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Dimensional Analysis^©

By: Patrick Williford

Dimensional analysis is a tool for the calculation of physical quantities. The idea is that all units must be divided out to unity when calculating or converting to different units. The concept is that used in algebra where only like-variables can be added (or subtracted) and multiplied (or divided).

When a conversion factor is used in dimensional analysis it does not change the meaning of the data, it changes the appearance of the data so that it is expressed in the desired units. A properly expressed conversion factor is similar to multiplying the data by unity and therefore does not change the data; it expresses it with different units.

One must be familiar with commonly used prefixes, units, and conversion factors.

Commonly Used Prefixes
Prefix	Long Spelling	Magnitude		Prefix	Long	Magnitude
G	giga	109 = 1,000,000,000		m	milli	10-3 = 0.001
M	mega	106 = 1,000,000		µ	micro	10-6 = 0.000 001
k	kilo	103 = 1,000		n	nano	10-9 = 0.000 000 001
- - -	unity1	100 = 1		p	pico	10-12 = 0.000 000 000 001
c	centi	10-2 = 0.01		f	femto	10-15 = 0.000 000 000 000 001

[1] unity = 1 (exactly) = 1.0000 . . . 0 = 1.00 (Where the underline implies that the digits underlined repeat for ever.)

Common Unit Abbreviations (both singular and plural forms are implied) g = gram m = meter L = liter N = newton y = yard gal = gallon lb = pound ft = foot qt = quart oz. = ounce (ADP) in = inch pt = pint s = second mol = mole fl. oz. = fluid ounce

Common Conversion Factors [the tilde ( ~ ) means that the value is not exact] 2.54 cm = 1 in (see note) 1 mile = 5280 ft 1 L = ~ 1.056718 qt 1 m = ~39.37 in 1 kg = ~2.2 lb 4 qt = 1 gal 1 ft = 12 in 1 lb = ~454 g 1 cm3 = 1 mL Note: The inch was redefined by the scientific community several years ago to be exactly 2.54 cm.

In Algebra:

• x + 5x = 6x, but 7y + 3x remains 7y + 3x;
• (5xy)(3y)(4x²) = 60x³y² and 5xy / 3y / 4x² = 5 / 12x

In Dimensional Analysis:

• 1 g + 5 g = 6 g, but 7 mol + 3 g remains 7 mol + 3 g;
• (5 g mol)(3 mol)(4 g²) = 60 g³ mol² and 5 g mol / 3 mol / 4 g² = 5 / 12 g

Another aspect to dimensional analysis when applied to chemistry is that one must consider the substance to which the value and unit refer as another variable. One should not divide grams of NaCl by grams of KCl without maintaining the notion of the mixture. For example: 15.0 g NaCl / 5.0 g KCl = 3.0 NaCl / KCl, meaning that the mass ratio of sodium chloride to potassium chloride is 3.0 to 1.0.

Conversion Factors:

Conversion factors (also called unity operators) can be definitions or empirical values.

1. The definition form of the conversion factor is unity (exactly one followed by an infinite number of significant figures).

   For example: 12 in = 1 ft is the exact definition of the number of inches in one foot, so anything times its ratio (12 in / 1 ft) is like multiplying it by unity. If one wanted to find out how many feet are in 100 in, then one would calculate it by dividing 100 by 12 and calling the answer eight and one-third feet. Dimensional analysis provides a means of writing down what you have done and when the conversions become difficult it simplifies the process.

   (100 in) (1 ft / 12 in) = (100 / 12) ( in / in) (ft) = (8.33) (1) (ft) = 8.33 ft

2. The empirical form of a conversion factor is close to unity, but not exact. The number of significant figures must be consider as that shown in the expression.

   For example: 1.00794 g H = 1 mol H is experimentally determined and therefore is an empirical conversion factor. To find out how many moles of hydrogen are in 5.06007031 grams of hydrogen; one would calculate it as such:

   (5.06007031 g H) (1 mol H / 1.00794 g H) = (5.06007031 / 1.00794) (g H / g H) (mol H) = (5.02021) (1) (mol H) = 5.02021 mol H

   In the last example the term "1 mol H / 1.00794 g H" has six significant figures and "5.06007031 g H" has nine significant figures. The number of significant figures is limited to the term with the least number of significant figures, so one is limited to six significant figures in that example. (Note that one mol is an exact value and generally 1 mol H / 1.00794 g H can be written as "mol H / 1.00794 g H," where the unity value is implied.)

   The best way to show the power of dimensional analysis is by example. One must practice dimensional analysis in order to become proficient at it. It is like riding the proverbial bicycle, once you learn it you do not forget it.

Example 1: