1. Numbers in Science
Chapter 2 2

2. Measurement What is measurement?
Quantitative Observation
Based on a comparison to an accepted scale.
A measurement has 2 Parts – the Number and the Unit
Number Tells Comparison
Unit Tells Scale
There are two common unit scales
English
Metric 2

3. The Unit

4. The measurement System units
English (US)
Length – inches/feet
Distance – mile
Volume – gallon/quart
Mass- pound
Metric (rest of the world)
Length – meter
Distance – kilometer
Volume – liter
Mass - gram

5. Related Units in the Metric System
All units in the metric system are related to the fundamental unit by a power of 10
The power of 10 is indicated by a prefix
The prefixes are always the same, regardless of the fundamental unit 6

6. Fundamental Unit 100

7. Fundamental SI Units
Physical Quantity
Name of Unit
Abbreviation
Mass
Kilogram kg
Length
Meter m
Time
Second s
Temperature
Kelvin K
Energy
Joules J
Pressure
Pascal Pa
Volume
Cubic meters m3
Established in 1960 by an international agreement to standardize science units
These units are in the metric system

8. Length….. SI unit = meter (m) About 3½ inches longer than a yard
1 meter = distance between marks on standard metal rod in a Paris vault or distance covered by a certain number of wavelengths of a special color of light
Commonly use centimeters (cm)
1 inch (English Units) = 2.54 cm (exactly) 7

9. Figure 2.1: Comparison of English and metric units for length on a ruler.

10. Volume
Measure of the amount of three-dimensional space occupied by a substance
SI unit = cubic meter (m3)
Commonly measure solid volume in cubic centimeters (cm3)
Commonly measure liquid or gas volume
in milliliters (mL)
1 L is slightly larger than 1 quart
1 mL = 1 cm3 8

11. Mass Measure of the amount of matter present in an object
SI unit = kilogram (kg)
Commonly measure mass in grams (g) or milligrams (mg)
1 kg = pounds, 1 lbs.. = g 9

12. Temperature Scales
Any idea what the three most common temperature scales are?
Fahrenheit Scale, °F
Water’s freezing point = 32°F, boiling point = 212°F
Celsius Scale, °C
Temperature unit larger than the Fahrenheit
Water’s freezing point = 0°C, boiling point = 100°C
Kelvin Scale, K (SI unit)
Temperature unit same size as Celsius
Water’s freezing point = 273 K, boiling point = 373 K 22

13. Thermometers based on the three temperature scales in (a) ice water and (b) boiling water.

14. The number

15. Scientific Notation
Technique Used to Express Very Large or Very Small Numbers
135,000,000,000,000,000,000 meters
liters
Based on Powers of 10
What is power of 10 Big?
0,10, 100, 1000, 10,000
100, 101, 102, 103, 104
What is the power of 10 Small?
0.1, 0.01, 0.001,
10-1, 10-2, 10-3, 10-4 3

16. Writing Numbers in Scientific Notation
1. Locate the Decimal Point : 1, Move the decimal point to the right of the non-zero digit in the largest place - The new number is now between 1 and Now, multiply this number by a power of 10 (10n), where n is the number of places you moved the decimal point - In our case, we moved 3 spaces, so n = 3 (103) 4

17. The final step for the number……
4. Determine the sign on the exponent n
If the decimal point was moved left, n is +
If the decimal point was moved right, n is –
If the decimal point was not moved, n is 0
We moved left, so 3 is positive
1.438 x 103 5

18. Writing Numbers in Standard Form
We reverse the process and go from a number in scientific notation to standard form…..
Determine the sign of n of 10n
If n is + the decimal point will move to the right
If n is – the decimal point will move to the left
Determine the value of the exponent of 10
Tells the number of places to move the decimal point
Move the decimal point and rewrite the number
Try it for these numbers: x 106 and 9.8 x 10-2

19. Let’s Practice….. Change these numbers to Scientific Notation:
1,340,000,000,000
697, 000
Change these numbers to Standard Form:
3.76 x 10-5
8.2 x 108
1.0 x 101
1.34 x 1012
6.97 x 105
9.12 x 10-12
820,000,000 10

