1. Chapter 3: Scientific Measurement
Jessica Contino

2. Scientific Notation
a way of expressing really big numbers or really small numbers.
Separates measurement digits from placeholder zeros
Helps in evaluating precision

3. Scientific notation consists of two parts:
Measurement digits
Number between 1 and 10
Decimal after first digit
Factor of Tens
Ten to a power
Provides order of magnitude
Example: Mass of the earth:
tons
6.592 x tons

4. To change standard form to scientific notation…
Place the decimal point so that there is one non-zero digit to the left of the decimal point.
Count the number of decimal places the decimal point has “moved” from the original number. This will be the exponent on the 10.
If the original number was less than 1, then the exponent is negative. If the original number was greater than 1, then the exponent is positive.

5. Examples Given: 289,800,000 Use: 2.898 (moved 8 places) Answer:

6. To change scientific notation to standard form…
Simply move the decimal point to the right for positive exponent 10.
Move the decimal point to the left for negative exponent 10.
(Use zeros to fill in places.)

7. (moved 6 places to the right) (moved 4 places to the left)
Example
Given: x 106
Answer:
Given: x 10-4
5,093,000
(moved 6 places to the right)
(moved 4 places to the left)

8. Learning Check Express these numbers in Scientific Notation: 4057
30000 2
0.4760
4.057 x 10 3
3.87 x 10 -3
3 x 10 4
2 x 10 0
4.760 x 10 -1

9. Can you hit the bull's-eye?
Three targets with three arrows each to shoot.
How do they compare?
Both accurate and precise
Precise but not accurate
Neither accurate nor precise
Can you define accuracy and precision?

10. Accuracy and percent error
Absolute error – how far a measured value is compared to an accepted value
Percentage – value per 100 units
Percentage error:
% error = measured value – accepted value x 100
Accepted value

11. % error = measured value – accepted value x 100
A student calculates the density of mercury to be 13.2 g/cm3. If the accepted value is 13.6 g/cm3, what is the percentage error?
% error = measured value – accepted value x 100
Accepted value
% error = x 100
13.6
NEGATIVE ERROR = MEASURED VALUE BELOW ACCEPTED!
= PERCENT

12. Significant Figures
The numbers reported in a measurement are limited by the measuring tool
Significant figures in a measurement include the known digits plus one estimated digit

13. Reading a Meterstick
. l I I I I4. . cm First digit (known) = 2 2.?? cm Second digit (known) = ? cm Third digit (estimated) between Length reported = 2.75 cm or 2.74 cm or 2.76 cm

14. Known + Estimated Digits
In 2.76 cm…
Known digits 2 and 7 are 100% certain
The third digit 6 is estimated (uncertain)
In the reported length, all three digits (2.76 cm) are significant including the estimated one

15. Learning Check Can it be 9.60?
. l I I I I cm
What is the length of the line?
1) cm
2) cm
3) cm
How does your answer compare with your neighbor’s answer? Why or why not?
Can it be 9.60?

16. Zero as a Measured Number
. l I I I I cm
What is the length of the line?
First digit ?? cm
Second digit ? cm
Last (estimated) digit is cm

17. zeros as placeholders Mass of the earth: 6592000000000000000000 tons 4
6.592 x tons 4
Significant
digits 18
Placeholder
zeros

18. zeros as placeholders Radius of a carbon atom: 77 picometers
77 x m m 2
Sig
digs 11
placeholder
zeros

19. zeros as placeholders Placeholders left after rounding: Ex: 734567
Rounded to nearest thousand:
735000
3 sig digs
3 placeholder zeros
Note: the zeros left after rounding are only placeholders
They drop out when not needed – as in scientific notation
7.35 x 10 5

20. Counting Significant Figures
RULE 1. All non-zero digits in a measured number are significant. Only a zeros can be used as placeholders.
Number of Significant Figures
38.15 cm 4
5.6 ft 2
65.6 lb m 3 5

21. Leading Zeros
RULE 2. Leading zeros in decimal numbers are NOT significant (they’re placeholders).
Number of Significant Figures
0.008 mm 1
oz 3
lb 2
mL 3

22. Sandwiched Zeros
RULE 3. Zeros between nonzero numbers are significant. (They can not be rounded or dropped, unless they are on an end of a number.)
Number of Significant Figures
50.8 mm 3
2001 min 4
0.702 lb 3
m 3

23. decimal means not rounded
Trailing Zeros
RULE 4. Trailing zeros in numbers without decimals are NOT significant. They are only serving as place holders (left after rounding).
Number of Significant Figures
25,000 in. 2
200. yr 3
48,600 gal 3
25,005,000 g 5
rounded to thousands
decimal means not rounded

24. Rounded to nearest thousand
Learning Check
A. Which answers contain 3 significant figures? 1) ) ) 4760 B. All the zeros are significant in 1) ) ) x 103 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x 105
Rounded to nearest thousand

25. Learning Check
In which set(s) do both numbers contain the same number of significant figures?
Hint: (Underline sig figs first)
1) and 22.00
2) and 40
3) and 150,000
______ _________
_________ __
____ ____

26. Final Learning Check – do all
State the number of significant figures in each of the following: A m B L C g D m E. 300,707 bees 1 3 6
infinite!
E is a count : which is exact – zeros extend to infinity!

