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You may find that you’re sometimes unclear on how to actually set up a particular chemistry problem in order to solve it. A scientific calculator handles the math, but it can’t tell you what you need to multiply or what you need to divide.

That’s why you need to know about the factor label method, which is sometimes called the unit conversion method or dimensional analysis. It can help you set up chemistry problems and solve them correctly. Two basic rules are associated with the factor label method:

✔ Rule 1: Always write the unit and the number associated with the unit. Rarely in chemistry will you have a number alone. Units are your guide to determining how to solve the problem, so always include them.

✔ Rule 2: Carry out mathematical operations with the units, canceling them until you end up with the unit you want in the final answer. In other words, treat a unit just like you would a number. For example, if you have a unit of g (grams) on top and g on the bottom of a fraction, as in , they can cancel out just like the 2s in . Both fractions end up equaling 1.

Here are some examples illustrating how to perform a conversion using the factor label method.

Examples

Q. A chemistry student measures a length of 423 mm, yet the lab she’s working on requires that it be in kilometers. What is the length in kilometers?

A. The length is . You can go about solving this problem in two ways. We first show you the slightly longer way involving two conversions and then shorten it to a nice, simple one-step problem.

This conversion requires you to move across the metric-system prefixes you find in Table 2-2. When you’re working on a conversion that passes through a base unit, it may be helpful to treat the process as two steps, converting to and from the base unit. In this case, you can convert from millimeters to meters and then from meters to kilometers:

You can see how millimeters cancels out and you’re left with meters. Then meters cancels out, and you’re left with your desired unit, kilometers.

The second way you can approach this problem is to treat the conversion from milli- to kilo- as one big step:

Notice the answer doesn’t change; the only difference is the number of steps required to convert the units. Based on Table 2-2 and the first approach we showed you, you can see that the total conversion from millimeters to kilometers requires converting 10⁶ mm to 1 km. You’re simply combining the two denominators in the two-step conversion (1,000 mm and 1,000 m) into one. Rewriting each 1,000 as 10³ may help you see how the denominators combine to become 10⁶.

Q. Suppose that you have an object traveling at 75 miles per hour. What is its speed in kilometers per second?

A. The speed is 0.034 km/s. To solve this problem using the factor label method, follow these steps:

1. Write down what you start with.

Note that per Rule 1, the expression shows the unit and the number associated with it.

2. Convert miles to feet, canceling the unit of miles per Rule 2.

3. Convert feet to inches.

4. Convert inches to centimeters.

5. Convert centimeters to meters.

6. Convert meters to kilometers.

7. Stop and stretch.

8. Now convert hours to minutes in the denominator of the original fraction.

9. Convert minutes to seconds.

10. Do the math to get the answer now that you have the units of kilometers per second (km/s).

The calculator gives you 0.033528 km/s. Round off that answer to the correct number of significant figures (see Chapter 1 for details on how to do so):

If you’d like to write the answer in scientific notation (again, see Chapter 1), the speed is

Tip: Note that although the setup of this example is correct, it’s certainly not the only correct setup. Depending on what conversion factors you know and use, there may be many correct ways to set up a problem and get the correct answer.

Q. Suppose that you have an object with an area of 35 inches squared, and you want to figure out the area in meters squared.

A. The area is 0.023 m². Follow these easy steps to make this calculation.

1. Write down what you start with.

2. Convert from inches to centimeters.

Remember: You have to cancel inches squared. You must square the inches in the new fraction, and if you square the unit, you have to square the number also. And if you square the denominator, you have to square the numerator, too:

3. Convert from centimeters squared to meters squared in the same way.

4. Now that you have the units of meters squared (m²), do the math to get your answer.

The calculator gives you 0.0225806 m². Rounded off to the correct number of significant figures, the answer is

With a little practice, you’ll really like and appreciate the factor label method. It ends up making things dramatically easier. Don’t give up on it. Pay attention to your units, and you’ll be just fine.

Practice Questions

1. How many meters are in 15 ft?

2. If Steve weighs 175 lb, what’s his weight in grams?

3. How many liters are in 1 gal of water?

4. If the dimensions of a solid sample are 3 in. x 6 in. x 1 ft, what’s the volume of that sample in cubic centimeters? Give your answer in scientific notation or use a metric prefix.

5. If there are 5.65 kg per every half liter of a particular substance, is that substance liquid mercury (density 13.5 g/cm³), lead (density 11.3 g/cm³), or tin (density 7.3 g/cm³)?

Practice Answers

1. 4.6 m. You have to convert all the way from feet to meters. Looking at the conversion factors in Table 2-4, you should see that you can convert feet into inches and then inches into centimeters. Then you can easily convert centimeters into meters.

2. . There’s no direct pound-to-gram conversion factor in Table 2-4, so you must determine the correct path to take. In this case, you can convert from pounds to kilograms and then from kilograms to grams:

3. 3.8 L. You must determine the correct pathway to get from gallons to liters using the conversions provided in Table 2-4. To do so, convert from gallons to cups, then to milliliters, and finally to liters:

4. . First convert all the inch and foot measurements to centimeters:

The volume is therefore , or .

5. The substance is lead. To set up this problem, be sure to begin with the correct initial amounts. The problem tells you there are 5.65 kg for every half liter of substance. This translates into 5.65 kg/0.5 L. After you’ve established the initial value, use conversion factors to find the density:

This answer is exactly the density of lead.