Chem Professor

Tutorial

04/25/09

Dimensional Analysis

�Before You Begin:

To be successful at this material, you should be familiar with units and measurement. You should also be able to round answers to the correct number of significant figures. You should review measurement section unit from the Prep Chem outline, before you begin.

Dimensional Analysis (also called Unit Analysis)

What is it?

l Dimensional analysis is a method for converting from one system of measure to another.

l Since measurements have units associated with them, we can use the information given by the unit to drive the calculation.

l It can also be used to perform simple calculations and stoichiometry.

How Does It Work?

l Dimensional analysis is based of the algebra of ratios (fractions).

l We use conversion factors as though they were fractions so that units in the numerator (top) and denominator (bottom) cancel one another.

Basic How to Do It:

l Examine the information given, including any units of measurement.

l Find appropriate conversion factors that can convert from the units given in the problem to the ones required in the answer.

l Start with the information given including the units.

l Align conversion factors so that the units cancel (top to bottom like common factors in fractions).

l Multiply the numerators and divide the denominators.

Example 1

l A sample of gas has a pressure of 630 Torr. What is its pressure in atmospheres?

l Conversion factor you will need: 1 atm = 760 Torr

630 Torr	1 atm	=	0.83 atm
	760 Torr

Example 2

l A cobalt atom has a mass of 58.933200 amu. What is the mass in grams of one cobalt atom?

l Conversion factor you will need: 1 amu = 1.6605402 x 10-24 g

58.933200 amu	1.6605402 x 10^-24 g	=	9.7860948 x 10^-21 g
	1 amu

Multiple Step Example

l One serving of candy is 233 calories. How many kJ of energy does it provide?

l Conversion factors: In the U.S., a nutritional calorie (Cal) is actually a thermodynamic kilocalorie (kcal) or 1000 cal, and 1 cal = 4.184 J.

233 Cal	1000 cal	4.184 J	1 kJ	=	975 kJ
	1 Cal	1 cal	1000 J

Rational Expression Units

l Some units are in the form of a rational expression. Speed, for example, is miles per hour or mi/hr.

l Write the original number as though it were a fraction with one in the denominator, so that you have a place for the rational part of the unit. Proceed as before but also include conversion factors to cancel the denominator.

Rational Expression Units Example

l A car gets 24 miles per gallon. What is the mileage in kilometers per liter?

l Conversion factors: 1 mi = 1.6093 km and 1 gal = 3.7854 gal.

24 mi	1.6093 km	3.7854 gal	=	150 km
1 gal	1 mi	1 L		1 L

l Note that the order in which you use the conversion factors doesn’t matter, as long as you write your ratios correctly (as opposed to ‘upside down’).

l Note also that the units of the final answer are km/L.

Units with Exponents

l Some units have exponents, area and volume for example.

l If a unit has an exponent and the conversion factor does not, raise the entire conversion factor to the same power as the given unit.

Example with Exponents

l The density of aluminum is 2.70 g/cm³. What is this in lbs/in³?

l Conversion factors: 1 lb = 453.59 g and 1 in = 2.54 cm.

l Note that the number 2.54 was cubed in this calculation.

2.70 g	1 lb	(2.54 cm)³	=	0.0975 lb
1 cm³	453.59 g	(1 in)³		1 in³

Stoichiometry

l Chemical calculations can be worked as though they were unit conversions.

l Conversion factors used in stoichiometery problems are the molar mass (or formula weight) and the mole ratio.

Stoichiometry Example

l 75 mL of 6.0 M copper (II) sulfate solution are heated until a copper (II) oxide precipitate forms. How many grams of copper (II) oxide are produced?

75 mL CuSO₄	1 L	6.0 mol CuSO4	1 mol CuO	79.55 g	=	36 g CuO
	1000 mL	1 L	1 mol CuSO₄	1 mol CuO

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