and Units Conversion Factors
Units are tags attached to numbers. They obey the algebraic rules of multiplication and division. Conversion factors allow one to change units. Example: How many meters tall is a 4.0 foot Emperor Penguin? Answer: 4.0 ft X 12in/ftX2.54cm/inX1m/100cm = 1.2 m
How do Conversion Factors Work?
Note that the formula 12 in = 1 ft can be arranged to read 12 in/1 ft = 1 or 1 ft/12 in = 1. Multiplying anything by 1 does not change it. Thus, for example a 39.0 in stick is, 39.0 in X 1 ft/ 12 in = 3.25 ft A series of such multiplications by 1 converts from one set of units to another
The Standard Units
Scientists have agreed on a set of international standard units for comparing all our measurements called the SI units
Systme International = International System
Quantity length mass time temperature
Unit meter kilogram second kelvin
Symbol m kg s K
Some Standard Units in the Metric System
Quantity Measured Mass
Length
Name of Unit gram
meter
Abbreviation
g
m
Volume
Time
liter
seconds
L
s
Temperature
Kelvin
K
Metric to English Units and Conversions
Length
Measure of the two-dimensional distance an object covers SI unit = meter
About 3 inches longer than a yard
1 meter = one ten-millionth the distance from the North Pole to the Equator = distance between marks on standard metal rod in a Paris vault = distance covered by a certain number of wavelengths of a special color of light
Commonly use centimeters (cm)
1 m = 100 cm 1 cm = 0.01 m = 10 mm 1 inch = 2.54 cm (exactly)
Mass
Measure of the amount of matter present in an object SI unit = kilogram (kg)
about 2 lbs. 3 oz.
Commonly measure mass in grams (g) or milligrams (mg)
1 kg = 2.2046 pounds, 1 lbs. = 453.59 g 1 kg = 1000 g = 103 g, 1 g = 1000 mg = 103 mg 1 g = 0.001 kg = 10-3 kg, 1 mg = 0.001 g = 10-3 g
Related Units in the SI System
All units in the SI system are related to the standard unit by a power of 10 The power of 10 is indicated by a prefix The prefixes are always the same, regardless of the standard unit
kilo = 1000 times base unit = 10 1 kg = 1000 g = 103 g deci = 0.1 times the base unit = 10-1 1 dL = 0.1 L = 10-1 L; 1 L = 10 dL centi = 0.01 times the base unit = 10-2 1 cm = 0.01 m = 10-2 m; 1 m = 100 cm milli = 0.001 times the base unit = 10-3 1 mg = 0.001 g = 10-3 g; 1 g = 1000 mg micro = 10-6 times the base unit 1 m = 10-6 m; 106 m = 1 m nano = 10-9 times the base unit 1 nL = 10-9L; 109 nL = 1 L
Prefixes Used to Modify Standard Unit 3
Prefixes
Volume
Volume
Measure of the amount of three-dimensional space occupied SI unit = cubic meter (m3)
a Derived Unit
Commonly measure solid volume in cubic centimeters (cm3)
1 m3 = 106 cm3 1 cm3 = 10-6 m3 = 0.000001 m3
Commonly measure liquid or gas volume in milliliters (mL)
1 L is slightly larger than 1 quart 1 L = 1 dL3 = 1000 mL = 103 mL 1 mL = 0.001 L = 10-3 L 1 mL = 1cm3
Which is Larger?
1 yard or 1 meter? 1 mile or 1 km? 1 cm or 1 inch? 1 kg or 1 lb? 1 mg or 1 g? 1 qt or 1 L? 1 L or 1 gal? 1 gal or 1000 cm3?
Which is Larger?
1 yard or 1 meter? 1 mile of 1 km? 1 cm or 1 inch? 1 kg or 1 lb? 1 mg or 1 g? 1 qt or 1 L? 1 L or 1 gal? 1 gal or 1000 cm3?
