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Measurement in Scientific Study

andUncertainty in Measurement

Lecture 3

- Chemistry 142 B
- James B. Callis, Instructor
- Autumn Quarter, 2004

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Precision and Accuracy Errors in

Scientific Measurements

Precision - Refers to reproducibility or how

close the measurements are to

each other. Accuracy - Refers to how close a

measurement is to the true

value. Systematic Error - produces values that

are either all higher

or all lower than the actual value. Random

Error - in the absence of systematic error,

produces some

values that are higher and some that

are lower than the actual value.

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Rules for Determining Which Digits Are

Significant

All digits are significant, except zeros that are

used only to position the decimal point.

1. Make sure that the measured quantity has a

decimal point. 2. Start at the left of the number

and move right until you reach the first

nonzero digit. 3. Count that digit and every

digit to its right as significant. Zeros that end

a number and lie either after or before the

decimal point are significant thus 1.030 mL

has four significant figures, and 5300. L has

four significant figures also. Numbers such as

5300 L have 2 sig. figs., but 5.30x103 L has 3.

A terminal decimal point is often used to clarify

the situation, but scientific notation is clearer

(best).

Examples of Significant Digits in Numbers

Number - Sig digits Number

- Sig digits

0.0050 L 1.34000 x

107 nm six 18.00 g four

5600 ng 0.00012 kg two

87,000 L two 83.0001 L

six 78,002.3 ng

six 0.006002 g four

0.000007800 g four 875,000 oz

1.089 x 106 L 30,000 kg one

0.0000010048 oz five 5.0000 m3

five 6.67000 kg

six 23001.00 lbs seven 2.70879000

mL nine 0.000108 g 1.0008000 kg

eight 1,470,000 L three

1,000,000,000 g

Examples of Significant Digits in Numbers

Number - Sig digits Number

- Sig digits

0.0050 L two 1.34000 x 107

nm six 18.00 g four

5600 ng two 0.00012 kg

two 87,000 L

two 83.0001 L five

78,002.3 ng six 0.006002 g

four 0.000007800 g

four 875,000 oz three 1.089 x

10 -6L four 30,000 kg one

0.0000010048 oz five 5.0000 m3

five 6.67000 kg

six 23,001.00 lbs seven 2.70879000

mL nine 0.000108 g three

1.0008000 kg eight 1,470,000 L

three 1,000,000,000 g one

Rules for Significant Figures in answers

1. For multiplication and division. The number

with the least certainty limits the certainty of

the result. therefore, the answer contains the

same number of significant figures as there are

in the measurement with the fewest significant

figures. Multiply the following numbers

9.2 cm x 6.8 cm x 0.3744 cm

2. For addition and subtraction. The answer has

the same number of decimal places as there are

in the measurement with the fewest decimal

places. Example, adding two volumes 83.5 mL

23.28 mL Example subtracting two volumes

865.9 mL - 2.8121393 mL

Rules for Significant Figures in answers

1. For multiplication and division. The number

with the least certainty limits the certainty of

the result. therefore, the answer contains the

same number of significant figures as there are

in the measurement with the fewest significant

figures. Multiply the following numbers

9.2 cm x 6.8 cm x 0.3744 cm 23.4225 cm3 23 cm3

2. For addition and subtraction. The answer has

the same number of decimal places as there are

in the measurement with the fewest decimal

places. Example, adding two volumes 83.5 mL

23.28 mL 106.78 mL 106.8 mL Example

subtracting two volumes 865.9 mL - 2.8121393

mL 863.0878607 mL 863.1 mL

Rules for Rounding Off Numbers

- (1) In a series of calculations, carry the extra

digits through to the final result, then round

off. - (2) If the digit to be removed
- is less than 5, the preceding digit stays the

same. For example, 1.33 rounds to 1.3. - is equal to or greater than 5, the preceding

digit is increased by one. For example, 1.36

rounds to 1.4. - (3) When rounding, use only the first number to

the right of the last significant figure. Do not

round off sequentially. For example, the number

4.348 when rounded to two significant figures is

4.3, not 4.4. - Notes
- Your TI-93 calculator has the round function

which you can use to get the correct result. Find

round by pressing the math key and moving to NUM.

