Dimensional Analysis Objective: Learn how to use dimensional analysis to solve problems
Mrs. Baldessari Chemistry
Conversion Problems
Because each country’s currency compares differently with the U.S. dollar, knowing how to convert currency units correctly is very important. Conversion problems are readily solved by a problemsolving approach called dimensional analysis.
What is Dimensional Analysis?
Ex: 3 cm = 50 km
Since the map is a small-scale representation of a large area, there is a scale that you can use to convert from small-scale units to large-scale units—for example, going from inches to miles or from cm to km.
What is Dimensional Analysis? Whenever you use a map or exchange currency, you are utilizing the scientific method of dimensional analysis.
A conversion factor is a ratio of equivalent measurements.
The ratios 100 cm/1 m and 1 m/100 cm are examples of 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.
Examples of Conversions
You can write any conversion as a fraction.
Be careful how you write that fraction. For example, you can write 1 m = 100cm as 1m 100 cm
or
100 cm 1m
Examples of Conversions
Again, just be careful how you write the fraction.
The fraction must be written so that like units cancel. For example. How many liters do you have with 50 mL?
Steps 1. Start with the given value. 2. Write the multiplication symbol. 3. Choose the appropriate conversion factor. 4. The problem is solved by multiplying the given data & their units by the appropriate unit factors so that the desired units remain.
5. Remember, cancel like units.
Let’s try some examples together…
1. Suppose there are 12 slices of pizza in one pizza. How many slices are in 7 pizzas? 1.
2.
3.
Start with the given value. Write the multiplication symbol. Choose the appropriate conversion factor.
Given: 7 pizzas Want: # of slices Conversion: 12 slices = one pizza
Solution
Check your work…
7 pizzas 1
X
12 slices 1 pizza
=
84 slices
Let’s try some examples together…
2.
How old are you in days? Given: 17 years Want: # of days Conversion: 365 days = one year
Solution
Check your work…
17 years 1
X
365 days 1 year
=
6052 days
Let’s try some examples together…
3.
There are 2.54 cm in one inch. How many inches are in 17.3 cm?
Given: 17.3 cm Want: # of inches Conversion: 2.54 cm = one inch
Solution
Check your work…
17.3 cm 1
X
1 inch 2.54 cm
=
6.81 inches
Be careful!!! The fraction bar means divide.
How many quarts is 9.3 cups?
9.3 cups
=
? quarts
9.3 cups
1
x
quart
9.3 x 1 =
4
cups
1 x 4
9.3 =
= 4
2.325
s
Now, you try…
1. Determine the number of eggs in 23 dozen eggs.
Answer: 276 eggs
Now, you try…
2. If one package of gum has 10 pieces, how many pieces are in 0.023 packages of gum?
Answer: 0.23 pieces
Multiple-Step Problems
Most problems are not simple one-step solutions. Sometimes, you will have to perform multiple conversions. Example: How old are you in hours? Given: 17 years Want: # of days Conversion #1: 365 days = one year Conversion #2: 24 hours = one day
Solution
Check your work…
17 years 1
X
365 days 1 year
X
24 hours 1 day
148,920 hours
=
Combination Units
Dimensional Analysis can also be used for combination units.
Like converting km/h into cm/s. Write the fraction in a “clean” manner:
km/h becomes km h
Combination Units
Example: Convert 0.083 km/h into m/s. Given: 0.083 km/h Want: # m/s Conversion #1: 1000 m = 1 km Conversion #2: 1 hour = 60 minutes Conversion #3: 1 minute = 60 seconds
Solution
Check your work…
0.083 km 1 hour
83 m 1 hour
X
1000 m 1 km
X
1 hour 60 min
=
X
0.023 m sec
83 m 1 hour
1 min 60 sec
=
Section Quiz
1. 1 Mg = 1000 kg. Which of the following would be a correct conversion factor for this relationship? 1000. 1/1000. ÷ 1000. 1000 kg/1Mg.
Now, you try…
Complete your assignment by yourself. If you have any questions, ask me as I will be walking around the room.
The following slides provide more worked examples for students who need more practice.
Extra practice # 1
Extra practice # 1 Start with what value is known, proceed to the unknown.
Extra practice # 1
Extra practice # 1
Extra practice # 2
Extra practice #2
extra practice # 2
Extra practice # 2
Extra practice # 3
Extra practice # 3
Extra practice # 3
Extra practice # 3
Extra practice # 4
Hint: When converting between units, it is often necessary to use more than one conversion factor.
Extra practice # 4
Extra practice # 4
Extra practice # 4
Converting Between Units
Converting Complex Units
Many common measurements are expressed as a ratio of two units. If you use dimensional analysis, converting these complex units is just as easy as converting single units. It will just take multiple steps to arrive at an answer.