Units
Always write every number with its associated unit Always include units in your calculations
you can do the same kind of operations on units as you can with numbers
cm cm = cm2 cm + cm = cm cm cm = 1
using units as a guide to problem solving is called dimensional analysis
Problem Solving and Dimensional Analysis
Many problems in Chemistry involve using relationships to convert one unit of measurement to another Conversion Factors are relationships between two units
May be exact or measured Both parts of the conversion factor have the same number of significant figures
e.g. 1 inch = 2.54 cm can give
2.54cm or 1in 2.54cm 1in
Conversion factors generated from equivalence statements
Problem Solving and Dimensional Analysis
Arrange conversion factors so starting unit cancels
Arrange conversion factor so starting unit is on the bottom of the conversion factor
May string conversion factors
So we do not need to know every relationship, as long as we can find something else the beginning and ending units are related to
unit 1 x
unit 2 unit 1
= unit 2
Solution Maps
a solution map is a visual outline that shows the strategic route required to solve a problem for unit conversion, the solution map focuses on units and how to convert one to another for problems that require equations, the solution map focuses on solving the equation to find an unknown value
Systematic Approach
1) Write down Given Amount and Unit 2) Write down what you want to Find and Unit 3) Write down needed Conversion Factors or Equations
a) Write down equivalence statements for each relationship b) Change equivalence statements to Conversion Factors with starting unit on the bottom
Systematic Approach
4) Design a Solution Map for the Problem
order conversions to cancel previous units or arrange Equation so Find amount is isolated check that units cancel properly multiply terms across the top and divide by each bottom term
correct size and unit
5) Apply the Steps in the Solution Map
6) Check the Answer to see if its Reasonable
Solution Maps and Conversion Factors
Convert Inches into Centimeters
1) Find Relationship Equivalence: 1 in = 2.54 cm 2) Write Solution Map
in cm
3) Change Equivalence into Conversion Factors with Starting Units on the Bottom 2.54 cm 1 in
1. Write down the Given quantity and its unit 2. Write down the quantity you want to Find and unit 3. Write down the appropriate Conversion Factors 4. Write a Solution Map
Given: Find: Conversio n Factors: Solution Map: Solution:
7.8 km ? miles 1 km = 0.6214 mi
km
0.6214 mi 1 km
7.8 km 0.6214 mi 1 km
mi
5. Follow the Solution Map to Solve the problem
4.84692 mi
6. Sig. Figs. and Round 7. Check
Round: 4.84692 mi = 4.8 Check: mi Units & Magnitude are correct
Solution Maps and Conversion Factors
Convert Cups into Liters
1) Find Relationship Equivalence: 1 L = 1.057 qt, 1 qt = 4 c 2) Write Solution Map c qt L
3) Change Equivalence into Conversion Factors with Starting Units on the Bottom
1 qt 4c
1L 1.057 qt
How many cups of cream is 0.75 L?
1. Write down the Given Given: quantity and its unit 2. Write down the quantity Find: you want to Find and unit 3. Write down the Conversio appropriate Conversion n Factors: Factors 4. Write a Solution Map Solution Map: 5. Follow the Solution Map to Solve the problem 6. Sig. Figs. and Round 7. Check Solution:
Round: 3.171 cu = 3.2 cu
0.75 L ? cu
1 L = 1.057 qt 1 qt = 4 cu L qt cu
1.057 qt 1L
0.75 L
4 cu 1 qt
3.171 cu
1.057 qt 4 cu 1L 1 qt
Check: Units & Magnitude
Convert 2,659 cm2 into square meters
1. Write down the Given Given: quantity and its unit 2. Write down the quantity Find: you want to Find and unit 3. Write down the Conversio appropriate Conversion n Factors: Factors 4. Write a Solution Map Solution Map: 5. Follow the Solution Map to Solve the problem 6. Sig. Figs. and Round 7. Check Solution:
Round: 0.2659 m2
2,659 cm2 ? m2
1 cm = 0.01 m
cm2
0.01 m 1 cm
2,659 cm
2
m2
2
1 10
4
m2
2
0.2659 m 2
1 cm
Check: Units & Magnitude
Mass & Volume
two main characteristics of matter cannot be used to identify what type of matter something is
if you are given a large glass containing 100 g of a clear, colorless liquid and a small glass containing 25 g of a clear, colorless liquid - are both liquids the same stuff?
even though mass and volume are individual properties - for a given type of matter they are related to each other!
Volume vs Mass of Brass
160
y = 8.38x
140
120
100
Mass, g
80
60
40
20
0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0
Volume, cm3
Density
Ratio of mass:volume Solids = g/cm3
1 cm3 = 1 mL
Liquids = g/mL Gases = g/L Volume of a solid can be determined by water displacement Archimedes Principle Density : solids > liquids >>> gases
except ice is less dense than liquid water!