20. Are you sure about that number?

21. Uncertainty in Measured Numberscm
A measurement always has some amount of uncertainty, you always seem to be guessing what the smallest division is…
To indicate the uncertainty of a single measurement scientists use a system called significant figures
The last digit written in a measurement is the number that is considered to be uncertain 10

22. Rules, Rules, Rules….
We follow guidelines (i.e. rules) to determine what numbers are significant
Nonzero integers are always significant
2753
89.659
.281
Zeros
Captive zeros are always significant (zero sandwich)
1001.4

23. Significant Figures – Tricky Zeros
Leading zeros never count as significant figures
Trailing zeros are significant if the number has a decimal point
22,000
63,850.
100,000

24. Significant Figures Scientific Notation
All numbers before the “x” are significant. Don’t worry about any other rules.
7.0 x 10-4 g has 2 significant figures
2.010 x 108 m has 4 significant figures
How many significant figures are in these numbers?
102, ,017
1.0 x ,
1,908, x 1014

25. Have a little fun remembering sig figs

26. Exact Numbers Exact Numbers are numbers known with certainty
Unlimited number of significant figures
They are either
counting numbers
number of sides on a square
or defined
100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm
1 kg = 1000 g, 1 LB = 16 oz
1000 mL = 1 L; 1 gal = 4 qts.
1 minute = 60 seconds 14

27. Calculations with Significant Figures
Exact numbers do not affect the number of significant figures in an answer
Answers to calculations must be rounded to the proper number of significant figures
round at the end of the calculation
For addition and subtraction, the last digit to the right is the uncertain digit.
Use the least number of decimal places
For multiplication, count the number of sig figs in each number in the calculation, then go with the smallest number of sig figs
Use the least number of significant figures 15

28. Rules for Rounding Off If the digit to be removed
is less than 5, the preceding digit stays the same
Round to 4 sig figs.
is equal to or greater than 5, the preceding digit is increased by 1
Round to 3 sig figs.
In a series of calculations, carry the extra digits to the final result and then round off
Don’t forget to add place-holding zeros if necessary to keep value the same!! Round 80,150,000 to 3 sig figs.

29. Examples of Sig Figs in Math
Answers must be in the proper number of significant digits!!!
5.18 x
116.8 – 0.33

30. Solutions: 0.107744 round to proper # sig fig 165.47 116.47
5.18 has 3 sig figs, has 3 sig figs so answer is 0.108
165.47
Limiting number of sig figs in addition is the smallest number of decimal places = 12 (no decimals) answer is 165
116.47
Same rule as above so answer is 116.5

31. Moving unit to unit: Conversion

32. Exact Numbers Exact Numbers are numbers known with certainty
They are either
counting numbers
number of sides on a square
or defined
100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm
1 kg = 1000 g, 1 LB = 16 oz
1000 mL = 1 L; 1 gal = 4 qts.
1 minute = 60 seconds 14

33. The Metric System
Fundamental Unit 100

34. Movement in the Metric system
In the metric system, it is easy it is to convert numbers to different units.
Let’s convert 113 cm to meters
Figure out what you have to begin with and where you need to go..
How many cm in 1 meter?
100 cm in 1 meter
Set up the math sentence, and check that the units cancel properly.
113 cm [1 m/100 cm] = m

35. Let’s Practice converting metric units
250 mL to Liters
0.250 mL
1.75 kg to grams
1,750 grams
88 µL to mL
0.088 mL
475 cg to kg
47,500,000 or
4.75 x 107
328 mm to dm
3.28 dm
nL to µL
0.75 µL

36. Converting Between Metric and non-Metric (English) units

37. Converting non-Metric Units
Many problems involve using equivalence statements to convert one unit of measurement to another
Conversion factors are relationships between two units
Conversion factors are generated from equivalence statements
e.g. 1 inch = 2.54 cm can give or 18

