Dimensional Analysis

Table of Contents

• Introduction
• Scientific Notation
• SI
• Derived Units
• Composite Units
• Unit Conversion
• The Factor-Label Method
• Why All This?
• Practice Exercises

Introduction

    If you are enrolled in a Basic (or even AP) Chemistry class in high school, then there is a very good chance that your teacher has what seems to be an obsession with something called Dimensional Analysis.  The first problem that most students seem to have is that no one says what this means.  It sounds like something right out of quantum physics, but it is actually an exceedingly simple topic.

Dimensional Analysis is a fancy way of saying "converting units."

    If this is all that it is, why make such a fuss about it?  Very simple.  Wrong units lead to wrong answers.  Scientists have thus evolved an entire system of unit conversion.  Before we can discuss the system, though, we must first talk briefly about the metric system or, more properly, SI.  We also must understand scientific notation.  

    From here on, it is assumed that you have some basic algebra skills.  Other than the four elementary arithmetic operations (addition, subtraction, multiplication, and division), the only other things expected is knowledge of working with division bars and numerical cancellation, and use of exponents.  If you need a primer on math, we highly recommend http://www.sosmath.com/

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Scientific Notation

    When scientists wish to express really small or really large numbers, it is inconvenient to write out the numbers as with all their zeroes.  Thus they use scientific notation, a method of expressing numbers as a multiplier times a factor of 10.

Scientific Notation allows scientists to easily express very large or very small numbers in exponential form.

    When one chooses to use scientific notation, the first step is to choose the multiplier.  The multiplier is always going to be between 1 and 10.  If the number we wish to express in scientific notation is less than 1, you move the decimal to the right until the number resulting between 1 and 10.  If the number we wish to express is greater than one, we move the decimal (which may be implied at the end of the number if there are no decimal places, i.e. 6,285 has an implied decimal after the 5) to the left until the number resulting is between 1 and 10.

    We next need to choose the exponent that we will raise 10 to in order to make our factor of 10.  This is simple.  If we moved the decimal to the left, the exponent will be the negative of the number of places we moved the decimal.  For example, if our original number was 15,789,000, we moved the decimal seven places to the left  to get a multiplier of 1.5789.  Thus, our exponent will be a positive seven.  Note that since we moved the decimal to the left the exponent will be positive.  Likewise, if our original number was .00045863, we moved the decimal four places to the right to get a multiplier of 4.5863, so our exponent will be a negative four.  Note that since we moved the decimal to the right, the exponent will be negative.

    Let us now put it all together.  We wish to convert 40,257,892,000,000 into scientific notation.  Starting at the decimal (in this case, implied at the end), we count as we move the decimal 13 places to the left, giving us a multiplier of 4.0257892.  We also now know that our exponent will be 13 (positive).  Our final answer is then the multiplier times ten to the exponent, or: 4.0257892 x 10¹³.

    It should be relatively simple to figure out how you would go back from scientific notation to a regular number.  You simply reverse the process!  Let's say that we want to convert 3.6879 x 10^-3 back into a normal number.  We look at the exponent and see that it is negative.  We therefore know that when the number was made into scientific notation, its decimal was moved to the right.  Reversing the process, we will move the decimal to the left.  So, we simply move the decimal in 3.6879 three places to the left, and, eureka, we get .0036879.

    There are plenty of opportunities to practice bringing things in and out of scientific notation in the practice exercises section.

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SI

    The Système International d'Unités, or International System of Units, is the set of measurement standards used by virtually all scientists. 

Basically, SI is the metric system with a few improvements.

    Let's review a little.  The following table contains the base SI units, what they represent, and what their abbreviations are.  The abbreviations can be used to shorten lengthy expressions, as visible in the examples below.  Note that you will hear many other units, discussed later.  These units are generally derived from the base units below.

    You may already know that SI is based on powers of 10.  SI uses prefixes (things you attach before the unit), shown below, to indicate a factor of 10.  Not all the prefixes are shown; only the common ones are listed.

