1. Chapter 2
Measurements

2. CHAPTER OUTLINE Scientific Notation Error in Measurements
Significant Figures
Rounding Off Numbers
SI Units
Conversion of Factors
Conversion of Units
Volume & Density

3. What is a Measurement? quantitative observation
comparison to an agreed upon standard
every measurement has a number and a unit

4. A Measurement
the unit tells you what standard you are comparing your object to
the number tells you
what multiple of the standard the object measures
the uncertainty in the measurement

5. Scientists have measured the average global temperature rise over the past century to be 0.6°C
°C tells you that the temperature is being compared to the Celsius temperature scale
0.6 tells you that
the average temperature rise is 0.6 times the standard unit
the uncertainty in the measurement is such that we know the measurement is between 0.5 and 0.7°C

6. A x 10n SCIENTIFIC NOTATION
Scientific Notation is a convenient way to express very large or very small quantities.
Its general form is
A x 10n
n = exponent
coefficient
1  A < 10

7. SCIENTIFIC NOTATION
To convert from decimal to scientific notation:
Move the decimal point in the original number so that it is located after the first nonzero digit.
Follow the new number by a multiplication sign and 10 with an exponent (power).
The exponent is equal to the number of places that the decimal point was shifted.
7.5 x 10 7

8. Scientific Notation: Writing Large and Small Numbers
A positive exponent means 1 multiplied by 10 n times.
A negative exponent (–n) means 1 divided by 10 n times.

9. SCIENTIFIC NOTATION
For numbers smaller than 1, the decimal moves to the left and the power becomes negative. 3
6.42 x 10

10. 1. Write 6419 in scientific notation.
Examples:
1. Write 6419 in scientific notation.
decimal after first nonzero digit
power of 10
64.19x102
6.419 x 103
641.9x101
6419.
6419

11. 2. Write 0.000654 in scientific notation.
Examples:
2. Write in scientific notation.
power of 10
decimal after first nonzero digit
x 10-1
x 10-2
0.654 x 10-3
6.54 x 10-4

12. CALCULATIONS WITH SCIENTIFIC NOTATION
To perform multiplication or division with scientific notation:
Change numbers to exponential form.
Multiply or divide coefficients.
Add exponents if multiplying, or subtract exponents if dividing.
If needed, reconstruct answer in standard exponential form.

13. Multiply coefficients Convert to exponential form
Example 1:
Multiply 30,000 by 600,000
Add exponents
Reconstruct answer
Multiply coefficients
Convert to exponential form 9
(3 x 104)
(6 x 105) =
18 x 10
1.8 x 1010

14. Convert to exponential form
Example 2:
Divided 30,000 by 0.006
Subtract exponents
Reconstruct answer
Divide coefficients
Convert to exponential form
4 – (-3)
(3 x 104) 7
= 0.5 x 10
(6 x 10-3)
5 x 106

15. Follow-up Problems: (5.5x103)(3.1x105) = 17.05x108 = 1.7x109

16. Follow-up Problems: (8.75x1014)(3.6x108) = 31.5x1022 = 3.2x1023

17. Is this measurement precise?
ACCURACY & PRECISION
Precision is the reproducibility of a measurement compared to other similar measurements.
Precision describes how close measurements are to one another.
Precision is affected by random errors.
Is this measurement precise?
Avg mass = 3.12± 0.01 g
This measurement has high precision because the deviation of multiple trials is small.

18. ACCURACY & PRECISION
Is this measurement accurate?
Accuracy is the closeness of a measurement to an accepted value (external standard).
Accuracy describes how true a measurement is.
Accuracy is affected by systematic errors.
Avg mass = 3.12± 0.01 g
True mass = 3.03 g
Accuracy cannot be determined without knowledge of the accepted value.
This measurement has low accuracy because the deviation from true value is large.

19. ACCURACY & PRECISION Poor precision Good precision Good accuracy
Poor accuracy
Poor precision
Good precision
Poor accuracy
Good accuracy

20. ACCURACY & PRECISION Two types of error can affect measurements:
Systematic errors: those errors that are controllable, and cause measurements to be either higher or lower than the actual value.
Random errors: those errors that are uncontrollable, and cause measurements to be both higher and lower than the average value.

21. ERROR IN MEASUREMENTS Two kinds of numbers are used in science:
Counted or defined:
exact numbers; have no uncertainty
Measured:
are subject to error; have uncertainty
Every measurement has uncertainty because of instrument limitations and human error.

22. ERROR IN MEASUREMENTS certain uncertain 8.65 8.6 uncertain certain
The last digit in any measurement is the estimated one.
What is this measurement?
What is this measurement?

23. RECORDING MEASUREMENTS TO THE PROPER NO OF DIGITS
What is the correct value
for each measurement?
a) 28ml (1 certain, 1 uncertain)
b) 28.2ml (2 certain, 1 uncertain)
c) 28.31ml (3 certain, 1 uncertain)

24. SIGNIFICANT FIGURES RULES
Significant figures rules are used to determine which digits are significant and which are not.
Significant figures are the certain and uncertain digits in a measurement.
All non-zero digits are significant.
All sandwiched zeros are significant.
Leading zeros (before or after a decimal) are NOT significant.
Trailing zeros (after a decimal) are significant.

