Dimensional Analysis

Name:

Loree Campbell 

Title of lesson:

Dimensional Analysis

Date of lesson:

01/04/06 – 01/05/06

Length of lesson:

2 days

Description of the class:

High School Chemistry

Source of the lesson:

Wade Green (SPR 2005)

           Original idea from Robert Krulwich, ABC News

TEKS addressed:

112.45 (c) Knowledge and skills:

(2)  Scientific processes. The student uses scientific methods during field and laboratory investigations. The student is expected to:

(C)  express and manipulate chemical quantities using scientific conventions and mathematical procedures such as dimensional analysis, scientific notation, and significant figures.

I. Overview

Dimensional Analysis is just a fancy way of saying unit conversion.  Scientists regularly have the need to convert a piece of data given in one unit system to another.  Using a simple technique of multiplying fractions and canceling units converts units between measurement systems and applies the necessary prefixes for metric measurement units.

II.  Performance or learner outcomes

            Students will be able to:

1. Convert between units of different measurement systems.

2. Properly apply unit prefixes to measurement figures.

3. Convert between types of measurements (i.e. volume to mass)

III. Resources, materials and supplies needed

 none

IV. Supplementary materials, handouts.

1. Conversion Factors Handout (1 per student)

2. Dimensional Analysis Worksheet   

Five-E Organization

Teacher Does           Probing Questions                              Student Does   

Engage: How much does a hurricane weigh? To begin with, you would need to know how much water is in a certain volume of air within a cloud. For that, you could find the mass of a cubic meter of cloud air and then subtract the mass of the non-water parts. Then all you need is the volume of the hurricane and you’re set. This might sound complicated, but the math is fairly easy and Chemists do the same type of calculations all the time. Somewhere in every Chemistry textbook is a list of constants and conversion factors for all sorts of units.  Some of these are familiar and some are not, but all of them are given with units. Some constants are even given more than once with different units. If we have the units, it is easy to convert one measurement to another with Dimensional Analysis.

If you wanted to know how much a hurricane weighs, how would you go about it?

Likely ideas: Measure the density of the air and multiply by the volume of a hurricane.

Explore: Dimensional analysis might sound like a big scary topic, but it is just a fancy way of saying, “convert units” like from centimeters to meters. Not all of these will be easy enough to do in your head, so there is a technique for converting whatever you have to whatever you need. All that’s needed is multiplying fractions, canceling units, and a little imagination. To prove that this works for any units where you have a conversion factor, we’re going to use some unique units in our calculations. On the Conversion Factors Handout, you will see some of the common conversions between customary and metric measurements as well as a few non-customary units. For any dimensional conversion, you will multiply a series of fractions together to convert to the units you want.  You should always start off by writing the information you have on the left using the given units. On the right side, write the units you want to end up with.  Between these two, find a series of conversions that change what you have to what you want. Example: What is the mass of two elephants in kilograms? Example 2: What is the mass of 10 bathtubs of water in elephants? See below…

Who can tell me what 500 cm is in meters?  Can you multiply  3/2 x 1/3?

Students should be able to convert this to 5 m. Students should be able to easily find the answer is 3/6 or ½.

Explain:

In class worksheet

Dimensional Analysis Worksheet

Extend / Elaborate:

At the beginning of class, I asked how much a hurricane weighs.  A newsman asked a meteorology professor this question and she calculated that the average cloud contains an amount of water equal to 100 elephants. If you apply this to a large storm cloud, you get about 13,750 elephants. For a large hurricane, the number becomes 40 million elephants worth of water. How is that possible?  You have to realize the enormous size of a hurricane.

If a hurricane has a radius of 675 miles and reaches a height of 35,000 ft., this works out to a volume of 1.39x10¹⁵ ft³.  Using the professor’s calculation of 55 billion gallons of water in a hurricane, this converts to 0.15 mL of water per cubic foot.