    If you are new to the metric (SI) system, perhaps you are a little confused by the usage of the prefixes above.  The following are some examples of proper SI usage:

1) You wish to express the distance from Chemville to Sciencetown, which has been measured to be 1,031,000 meters (or 1,031,000 m).  You choose to express your answer in kilometers (note the attachment of the prefix "kilo" directly to the beginning of the unit "meter").  Since 1 kilometer = 1,000 meters, you divide the number of meters by 1,000 to get a result of 1,031 kilometers, or 1,031 km (again, note the direct attachment of the prefix abbreviation "k" to the unit abbreviation "m").

 

2) You find that a chemical reaction produces 42.1 grams (or 42.1 g) of a certain substance.  You choose to express your answer in milligrams.  Since 1 milligram = .001 grams, you divide the number of grams by .001.  Thus you can state on your lab report that you made 42,100 milligrams (or 42,100 mg) of a substance.

 

3) An ammeter (a device measuring electric current) reads a current of 12 cA (or 12 centiamperes) in a circuit powered by a certain battery.  You seek to know how many amperes this current is equal to.  You know that 1 centiampere = .01 amperes, so you multiply the number of milliamperes by .01 and arrive at a result of .12 A (or .012 amperes).

    It may not be immediately obvious why we are multiplying or dividing in each of the cases above.  That is not really important right now.  The important part is that you see how you write units in accordance with SI rules.  It should be apparent that one writes the number, followed by either the name or abbreviation of the unit preceded, if needed, by an appropriate prefix.

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Derived Units

    Obviously, not everything can be measured using just the unit listed earlier.  For example, how would we measure the volume of a liquid (something that happens frequently in chemistry)?  There is no base unit for volume.  So what do we do?  Simple.

To measure quantities that do not have base units in SI we create units that are composed of SI base units.

    How do we create a unit?  Let us again consider the topic of volume.  What is volume?  It is generally known that the volume of a cube is equal to the area of one of the surfaces times the height of the cube (or, rather, the length of one of the sides).  But the area of one of the surfaces is equal to the length of a side times the length of a side (since all the sides are the same length).  By substitution into our original volume equation, we find that the volume of a cube is equal to the length of a side cubed (that is, to the power of three).  Now, do we have a unit that measures length?  Indeed, the meter measures length.  Since the volume of a cube equals a length cubed, the unit of volume is the cubic meter (m³).  From this painful analysis it easily follows that the unit of area is the square meter (m²).

   Rather than repeat this thought process to find various common units, we simply list a table containing common units that are encountered in chemistry, what they measure, and what other false SI units they are equal to (if any).  In general, we do not bother to explain the derivation of the unit here.

    A brief note on degrees Celsius.  A change in temperature of 1 ^oC is equivalent to a change in temperature of 1 K.  However, 400 K does not mean the same thing as 400 ^oC.  The Kelvin scale begins at absolute zero, whereas the Celsius scale begins at the freezing point of water.  There is a difference of 273.15 K or 273.15 ^oC between the two starting points.  Specifically, 0 K = -273.15 ^oC, or, 273.15 K = 0 ^oC.  Note that to convert from degrees Celsius to Kelvin, one simply need subtract 273.15.  Likewise, one simply need add 273.15 to convert from Kelvin to degrees Celsius.  Also, note that one never uses the term "degree" when referring to the Kelvin scale.

    Another note on the units above.  There are many other pressure units other those listed above, and you will find many pressure units used in books.  Note that only the bar and the pascal can be used with SI prefixes, and only the pascal is a true SI unit.  Also, you may find the calorie listed as the accepted unit of energy in many books.  It is not part of SI, so we will not use it on this site except when demonstrating conversions.

    Again, though, there are still quantities that cannot be measured with any of the above.  For example, density (how much mass is in a given space), has no unit yet.  To complete our discussion of SI units we shall consider the concept of a composite unit.  This discussion will make the square and cubic meters seem out of place above, for they are really composite units, but we placed them above because of their importance in everyday measurement.

