1. Measurements and Calculations
Chapter 5
Measurements and Calculations

2. Goals of Chapter 5: Measurement & Calculations
Express numbers in scientific notation
Learn English, metric, & SI system of measurement
Use metric system to measure length, volume, and mass
Significant digits
Dimensional Analysis
Temperature Scales
Density/Specific Gravity
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Scientific Notation
Used to express very large or very small numbers
A number is expressed in scientific notation when it is in the form
a x 10n
where a is between 1 and 10
and n is an integer
use positive power for large numbers
use negative number for small numbers (decimals)
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4. Scientific Notation: Large Numbers
210,000,000,000,000,000,000,000
Where is the decimal point?
After the last zero.
Where would you put the decimal to make this number be between 1 and 10?
Between the 2 and the 1

5. 2.10,000,000,000,000,000,000,000.
How many decimal places did you move the decimal? 23
When the original number is more than 1, the exponent is positive.
The answer in scientific notation is
2.1 x 1023

6. Scientific Notation: Small Numbers
Where would the decimal go to make the number be between 1 and 10?
9.02
The decimal was moved how many places? 8
When the original number is less than 1, the exponent is negative.
9.02 x 10-8

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Learning Check
Convert the following to scientific notation
100,000 b
c. 45,000,000 d
e f
Convert to standard notation
g. 8.9 x h. 7.7 x 10-4
i x j x 10-8
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8. Copyright © Houghton Mifflin Company
Units
Tell us what scale or standard is being used to represent measurement
Scientists need common units to represent quantities like mass, length, time, and temperature
If everyone had own set of units – chaos would result
US uses English system, most of world (& scientists) use metric system, also SI system
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9. Units of Measurement

10. Figure 5.1: Comparison of English and metric units.
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11. Figure 5.2: Cube representations.
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12. Figure 5.3: A 100 mL graduated cylinder.
1 mL = 1 cm3
1 milliliter = 1 cubic centimeter
100 mL = 100 cm3
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Move decimal point left
Move decimal point right
Power of 10 between each increment
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14. Chapter 2
SI Conversions

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Learning Check
1000 meters = ___ km (decimal moved ___ places left)
1 meter = ___ cm (decimal moved ___ places right)
10 mm = ___ cm (decimal moved ___ place left)
1 Liter = ___ mL (decimal moved ___ places to right)
600 grams = ___ kg (decimal moved ___ places left)
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16. Derived Units

17. Uncertainty of Measurement
When using an instrument to measure (such as a ruler or graduated cylinder), we visualize divisions between markings and estimate
When making measurement, record all certain numbers and first uncertain number
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18. Figure 5.5: Measuring a pin.
Reading is between 2.8 cm & 2.9 cm
These divisions were visualized
2.85 cm is measurement
“5” is uncertain
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Significant Figures
Includes all numbers recorded in a measurement
For pin, length = 2.85 cm: 3 significant figures
All certain numbers plus first uncertain
Assume to be accurate to ± 1 in last #
Pin length is 2.85 ± 0.01 cm
Pin is somewhere between 2.84 & 2.86 cm
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20. Rules for Counting Significant Figures
All non-zero digits ARE significant
contains FIVE significant figures
Leading Zeros ARE NOT significant
contains TWO significant figures
3. Middle Zeros ARE significant
- 505 contains THREE significant figures
4. Ending Zeros ARE significant IF there is a decimal present
contains FIVE significant figures
5. Ending Zeros ARE NOT significant IF there is no decimal present
- 500 contains ONE significant figure

21. Rules for counting significant figures (continued)
Significant Figures only apply to measurements and calculations involving those measurements
Any number written in scientific notation is considered significant
An ending zero can be expressed as significant by placing a line above it
Remember! Just because a number is not significant, does not mean it’s not important! A non-significant zero acts as a placeholder to show the magnitude of the number!

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Learning Check
How many significant figures are in the following measurements?
0.0108
110
5.030 x 103
480
500
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23. To give answer with correct number of significant figures – round off
Look at number to right of last s.f.
If number is 4 or below, let it go!
If number is 5 or above, give it a shove!
Do not round off until end of calculations

24. Rules for Significant Figures in Calculations
Multiplication & Division
Answer should have same number of significant figures as measurement with smallest number of significant figures
Example: 4.56 x 1.4 = → 6.4
(3 s.f.) (2 s.f.)* (2 s.f.)*

25. Rules for Significant Figures in Calculations
Addition & Subtraction
Limited by smallest number of decimal places
Example: (2 decimal places)
(1 decimal place)*
(3 decimal places)
→ (1 decimal place)*
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Learning Check
Perform the following operations and round to the correct number of significant figures a
b – 7.25
c. 2.3 x 3.14
d /2.5
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27. Dimensional Analysis
Dimensional Analysis is a method of problem solving that allows for the conversion of one unit to another by way of a conversion factor.
A unit conversion factor is a fraction whose numerator and denominator are equivalent measures. Some common unit conversion factors are given below. You can also use the reciprocal of these.
Ask for some reciprocals so you know they understand.

