Dimensional Analysis

dimensional analysis examples

Core Concepts

In this tutorial, you will learn what dimensional analysis is in the field of chemistry, how to use it, see examples, and learn how it can be applied to chemistry. 

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What is Dimensional Analysis?

What is the definition of dimensional analysis? Dimensional analysis is an essential skill used widely in the field of chemistry. Using this technique can answer questions like: “How much of this chemical do I need in my reaction?” and “What is the concentration of my solution?” At its simplest form, dimensional analysis is the methodical canceling-out of units. Take the example below:

In more real-world applications, dimensional analysis is used to convert between different units of measurement, and find unknown characteristics from those that we do know. 

Unit Conversions

It is often necessary to convert between units of measurement. Units of measurement are used to define the qualities of something. 

  • A gold block weighs 12 kilograms
  • There are 150 mL of water in the container. 

You will encounter Imperial and SI (International System) Units. Imperial units are measurements like feet, inches, and pounds. SI units are measurements like meters, centimeters, and kilograms. SI units are most common in chemistry; furthermore, one of the most important units of measurement is the mole (mol).

Below are some of the relationships between these units you might see throughout various sources, such as textbooks, or online:

Imperial Units
Quantity Unit of Measurement Relationship
Length Mile, mi
Yard, yd
Feet, ft
Inches, in
1 mi = 1760 yd
1 yd = 3 ft 
1 ft = 12 in
WeightPound, lb
Ounce, oz
1 lb = 12 oz
VolumeGallon, gal
Quart, qt
Pint, pt
Cup, c
1 gal = 4 qt
1 qt = 2 pt
1 pt = 2 c
SI Units 
QuantityUnit of MeasurementRelationship
LengthMeters, m
Centimeters, cm
Millimeters, mm
Nanometers, nm
1 m = 100 cm = 102 cm
1 cm = 102 mm
1 mm = 106 nm
WeightKilograms, kg
Grams, g
Milligram, mg
Micrograms, ug
1 kg = 1000 g= 103 g
1 g = 103 mg
1 mg = 103 mg
VolumeLiters, L
Milliliters, ml
1 L = 1000 ml = 103 ml

Additionally, it is also possible to convert between the two systems of measurement. 

what is dimensional analysis

If the gold block weighs 12 kilograms, how many pounds does it weigh?

1.0 pound (lbs) is about 0.45 kilograms (kg), so we find …

The relationship 1.0 lbs = 0.45 kg is first rewritten as a ratio.

When writing the ratio, place kg in the denominator and lbs in the numerator so that kgs later cancels out and only lbs is left in our answer. 

Then, we multiply our known value, 12 kg, by the ratio 1.0 lbs/0.45 kg. 

We are left with the solution: For every 12 kilograms, there are 26 pounds. 

Finding Unknowns

Dimensional analysis is not only useful for converting between one unit to another, but can help in solving for a number of different properties. It becomes important to be aware of how quantities like mass, volume, and density are related:

Energy force✕distance
Volume area✕length
Adapted from WebAssign

It is also increasingly important to pay attention to units of measurement. 

If there are 150 ml of water in a container, and the water in the container weighs 150 grams, what is the density of the water?

In this case, mass is given in grams, and volume is given in milliliters. We define density as mass/volume, so for this case density is grams/milliliter.

Water has the density of 1g/ml.

Dimensional Analysis Examples in Chemistry

In chemistry, dimensional analysis can be used in the ways described above, but it can also be used to relate the quantities of chemicals in a reaction. Ratios can be created using stoichiometric relationships.

CaI2 (s) → Ca2+ (aq) + 2I (aq)

Since there are two mols of Iodide (I ) for every one mol of Calcium Iodide (CaI2), the following ratio can be written: 

How many Iodide ions are produced when 4 mols of CaI2 are fully dissolved?

For a more in depth tutorial, see Solving Stoichiometry Problems.

Further Reading

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