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Composite Units

    If there is no "special name" in SI for a unit of a given quantity we call it a composite unit.  Technically, even the above are composite units since they can be written in terms of base units, but we separate them because of their special names.

Composite units, written in terms of base units or derived units, allow us to measure any possible quantity.

   Let us once again consider density.  We said that density is the amount of mass in a given space, or more technically, mass divided by volume.  This definition, whether or not you realize it, gives us our unit.  Simply substitute units into the definition.  Since mass is measured in grams (or a multiple thereof) and volume is a derived unit (cubic meters), one unit for density is grams per cubic meter (g / m³).  Note, however, that this is not the only unit of density.  For example, another unit could be kilograms per cubic centimeter (kg/cm³).  As long as the units in the composite unit represent the same types of quantities, the composite unit is valid.

    Since virtually all units used in chemistry are composite units, we shall not even attempt to generate a table like those above for this category of units.

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Unit Conversion

    We now enter the heart of dimensional analysis: unit conversion.  Most would agree that adding and subtracting prefixes is relatively simple.  However, unit conversion can become difficult for even the most experienced scientists, especially when more than a few units are involved, and/or those involved are composite units.  Conversion can be further complicated if it is necessary to leave the SI realm and enter something less organized, such as the generally familiar English system (pounds, inches, et cetera).  To aid unit conversion, scientists and engineers have created a system, mentioned in the introduction, that greatly simplifies things.  Many teachers of different subjects call this system different things.  We shall call this system the Factor-Label Method.

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The Factor-Label Method

   The Factor-Label Method is based on the fact that, like numbers, a unit will cancel and disappear if it is present on both the top and bottom of a division bar.

    To convert units you need to know only one thing: the conversion factor of two units.  A conversion factor is nothing more than a ratio of one unit to another.  Let us consider a simple example.  Earlier we stated that there are 1000 grams in a kilogram.  So what is the conversion factor here?  The following are the two possible conversion factors:

    The two conversion factors express the same information, but only if the units are retained on the correct sides of the division bar.  In other words, you cannot, obviously, say that a 1000 kilograms are in a gram.  Another conversion factor is the ratio of inches to centimeters, or centimeters to inches, depending on which is on top of the division bar.  Consider:

    Again, these two reciprocal conversion factors provide the same information.  It should be apparent that two units always have two common conversion factors, and that the product of these factors, if multiplied together, always equal zero.

    How do we use conversion factors?  This is our last and probably most important topic.  Let us imagine we want to convert 52 meters per gram (an arbitrary composite unit) to centimeters per kilogram (another arbitrary composite unit that measures the same imaginary quantity).  How would we do this?  We set up a Factor-Label Chart.  This involves multiplying various conversion factors together on the left (being certain to multiply both the actual numbers and the units they are measured in) and writing the result on the right.  If one chooses good conversion factors on the left then the proper units will cancel, leaving only the desired units on the right, along with the desired answer.  If one does not end up with the desired units, the numerical answer is most certainly wrong as well.  Let us now walk through the conversion problem above.

1) What are the possible conversion factors for grams and kilograms?

See above.

2) What are the possible conversion factors for meters and centimeters?

 

1 m 100 cm

100 cm 1 m

3) How can we string together the right conversion factors so that they cancel, leaving us with centimeters per kilogram?

 

52 m 1 g

x

100 cm 1 m

x

1000 g 1 kg

=

5200000 cm 1 kg

=

5.2 x 106 cm kg

    You will note that to make things easier we cross out units that cancel above and below the division bar.  The units that are not canceled are multiplied and divided as appropriate to find our final unit.  If this unit matches what we set out to convert to, we know that we converted correctly.

    At the end of this document there are a small number of conversion problems to practice.  Each one has answers (but not the whole solution process, unlike above).

    It is important to realize that direct unit conversion is not the only use of the factor-label method.  For example, if one has a formula, one can use factor-label charts to determine whether one "plugged in" the right numbers for each of the variables by whether or not the units multiply and/or cancel to yield the correct unit.  Since the use of formulas is more specialized than unit conversion, this topic is discussed in other sections, such as stoichiometry.

