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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 d meaning of organic chemistry
Elizabeth Reply
it's a compound that comprises of hydrocarbon
what iz alkanol
icha Reply
alkanol there are organic compounds with the functional group of ROH and relative molecular formula (CnH2n+1+OH)
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Sam. A
What's alkaline soil
Amoo Reply
a system in which only energy is transferred between the system and the surrounding is called?
Ani Reply
which Element exhibit diagonal relationship with aluminum
following processes: Solid phosphorus pentachloride decomposes to liquid phosphorus trichloride and chlorine gas b. Deep blue solid copper(II) sulfate pentahydrate is heated to drive off water vapor to form white solid copper(II) sulfate
Hisham Reply
How to know periodic table oftend
Ahmed Reply
u can know it through singing it as song it simple
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Ogunleye Reply
how do you solve the examples in a much more explanatory way
it seems by multiplying d number of d element by 2
E.g like carbon 6*2=12 so d atomic number is 12
The reaction of aceto nitrile with propane in the presence of the acid
Explain this paragraph in short
Manish Reply
What is solid state?
Manish Reply
What is chemical reaction
transforming reactants to product(s)
solid state is composed of tightly particles and it has a definite shape and volume
Example of Lewis acid
Chidera Reply
Example of Lewis acid
Anything with an empty orbital... the hydrogen ion is the most common example. BH3 is the typical example, but any metal in a coordination complex can be considered a Lewis acid.
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aluminium and sulphur react to give aluminium sulfide.How many grams of Al are required to produce 100g of aluminium sulphide
Soni Reply
aluminium and sulphur react to give aluminium sulphide how many grams of Al are required to produce 100g of aluminium sulphide?
aluminium and sulphur react to give aluminium sulphide how many grams of Al are required to produce 100g of aluminium sulphide?
150 comes from?
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molar mass of Al2S3
what's periodic table
Why can't atom be created or destroyed
Jacaranda Reply
matter simply converts to pure energy
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explain how to distinguish ethanol from a sample of ethanoic acid by chemical test
Alice Reply
explain how ethanol can be distinguished from ethanoic acid by chemical test
Using a suitable experiment, describe how diffusion occurs in gases.
Melody Reply
when the excited energy which are in gaseous state collides with another to liberate from one place to another
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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