27. 0.000 77 m = 2 sf’s 6 592 000 tons = 4 sf’s Zeros as placeholders
Review
m = 2 sf’s
tons = 4 sf’s
Zeros as placeholders
= 7 sf’s
Zeros as sig digs
303,000 = 3 sf’s
= 4 sf’s

28. Significant Numbers in Calculations
A calculated answer cannot be more precise than the measuring tool.
A calculated answer must match the least precise measurement used in the calculation.
Significant figures are needed for final answers from
1) adding or subtracting
2) multiplying or dividing

29. Adding and Subtracting
The answer has the same number of “decimal places” as the measurement with the fewest decimal places one decimal place two decimal places answer 26.6 one decimal place ?

30. Learning Check
In each calculation, round the answer to the correct number of significant figures. A = 1) ) ) 257 B = 1) ) ) 40.7

31. Multiplying and Dividing
Round (or add zeros to) the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures.

32. Learning Check
A X 4.2 =
1) ) )
B ÷ =
1) ) ) 60
C X =
X 0.060
1) ) )

33. UNITS OF MEASUREMENT
Use SI units — based on the metric system Length Mass Volume Time Temperature
Meter , m
Kilogram, kg
Liter, L or dm3
Seconds, s
Celsius degrees, ˚C
kelvins, K

34. Mass vs. Weight
Mass: Amount of Matter (grams, measured with a BALANCE)
Weight: Force exerted by the mass, only present with gravity (pounds, measured with a SCALE)
Can you hear me now?

35. Metric Prefixes
Kilo- means 1000 of that unit ex: 1 kilometer (km) = 1000 meters (m) 1 kilo-unit = 1000 units Centi- means 1/100 of that unit ex: 1 meter (m) = 100 centimeters (cm) 1 unit = 100 centi-units Milli- means 1/1000 of that unit ex: 1 Liter (L) = 1000 milliliters (mL) 1 unit = 1000 milli-units

36. Metric Prefixes
103 = 1000
10 -3 = = 1
or 0.001

37. Metric Prefixes in action
A unit of Computer information is a byte
Older computers could work with 64,000 bytes at one time
64 thousand bytes
64 kilobytes
OR 64 kb
Later computers could work with 64,000,000 bytes at one time
64 million
64 megabytes
64 mb
The newest computers can work with
Billions of bytes = gigabytes
Trillions of bytes = terabytes
A 1 Terabyte hard disk drive holds
1 trillion bytes of info = 1,000,000,000,000

38. Atomic dimensions
A carbon atom has a radius of 91 picometers Since a pico = That’s 91 x meters OR 9 of a meter 1,000,000,000, m

39. Learning Check
m = 1 ___ a) mm b) km c) dm g = 1 ___ a) mg b) kg c) dg L = 1 ___ a) mL b) cL c) dL m = 1 ___ a) mm b) cm c) dm b
10 3 a
10 -3 b c

40. DERIVED UNITS: Cubic length = Volume
1 cm x 1cm x 1cm = 1cm3 10cm x 10 cm x 10 cm = 1000 cm3 1 dm x 1 dm x 1 dm = 1 dm3 1 dm3 = 1 liter 1 cm3 = 1 milliliter 1000 cm3 or ml = 1 liter or dm3

41. Converting between units
Uses the factor – label method
Also called dimensional analysis
Involves conversion factors (fractions)
Attention is paid to the labels (units)
Must know equalities between units

42. length 10.0 in. 25.4 cm Therefore: 1.00 in = 2.54 cm
Equalities
State the same measurement in two different units
length
10.0 in.
25.4 cm
Therefore: 1.00 in = 2.54 cm

43. Conversion Factors (fractions)
Fractions in which the numerator and denominator are EQUAL quantities expressed in different units
Example: in. = 2.54 cm
Factors: 1 in and cm
2.54 cm in.

