Back To CourseAP Biology: Homeschool Curriculum
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Sheila has a master's degree in geology and has taught middle school through university-level science courses.
Dimensional Analysis is a way chemists and other scientists convert units of measurement. We can convert any unit to another unit of the same dimension. This means we can convert some number of seconds into another unit of time, such as minutes, because we know that there are always 60 seconds in one minute. Or we can convert some amount of mass from grams to kilograms, knowing that there are always 1000 grams in one kilogram. Or we can convert lengths, say, from kilometers to miles, although metric units are most frequently used in chemistry. With a known conversion factor, it is sometimes possible to convert to a new dimension. For instance, 1 kilogram of pure water is equal to 1 liter. Using that knowledge, you could convert a volume of water to a mass of water, or vice versa.
Commonly used dimensions in chemistry include time, mass, length, and volume. The standard units of measurement (those used most frequently so that they can be easily shared with other scientists) are the SI units, from the International System of Units. The SI unit of time is seconds, and its unit symbol is s. The SI unit of mass is kilogram. It is written as kg, where the k is the metric prefix meaning kilo and the g stands for gram, the base unit of mass. The SI unit of length is the meter, with the symbol m. Chemists also often use moles, an SI unit describing an amount of a substance. A mole is equal to approximately 6.022 x 10^23 particles, such as atoms or molecules. Its symbol is mol.
In order to convert from one unit to another, we need to know how those units are related. Sometimes there is a well-established conversion factor that you can use to convert between dimensions, such as 1 kilogram of pure water equals 1 liter of pure water. You could also measure (or look up) conversion factors between metric and English units of the same dimension. For instance, 1 mile is approximately equal to 1.6 kilometers.
Most often in chemistry, your conversions will include either moles or metric units. To make your dimensional analysis life easier, it is important to work toward memorizing both an approximation of Avogadro's constant, the number of particles in a mole, and the metric prefixes. Equally important is that you have a sense of what these units mean.
We have said before that one mole is equal to approximately 6.022 x 10^23 particles (this can be anything, really, though most often it will be atoms or molecules). That's a really big number! Avogadro's constant sometimes scares new chemistry students, but it's just a number.
So if I were to ask you how many cookies are in a dozen, you'd hopefully tell me that there are 12 cookies in a dozen. And if I said I had a mole of cookies, you would know that means I have approximately 6.022 x 10^23 cookies.
Let's bring it back to chemistry. If you have 1 mole of carbon atoms, then that would mean you have approximately 6.022 x 10^23 carbon atoms. Or, if you have 1 mole of water molecules, that would be approximately 6.022 x 10^23 water molecules.
Our final step before we start practicing dimensional analysis is to learn about the metric prefixes so that we can make conversions between like base units. If we need to convert 1 kg to g, for instance, you will need to remember that g stands for grams, a unit of weight, and that the prefix k means kilo, or one thousand. So if we have 1 kg (or 1 kilogram), that means that we have one thousand grams, and our conversion is complete.
You might already recognize some of the biggest metric prefixes from your experience with computers, specifically, k (kilo), M (mega), G (giga) and T (tera). Of the prefixes smaller than 1, you might recognize deci, meaning one tenth, from the word decimal and realize that centi, one hundredth, has the same origins and meaning as the word cent. One penny, or one cent, is one one-hundredth of a dollar.
It's time to put our understanding of units and conversion factors to use. We will use dimensional analysis to set up and solve our unit conversion problems with known conversion factors.
Practice Problem #1
Convert 25.0 mL to L.
First, write down your starting measurements. Here, we have 25.0 mL. Sometimes the units will have both a numerator and a denominator, like meters per second, but if not, then you can put your measurement over 1.
In order to not change our initial measurement of 25.0 mL but only change the units, we want to find a way to multiply it by a value of 1. We'll need to find a conversion factor that will cancel out the mL units we already have, since we're moving away from those units, and one that will introduce Liters, the final units we're moving towards. We already know that mL means milliliters and that the prefix milli means one thousandth. That means we'd have to have 1000 mL to equal 1 L. We'll plug in that conversion factor.
Looking at our new dimensional analysis setup, we can cancel the mL units because we have one in the numerator and one in the denominator. The only units remaining are L, which is our final destination in this unit conversion. Finally, all we need to do is to follow through with the multiplication of the numbers, and we get 0.025 L.
Practice Problem #2
Convert 5.0 kg to cg.
In this problem, we'll be converting from kg (kilograms, or a thousand grams) to mg (milligrams, or thousandth of a gram). Before beginning the dimensional analysis, it's helpful to consider what the relative sizes of the units mean. We'll be moving from a relatively large unit of mass to much smaller units. This means we should end up with a much larger number in the end.
It might be helpful to first think about it with made up units that we already understand, so let's just pretend the grams are dollars. We'll pretend we're converting from kilodollars, or thousands of dollars, to centidollars, or pennies. If you brought some amount of thousands of dollars to the bank and asked to exchange it for pennies, you'd come home with many, many more pennies than the number of thousand dollar bills that you exchanged.
Let's get back to the problem. We're starting with 5.0 kg (kilograms, or a thousand grams) and converting that same measurement to units of cg (centigrams, or hundredths of grams). First, write down the starting measurement of 5.0 kg and put it over 1. The 1 keeps it the exact same measurement and units, and it can just be helpful for keeping track of our units on some of the longer dimensional analysis problems. Next, find a conversion factor that will bring us to the base units of g. We will need to cancel out the kg units, so we'll use 1000 g equals 1 kg. So far, we don't yet have our final units, so we'll need one more conversion to cancel out the units of g and introduce cg.
We know that 100 cg equals 1 g. We also know that, in order to cancel out our units of g, we will need to put the g in the denominator, and, in order to introduce the final units of cg, those units will go in the numerator. Check that your units have cancelled so that you end up only with the final units asked for in the problem.
Here, kg and g units cancelled appropriately, and we end up with 500,000 cg or 5.0 x 10^5 cg.
Practice Problem #3
Convert 12.0 in to cm, given that 1 in is approximately equal to 2.54 cm.
When converting between systems of measurement, we still set up our dimensional analysis problem the same way. The starting measurement of 12.0 in gets plugged in first. We multiply that measurement by 2.54 cm over 1 in, a fraction that equals 1 but allows us to convert units. The units of in gets cancelled, and we are left with only units of cm. Multiplying across, we get that 12.0 in is equal to 30.5 cm.
Dimensional analysis is a way chemists and other scientists convert unit of measurement. We can convert any unit to another unit of the same dimension which can include things like time, mass, length and volume. Generally speaking though, chemists usually use moles which is an SI unit describing an amount of a substance. Also, chemists need to be well-versed in metric prefixes, some of which are probably familiar to you. Like k (kilo), M (mega), G (giga), and T (tera), as well as how many of each prefix make up another prefix. Once familiar with all of this, you should be ready to tackle your own dimensional analyses.
When this lesson is over, students should be able to:
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Back To CourseAP Biology: Homeschool Curriculum
27 chapters | 270 lessons