Dimensional Analysis I A Year-Long (and Hopefully Longer) Tool for Problem Solving

What’s the Rationale/Trick? As chemistry students, you have two goals with calculation-type problems. First, to get the correct answer. Second, to be able to show others WHY your answer is correct. Dimensional Analysis meets both of these goals. In using dimensional analysis to solve problems, you will need to identify a GIVEN value (or quantity) and one or more CONVERSION FACTORS that allow you to determine the DESIRED value (or quantity).

Some Basic Principles Physical quantities are always made up of a number and a unit. Examples: 1.34 km, 14.23 mL, 25.0 °C. Conversion factors are valid relationships or equalities expressed as a fraction. For 1 km = 0.6 miles, the CONVERSION FACTOR is either:

Conversion Factors and What “Per” Means When one sees “something per something” or “something/something,” which is read as “something per something,” the phrase or expression is usually a CONVERSION FACTOR. 60 miles per hour or 60 mile/hr means 60 miles = 1 hour. The 60 stays with the miles and a 1 is associated with the “per hour” part. A value of 1 always goes with the second unit in a “per” expression.

More “Per” Conversion Factors hr 1 miles 45 or 45 miles/hr: 12.01 grams/mol: 106 Yen per Dollar: mol 1 g 12.01 or $1.00 106 Yen or Yen 106

Basic Steps in Dimensional Analysis Identify the GIVEN value/quantity. Identify the DESIRED value/quantity. Identify the CONVERSION FACTOR(S). Set up the problem with the GIVEN first and the DESIRED last--this will just be a blank space with an attached unit or units. Put in “bridging” conversion factors which will cancel units UP/DOWN such that the DESIRED unit(s) will come through un-cancelled. Do the math: multiply factors together that are in the numerators and divide by factors that are in the denominators.

Simple Conversion Problems: Two Forms of the Same Thing “Convert 5 cm to inches.” Here the first value, 5 cm, is the GIVEN and the part after the “to” is the DESIRED, an unknown value with associated units. GIVEN: 5 cm DESIRED: ?? Inches “How many inches in 5 cm?” Here the DESIRED comes first as part of the how many phrase. The GIVEN comes after the “in.” GIVEN: 5 cm DESIRED: ?? inches

Example #1: How many cm are in 1.32 meters? equality: 1 m = 100 cm applicable conversion factors: ______ 1 m 100 cm ______ 1 m 100 cm or ( ) _____ 1 m 100 cm 132 cm 1.32 m = cm One uses UP/DOWN unit cancellation to decide correctly which of the two forms of conversion factors must be used to solve the problem

( ) _____ Example #2: Convert 8.72 cm to meters? equality: 1 m = 100 cm applicable conversion factors: ______ 1 m 100 cm ______ 1 m 100 cm or ( ) _____ 1 m 100 cm 0.0872 m 8.72 cm = m Again, the units must cancel UP/DOWN!

Again, the units must cancel UP/DOWN! Example #3: How many feet is 39.37 inches? equality: 1 ft = 12 in applicable conversion factors: ______ 1 ft 12 in ______ 1 ft 12 in or ( ) ____ 1 ft 12 in 3.28 ft 39.37 in = ft Again, the units must cancel UP/DOWN!

Example #4: Convert 4.38 days to seconds? ___ ( ) ( ) ____ ( ) ____ 24 h 1 d 1 h 60 min 1 min 60 s 4.38 d = s 378,432 s = If we are accounting for significant figures, we would change this to… 3.78 x 105 s

Summary of Dimensional Analysis Identify the GIVEN, the DESIRED, and the CONVERSION FACTOR(S). Set up the problem with the GIVEN first and the DESIRED last--this will just be a blank space with an attached unit or units. Put in “bridging” conversion factors which will cancel units UP/DOWN such that the DESIRED unit(s) will come through un-cancelled. Do the math: multiply factors together that are in the numerators and divide by factors that are in the denominators.