44. Learning Check 1. Liters and mL 2. Hours and minutes
Write conversion factors that relate each of the following pairs of units:
1. Liters and mL
2. Hours and minutes
3. Meters and kilometers
1 L
1000 mL OR
1 liter = 1000 milliliters
1 hr
60 min OR
1 hour = 60 minutes
1 Km
1000 m OR
1000 meters = 1 kilometer

45. Setting up conversions
How many minutes are in 2.5 hours?
Identify the starting and ending units:
hours to minutes
Recognize the equality between the unit:
1 hour = 60 minutes
Build conversion factor with ending unit over starting unit:
60 min
1 hour
Ending: ? minutes
Starting: 2.5 hours

46. How many minutes are in 2.5 hours?
2.5 hr x =
Conversion factor
60 min
1 hr
Set up to Cancel
150 min
using dimensional analysis / factor-label method, the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the numbers!

47. Sample Problem You have $7.25 in your pocket in quarters.
How many quarters do you have?
4 quarters = 1 dollars
How many quarters = 1 dollar?
4 quarters
1 dollar
How will we set up our conversion factor?
(Ending over starting unit)
4 quarters 1 dollar
7.25 dollars
= 29 quarters X

48. Learning Check a) 2440 cm b) 244 cm c) 24.4 cm
A rattlesnake is 2.44 m long. How long is the snake in cm?
a) cm
b) 244 cm
c) 24.4 cm
1 meter = 100 centimeters
2.44 m x 100 cm
1 m

49. 90 km x 6.2 miles 10 km = (90)(6.2) (10) = 55.8 miles
A practical problem: Driving in Canada, the speed limit says 90 km per hour, how fast is that in miles per hour? (Don’t want to get a speeding ticket, do I?)
90 km
x 6.2 miles
10 km
= (90)(6.2)
(10)
= 55.8 miles
km to miles
Step 1: ID Starting and ending unit
10 km = 6.2 miles
Step 2: Do I know an equality?
Step 3: Write down starting value and unit
Step 4: Write X conversion fraction Ending unit
Starting unit
Step 5: cancel units and calculate

50. Calculating fractions review12 6
3 x =
= 2 16 2
= 8
4 x = 2
____ 1
8 x = 2 8 2
= 4

51. Worksheet # 8 Convert the following into new units
Worksheet # 8 Convert the following into new units. Show labeled factor label setup.
340 ml = L (b) 235 g = Kg
(c) 1.4 m = cm (d) 4.7 Kilocalories = calories
(e) 3.5 dm3 = cm3 (f) 32 Joules = calories
(g) 2300mm = cm (h) 1.0 oz = grams
340 ml x 1 liter =
1000 ml
235 g x 1 kg =
1000 g
1.4 m x 100 cm =
1 m
4.7 kcal x 1000 cal =
1 kcal
32 J x 1 cal =
4.2 J
3.5 dm3 x 1000 cm3 =
1 dm3

52. Sample Problem – multi-step problems
How many seconds are in 1.4 days?
Unit plan: days  hr  min  seconds
x 60 min x
1 hr
60 sec =
1 min
1.4 days
x 24 hr
1 day
Check
ending
unit
= 120,000 sec

53. Sample Problem – multi-step problems
How many feet is a meter?
Unit plan: 1 m  cm  in  feet
1 inch x
2.5 cm
1 foot =
12 inches
1 meter x 100 cm x 1 m
Check
ending
unit
How do we plug this into a calculator?
1 x 100 ÷ 2.5 ÷ 12 =
3.3 ft
Multiply tops, divide by bottoms

54. Using Table C 45 10-3 = 45000 pm 45 nm = _____pm 45 nm X 10-9 m 1 nm
X pm m =
0.001
= pm

55. Three Steps to Problem Solving
Analyze the problem
List the knowns and the unknown
Identify Equalities for conversion problems
Do we need outside information? (Reference Tables)
Is there a formula ?
II. Calculate
Create factor label or formula setup
Cancel units and check setup for ending unit
Do the arithmetic on the calculator
Round appropriately and write answer with final unit
III. Evaluate
Reasonable answer? Unit correct? Check math?

56. Dealing with Two Units
If your pace on a treadmill is 65 meters per minute, how many seconds will it take for you to walk a distance of feet?

57. Knowns Starting: distance 8450 ft Rate (equality): 65 m / 1 min
If your pace on a treadmill is 65 meters per minute, how many seconds will it take for you to walk a distance of feet?
Knowns
Starting: distance 8450 ft
Rate (equality): 65 m / 1 min
Unknown
Ending unit :
seconds
Unit plan:
ft  inches  cm  meters
Since rate is 65 m per minute
Then 65 m = 1 minute
Meters  minutes  seconds

