1. Chapter 3 Observation, Measurement and Calculations

2. Measurement
Measurement – a quantity that has both a number and a unit.
Measurements are fundamental to the experimental sciences
Units typically used in the sciences are the International System of Measurements (SI)

3. Scientific Notation In science, we deal with some very LARGE numbers:
1 mole =
In science, we deal with some very SMALL numbers:
Mass of an electron = kg

4. Scientific Notation
Imagine the difficulty of calculating the mass of 1 mole of electrons! kg x
???????????????????????????????????

5. Scientific Notation:
A method of representing very large or very small numbers in the form:
M x 10n
M is a number between 1 and 10
n is an integer

6. . 9 8 7 6 5 4 3 2 1
Step #1: Insert an understood decimal point
Step #2: Decide where the decimal must end
up so that one number is to its left
Step #3: Count how many places you bounce
the decimal point
Step #4: Re-write in the form M x 10n

7. 2.5 x 109
The exponent is the number of places we moved the decimal.

8. 0.0000579 1 2 3 4 5 Step #2: Decide where the decimal must end
up so that one number is to its left
Step #3: Count how many places you bounce
the decimal point
Step #4: Re-write in the form M x 10n

9. 5.79 x 10-5
The exponent is negative because the number we started with was less than 1.

10. PERFORMING CALCULATIONS IN SCIENTIFIC NOTATION
ADDITION AND SUBTRACTION

11. Review: M x 10n Scientific notation expresses a number in the form:
n is an integer
1  M  10

12. IF the exponents are the same, we simply add or subtract the numbers in front and bring the exponent down unchanged.
4 x 106
+ 3 x 106 7
x 106

13. The same holds true for subtraction in scientific notation.
4 x 106
- 3 x 106 1
x 106

14. If the exponents are NOT the same, we must move a decimal to make them the same.

15. 4.00 x 106
4.00 x 106
x 105
+ .30 x 106
4.30
x 106
Move the decimal on the smaller number!

16. A Problem for you…
2.37 x 10-6
x 10-4

17. Solution…
x 10-6
2.37 x 10-6
x 10-4

18. Solution…
x 10-4
x 10-4
x 10-4

19. Scientific Notation Calculation Summary
Adding and Subtracting
You must express the numbers as the same power of 10. This will often involve changing the decimal place of the coefficient.
(2.0 x 106) + ( 3.0 x 107)
(0.20 x 107) + (3.0 x 107) = 3.20 x 107
(4.8 x 105) - ( 9.7 x 104)
(4.8 x 105) - ( 0.97 x 105) = 3.83 x 105

20. Multiply the coefficients and add the exponents
Scientific Notation
Multiplying
Multiply the coefficients and add the exponents
(xa) (xb) = x a + b
(2.0 x 106) ( 3.0 x 107) = 6.0 x 1013
Dividing
(xa) / (xb) = x a - b
(2.0 x 106) / ( 3.0 x 107) = 0.67 x 10-1
Divide the coefficients and subtract the exponents

21. Nature of Measurement Measurement - quantitative observation
consisting of 2 parts
Part 1 - number
Part 2 - scale (unit)
Examples:
20 grams
6.63 x Joule seconds

22. Uncertainty in Measurement
A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty.

23. Precision and Accuracy
Accuracy – measure of how close a measurement comes to the actual or true value of whatever is being measured.
Precision – measure of how close a series of measurements are to one another.
Neither accurate nor precise
Precise but not accurate
Precise AND accurate

24. Why Is there Uncertainty?
Measurements are performed with instruments
No instrument can read to an infinite number of decimal places
Which of these balances has the greatest uncertainty in measurement?

25. Determining Error
Accepted Value – the correct value based on reliable references
Experimental Value – the value measured in the lab
Error(can be +or-)=experimental value – accepted value
Percent error = absolute value of error x 100%
accepted value

26. Significant Figures in Measurement
In a supermarket, you can use the scales to measure the weight of produce.
If you use a scale that is calibrated in 0.1 lb intervals, you can easily read the scale to the nearest tenth of a pound.
You can also estimate the weight to the nearest hundredth of a pound by noting the position of the pointer between calibration marks.

27. Significant Figures in Measurement
If you estimate a weight that lies between 2.4 lbs and 2.5 lbs to be 2.46 lbs, the number in this estimated measurement has three digits.
The first two digits (2 and 4 ) are known with certainty.
The rightmost digit (6) has been estimated and involves some uncertainty.

28. Significant Figures in Measurement
Significant figures in a measurement include all of the digits that are know, plus a last digit that is estimated.
Measurements must always be reported to the correct number of significant figures because calculated answers often depend on the number of significant figures in the values used in the calculation.

