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By the end of this section, you will be able to:
  • Explain the dimensional analysis (factor label) approach to mathematical calculations involving quantities
  • Use dimensional analysis to carry out unit conversions for a given property and computations involving two or more properties

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. For example, consider measuring the average speed of an athlete running sprints. This is typically accomplished by measuring the time required for the athlete to run from the starting line to the finish line, and the distance between these two lines, and then computing speed from the equation that relates these three properties:

speed = distance time

An Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of

100 m 10 s = 10 m/s

Note that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100/10 = 10) and likewise dividing the units of each measured quantity to yield the unit of the computed quantity (m/s = m/s). Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m. The same relation between the three properties is used, but in this case, the two quantities provided are a speed (10 m/s) and a distance (25 m). To yield the sought property, time, the equation must be rearranged appropriately:

time = distance speed

The time can then be computed as:

25 m 10 m/s = 2.5 s

Again, arithmetic on the numbers (25/10 = 2.5) was accompanied by the same arithmetic on the units (m/m/s = s) to yield the number and unit of the result, 2.5 s. Note that, just as for numbers, when a unit is divided by an identical unit (in this case, m/m), the result is “1”—or, as commonly phrased, the units “cancel.”

These calculations are examples of a versatile mathematical approach known as dimensional analysis    (or the factor-label method ). Dimensional analysis is based on this premise: 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.

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 (by definition), and so a unit conversion factor may be derived from the ratio,

2.54 cm 1 in. (2.54 cm = 1 in.) or 2.54 cm in.

Several other commonly used conversion factors are given in [link] .

Common Conversion Factors
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 3 = 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

Questions & Answers

What is stoichometry
ngwuebo Reply
what is atom
yinka Reply
An indivisible part of an element
the smallest particle of an element which is indivisible is called an atom
An atom is the smallest indivisible particle of an element that can take part in chemical reaction
is carbonates soluble
Ebuka Reply
what is the difference between light and electricity
Joshua Reply
What is atom? atom can be defined as the smallest particles
what is the difference between Anode and nodes?
What's the net equations for the three steps of dissociation of phosphoric acid?
Lisa Reply
what is chemistry
Prince Reply
the study of matter
what did the first law of thermodynamics say
Starr Reply
energy can neither be created or distroyed it can only be transferred or converted from one form to another
Graham's law of Diffusion
Ayo Reply
what is melting vaporization
Anieke Reply
melting and boiling point explain in term of molecular motion and Brownian movement
Scientific notation for 150.9433962
Steve Reply
what is aromaticity
Usman Reply
aromaticity is a conjugated pi system specific to organic rings like benzene, which have an odd number of electron pairs within the system that allows for exceptional molecular stability
what is caustic soda
Ogbonna Reply
sodium hydroxide (NaOH)
what is distilled water
is simply means a condensed water vapour
advantage and disadvantage of water to human and industry
Abdulrahman Reply
a hydrocarbon contains 7.7 percent by mass of hydrogen and 92.3 percent by mass of carbon
Timothy Reply
how many types of covalent r there
JArim Reply
how many covalent bond r there
they are three 3
TYPES OF COVALENT BOND-POLAR BOND-NON POLAR BOND-DOUBLE BOND-TRIPPLE BOND. There are three types of covalent bond depending upon the number of shared electron pairs. A covalent bond formed by the mutual sharing of one electron pair between two atoms is called a "Single Covalent bond.
Practice Key Terms 3

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Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
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