Dimensional Analysis

1. Introduction to Dimensional Analysis

It is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate mathematical relationships. This is a versatile mathematical approach known as dimensional analysis.

key concept

Dimensional analysis is based on the premise that the units of quantities must be subjected to the same mathematical operations as their associated numbers. This method can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities.

1a. Conversion Factors and Dimensional Analysis

A ratio of two equivalent quantities expressed with different measurement units can be used as a unit conversion factor. For example, the lengths of 2.54 cm and 1 in. are equivalent, and so a unit conversion factor may be derived from the ratio,

Several other commonly used conversion factors are given in the table below.

Length	Volume	Mass
1 m = 1.0936 yd	1 L = 1.0567 qt	1 kg = 2.2046 lb
1 in. = 2.54 cm (exact)	1 qt = 0.94635 L	1 lb = 453.59 g
1 km = 0.62137 mi	1 ft = 28.317 L	1 (avoirdupois) oz = 28.349 g
1 mi = 1609.3 m	1 tbsp = 14.787 mL	1 (troy) oz = 31.103 g

When a quantity (such as distance in inches) is multiplied by an appropriate unit conversion factor, the quantity is converted to an equivalent value with different units (such as distance in centimeters).

EXAMPLE

A basketball player’s vertical jump of 34 inches can be converted to centimeters as follows:

34 in. x (2.54 cm/1 in.) = 86 cm

The numbers of these two quantities are multiplied to yield the number of the product quantity, 86, whereas the units are multiplied to yield cm. Just as for numbers, a ratio of identical units is also numerically equal to one, (in./in.)= 1, and the unit product thus simplifies to cm. (When identical units divide to yield a factor of 1, they are said to “cancel.”)

Beyond simple unit conversions, dimensional analysis can be used to solve more complex problems involving computations. Regardless of the details, the basic approach is the same—all the factors involved in the calculation must be appropriately oriented to ensure that their labels (units) will appropriately cancel and/or combine to yield the desired unit in the result. As your study of chemistry continues, you will encounter many opportunities to apply this approach.

EXAMPLE

If 4.00 qts of antifreeze weighs 9.26 lbs, what is the density in kilograms/L?

Density = mass/volume = 9.26 lbs/4.00 qts = 2.315 lbs/qt

Now, you need to convert lbs/qt to kg/L. To do this, you must convert lbs to kg (1 kg = 2.2046 lbs) and qts to L (1 qt = 0.94635 L) using dimensional analysis

2.315 lbs/qt x (1 kg/2.2046 lbs) x (1 qt/0.94635 L)

Mathematically this yields: (2.315 x 1 x 1)/(2.2046 x 0.94635) = 2.315/2.0863 = 1.1096

Units yield: (lbs/qt) x (kg/lbs) x (qt/L) = kg/L

Therefore, 1.1096 kg/L is the correct answer.

1b. Temperature Conversions

Temperature is an intensive property. An intensive property is a property that does not depend on the quantity of matter. We use the word temperature to refer to the hotness or coldness of a substance.

Temperature scales are defined relative to selected reference temperatures. Two of the most commonly used are the freezing and boiling temperatures of water at a specified atmospheric pressure. On the Celsius scale, 0 °C is defined as the freezing temperature of water and 100 °C as the boiling temperature of water. On the Fahrenheit scale, the freezing point of water is defined as 32 °F and the boiling temperature as 212 °F.

The relationship between the Celsius and Fahrenheit temperature does not have a simple conversion like some of the earlier examples, such as 1 in. = 2.54 cm. The relationship between these temperature scales can be written as a formula as seen below.

formula to know

°C = (°F − 32)/1.8

°F = (1.8 × °C) + 32

Unlike the Celsius and Fahrenheit scales, the Kelvin scale is an absolute temperature scale in which 0 (zero) K corresponds to the lowest temperature that can theoretically be achieved. Since the Kelvin temperature scale is absolute, a degree symbol is not included in the unit abbreviation, K. There is a constant value (273.15) between kelvin and Celsius, which gives the following relationship.

formula to know

K = °C + 273.15

°C = K − 273.15

terms to know

Temperature

The hotness or coldness of a substance.

Intensive Property

A property that does not depend on the quantity of matter.