25. Examples:
Determine the number of significant figures in each of the following measurements. g
461 cm
3 sig figs
5 sig figs
0.006 m
1025 g
1 sig fig
4 sig figs
0.705 mL
5500 km
2 sig figs
3 sig figs

26. ROUNDING OFF NUMBERS 51.234 Round to 3 sig figs 1.875377
If rounded digit is less than 5, the digit is dropped.
51.234
Round to 3 sig figs
Round to 4 sig figs
Less than 5
Less than 5

27. ROUNDING OFF NUMBERS 51.369 Round to 3 sig figs 4 5.4505 1
If rounded digit is equal to or more than 5, the digit is increased by 1.
51.369
Round to 3 sig figs 4
5.4505 1
Round to 4 sig figs
More than 5
Equal to 5

28. SIGNIFICANT FIGURES & CALCULATIONS
The results of a calculation cannot be more precise than the least precise measurement.
In multiplication or division, the answer must contain the same number of significant figures as in the measurement that has the least number of significant figures.
For addition and subtraction, the answer must have the same number of decimal places as there are in the measurement with the fewest decimal places.

29. MULTIPLICATION & DIVISION
Calculator answer
4 sig figs
3 sig figs
(9.2)(6.80)(0.3744) =
2 sig figs
The answer should have two significant figures because 9.2 is the number with the fewest significant figures.
The correct answer is 23

30. ADDITION & SUBTRACTION
Add 83.5 and 23.28
83.5
23.28
Least precise number
Calculator answer
106.78
106.8
Correct answer

31. Example 1: 5.008 + 16.2 + 13.48 = 34.688 34.7 Least precise number
Round to

32. Example 2:
3 sig figs
6.1788
6.2
2 sig figs
Round to

33. SI UNITS
Measurements are made by scientists to determine size, length and other properties of matter.
For measurements to be useful, a measurement standard must be used.
A standard is an exact quantity that people agree to use for comparison.
SI is the standard system of measurement used worldwide by scientists.

34. SI (METRIC) BASE UNITS Quantity Measured Metric Units Symbol English
Length
Meter m yd
Mass
Kilogram kg lb
Time
Seconds s
Temperature
Kelvin K F
Amount of substance
Mole
mol

35. Basic Units of Measurement
The kilogram is a measure of mass, which is different from weight.
The mass of an object is a measure of the quantity of matter within it.
The weight of an object is a measure of the gravitational pull on that matter.
Consequently, weight depends on gravity while mass does not.

36. Derived Units A derived unit is formed from other units.
Many units of volume, a measure of space, are derived units.
Any unit of length, when cubed (raised to the third power), becomes a unit of volume.
Cubic meters (m3), cubic centimeters (cm3), and cubic millimeters (mm3) are all units of volume.
Change to equation objects

37. DERIVED UNITS
In addition to the base units, several derived units are commonly used in SI system.
Quantity Measured
Units
Symbol
Volume
Liter L
Density
grams/cc
g/cm3

38. SI PREFIXES
Common prefixes are used with the base units to indicate the multiple of ten that the unit represents.
The SI system of units is easy to use because it is based on multiples of ten.
SI Prefixes
Prefixes
Symbol
Multiplying factor
mega- M
1,000,000
kilo- k
1000
centi- c
0.01
milli- m
0.001
micro- 
0.000,001
106
103
10-2
10-3
10-6

39. SI UNITS & PREFIXES
SI system used a common set of prefixes for use with the base units.
Base Unit
micro
106
milli
103
kilo
103
mega
106 10 10 10 10 10 10
deci
centi
Smaller units
Larger units

40. SI CONVERSION FACTORS Base Unit micro 106 kilo 103 mega 106 milli
103 10 10 10 10 10 10
deci
centi
1 m =
103 mm or
1 mm =
103 m
1 mm =
103 m or
1 m =
103 mm

41. SI PREFIXES 100000 or 105 How many mm are in a cm?
How many cm are in a km?
10x10x10x10x10 10

42. Prefix Multipliers
Choose the prefix multiplier that is most convenient for a particular measurement.
Pick a unit similar in size to (or smaller than) the quantity you are measuring.
A short chemical bond is about 1.2 × 10–10 m. Which prefix multiplier should you use?
The most convenient one is probably the picometer. Chemical bonds measure about 120 pm.
Change to equation object

43. CONVERSION FACTORS
Many problems in chemistry and related fields require a change of units.
Any unit can be converted into another by use of the appropriate conversion factor.
Any equality in units can be written in the form of a fraction called a conversion factor. For example:
Metric-Metric Factor
Equality
1 m = 100 cm
1 m
100 cm
100 cm
1 m
Conversion Factors or

