Dimensional Analysis (a.k.a “Conversions”)

Metric System Advantage One advantage of the metric system is that it is a decimal system. It uses prefixes to enlarge or reduce the basic units. For example: A kilometer is 1000 meters. A millimeter is 1/1000 of a meter.

Metric System Prefixes The following table lists the common prefixes used in the metric system:

Metric Prefixes, continued For example, the prefix kilo- increases a base unit by 1000: 1 kilogram is 1000 grams The prefix milli- decreases a base unit by a factor of 1000: 1 millimeter is 0.001 meters

Metric Symbols The names of metric units are abbreviated using symbols. Use the prefix symbol followed by the symbol for the base unit, so: nanometer is abbreviated nm microgram is abbreviated mg

Metric Equivalents We can write unit equations for the conversion between different metric units. The prefix kilo- means 1000 basic units, so 1 kilometer is 1000 meters. The equivalent is: 1 km = 1000 m. Similarly, a millimeter is 1/1000 of a meter. The equivalent is : 1 mm = 0.001 m.

Metric Unit Factors Since 1000 m = 1 km, we can write the following unit factors for converting between meters and kilometers: 1 km or 1000 m 1000 m 1 km Since 1 m = 0.001 mm, we can write the following unit factors. 1 mm or 0.001 m 0.001 m 1 mm

Metric-Metric Conversions We will use the dimensional analysis method to work through conversion problems. Here are the steps to this method: Write down the given value related to the answer. This will be your starting point. Also make note of your desired ending point. Apply unit factor(s) to convert the given unit to the units desired in the answer. Place the equivalent in such a way that the units cancel out, leaving you with your desired units for your ending point.

One-step Conversion Problem Example Determine the mass in grams of a 325 mg aspirin tablet Step 1: We write down the given: 325 mg. We want grams for the final answer. Step 2: Apply unit factors to solve the problem. In this case, use 1 mg = 0.001 g to solve, and then round to your answer to three significant figures. 325 mg × = 0.325 g 1 mg 0.001 g

Multi- Step Dimensional Analysis Conversions The average American students is in class 330 minutes per day. How many seconds per school year is this? (1 school year = 182 days of classes) Step 1: We write down the given: 330 minutes per day or 330 minutes/ 1 day. We want seconds/ 1 school year for the final answer. Step 2: Apply unit factors to solve the problem. In this case, use you need two equivalents to change both units of your given. Finally, round to your answer to two significant figures. 75 mi/hr  m/s

Problem, continued 330 min = 3,564,000 sec/school year × 1 min 60 s You will need to use the following equivalents to solve this problem: 1 minute = 60 seconds; 1 school year = 180 days 330 min = 3,564,000 sec/school year Rounded for significant figures 3,600,000 sec/school year × 1 min 60 s 180 days 1 school yr × 1 day

Another Example The mass of the Earth’s moon is 7.35 × 1022 kg. What is the mass expressed in megagrams, Mg? We want Mg; we have 7.35 × 1022 kg. Convert kilograms to grams, and then grams to megagrams. 7.35 × 1022 kg × = 5.98 × 1019 Mg × 1 kg 1000 g 1 Mg 1,000,000 g