Calculations: Unit Conversions

Now, let's deal with conversion problems. If you are already familiar with doing calculations using conversion factors, feel free to move on to the next topic ("Density Conversions"). Your score on the "Conversion Calculations Pretest" should give you a good idea of your ability to do these kinds of calculations. Or try the Study Checks on "Metric Conversions" and "Conversions between Metric and American Units" and then decide whether to move on or not.

A Simple Example

You've done conversion factor calculations before you even came to a chemistry class. For example, how many inches are there in two feet? The answer, or course, is 24 inches. I'm sure you could figure that out quickly, almost without thinking about it, simply because you have dealt with that kind of question before.

But now, I'd like you to think about it, not just do it automatically. How is it you came up with 24 inches from 2 feet? Well, you multiply 2 by 12. That gives you 24. But why multiply by 12? You multiply by 12 because there are 12 inches in one foot. This is set up in a rather formal fashion here. Let's go through the process step by step.

Starting Information Conversion Factor Answer

2 ft * 12 in/1 ft = 24 in

This is what we call the conversion factor method. It also goes by other names. We start with "2 feet." We are going to change this to "inches," that is find out what 2 feet is equal to in inches. To do this we can multiply by a conversion factor that will change from feet (that goes on the bottom) to inches (that goes on the top). The equivalence between feet and inches goes into the conversion factor. The top part (the numerator) of the conversion factor has to be equivalent to the bottom part (the denominator). The relationship, of course, is that 12 inches is the same as 1 foot. Because the unit "foot" is in the denominator, it will cancel with the feet that we started with, leaving us with the unit "inches." Then calculating 2 x 12 ÷ 1 tells us that we have 24 inches.

This has been a somewhat ridiculous example. To work out this problem in this way makes a difficult problem out of a simple one. If you can work out a problem simply, do so. The purpose of the example is to use something very familiar to introduce what may be a new technique.

Another Simple Example

Let's go through another similar example. How many feet are there in 30 inches?

Starting Information Conversion Factor Answer

30 in * 1 ft/12 in = 2.5 ft

Here we want to change inches into feet. So, let's find out how many feet there are in 30 inches. Write down 30 inches first and then the conversion factor with inches on the bottom and feet on the top. The reason for this is that we are changing from inches to feet. then put the right numbers with the units. So we have to multiply by 1 foot over 12 inches to get the correct answer. The inches cancel out, and 30 times 1 divided by 12 gives 2.5 feet as the final answer.

Purpose of These Examples

The purpose of these examples has been to show you a method for doing certain types of calculations, even though you might be able to solve these particular examples without using the method.

In these examples a measurement made in one unit (such as feet) was converted into another unit (such as inches). To work these kinds of problems, you need to have a conversion factor--a relationship between the units. In these examples you needed to know that one foot equals 12 inches.

This approach can be used anytime you are converting from one thing to another if they are proportional to one another.

Metric Conversions

In this course you must be able to do conversions from one unit to another within the metric system. For example you will have to be able to change a measurement made in milliliters into liters, or a measurement made in meters into centimeters, or a measurement made in kilograms into grams or milligrams. To do such things, you need to know conversion factors that relate the different units.

In the metric system, all the units are related to one another by factors of ten as indicated by the prefixes. Factors of ten include 10, 100, 1,000, 1 million and so forth, as well as 1/10, 1/100, or 1/1,000. You should have already learned the most common of these prefixes. The unit can be meters, liters or grams (or any other unit, for that matter).

The definitions of the prefixes give you the conversion relationships you need. One milliliter is 1/1,000th of a liter, or .001 liter. One centimeter is 1/100th of a meter or .01 meter. One kilogram is 1,000 grams. Since people usually talk about how many of the smaller units there are in the larger one, this example also shows that you can look at one liter as 1,000 milliliters, and one meter as 100 centimeters. You can use those relationships and often they will be easier.

You might prefer to think of these prefixes lined up from smallest to largest. Whatever unit you are working with (meters, liters or grams), the prefixes to the left on this scale represent fractions of that unit. Deci- is one-tenth, centi- is one-hundredth, milli- is one-thousandth, and micro- is one-millionth. The prefixes on the right side of this scale represent multiples of that unit. Deka- is ten times, hecto- is one hundred times, kilo- is one thousand times, and mega- is one million times.