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Why All This?

Chemistry is fun, but fun does require math.

    "Why do chemists bother with all this math nonsense, creating measurement systems and developing complex ways to do something as simple as unit conversion?  I thought that chemistry was about wearing lab coats, making things blow up, and seeing cool colors."

    Well, sorry, but chemistry involves a lot of mathematics, measurement, and calculation.  Even if you want to make the cool explosions and colors you still need careful measurement.  Not so long ago there was a nearly catastrophic accident at a nuclear reactor facility in Japan that was caused by non other than a unit conversion error.  If the employee mixing up the fuel rods for the reactor had done his conversion properly, perhaps a small-scale nuclear meltdown could have been averted.

    Learn the math but have fun with it.  This may be the most boring part of this entire site, but once you have mastered SI and unit conversion, you can take off with chemistry, working in the lab and on paper like a pro.

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Practice Exercises

Scientific Notation

1. Express .671230 g in scientific notation.  Answer
2. Express 452.5300 mol in scientific notation.  Answer
3. Express 42,000,000 m in scientific notation.  Answer
4. Express .00829 s in scientific notation.  Answer
5. Express 1.257 x 10² K in standard (normal) notation.  Answer
6. Express 2.013 x 10⁵ A in standard (normal) notation.  Answer
7. Express 7.10 x 10^-1 cd in standard (normal) notation.  Answer
8. Is .722 x 10³ correct scientific notation?  Why or why not?  Answer

SI

Try to do these without looking at the tables above.

1. What base unit would you use to measure the distance from one end of a lab to the other?  Answer
2. What base unit would you use to measure the time a chemical reaction takes to complete?  Answer
3. What base unit would you use to measure the number of atoms of helium in a blimp?  Answer
4. What base unit and prefix, if any, would you use to measure the distance from Los Angeles, California to Seattle, Washington?  Answer
5. What base unit and prefix, if any, would you use to measure the mass of a paperclip?  Answer
6. How many meters are in a km?  Answer
7. How many grams are in a mg?  Answer
8. How many amperes are in a MA?  Answer
9. What does the prefix µ mean?  Answer
10. Convert 61 mA into A.  Answer
11. Convert 492.3 nm into m.  Answer
12. Convert 5 mol into kmol.  Answer

Derived Units

Try to do these without looking at the tables above.

1. One measures energy with what unit?  Answer
2. What is the abbreviation for the watt?  What does the watt measure?  Answer
3. One dalton means the same thing as what (abbreviation and name)?  Answer
4. Convert 51 K into ^oC.  Answer
5. Convert 51 ^oC into K.  Answer
6. How many liters are in 2.5 m³?  Answer

Composite Units

1. Name three composite units that can measure density (other than g/cm³).  Answer
2. What is one possible composite unit for velocity (distance per unit time)?  Answer
3. What is one possible unit for molar mass (mass per unit quantity)?  Answer
4. What are two units that can measure flow rate (volume per unit time)?  Answer
5. Is pounds per square centimeter a good composite unit for pressure?  Explain.  Answer
6. Does grams per meter measure the same thing as meters per gram?  Explain.  Answer

The Factor-Label Method

1. What are the two conversion factors for grams and centigrams?  Answer
2. What are the two conversion factors for µA and A?  Answer
3. Convert 100 mm into cm.  Answer
4. Convert 231 Mg into mg.  Answer
5. Approximately how many times larger is 453.5 dag than 262.8 mg?  Answer
6. Approximately how many times larger is 4.2 x 10³ mol than 2.15 x 10^-2 mol?  Answer
7. Convert .371 gram-newtons per square meter into kilogram-newtons per square centimeter.  Answer
8. Convert 45.9 amperes per cubic meter into milliamperes per cubic centimeter.  Answer
9. Convert 12.3 kilogram-meters per mole-second into gram-millimeters per millimole-minute.  Answer
10. Convert 90.004 (kg*m)/s² into (mg*km)/s².  Answer

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