58. Initial 8450 ft Solution x 12 in 1 ft x 2.54 cm 1 in x 1 m 100 cm
x 1 min
65 m
x 60 sec
1 min
2400 sec =

59. oz -> lb -> kg -> g
Worksheet # 8 Convert the following into new units. Show labeled factor label setup.
340 ml = L (b) 235 g = Kg
(c) 1.4 m = cm (d) 4.7 Kilocalories = calories
(e) 3.5 dm3 = cm3 (f) 32 Joules = calories
(g) 2300mm = cm (h) 1.0 oz = grams
340 ml x 1 liter =
1000 ml
235 g x 1 kg =
1000 g
1.4 m x 100 cm =
1 m
4.7 kcal x 1000 cal =
1 kcal
32 J x 1 cal =
4.2 J
3.5 dm3 x 1000 cm3 =
1 dm3
16 oz = 1 lb
2.2 lb = 1 kg
1 kg = 1000 g
2300 mm x 1 m x 100 cm =
1000 mm m
Unit Plan ?:
oz -> lb -> kg -> g
1 oz x 1 lb x 1 kg x g =
16 oz lb kg

60. What about Square and Cubic units?
Use the conversion factors you already know, but when you square or cube the unit, don’t forget to cube the number also!
Best way: Square or cube the ENTIRE conversion factor
Example: Convert 4.3 cm3 to mm3
( )3
4.3 cm mm
1 cm
4.3 cm mm3
1 cm3 =
= 4300 mm3

61. Temperature Scales Fahrenheit Celsius Kelvin Anders Celsius 1701-1744
Lord Kelvin
(William Thomson)

62. Temperature Scales Fahrenheit Celsius Kelvin 32 ˚F 212 ˚F 180˚F 100 ˚C
Boiling point of water
32 ˚F
212 ˚F
180˚F
100 ˚C
0 ˚C
100˚C
373 K
273 K
100 K
Freezing point of water
Notice that a change of 1 0C = 1 K
But that 0 0C = 273 K

63. Celsius
is based on water
Is a relative scale
0 0C = normal melting point
100 0C = normal boiling point
Kelvin
is based on molecular speed energy
Is an absolute scale
0 K – absolute zero
- molecules stop moving

64. K = 100 0C
K = 0 0C
0 K = C

65. Calculations Using Temperature: Using a formula
K = ˚C
Body temp = 37 ˚C = 310 K
Liquid nitrogen = ˚C = 77
Or ˚C = K

66. ˚C = K - 273 ˚C = 298 - 273 ˚C = 25 Learning Check
Room temperature is 298 Kelvins. What is this expressed in Celsius?
˚C = K
1) Write down the relevant formula
˚C =
2) substitute given values
˚C = 25
3) Calculate
4) Evaluate: is answer reasonable?

67. DENSITY - an intensive physical property (specific to a substance)
Compactness of its matter (mass)
Ratio of mass compared to volume
Aluminum
Platinum
Mercury
13.6 g/cm3
21.5 g/cm3
2.7 g/cm3

68. Problem A piece of copper has a mass of g. It displaces 6.4 cm3 . Calculate its density (g/cm3).

69. what should be the mass of 1 of the 6 cm3’s?
Using a formula
Analyze: List knowns and unknowns:
Known:
Unknown:
write the formula:
Setup problem by substituting values
Evaluate answer: is it reasonable?
Mass = grams Volume = 6.4 cm3
Density
D = M V
D = grams
6.4 cm3
Solve,
Round,
And label
D =
D = 9.0
g/cm3
what should be the mass of 1 of the 6 cm3’s?

70. A minimum proper math setup
D = M V
D = grams
6.4 cm3
D = 9.0
g/cm3

71. PROBLEM: Mercury (Hg) has a density of 13. 6 g/cm3
PROBLEM: Mercury (Hg) has a density of 13.6 g/cm3. What is the mass of 95 mL of Hg in grams?
Solve the problem using DIMENSIONAL
ANALYSIS.

72. 3. Evalute: Is our answer reasonable?
PROBLEM: Mercury (Hg) has a density of 13.6 g/cm3. What is the mass of 95 mL of Hg?
Analyze: knowns -
V = 95 mL
D = 13.6 g/cm3 or g = 1 cm3
1 cm3 = 1 mL
unknowns - mass
2. Setup, calculate, round
95 cm3
x g =
1 cm3
1292 g
=1300 g
3. Evalute: Is our answer reasonable?

73. Learning Check
Osmium has a density of 22.5 g/cm3. What is the volume of a g sample? 1) 0.45 cm3 2) 2.22 cm3 3) 11.3 cm3
50.00 g x 1 cm3
22.5 g

74. Volume Displacement 25 mL
A solid displaces a matching volume of water when the solid is placed in water.
33 mL
25 mL

75. Learning Check
What is the density (g/cm3) of 48 g of a metal if the metal raises the level of water in a graduated cylinder from 25 mL to 33 mL?
1) 0.2 g/ cm ) 6 g/cm ) g/cm3
33 mL
25 mL