1. Chapter 3 Measurement and Chemical Calculations

2. 3.1 Introduction to Measurement
Measurement of a quantity is made by comparison with a standard that serves to represent the magnitude of a unit.
Result of a measurement = number x unit

3. 3.1 Introduction to Measurement
Measurements in the U.S. are made in the United States Customary System.
Unit of mass is 1 pound
Unit of length is 1 inch
Measurements in science are made in the International System (SI).
Unit of mass is 1 kilogram
Unit of length is 1 meter

4. 3.1 Introduction to Measurement
Mass of a notebook = 2.4 kg
The mass of the notebook is 2.4 times the
unit of mass, which is 1 kilogram

5. 3.1 Introduction to Measurement
The SI system is defined by seven base units.
Examples of base units include:
Quantity Base Unit
Mass Kilogram
Length Meter
Temperature Kelvin
Time Second
Other measurement units are derived from the base units; accordingly, they are called derived units.

6. E is the power or exponent
3.2 Exponential Notation
Exponentials BE
B is the base
E is the power or exponent
104 = 10 × 10 × 10 × 10 = 10,000
10–4 = = × × × = =

7. Usually 1 ≤ coefficient < 10
3.2 Exponential Notation
Exponential Notation
a.bcd × 10E
Coefficient: a.bcd
Usually 1 ≤ coefficient < 10
Exponent: E
A whole number

8. Conversion Between Decimal Numbers and
3.2 Exponential Notation
Conversion Between Decimal Numbers and
Standard Exponential Notation
Example:
Convert 724,000 to standard exponential notation
7.24 × 10e
Five places
7.24 × 105

9. Conversion Between Decimal Numbers and
3.2 Exponential Notation
Conversion Between Decimal Numbers and
Standard Exponential Notation
Example:
Convert to standard exponential notation
4.27 × 10e
Four places
4.27 × 10–4

10. 3.3 Dimensional Analysis Dimensional Analysis
A problem-solving technique when quantities are directly proportional.
A conversion factor is a mathematical statement that two quantities are directly proportional to one another
Examples:
7 Days = 1 week
7 Days/1 week
7 Days per week

11. 3.3 Dimensional Analysis Direct Proportionality
Two variables, X and Y,
are directly proportional if there is a nonzero constant m
that relates them in the form
Y = m X
This relationship can also be expressed as a proportionality:
Y X

12. How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 1:
Identify and write down the
GIVEN quantity, including GIVEN: 23 weeks
units.

13. How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 2:
Identify and write down the GIVEN: 23 weeks
units of the WANTED WANTED: days
quantity.

14. How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 3: GIVEN: 23 weeks
Write down the WANTED: days
PER/PATH. PER: days/week
PATH: wk days

15. How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 4: GIVEN: 23 weeks
Write the calculation WANTED: days
setup. Include units. PER: days/week
PATH: wk days
23 weeks × =

16. How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 5: GIVEN: 23 weeks
Calculate the answer. WANTED: days PER: days/week
PATH: wk days
23 weeks × = 161 days

17. How to Solve a Problem by Dimensional Analysis
Sample Problem:
How many days are in
23 weeks?
Step 6: GIVEN: 23 weeks
Check the answer to be sure WANTED: days
both the number and the PER: days/week
units make sense. PATH: wk days
23 weeks × = 161 days
More days (smaller unit) than weeks (larger unit).
OK.

18. 3.4 Metric Units: Mass and Weight
Mass is a measure of quantity of matter.
Unit for mass is 1 kilogram
Weight is a measure of the force of gravitational attraction.
Unit for force is 1 Neuton.
Mass and weight are directly proportional to each other.
Same weight, same mass.
Weight = mass x acceleration of gravity
Acceleration of gravity on the earth = 9.80 m/s2
Weight of one apple (0.1 kg) is about 1 Neuton

19. 3.4 Metric Units. Kilogram as unit
The SI unit of mass is the
kilogram, kg.
It is defined as the mass of a platinum-iridium cylinder stored in a vault in France.
1 lb  kg (definition)
A kilogram is about 2.2 pounds.

20. For most laboratory work, the basic metric mass unit is used:
3.4 Metric Units
For most laboratory work, the basic metric mass unit is used:
the gram, g.