Density
Mass Volume
Density
Density
Mass Volume
For equal volumes, denser object has larger mass For equal masses, denser object has smaller volume Heating objects causes objects to expand
does not effect their mass!! How would heating an object effect its density?
In a heterogeneous mixture, the denser object sinks
Using Density in Calculations
Solution Maps:
Density
Mass Volume
m, V
D
Mass Volume Density
m, D
V
Mass Density Volume
V, D
m
Platinum has become a popular metal for fine jewelry. A man gives a woman an engagement ring and tells her that it is made of platinum. Noting that the ring felt a little light, the woman decides to perform a test to determine the ring's density before giving him an answer about marriage. She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds that the ring displaces 0.556 cm3 of water. Is the ring made of platinum? (Density Pt = 21.4 g/cm3)
She places the ring on a balance and finds it has a mass of 5.84 grams. She then finds that the ring displaces 0.556 cm3 of water. Is the ring made of platinum? (Density Pt = 21.4 g/cm3) Given: Mass = 5.84 grams Volume = 0.556 cm3 Find: Density in grams/cm3 Equation: m
V
Solution Map: m and V d
D
Density as a Conversion Factor
can use density as a conversion factor between mass and volume!!
density of H2O = 1 g/mL 1 g H2O = 1 mL H2O density of Pb = 11.3 g/cm3 11.3 g Pb = 1 cm3 Pb
How much does 4.0 cm3 of Lead weigh? 4.0 cm3 Pb x 11.3 g Pb = 45 g Pb 1 cm3 Pb
Measurement and Problem Solving Density as a Conversion Factor
The gasoline in an automobile gas tank has a mass of 60.0 kg and a density of 0.752 g/cm3. What is the volume? Given: 60.0 kg Find: Volume in L Conversion Factors: 0.752 grams/cm3 1000 grams = 1 kg
Measurement and Problem Solving Density as a Conversion Factor
Solution Map: kg g cm3
1000 g 1 cm 60.0 kg 1 kg 0.752 g
3
7.98 10 cm
4
3
Example: A 55.9 kg person displaces 57.2 L of water when submerged in a water tank. What is the density of the person in g/cm3?
Information: Given: m = 55.9 kg V = 57.2 L Find: density, g/cm3 Solution Map: m,V D m Equation: D V Conversion Factors: 1 kg = 1000 g 1 mL = 0.001 L 1 mL = 1 cm3
Write a Solution Map for converting the Mass units
kg g
1000 g 1 kg Write a Solution Map for converting the Volume units L mL cm3
1 mL 0.001L
1 cm3 1 mL
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Example: A 55.9 kg person displaces 57.2 L of water when submerged in a water tank. What is the density of the person in g/cm3?
Information: Given: m = 55.9 kg V = 57.2 L Find: density, g/cm3 Solution Map: m,V D Equation: D m V
Apply the Solution Maps
1000 g 55.9 kg 1 kg
= 5.59 x 104 g
g
Example: A 55.9 kg person displaces 57.2 L of water when submerged in a water tank. What is the density of the person in g/cm3?
Information: Given: m = 5.59 x 104 g V = 57.2 L Find: density, g/cm3 Solution Map: m,V D Equation: D m V
Apply the Solution Maps
1 mL 1 cm 57.2 L 0.001L 1 mL
= 5.72 x 104 cm3
3
cm
3
Example: A 55.9 kg person displaces 57.2 L of water when submerged in a water tank. What is the density of the person in g/cm3?
Information: Given: m = 5.59 x 104 g V = 5.72 x 104 cm3 Find: density, g/cm3 Solution Map: m,V D Equation: D m V
Apply the Solution Maps - Equation
D
m V
5.59 x 104 g 5.72 x 104 cm3
= 0.9772727 g/cm3
= 0.977 g/cm3
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Example: A 55.9 kg person displaces 57.2 L of water when submerged in a water tank. What is the density of the person in g/cm3?
Information: Given: m = 5.59 x 104 g V = 5.72 x 104 cm3 Find: density, g/cm3 Solution Map: m,V D Equation: D m V
Check the Solution D = 0.977 g/cm3 The units of the answer, g/cm3, are correct. The magnitude of the answer makes sense. Since the mass in kg and volume in L are very close in magnitude, the answer's magnitude should be close to 1.
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