogarithm should equal the number of digits in the mantissa.

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Significant Figures and in Arithmetic

Logarithms & antilog, see p64-65[H+]=2.010-3

pH=-log(2.010-3) = -(-3+0.30)=2.70

antilogarithm of 0.072 1.18

logarithm of 12.1 1.083

log 339 = 2.5301997… = 2.530

antilog (-3.42) = 10-3.42 = 0.0003802

= 3.8x10-4

3.3 Types of Errors

Every measurement has some uncertainty experimental error.

Experimental error is classified as either systematic or random.

Maximum error v.s. time required

3.3 Types of Errors

1) Systematic error

= Determinate error = consistent error

- Errors arise: instrument, method, & person - Can be discovered & corrected- Is from fixed cause, & is either high (+) or lo

w (-) every time.- Ways to detect systematic error:

examples (a) pH meter (b) buret

3.3 Types of Errors

2) Random error = Indeterminate error

Is always present & cannot be corrected

Has an equal chance of being (+) or (-).

(a) people reading the scale

(b) random electrical noise in an instrument.

(c) pH of blood (actual variation: time, or part)

3) Precision & Accuracy

reproducibility

confidence of nearness to the truth

Precision ? Accuracy ?

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3.3 Types of Errors

4) Absolute & Relative uncertaintya) Absolute : the margin of uncertainty

0.02(the measured value - the true value)

b).

0.2%0.00212.35

0.02

mL 0.0212.35 (ex)

tmeasuremen of magnitude

yuncertaint absoluteRelative

3-4 Propagation of Uncertainty

• The uncertainty might be based on how well we can read an instrument or on experience with a particular method. If possible, uncertainty is expressed as the standard deviation or as a confidence interval.

Addition and Subtraction

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3.4 Propagation of uncertainty

1) Addition & Subtraction

(ex) p.69

)e( 3.06

e 0.02)( 0.59

e 0.02)( 1.89

e 0.03)( 1.76

4

3

2

1

1%)( 3.06

0.04)( 3.06

%1.

0.04

eeee

3

1

23

22

214

3.4 Propagation of uncertainty

23

22

214 %e%e%ee

2) Multiplication & Division

use % relative uncertainties.

3.4 Propagation of uncertainty

4%)( 5.6

0.2)( 5.6

0.25.6%4.

%4.%3.%1.%1.e

%)3.( 0.59

%)1.( 1.89%)1.( 1.76

e5.640.02)( 0.59

0.02)( 1.890.03)( 1.76 (ex)

340

02

42

12

74

4

17

4

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Example ： Scientific Notation and Propagation of Uncertainty

Express the absolute uncertainty in

SOLUTION SOLUTION ：： (a) The uncertainty in the denominator is 0.04/2.11 = 1.896%. The uncer

tainty in the answer is (b)

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3.4 Propagation of uncertainty

3) Mixed Operations

3%)( 0.62

0.02)( 0.62

%)3.( 0.619

)0.02( 0.619

?0.6190.02)( 1.89

0.02)( 0.590.03)( 1.76

3

0

0

Example ： Significant Figures in Laboratory Work at p.73

3.4 Propagation of uncertainty

4) The real rule for significant figures

The 1st uncertain figure of the answer is the last significant figure.

3.4 Propagation of uncertainty

① .

② .

③ . 0.004)( 1.0220.002)( 0.803

0.002)( 0.821

0.0002)( 0.10660.00005)( 0.02500

0.000003)( 0.002664

0.0002)( 0.09460.00005)( 0.02500

0.000003)( 0.002364

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