29. Rules for Counting Significant
Figures
Nonzero integers always count as significant figures.
3456 has
4 sig figs.

30. Rules for Counting Significant Figures
Leading zeros do not count as
significant figures.
has
3 sig figs.

31. Rules for Counting Significant Figures
Zeros at the end of a number and to the right of a decimal point are always significant.
9.000 has
4 sig figs
1.010 has

32. Rules for Counting Significant Figures
Captive zeros always count as
significant figures.
16.07 has
4 sig figs.

33. Rules for Counting Significant Figures
Zeros at the rightmost end that lie at the left of an understood decimal point are not significant.
7000 has
1 sig fig
27210 has
4 sig figs

34. Rules for Counting Significant Figures
Exact numbers have an infinite number of significant figures.
1 inch = cm, exactly

35. Rules for Significant Figures in Mathematical Operations
Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation.
6.38 x 2.0 =
12.76  13 (2 sig figs)

36. Rules for Significant Figures in Mathematical Operations
Addition and Subtraction: The number of decimal places in the result equals the number of decimal places in the least precise measurement. =
 18.7 (3 sig figs)

37. Sig Fig Practice #1 1.0070 m  5 sig figs 17.10 kg  4 sig figs
How many significant figures in each of the following?
m 
5 sig figs
17.10 kg 
4 sig figs
100,890 L 
5 sig figs
3.29 x 103 s 
3 sig figs
cm 
2 sig figs
3,200,000 
2 sig figs

38. Sig Fig Practice #2 Calculation Calculator says: Answer 3.24 m x 7.0 m
100.0 g ÷ 23.7 cm3
g/cm3
4.22 g/cm3
0.02 cm x cm
cm2
0.05 cm2
710 m ÷ 3.0 s
m/s
240 m/s
lb x 3.23 ft
lb·ft
5870 lb·ft
1.030 g ÷ 2.87 mL
g/mL
2.96 g/mL

39. Sig Fig Practice #3 Calculation Calculator says: Answer 3.24 m + 7.0 m
100.0 g g
76.27 g
76.3 g
0.02 cm cm
2.391 cm
2.39 cm
713.1 L L L
709.2 L
lb lb lb lb
2.030 mL mL
0.16 mL
0.160 mL

40. Questions
78ºC, 76ºC, 75ºC 2) 77ºC, 78ºC, 78ºC
80ºC, 81ºC, 82ºC
The sets of measurements were made of the boiling point of a liquid under similar conditions. Which set is the most precise?
Set 2 – the three measurements are closest together.
What would have to be known to determine which set is the most accurate?
The accepted value of the liquid’s boiling point

41. Questions
How do measurements relate to experimental science?
Making correct measurements is fundamental to the experimental sciences.
How are accuracy and precision evaluated?
Accuracy is the measured value compared to the correct values. Precision is comparing more than one measurement.

42. Questions
Why must a given measurement always be reported to the correct number of significant figures?
The significant figures in a calculated answer often depend on the number of significant figures of the measurements used in the calculation.
How does the precision of a calculated answer compare to the precision of the measurements used to obtain it?
A calculated answer cannot be more precise than the least precise measurement used in the calculation.

43. Question Error = experimental value – accepted value
A technician experimentally determined the boiling point of octane to be 124.1ºC. The actual boiling point of octane is 125.7ºC. Calculate the error and the percent error.
Error = experimental value – accepted value
error = 124.1ºC – 125.7ºC = -1.6ºC
Absolute error / accepted value x 100%
% error = -1.6ºC / 125.7ºC x 100% = 1.3%

44. Question
Determine the number of significant figures in each of the following
11 soccer player
unlimited
meter 5
10,800 meters 3
5.00 cubic meters

45. Question
Solve the following and express each answer in scientific notation and to the correct number of significant figures.
(5.3 x 104) + (1.3 x 104)
6.6 x 104
(7.2 x 10-4) / (1.8 x 103)
4.0 x 10-7
(104)(10-3) (106)
107
(9.12 x 10-1) - (4.7 x 10-2)
8.65 x 10-1
(5.4 x 104) (3.5 x 109)
18.9 x or 1.9 x 10 14

46. End of section 3.1

47. International Systems of Units
The standards of measurement used in science are those of the metric system
All metric units are based on multiples of 10
Metric system was originally establish in France in 1795
The International System of Units (SI) is a revised version of the metric system.
The SI comes from the French name, le Systeme International d’Unites.
The SI was adopted by international agreement in 1960.

