Unit Conversion Dimensional Analysis

Transcription

1 Chemistry 210 Unit Conversion Dimensional Analysis T.J. Reinert It is often necessary to convert a measurement from one system of units to another, particularly for citizens and residents of the United States. In spite of the fact that all other countries of the world and all scientists use the metric system to express measured quantities, the U.S. still clings to the archaic British system of measurement. Even Great Britain herself has converted to the metric system. For example, when your physician prescribes medication, he or she needs to convert your body weight to kilograms because dosages are usually expressed as milligrams of medication per kilogram of body weight. To convert a quantity from one system of units to another, medical personnel, scientists, and engineers frequently use a procedure called dimensional analysis. Background Measured quantities are always represented by a number and its associated unit, such as 1.9 pounds or 3.5 inches. If you think of the number as a factor (or a scalar) that multiplies the unit, you can apply standard algebraic conventions when you convert a measured quantity from one system of units to another. Dimensional analysis works because the given unit is always multiplied by a conversion factor that is equal to one. The conversion factor comes from an equation that relates the given unit to the wanted, or desired, unit. For example, = lb defines the relationship between kilograms and pounds. If we divide both sides of this equation by, we get a fraction that is equivalent to one: = 1 = lb The expression lb/ is a conversion factor that converts the measurement in kilograms to pounds or vice versa. The quantity in this conversion factor is exactly 1 kilogram. Therefore when you use this conversion factor, the number of significant figures is determined by the number of significant figures in lb. Dimensional Analysis in Simple Steps You can use the following two-step process to convert any given measurement to the unit you need: 1. Build the concept map which connects the two units or ideas. Changes may require one or more intermediates to connect the two. 2. For each identified step in the concept map, write a conversion factor that relates the given unit to the wanted unit. (Be certain to follow the algebraic rules.) Unit Conversion Dimensional Analysis page 1 of 5

2 Here is an example of a conversion that requires an intermediate step: Ex. Convert feet to centimeters. 1 foot = 12 inches; 1 inch 2.54 centimeters (exact). Concept feet inches centimeters map Execution ft 12 inches 1 ft 2.54 cm 1 inch = cm {Note: the number of significant digits in the answer is determined by the initial measurement, not by the conversions (both are exact)} The idea of concept maps isn't just useful for changing units of measurement, but can be more broadly applied to a number of relationships: speed and time can be related to distance; diameter of a sphere can be related volume; etc. Later in the course, we will relate masses of compounds to one another through chemical reactions using concept maps. Dimensional Analysis Problems In this activity, we will use Round Robin group problem-solving method whichhelps people to work together and feel comfortable with group problem solving. Round Robin Instructions 1. At your table, the facilitator will read the first question aloud. 2. For one minute, each student will plan the concept map for the question. 3. Going around the table (clockwise), the facilitator will state his/her first step in the concept map, the second student states the second step, etc., until you arrive at the desired unit of measure. 4. After agreeing on the concept map, each student does the calculation and shares his/her result with the group. 5. The next student in the circle will become the facilitator for the next question and will start the question by reading it aloud. Questions Use concept maps and dimensional analysis to answer each question by group round robin. Record your solutions and notes in the spaces provided. 1. Find the number of centimeters in 1.00 x 10²yards. Unit Conversion Dimensional Analysis page 2 of 5

3 2. Determine the number of meters in 1.00 mile. 3. The speed of light is 1.86 x 10⁵ miles per second. How many meters will light travel in 5.0 seconds? 4. A light year is the distance that light travels in one year. Determine the number of miles, meters, and kilometers in one light year. Unit Conversion Dimensional Analysis page 3 of 5

4 5. The units of the chain system of measure, used by surveyors, are as follows: 7.92 inches = 1 link 100 links = 1 chain 10 chains = 1 furlong 80 chains = 1 mile The distance of the Kentucky Derby, a classic horse race, is 1.25 miles. How is this distance expressed in furlongs? 6. A cube that has a length of 1 cm on each side has a volume of 1 cm³. How many cubic centimeters are in 1 cubic meter? (Hint: The answer is not 100.) Unit Conversion Dimensional Analysis page 4 of 5

5 7. A single layer of gold atoms forms a surface whose dimensions are ³angstroms by ³ angstroms. (1 angstrom (Å) 10 ¹⁰ meter). a) What is the area of this surface in square angstroms? b) What is the surface area in square centimeters? 8. The displacement (total volume of the cylinders) of the engine in a Ford Mustang is 5.0 L. Convert this volume to cubic inches. (1 ml is 1 cm³) Unit Conversion Dimensional Analysis page 5 of 5