21. 3.4 Metric Units In the metric system, units that are larger than the
basic unit are larger by multiples of 10.
For example, the kilo- unit is 1000 times
larger than the basic unit.
Units that are smaller than the basic unit are smaller
by fractions that are also multiples of 10.
For example, the milli- unit is 1/1000 times
smaller than the basic unit.

22. 3.4 Metric Units: Metric Prefixes
Large Units Small Units
Metric Metric Metric Metric
Prefix Symbol Multiple Prefix Symbol Multiple
tera- T Unit 1
giga- G 109 deci- d 0.1
mega- M 106 centi- c 0.01
kilo- k milli- m 0.001
hecto- h 100 micro- µ 10–6
deca- da 10 nano- n 10–9
Unit 1 pico- p 10–12

23. 3.4 Metric Units. Length The SI unit of length is the meter, m.
It is defined as the distance light travels in a vacuum in 1/299,792,458 second
One inch is defined as 2.54 centimeters.(1 in. ≡ 2.54 cm)
One meter is about forty inches:

24. One inch is defined as 2.54 centimeters.
Metric Units
One inch is defined as 2.54 centimeters.

25. 3.4 Metric Units. Volume The SI unit of volume is the cubic meter, m3.
A more practical unit for laboratory work is the
cubic centimeter, cm3.

26. One liter (L) is defined as exactly 1000 cubic centimeters.
3.4 Metric Units
One liter (L) is defined as exactly 1000 cubic centimeters.
1 mL ≡ L ≡ 1 cm3

27. 3.4 Metric Units Metric Relationship Example
1000 units per kilounit meters per kilometer
1000 m/km
100 centiunits per unit 100 centigrams per gram
100 cg/g
1000 milliunits per unit milliliters per liter
1000 mL/L

28. 3.4 Metric Units Example: How many centigrams are in 0.87 gram?
Solution:
Use dimensional analysis.
GIVEN: 0.87 g
WANTED: cg
PER: cg/g
PATH: g cg
0.87 g × = 87 cg
More centigrams (smaller unit) than grams (larger unit). OK.

29. 3.4 Metric Units Example: How many kilometers are in 2,335 meters?
Solution:
Use dimensional analysis.
GIVEN: 2335 m
WANTED: km
PER: m/km
PATH: m km
2335 m × = km
More meters (smaller unit) than kilometers (larger unit). OK.

30. 3.4 Metric Units Example: How many milliliters are in 0.00339 liter?
Solution:
Use dimensional analysis.
GIVEN: L
WANTED: mL
PER: mL/L
PATH: L mL
L × = 3.39 mL
More milliliters (smaller unit) than liters (larger unit). OK.

31. 3.5 Significant Figures Uncertainty in Measurement
No measurement is exact.
In scientific writing, the uncertainty associated with a measured quantity is always included.
By convention, a measured quantity is expressed by stating all digits known accurately plus one uncertain digit.

32. 3.5 Significant Figures

33. 3.5 Significant Figures The bottom board is one meter long.
How long is the top board?
More than half as long as the meter stick,
but less than one meter—about 6/10 of a meter.
The uncertain digit is the last digit written: 0.6 m

34. 3.5 Significant Figures Now the meter stick has marks every 0.1 m,
numbered in centimeters. How long is the board?
Between 0.6 m and 0.7 m with certainty, and the uncertain digit must be estimated—the board is about 4/10 of the way between 0.6 m and 0.7 m: 0.64 m.

35. 3.5 Significant Figures
The measuring instrument now has centimeter marks.
How long is the board?
Between 0.64 m and 0.65 m with certainty.
It is about 3/10 of the way between the two marks,
so we record m as the length of the board.

36. 3.5 Significant Figures
The measuring instrument now has millimeter marks.
We could estimate between the millimeter marks, but the alignment of the board and the meter stick has an uncertainty of a millimeter or so.
We have reached the limit of this measuring instrument: m.