38. Converting non-Metric Units
Arrange conversion factor so starting unit is on the bottom of the conversion factor
Convert kilometers to miles
You may string conversion factors together for problems that involve more than one conversion factor.
Convert kilometers to inches
Find the relationship(s) between the starting and final units.
Write an equivalence statement and a conversion factor for each relationship.
Arrange the conversion factor(s) to cancel starting unit and result in goal unit. 19

39. Practice Convert 1.89 km to miles Convert 5.6 lbs to grams
Find equivalence statement 1mile = km
1.89 km (1 mile/1.609 km)
1.17 miles
Convert 5.6 lbs to grams
Find equivalence statement 454 grams = 1 lb
5.6 lbs(454 grams/1 lb)
2500 grams
Convert 2.3 L to pints
Find equivalence statements: 1L = 1.06 qts, 1 qt = 2 pints
2.3 L(1.06 qts/1L)(2 pints/1 qt)
4.9 pints

40. Temperature Conversions
To find Celsius from Fahrenheit
oC = (oF -32)/1.8
To find Fahrenheit from Celsius
oF = 1.8(oC) +32
Celsius to Kelvin
K = oC + 273
Kelvin to Celsius
oC = K – 273

41. Temperature Conversion Examples
180°C to Kelvin
To convert Celsius to Kelvin add 273
= 453 K
23°C to Fahrenheit
Use the conversion factor: F = (1.80)C + 32
F = (1.80)
F=73.4 or 73°F
87°F to Celsius
Use the conversion factor C=5/9(F-32)
C = 5/9(87-32)
C = … or 31°C
694 K to Celsius
To convert K to C, subtract 273
= 421°C

42. Measurements and Calculations

43. Density
Density is a physical property of matter representing the mass per unit volume
For equal volumes, denser object has larger mass
For equal masses, denser object has small volume
Solids = g/cm3
Liquids = g/mL
Gases = g/L
Volume of a solid can be determined by water displacement
Density : solids > liquids >>> gases
In a heterogeneous mixture, denser object sinks 23

44. Using Density in Calculations24

46. Density Example Problems
What is the density of a metal with a mass of g whose volume occupies 6.30 cm3?
What volume of ethanol (density = g/mL) has a mass of 2.04 lbs?
What is the mass (in mg) of a gas that has a density of g/L in a 500. mL container?

47. How could you find your density?

48. Volume by displacement
To determine the volume to insert into the density equation, you must find out the difference between the initial volume and the final volume.
A student attempting to find the density of copper records a mass of 75.2 g. When the copper is inserted into a graduated cylinder, the volume of the cylinder increases from 50.0 mL to mL. What is the density of the copper in g/mL?

49. A student masses a piece of unusually shaped metal and determines the mass to be grams. After placing the metal in a graduated cylinder, the water level rose from 50.0 mL to 60.2 mL. What is the density of the metal?
A piece of lead (density = g/cm3) has a mass of g. If a student places the piece of lead in a graduated cylinder, what is the final volume of the graduated cylinder if the initial volume is 10.0 mL?

50. Percent Error
Percent error – absolute value of the error divided by the accepted value, multiplied by 100%.
% error = measured value – accepted value x 100%
accepted value
Accepted value – correct value based on reliable sources.
Experimental (measured) value – value physically measured in the lab.

51. Percent Error Example
In the lab, you determined the density of ethanol to be g/mL. The accepted density of ethanol is g/mL. What is the percent error?
The accepted value for the density of lead is g/cm3. When you experimentally determined the density of a sample of lead, you found that a 85.2 gram sample of lead displaced 7.35 mL of water. What is the percent error in this experiment?

52. Joe measured the boiling point of hexane to be 66. 9 °C
Joe measured the boiling point of hexane to be 66.9 °C. If the actual boiling point of hexane is 69 °C , what is the percent error?
A student calculated the volume of a cube to be cm3. If the true volume is cm3, what is the student’s percent error?
Tom used the density of copper and the volume of water displaced to measure the mass of a copper pipe to be g. When he actually weighed the sample, he found a mass of g. What was his percent error?