Evaluate:

Quiz

Original idea source:

Elephants in the Sky
Using the Largest Living Land Mammal to Calculate Cloud Mass

By Robert Krulwich

Sept. 3— Ever wonder how much a cloud weighs? What about a hurricane? A meteorologist has done some estimates and the results might surprise you.

Let's start with a very simple white puffy cloud — a cumulus cloud. How much does the water in a cumulus cloud weigh? Peggy LeMone, senior scientist at the National Center for Atmospheric Research in Boulder, Colorado, did the numbers.

"The water in the little cloud weighs about 550 tons," she calculates. "Or if you want to convert it to something that might be a little more meaningful … think of elephants."

Floating Masses

Assume an elephant weighs about six tons, she says, that would mean that water inside a typical cumulous cloud would weigh about one hundred elephants.

The thought of a hundred elephants-worth of water suspended in the sky begs another question — what keeps it up there?

"First of all, the water isn't in elephant sized particles, it's in tiny tiny tiny particles," explains LeMone.

And those particles float on the warmer air that's rising below. But still, the concept of so much water floating in the sky was surprising even to a meteorologist like LeMone.

"I had no idea how much a cloud would weigh, actually, when I started the calculations," she says.

Outweighing Elephant Populations

So how many elephant units of water are inside a big storm cloud … 10 times bigger all the way around than the "puffy" cumulus cloud? Again, LeMone did the numbers: About 200,000 elephants.

Now, ratchet up the calculations for a hurricane about the size of Missouri and the figures get really massive.

"What we're doing is weighing the water in one cubic meter theoretically pulled from a cloud and then multiplying by the number of meters in a whole hurricane," she explains.

The result? Forty million elephants. That means the water in one hurricane weighs more than all the elephants on the planet. Perhaps even more than all the elephants that have ever lived on the planet.

And that is a lot of water.

Type of Cloud	Gallons	Tons of water	Elephants	Typical home pool
Light cloud	137,500	550	100	6.75
Puffy cumulus cloud	275,000,000	1,100,000	200,000	13,750
Hurricane	55,000,000,000	220,000,000	40,000,000	2,750,000
In other words a normal hurricane will have enough water to fill 46% of Lake Erie.

Conversion Factors

Symbol	Unit	=	Unit	Symbol
lb.	1 pound	=	453.59 grams	g
oz.	1 ounce	=	28.35 grams	g
gal.	1 gallon	=	4 quarts	qt.
gal.	1 gallon	=	3.785 liters	L
BT	1 bathtub	=	55 gallons	Gal.
ft.	1 foot	=	30.48 centimeters	cm
EL	1 elephant	=	12,000 pounds	lb.
BB	1 basketball court length	=	94 feet	ft.
mi.	1 mile	=	5280 feet	ft.

Metric system prefixes                    

Prefix	Multiplication factor	Symbol
femto	x 10^-15	f
pico	x 10^-12	p
nano	x 10^-9	n
micro	x 10^-6	μ
milli	x 10^-3	m
centi	x 10^-2	c
kilo	x 10^3	k
mega	x 10^6	M
giga	x 10^9	G

Useful Extras:

1 L of water weighs 1 kg.

1 m³ of water contains 10 L

Name__________________________

Date___________________________

Dimensional Analysis Worksheet

Do the following conversion problems.  Be sure to show your work.

1. Mr. Green’s dog weighs about 43 kg.  What is the dog’s weight in elephants (EL)?

2. A car is traveling at 55 miles per hour, how many basketball courts (BB) does it cross per second?

3. The average home pool holds 20,000 gallons of water.  How many bathtubs (BT) is this?

4. How much does this water weigh in elephants (EL)?

Dimensional Analysis Worksheet Answer Key

Do the following conversion problems.  Be sure to show your work.

1. Mr. Green’s dog weighs about 43 kg.  What is the dog’s weight in millielephants (mEL)?

2. A car is traveling at 55 miles per hour, how many basketball courts (BB) does it cross per second?

3. The average home pool holds 20,000 gallons of water.  How many bathtubs (BT) is this?

4. How much does this water weigh in elephants (EL)?