To make sure you have the relationships figured out correctly, try the following study check on metric units.

Determine the value that should fill in the blank in each of the following relationships.

1 L = __ mL 1 kg = __ g 1 m = __ cm 1 g= __ mg 1 mL = __ L 1 g = __ kg

Answers:

1000 mL 1000 g 100 cm 1000 mg 0.001 L 0.001 kg

The relationships between the different metric units can be used as conversion factors. For example, if you need to change a measured value from mg to g, you can multiply by the conversion factor of 1 g over 1000 mg.

45 mg * 1 g/1000 mg = 0.045 g

To change a value from g to mg multiply by the conversion factor of 1000 mg over 1 g.

3.25 g * 1000 mg/1 g = 3250 mg

As you become familiar with metric units, you will be able to move the decimal point back and forth the proper number of places to make metric conversions, rather than using this longer method of multiplying by 100 or 1,000 or dividing by those numbers.

45 mg → move decimal point three places left → 0.045 g

3.25 g → move decimal point three places right → 3250 mg

Now, try the study check on metric conversions.

43 mL = __ L 8.43 g = __ mg 35 mg = __ kg 5.2 L = __ mL 0.15 cm = __ mm

Answers:

0.043 L 8430 mg 0.000035 kg 5200 mL 1.5 mm

Metric-American Conversions

Making conversions between the metric and American units is very similar to making metric conversions or any other type of conversion. For example you might be asked how many kilometers are equal to 15 miles or how many feet are there in 100 meters. These problems are solved in the same way as other conversion problems. One difference from metric conversions is that you will not be required to memorize the conversion factors. We will give you a relationship between metric and current American units when you need to solve a problem, or have you look it up. A number of these relationships are listed here.

1 qt = 946 mL 1 mile = 1.6 km 1 in = 2.54 cm 1 lb = 454 g

Sometimes, the relationship available to you may not have the exact units you want. For example, in relating kilometers to miles, you might be given the relationship or conversion factor between meters and miles. But you can make adjustments within the metric system fairly easily. It just makes two problems instead of one -- change miles to meters and then meters to kilometers.

How many centimeters are in 4.0 yards?

4.0 yard * 36 in/1 yd * 2.54 cm/1 in = 365.76 cm (unrounded) = 370 cm (correctly rounded)

If you happened to know how many centimeters there are in a yard, then this would not be the way to do this problem. You would need just the one conversion factor for yards-to-centimeters. However, we don't know the relationship between centimeters and yards, so we'll need to do this in two steps so that we can use the relationships that we do know. The first conversion factor changes yards to inches. The second conversion factor changes inches to centimeters. In the calculation yards cancel and inches cancel, leaving us with centimeters.

*Let's take a minute to think about significant digits. Since we are multiplying and dividing when we do these conversion problems, we round off the answer to the same number of significant digits that are in the number with the fewest significant digits. Metric unit relationships are exact, so they will never limit your significant digits; you can think of them as having an infinite number of significant digits, if that makes it easier for you. For this course, we will also treat the given metric-American unit relationships as exact so they will also not limit your significant digits. (They aren't actually exact, with a few exceptions, but we will treat them as exact to simplify things a little bit.) Looking at our problem above, the 4.0 yards has two significant digits; the conversion factors are unit equalities so they will not limit our significant digits. That means that we must round our final answer to two significant digits, so that it matches the least number of significant digits in the problem.

Now take some time to work through the problems in the next study check.

Study Check: Metric-American Conversions

Determine the correct value for each of the following conversions

Question 1

Change 750 mL to quarts.

Question 3

Change 13 mm to inches.

Question 2

Change 27 miles to kilometers.

Question 4

Change 172 pounds to grams.

Answers

Question 1

750 mL = 0.79 quarts

Question 3

13 mm = 0.51 inches

Question 2

27 miles = 43 kilometers or 44 kilometers (depending on which Metric-American conversion factor you choose to use)

Question 4

172 pounds = 7.81x10⁴ grams or 7.82x 10⁴ grams