Its use is round(num, no of decimal places

desired), e.g. round(2.746,1) 2.7. - Your book will show intermediate results

rounded off. Dont use these rounded results to

get the final answer.

Rounding Off Numbers Problems

(3-1a) Round 5.379 to three significant figures

Ans (3-1b) Round 5.379 to two significant

figures Ans We used the rule If the digit

removed is greater than or equal to 5, the

preceding number increases by 1. (3-2a) Round

0.2413 to three significant figures Ans (3-2b)

Round 0.2413 to two significant figures Ans We

used the rule If the digit removed is less than

5, the preceding number is unchanged

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Sample Problem 3-3

Lithium (Li) is a soft, gray solid that has the

lowest density of any metal. If a slab of Li

weighs 1.49 x 103 mg and has sides that measure

20.9 mm by 11.1 mm by 12.0 mm, what is the

density of Li in g/ cm3 ?

Sample Problem 3-3(cont.)

Mass (g) of Li 1.49 x 103 mg Length (cm) of

one side 20.9 mm Similarly, the other side

lengths are Volume (cm3) Density

mass/volume Density of Li

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Problem 3-4 Volume by Displacement

Problem Calculate the density of an irregularly

shaped metal object that has a mass of 567.85

g if, when it is placed into a 2.00 liter

graduated cylinder containing 900.00 mL of

water, the final volume of the water in the

cylinder is 1277.56 mL ? Plan Calculate the

volume from the different volume readings,

and calculate the density using the mass that

was given. Solution

Volume

mass

Density

volume

Definitions - Mass Weight

Mass - The quantity of matter an object contains

kilogram - ( kg ) - the SI base unit of mass, is

a platinum -

iridium cylinder kept in

Paris as a standard!

Weight - depends upon an objects mass and the

strength of the gravitational

field pulling on it, i.e. w f ma.

Problem 3-5 Computer Chips

Future computers might use memory bits which

require an area of a square with 0.25 mm sides.

(a) How many bits could be put on a 1 in x 1 in

computer chip? (b) If each bit required that 25

of its area to be coated with a gold film 10 nm

thick, what mass of gold would be needed to make

one chip?

- Approach
- use Achip
- (b) use r m/V

Solution to Chip Problem (3-7)

Solution to Chip Problem (3-7)

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Temperature Scales and Interconversions

Kelvin ( K ) - The Absolute temperature scale

begins at absolute zero

and only has positive values.

Celsius ( oC ) - The temperature scale used by

science, formally

called centigrade and most

commonly used scale around the world,

water freezes at 0oC, and boils

at 100oC.

Fahrenheit ( oF ) - Commonly used scale in

America for our

weather reports, water freezes at 32oF,

and boils at 212oF.

T (in K) T (in oC) 273.15 T (in oC) T (in

K) - 273.15

T (in oF) 9/5 T (in oC) 32 T (in oC) T

(in oF) - 32 5/9

Problem 3-6Temperature Conversions

(a) The boiling point of Liquid Nitrogen is

-195.8 oC, what is the temperature in Kelvin and

degrees Fahrenheit?

T (in K) T (in oC) 273.15 T (in K)

T (in oF) 9/5 T (in oC) 32 T (in oF)

(b)The normal body temperature is 98.6oF, what is

it in Kelvin and degrees Celsius?

T (in oC) T (in oF) - 32 5/9 T (in oC)

T (in K) T (in oC) 273.15 T (in K)

Answers to Problems in Lecture 3

- (a)5.38 (b) 5.4
- (a) 0.241 (b) 0.24
- 0.536 g/cm3
- 1.5040 g / mL
- 31 mg gold
- (a) 77.4 K -320.4 oF (b) 37.0 oC 310.2 K