44. Metric-English Factor
CONVERSION FACTORS
Metric-English Factor
Equality
1 kg = 2.20 lb
1 kg
2.20 lb
2.20 lb
1 kg
Conversion Factors or
Percentage Factor
Sometimes a conversion factor is given as a percentage. For example:
Percent quantity:
18% body fat by mass
Conversion Factors
18 kg body fat
100 kg body mass
100 kg body mass
18 kg body fat or

45. CONVERSION OF UNITS final unit beginning unit Conversion factor
Problems involving conversion of units and other chemistry problems can be solved using the following step-wise method:
3. Write the conversion factor for each units change in your plan.
4. Set up the problem by arranging cancelling units in the numerator and denominator of the steps involved.
Plan a sequence of steps to convert the initial unit to the final unit.
1. Determine the intial unit given and the final unit needed.
final unit
beginning unit
Conversion factor

46. Metric-English factor
Example 1:
Convert 164 lb to kg (1 kg = 2.20 lb)
Step 1:
Given: 164 lb Need: kg
Metric-English factor lb kg
Step 2:
1 kg
2.20 lb
2.20 lb
1 kg
Step 3: or
Step 4:

47. Example 2:
The thickness of a book is 2.5 cm. What is this measurement in mm?
Step 1:
Given: 2.5 cm Need: mm
Metric-Metric factor cm mm
Step 2:
1 cm
10 mm
10 mm
1 cm
Step 3: or
Step 4:

48. English-English factor Metric-English factor
Example 3:
How many centimeters are in 2.0 ft? (1 in=2.54 cm)
Step 1:
Given: 2.0 ft Need: cm
English-English factor
Metric-English factor ft in cm
Step 2:
1 ft
12 in
1 in
2.54 cm
Step 3:
and
61 cm
60.96 cm
Step 4:

49. Given: 1.75 lb bronze Need: g of copper English-Metric factor
Example 4:
Bronze is 80.0% by mass copper and 20.0% by mass tin. A sculptor is preparing to case a figure that requires 1.75 lb of bronze. How many grams of copper are needed for the brass figure (1lb = 454g)?
Step 1:
Given: 1.75 lb bronze Need: g of copper
lb brz
English-Metric factor
g brz
Percentage factor
g Cu
Step 2:

50. Example 4: 1 lb 454 g 80.0 g Cu 100 g brz 454 g 1 lb 80.0 g Cu
Step 3:
and
1.75 lb brz x
454 g
1 lb
80.0 g Cu
100 g brz
Step 4: x
= g
= 636 g

51. VOLUME
Volume is the amount of space an object occupies.
Common units are cm3 or liter (L) and milliliter (mL).
1 L = 1000 mL
1 mL = 1 cm3

52. VOLUME Volume of various regular shapes can be calculated as follows:
Cube V = s x s x s
Rect. V = l x w x h
Cylinder V = π x r2 x h
Sphere V =4/3 πr3

53. Which has greatest density?
Density is mass per unit volume of a material.
Common units are g/cm3 (solids) or g/mL (liquids).
Density is directly related to the mass of an object.
Density is indirectly related to the volume of an object.
Which has greatest density?

54. Example 1:
A copper sample has a mass of g and a volume of 5.0 mL. What is the density of copper?
m = g
v = 5.0 mL
d = ???
= 8.93 g/mL
= 8.9 g/mL
Round to 2 sig figs

55. Example 2:
A silver bar with a volume of 28.0 cm3 has a mass of 294 g. What is the density of this bar?
m = 294 g
v = 28.0 mL
d = ???
= 10.5 g/mL
3 sig figs

56. Example 3:
If the density of gold is 19.3 g/cm3, how many grams does a 5.00 cm3 nugget weigh?
Step 1:
Given: 5.00 cm Need: g
density
Step 2:
cm3 g
Step 3:
5.00 cm3
= 96.5 g

57. English-Metric factor
Example 4:
If the density of milk is 1.04 g/mL, what is the mass of 0.50 qt of milk? (1L = 1.06 qt)
Step 1:
Given: 0.5 qt Need: g
English-Metric factor
Step 2: qt
Metric-Metric factor
density L mL g
1 L
1.06 qt
103 mL
1 L
1.04 g
1 mL
Step 3:
and
and
Step 4:
= g
= 490 g

58. Example 5:
What volume of mercury has a mass of 60.0 g if its density is 13.6 g/mL? 1
13.6 mL
4.41
60.0 g g
inverse of density

59. IS UNIT CONVERSION IMPORTANT?
In 1999 Mars Climate orbiter was lost in space because engineers failed to make a simple conversion from English units to metric units, an embarrassing lapse that sent the $125 million craft fatally close to the Martian surface.
Further investigation showed that engineers at Lockheed Martin, which built the aircraft, calculated navigational measurements in English units. When NASA’s JPL engineers received the data, they assumed the information was in metric units, causing the confusion.

60. THE END