48. International Systems of Units
There are seven SI base units
SI Base Units
Quantity
SI base unit
Symbol
Length
Meter m
Mass
kilogram kg
Temperature
kelvin K
Time
second s
Amount
mole
mol
Luminous intensity
candela cd
Electric current
ampere A

49. Metric Prefixes Mega (M) Kilo (k) 103 Hecto (hm) 102 Deka(da) 101
Meter (m)
Deci (d) 10-1
left
Centi (c) 10-2
Milli (m) 10-3
right
Micro (µ) 10-6
Nano (nm) 10-9
Pico (pm) 10-12

50. Metric Conversions 1.0 decimeter (dm) = ? hectometers
0.001 hectometer (hm)
2.5 hectometer (hm) = ? millimeters
250,000 millimeters (mm)
9.7 centimeters (cm) = ? kiometers
kilometers (km)
7.4 grams (g) = ? Milligrams (mg)
7400 milligrams (mg)

51. Other Common Conversions
1 cm3 = 1ml
1dm3 = 1L
1 inch = 2.54 cm
1kg = 2.21 lb
454 g = 1 lb
4.18 J = 1 cal
1 mol = 6.02 x 1023 pieces
1 GA = 3.79 L

52. Units of Length meter – the basic SI unit of length or linear measure
Common metric units of length include the centimeter (cm), meter (m), and kilometer (km)

53. Units of Volume Volume -the space occupied by any sample of matter
Volume (cube or rectangle) = length x width x height
The SI unit of volume is the amount of space occupied by a cube that is 1m along each edge. (m3)
Liter (L) – non SI unit – the volume of a cube that is 10cm along each edge (1000cm3)
The units milliliter and cubic centimeter are used interchangeably.
1 cm3 = 1ml
1dm3 = 1L

54. Objects can become weightless, but not massless
Units of Mass
Common metric units of mass include the kilogram, gram, milligram and microgram.
Weight – is a force that measures the pull on a given mass by gravity.
Weight is a measure of force and is different than mass.
Mass – measure of the quantity of matter.
Although, the weight of an object can change with its location, its mass remains constant regardless of its location.
Objects can become weightless, but not massless

55. Units of Temperature
Temperature – measure of how hot or cold an object is.
The objects temperature determines the direction of heat transfer.
When two objects at different temperatures are in contact, heat moves from the object at the higher temperature to the object at the lower temperature.
Scientist use two equivalent units of temperature, the degree Celsius and the Kelvin.

56. Units of Temperature
The Celsius scale of the metric system is named after Swedish astronomer Anders Celsius.
The Celsius scale sets the freezing point of water at 0ºC and the boiling point of water at 100ºC
The Kelvin scale is named for Lord Kelvin, a Scottish physicist and mathematician.
On the Kelvin scale, the freezing point of water is kelvins (K), & the boiling point is K.
With the Kelvin scale the degree (º) sign is not used

57. Units of Temperature
A change of 1 º on the Celsius scale is equivalent to one kelvin on the Kelvin scale.
The zero point on the Kelvin scale, 0K, or absolute zero, is equal to º C.
K = ºC
ºC = K .

59. Units of Energy Energy – the capacity to do work or to produce heat.
The joule and the calorie are common units of energy.
The joule (J) is the SI unit of energy named after the English physicist James Prescott Joule.
1 calorie (cal) - is the quanity of heat that raises the temperature of 1 g of pure water by 1ºC.
1 J = cal
1 cal = J

60. End of Section 3.2

61. Conversion Factors Different ways to express length
1 dollar = 4 quarters = 10 dimes = 20 nickels = 100 pennies
Different ways to express the same amount of money
1 meter =10 decimeters =100 centimeters =1000 millimeters
Different ways to express length
Whenever two measurements are equivalent, a ratio of the two measurements will equal 1.
1 m = 100 cm = 1 1m
Conversion factor

62. Conversion Factors
Conversion factor – a ratio of equivalent measurements
100 cm / 1 m
1000 mm / 1 m
The measurement on the top is equivalent to the measurement on the bottom
Read “one hundred centimeters per meter” and “1000 millimeters per meter”
Smaller number m larger unit
Larger number cm smaller unit

63. Conversion Factors
When a measurement is multiplied by a conversion factor, the numerical value is generally changed, but the actual size of the quantity measured remains the same.
Conversion factors within a system of measurements are defined quantities or exact quantities.
Therefore, they have an unlimited number of significant figures and do not affect the rounding of a calculated answer.
How many significant figures does a conversion factor within a system of measurements have?