37. 3.5 Significant Figures Significant Figures
Significant figures are applied to measurements
and quantities calculated from measurements.
They do not apply to exact numbers.
An exact number has no uncertainty.
Types of exact numbers:
Counting numbers
Numbers fixed by definition

38. 3.5 Significant Figures Significant Figures
The number of significant figures in a quantity is the number of digits that are known accurately plus the one that is uncertain—the uncertain digit.
The uncertain digit is the last digit written
when expressing a scientific measurement.

39. 3.5 Significant Figures Significant Figures
The measurement process, not the unit in which
the result is expressed, determines the
number of significant figures in a quantity.
The length of the board in the previous illustrations
was m. Expressed in centimeters, it is 64.3 cm.
They are the same measurement with the same uncertainty.
Both must have the same number of significant figures.

40. 3.5 Significant Figures Significant Figures
The location of the decimal point has nothing
to do with significant figures.
The same m board is km.
The three zeros before the decimal point are not significant.
Begin counting significant figures at the first nonzero digit,
not at the decimal point.

41. 3.5 Significant Figures Significant Figures
The uncertain digit is the last digit written.
If the uncertain digit is a zero to the right of the decimal point,
that zero must be written.
If the mass of a sample on a triple beam balance is g, and the balance is accurate to ±0.01 g, the last digit recorded must be zero to indicate the correct uncertainty.

42. 3.5 Significant Figures Significant Figures
Exponential notation must be used for very large numbers to show if final zeros are significant.
If the length of the m board is expressed
in micrometers, its length is 643,000 µm.
The uncertainty is ±1,000 µm.
The ordinary decimal number makes this ambiguous.
Writing 6.43 × 105 µm shows clearly the correct
location of the uncertain digit.

43. 3.5 Significant Figures Rounding a Calculated Number
If the first digit to be dropped is less than 5,
leave the digit before it unchanged.
Examples:
Round to three significant figures. Answers
1.743 m m
kg kg

44. 3.5 Significant Figures Rounding a Calculated Number
If the first digit to be dropped is 5 or more,
increase the digit before it by 1.
Examples:
Round to three significant figures. Answers
32.88 mL mL
km km

45. Significant Figure Rule
3.5 Significant Figures
Significant Figure Rule
for Addition and Subtraction
Round off the answer to the first column
that has an uncertain digit.

46. 3.5 Significant Figures Example:
The following is a list of masses of items to be shipped. What is the total mass of the package?
Carton: 226 g; Item 1: 33.5 g; Item 2: 589 g; Packaging: g
Answer: g g g g
g = 860 g = × 102 g

47. Significant Figure Rule
3.5 Significant Figures
Significant Figure Rule
For Multiplication and Division
Round off the answer to the same number of significant figures as the smallest number of significant figures in any factor.

48. 3.5 Significant Figures Example:
What is the volume of a cube that is cm long, 23.0 cm wide and 15 cm high?
Solution:
Use algebra because the GIVENS and WANTED are related by a formula, volume = length × width × height.
V = l × w × h = cm × 23.0 cm × 15 cm
4 sf sf sf
= 11, cm3 (unrounded)
The answer is rounded to 2 sf, 1.2 × 104 cm3

49. 3.6 Metric–USCS Conversions
Conversions between the United States Customary System (USCS) and the metric system are made by applying dimensional analysis.
Length
1 in cm (definition of an inch)
Mass
1 lb g (definition of a pound)
Volume
1 gal L (definition of a gallon)

50. 3.6 Metric–USCS Conversions
Some Useful Definitions
Length
1 ft in.
1 yd 3 ft
1 mi ft
Mass
1 lb oz
Volume
1 qt fl oz
1 gal 4 qt

51. 3.6 Metric–USCS Conversions
Example:
How many milliliters are in 1.0 quart?
Solution:
PER: gal/4 qt L/gal mL/L
PATH: qt gal L mL
1.0 qt × × × = 9.4 × 102 mL
CHECK: More milliliters (smaller unit) than quarts (larger unit). OK.

52. 3.7 Temperature There are three scales: Celsius, Fahrenheit and Kelvin
Given a temperature in Celsius or Kelvin or Fahrenheit degrees, convert it to the other scales.

