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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 atomicity
Simbiat Reply
A 45 ml of ph=1,hcl was reacted with a 55l ml of ph=13, naoh solution . what is the final ph
chamini Reply
what is coordination number
YERUMAKULA Reply
coordination number is the number of atoms or ions immediately surrounding a central atom in a complex or crystal
Chidera
what is isotope
Bukar
who is the father of chemistry
Roland Reply
Antoine Lavoisier. Father of modern chemistry
Yapi
Lavoisier
Simbiat
What is geometric isomerism
Imoh Reply
jo
Lakshmi
pls I don't really know teach me
Joel
geometric isomerism are molecules that are locked into their spatial position with respect to one another due to a double Bond or ring structure
Chidera
Chromatography is a physical method of seperation where by mixtures that are in two phrases are separated
Lexzzy Reply
What is chromatography?
Chukwudi Reply
meaning of electrode
DJKRANKY Reply
is ion exchange and ion chromatography the same process?
Vivian
define boyes law and give the formula
Esther Reply
define chemistry and it's properties
Esther
Boyle's law state that at a constant temperature, the pressure of a fixed mass of gas is inversely proportionally to it's volume. P=k/v P1v1=p2v2
MAUREE
Chemistry is the study of matter and it's properties..... It's deals with the composition that governors d use of matter..... Chemistry is also d study of chemical reaction of a substance and it's matter. Properties of chemistry Atom Elements Compound Molecules
MAUREE
chemistry seems more accurately defined as the measurement and study of the interaction / behavior that occurs between any given person place or thing or unit of measure, ie time and energy correlation
Elaine
its the given mass of gas that is inversely proportional to it pressure
Simbiat
Reactive metals are extracted by _______?
Joseph Reply
electrolytic reduction
Amisu
what is ionxide
Wisdom Reply
What is isometrics
Kamaluddeen Reply
Molar mass of aluminum
Victor Reply
why an equation should balanced?
Kwanele Reply
in order to obey the law of conservation of mass
Ibrahim
why does the atomic radius increase across the period
Timothy Reply
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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