64. Dimensional Analysis
Dimensional analysis – a way to analyze and solve problems using the units, or dimensions, of the measurements.
How many minutes are there in exactly one week?
60 minutes = 1 hour 24 hours = 1 day
7 days = 1 week
1 week 7 days 24 hours minutes = 10,080 min
1 week day hour
x 104 min

65. Dimensional Analysis
How many seconds are in exactly a 40-hr work week?
60 minutes = 1 hour 24 hours = 1 day
7 days = 1 week 60 seconds = 1 minute
40 hr 60 min 60 sec = 144,000 s
1 hr min
x 105 s

66. Dimensional Analysis
An experiment requires that each student use an cm length of Mg ribbon. How many students can do the experiment if there is a 570 cm length of Mg ribbon available?
570 cm ribbon student = 67 students
8.5 cm ribbon
2 sig figs

67. Dimensional Analysis 48.0ºC 1.80ºF = 86.4ºF 1.00ºC
A 1.00º increase on the Celsius scale is equivalent to a 1.80º increase on the Fahrenheit scale. If a temperature increases by 48.0ºC, what is the corresponding temperature increase on the Fahrenheit scale?
48.0ºC ºF = 86.4ºF
1.00ºC
A chicken needs to be cooked 20 minutes for each pound it weights. How long should the chicken be cooked if it weighs 4.5 pounds?
4.5 lb 20 min = 90 min lb

68. Dimensional Analysis
Gold has a density of 19.3 g/cm3. What is the density in kg/m3
19.3 g 1 kg x 106 cm3 = x 104 kg / m cm g m3
There are 7.0 x 106 red blood cell (RBC) in 1.0 mm3 of blood. How many red blood cells are in 1.0 L of blood?
7.0 x 106 RBC 1 x 106 mm3 1 dm3 = 7.0 x 1012
1.0 mm dm L

69. Dimensional Analysis 2.69 x 1022 atoms 1 L dm3
1.00 L of neon gas contains 2.69 x 1022 neon atoms. How many neon atoms are in 1.00mm3 of neon gas under the same conditions?
2.69 x 1022 atoms L dm3
1.00 L dm x 106 mm3
2.69 x 1016 atoms in 1.00mm3 of gas

70. Cubic decimeters to milliliters
Questions
What conversion factor would you use to convert between these pairs of units?
Minutes to hours
1 hour / 60 minutes
grams to milligrams
1000 mg / 1 g
Cubic decimeters to milliliters
1000 ml / 1 dm3

71. Questions 1.53 x 1022 atoms of gold
An atom of gold has a mass of x 10-22g. How many atoms of gold are in 5.00 g of gold?
1.53 x 1022 atoms of gold
Light travels at a speed of 3.00 x 1010 cm/sec. What is the speed of light in km/hour?
1.08 x 109 km/hr

72. Questions 7.5 x 104 J to kJ 7.5 x 101 kJ 3.9 x 105 mg to dg
Convert the following. Express your answers in scientific notation.
7.5 x 104 J to kJ
7.5 x 101 kJ
3.9 x 105 mg to dg
3.9 x 103dg
2.21 x 10-4 dL to µL
2.21 x 101µL

73. Questions 14.8 g to µg 1.48 x 107 µg 3.75 x 10-3 kg to g 3.72 g
Make the following conversions. Express your answers in standard exponential form.
14.8 g to µg
1.48 x 107 µg
3.75 x 10-3 kg to g
3.72 g
66.3 L to cm3
6.63 x 104 cm3

74. End of Section 3.3

75. Density
If a piece of led and a feather of the same volume are weighted, the lead would have a greater mass than the feather.
It would take a much larger volume of feather to equal the mass of a given volume of lead.
Density = mass / volume
D = m / v
Mass is a extensive property (a property that depends on the size of the sample)
Density is an intensive property (depends on the composition of a substance, not on the size of the sample)

76. Density
A helium filled balloon rapidly rises to the ceiling when released.
Whether a gas-filled balloon will sink or rise when released depends on how the density of the gas compares with the density of air.
Helium is less dense than air, so a helium filled balloon rises.

77. Density and Temperature
The volume of most substances increase as the temperature increases.
The mass remains the same despite the temperature and volume changes.
So if the volume changes with temperature while the mass remains constant, then the density must also change with temperature.
The density of a substance generally decreases as its temperature increases. (water is the exception: ice floats because it is less dense than liquid water)

78. Questions
A student finds a shiny piece of metal that she thinks is aluminum. In the lab, she determines that the metal has a volume of 245cm3 and a mass of 612g. Was is the density? Is it aluminum?
D = 612g / 245cm3 = 2.50g/cm3
D of aluminum is 2.70 g/cm3; no it is not aluminum
A bar of silver has a mass of 68.0 g and a volume of 6.48 cm3. What is the density?
D = 68.0g / 6.48 cm3 = 10.5 g/cm3

79. Questions
The density of boron is 2.34 g/cm3. Change 14.8 g of boron to cm3 of boron.
D = m / v or v = m / D
V = 14.8 g cm3 = 6.32 cm 3
2.34 g
Convert 4.62 g of mercury to cm3 by using the density of mercury g/cm3.
V = 46.2 g cm3 = cm 3
13.5 g

80. Density
D = m / v
v = m / D
m = D · v

81. End of Chapter 3