53. 3.7 Temperature

54. 3.7 Temperature Fahrenheit Temperature Scale
Water freezes at 32°F and boils at 212°F.
There are 180 Fahrenheit degrees between freezing and boiling.
Celsius Temperature Scale
Water freezes at 0°C and boils at 100°C.
There are 100 Celsius degrees between freezing and boiling.
T°F – 32 = T°C
T°F – 32 = 1.8 T°C

55. Kelvin (Absolute) Temperature Scale
The degree is the same size as a Celsius degree,
but 0 on the Kelvin scale is set at the lowest
temperature that can be approached, which is –273°C.
TK = T°C + 273

56. 3.7 Temperature
Example:
Convert 65°F to its equivalent in degrees Celsius and Kelvins.
Solution:
GIVEN: 65°F WANTED: °C and K
EQUATIONS: T°F – 32 = 1.8 T°C and TK = T°C + 273
T°C = = = 18°C
TK = T°C = = 291 K

57. 3.8 Proportionality and Density
A direct proportionality exists between two quantities when they increase or decrease at the same rate.
If a graph of two related measurements is a straight line that passes through the origin, the measured quantities are directly proportional to each other.
Y X
Y is proportional to X
Y = m X
m is the proportionality constant, which is the slope of the graph.

58. 3.8 Proportionality and Density
Direct proportionalities between measured quantities yield two conversion factors between the quantities.
Given either quantity in a direct proportionality and the conversion factor between the quantities, we can calculate the other quantity with dimensional analysis.

59. 3.8 Proportionality and Density
The mass and volume of any pure substance at a given temperature are directly proportional:
mass is proportional to volume
mass volume
m V
A proportionality is changed into an equation by inserting a multiplier called a proportionality constant.
Let D be the proportionality constant:
m = D × V

60. 3.8 Proportionality and Density
Solving for the proportionality constant yields the defining equation for a physical property of a pure substance called density:
D =
In words, density is the mass per unit volume of a substance:
Density =

61. 3.8 Proportionality and Density
The definition of density establishes its common units:
Density =
The common laboratory unit for mass is grams.
The common laboratory unit for volume is
milliliters or cubic centimeters.
Density is therefore typically expressed in g/mL or g/cm3.
Since volume varies with temperature,
density is temperature dependent.

62. 3.8 Proportionality and Density
Densities of Some Common Substances
(g/cm3 at 20°C and 1 atm)
Substance Density Substance Density
Helium Aluminum
Air Iron
Pine lumber Copper
Maple lumber 0.6 Silver
Oak lumber Lead
Water Mercury 13.6
Glass Gold

63. 3.8 Proportionality and Density
Example:
What is the volume of 15 g of silver?
Solution:
GIVEN: 15 g silver
WANTED: volume (assume cm3)
PER: g/cm3 (from Densities of Some Common Substances table)
PATH: g cm3
15 g × = 1.4 cm3

64. 3.9 Strategy for Solving Problems
A problem can be solved by dimensional analysis if the
GIVEN and WANTED can be linked by one or more
PER expressions and you know or can find the
conversion factor for each expression.
A problem can be solved by algebra if the GIVEN and
WANTED appear in an algebraic equation
in which the WANTED is the only unknown.

65. 3.9 Strategy for Solving Problems
The density of an oil is g/mL. Find the volume occupied by 196 g of that oil.
Dimensional analysis method
Given : mass of oil g
Wanted Volume of oil in ml
Path g ( mass of oil)  mL ( volume of oil)
Factors g/1 mL and mL / 0.862g
Set up and calculation g x 1 mL/ g = 227 mL

66. 3.9 Strategy for Solving Problems
The density of an oil is g/mL. Find the volume occupied by 196 g of that oil.
Algebraic method
Given : mass of oil g
Wanted Volume of oil in mL
Equation V = m / D = 196 g / (0.862g/mL) = 227 mL

67. 3.9 Strategy for Solving Problems

68. Practice
To become a better problem solver, you must practice solving problems. When you solve a problem and check the answer section of the textbook, reflect on both the chemistry concept and the strategy you used in solving the problem.
The reason for solving problems is not to copy the answers from the textbook, but to learn how to solve the problem.
Homework for chapter 3:
1, 5, 7, 9, 11, 13, 35, 51, 53, 71, 73, 89, 91, 93, 95.