28. Choose a unit conversion factor that…
Introduces the unit you want in the answer
Cancels out the original unit so that the one you want is all that is left.

29. “Canceling” out Words
Which conversion factor will convert feet to inches?
? Feet 12 inches 1 foot
1 foot 12 inches
You can cancel a unit if it appears in both the numerator and the denominator
Feet x inches
1 foot

30. Learning Check: Choose the appropriate conversion factor.
Inches to feet
Minutes to hours
Meters to centimeters

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Single-Step Problems
Choose a conversion factor that will get you from the starting unit to the ending unit
Multiply the given number by the conversion factor that will cancel out the original unit
Solve the problem by multiplying all number on top of the fraction and divide by the number on the bottom of the fraction
Be sure to include a UNIT in your answer!
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32. Example 1: Convert 8 yards to feet
Make a decision: What conversion factor will you use?
1 yard 3 feet
3 feet 1 yard
Set up the problem: Multiply the measurement by the conversion factor.
8 yards x 3 feet = 24 feet
1 yard
Solve the problem: Perform the multiplication
Multiply all numbers on the top
Divide by all numbers on the bottom

33. Example 2: Converts 16 quarts to gallons
What are the two conversion factors comparing quarts and gallons?
Which one will “cancel” quarts?
16 quarts x 1 gallon = gallons
4 quarts

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Learning Check
Water is often bottled in L containers. Convert L into quarts.
1 L = 1.06 quarts
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35. Multiple-Step Problems
Problem that requires the use of multiple conversion factors
Determine which conversion factors are needed to get from starting to ending unit
Develop plan to put the required conversion factors in order to cancel them out
3. Solve the problem
Multiply all numbers on top of fractions and divide by all numbers on the bottom of fractions
Include a UNIT in your answer!
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36. Example 3 : Convert 26.2 miles to kilometers
Determine conversion factors
1 mile = 1760 yards
1 meter = yards
1000 meter = 1 kilometer
Outline plan
Miles  yards  meters  kilometers
Solve the Problem
26.2 miles x yards x 1 meter x kilometer
1 mile yards meters
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37. Copyright © Houghton Mifflin Company
Learning Check
If a race car travels around a track at the speed of 225 miles/hour, what is the speed of the car in kilometers/hour?
Use conversion factors from Example 3
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38. Temperature Conversions
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39. Figure 5.6: The three major temperature scales.
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40. Converting between Kelvin & Celsius
To convert from Kelvin to Celsius:
T°C = TK – 273
Liquid Nitrogen boils at 77K, what is this in Celsius?
T°C = 77 – 273 = -196 °C
To convert from Celsius to Kelvin:
TK = T°C
The bp of water on top of Mt. Everest is 70 °C. Convert to K.
TK = = 343 K

41. Fahrenheit/Celsius Conversions
To convert from Celsius to Fahrenheit:
T°F = 1.80(T°C) + 32
If the temperature is 28°C, what is this in °F?
T°F = 1.80(28) + 32 = = 82°F (2 s.f.)
To convert from Fahrenheit to Celsius:
T°C = (T°F – 32)/1.80
If you have a temperature of 101°F, what is this in °C?
T°C = (101 – 32)/1.80 = 69/1.8 = 38°C
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Learning Check
a. Which temperature is colder, 172 K or -75 oC?
b. Hot tubs are often maintained at 41 oC, what is the temperature in Fahrenheit?
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43. Density Units are in g/cm3 or g/mL
Defined as the amount of matter present in a given volume of a substance
Density = mass / volume
Units are in g/cm3 or g/mL
Volume is dependent on temperature, so density is as well

44. Densities of Common Substances @ 20oC
Density (g/cm3)
Oxygen
Aluminum
2.70
Hydrogen
Iron
7.87
Ethanol
0.785
Copper
8.96
Benzene
0.880
Silver
10.5
Water
1.000
Lead
11.34
Magnesium
1.74
Mercury
13.6
Salt (sodium chloride)
2.16
Gold
19.32
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45. Determining volume by water displacement
Place water in graduated cylinder & record level
Add object
Record volume after addition of object
Volume is difference between second volume and first volume
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Figure 5.9: Tank of water.
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47. Figure 5.9: Person submerged in the tank.
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Learning Check
A student wants to identify the main component in a liquid cleaner. He finds that 35.8 mL of the liquid weighs 28.1 g. Which is the main component?
a. chloroform g/mL
b. diethyl ether g/mL
c. isopropyl alcohol g/mL
d. toluene g/mL
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