Chemistry · General chemistry I · Concept

Dimensional analysis and unit conversion

Dimensional analysis converts a quantity to new units by multiplying it by conversion factors, ratios equal to one, each arranged so that a unit you do not want cancels. The units left at the end tell you whether the setup was right.

A concrete case

Hours to seconds, with the units written in

How many seconds are in $2.50 hours$ ? Most people multiply by $60$ to get minutes and by $60$ again to get seconds, $2.50 \times 60 \times 60 = 9000 s$ . Dimensional analysis is that same arithmetic with the units written next to every number.

Start from $2.50 h$ . An hour is $60 minutes$ , so the fraction $60 \min$ over $1 h$ compares two equal times, and its value is one. Multiplying by it changes the units without changing the amount of time. The h on top of $2.50 h$ meets the h on the bottom of the fraction, and they cancel like a number divided by itself.

A minute is $60 seconds$ , so a second fraction, $60 s$ over $1 \min$ , cancels the minutes. Only seconds are left, which is what the question asked for. The answer is $9.00 \times 10^{3} s$ , three significant figures like $2.50$ , because $60 \min = 1 h$ and $60 s = 1 \min$ are definitions.

2.50 h \times \frac{60 \min}{1 h} \times \frac{60 s}{1 \min} = 9.00 \times 10^{3} s

\begin{array}{c} 2.50 h \times \frac{60 \min}{1 h} \times \frac{60 s}{1 \min} \\ = 9.00 \times 10^{3} s \end{array}

The rule

Conversion factors

A conversion factor is a ratio of two equal quantities written in different units. Because the top and the bottom are the same amount, the ratio equals one, and multiplying by it changes only the units.

Every equality gives two factors, one the upside-down of the other. $1 L = 1000 mL$ gives $1 L$ over $1000 mL$ and $1000 mL$ over $1 L$ . Choose the one with the unit you want to remove on the opposite side from where that unit sits now.

Units behave like letters in algebra. They multiply, divide and cancel. A chain of factors is one long product, and the units that survive the cancelling are the units of the answer.

\frac{1 L}{1000 mL} = 1 = \frac{1000 mL}{1 L}

\begin{array}{c} \frac{1 L}{1000 mL} = 1 \\ \frac{1000 mL}{1 L} = 1 \end{array}

A property used as a factor

A density, a speed or a molar mass can be written as a factor too. $1.20 g / mL$ means $1.20 g$ for every $1 mL$ , which gives $1.20 g$ over $1 mL$ and $1 mL$ over $1.20 g$ .

Such a factor is not equal to one. It connects two different quantities, mass and volume, for one particular substance under stated conditions. It is a measurement, so its digits count toward the significant figures of the answer. A definition such as $1 L = 1000 mL$ never does.

\frac{1.20 g}{1 mL} \frac{1 mL}{1.20 g}

Why it works

Why the units cancel, and what a flipped factor does

Writing $2.50 h$ means $2.50$ times one hour. A unit is a factor of the quantity, just as $x$ is a factor of $2.50 x$ . An hour on top divided by an hour on the bottom is one, for the same reason that $x$ over $x$ is one.

Multiplying by a factor equal to one never changes the quantity. That is why a flipped factor is not an arithmetic error at first. $250.0 mL$ times $1000 mL$ over $1 L$ is still the same volume, now written as $250, 000 {mL}^{2} / L$ . The statement is true and useless, because ${mL}^{2} / L$ is not a unit anyone wants.

The leftover units are the signal. If the units at the end are not the units the question asked for, a factor is upside down or missing. A physical property works the same way. Volume divided by density leaves ${mL}^{2} / g$ , not grams.

250.0 mL \times \frac{1000 mL}{1 L} = 2.500 \times 10^{5} \frac{{mL}^{2}}{L}

\begin{array}{c} 250.0 mL \times \frac{1000 mL}{1 L} \\ = 2.500 \times 10^{5} \frac{{mL}^{2}}{L} \end{array}

The method

The method in six steps

The table converts $250.0 mL$ to liters. Write every number with its unit, including the numbers inside each factor. A bare number is how a flipped factor hides.

250.0 mL \times \frac{1 L}{1000 mL} = 0.2500 L

\begin{array}{c} 250.0 mL \times \frac{1 L}{1000 mL} \\ = 0.2500 L \end{array}

Converting $250.0 mL$ to liters
Step	What to do	For $250.0 mL$ to liters
1	Write the given quantity with its unit	$250.0 mL$
2	Write the unit the answer needs	L
3	Find an equality that links them	$1 L$ $= 1000 mL$
4	Write the factor with the unwanted unit on the opposite side	$1 L$ over $1000 mL$ , so $mL$ is on the bottom
5	Cancel, then multiply and divide the numbers	$250.0 \div 1000$ $= 0.2500$
6	Round by the measured values only, then check the size	$0.2500 L$ , four figures, and a liter is bigger than a milliliter, so the number got smaller

Prefixes and longer chains

A metric prefix is a power of ten. Write each prefix factor with a $1$ in front of the larger unit, and the arithmetic stays clear. $1 km = 1000 m$ and $1 m = 10^{9} nm$ , because many tiny nanometers make up one meter.

When no single equality links the two units, chain several factors. To turn kilometers into centimeters, go through meters. $3.25 km$ $\times 1000 m / 1 km$ $\times 100 cm / 1 m$ $= 3.25 \times 10^{5} cm$ . Each factor cancels the unit the one before it left behind.

3.25 km \times \frac{1000 m}{1 km} \times \frac{100 cm}{1 m} = 3.25 \times 10^{5} cm

\begin{array}{c} 3.25 km \times \frac{1000 m}{1 km} \times \frac{100 cm}{1 m} \\ = 3.25 \times 10^{5} cm \end{array}

Prefixes used most in chemistry
Prefix	Symbol	Power of ten	Factor
kilo	k	$10^{3}$	$1 km$ $= 1000 m$
centi	c	$10^{- 2}$	$1 m$ $= 100 cm$
milli	m	$10^{- 3}$	$1 L$ $= 1000 mL$
micro	μ	$10^{- 6}$	$1 g$ $= 10^{6} µg$
nano	n	$10^{- 9}$	$1 m$ $= 10^{9} nm$

Rates, areas and volumes

A unit with a slash has two parts, and each part may need converting. In $mi / h$ , miles are on top and hours are on the bottom. To cancel the hours, the time factor must put hours on top, $1 h$ over $3600 s$ .

A squared or cubed unit needs its factor squared or cubed. A meter is $100 cm$ , so a square meter is $100 \times 100 = 10^{4} {cm}^{2}$ , and a cubic meter is $10^{6} {cm}^{3}$ . A cubic centimeter is exactly one milliliter, and a cubic meter is exactly $1000 L$ .

1 m^{3} = {(100 cm)}^{3} = 10^{6} {cm}^{3} = 1000 L

\begin{array}{c} 1 m^{3} = {(100 cm)}^{3} \\ = 10^{6} {cm}^{3} = 1000 L \end{array}

Where it breaks

Temperature scales do not convert by a factor alone, because their zeros differ. $25.0 ° C$ is $298.2 K$ by adding $273.15$ , not by multiplying. A Fahrenheit temperature needs both a factor and a shift, $T_{F} = 1.8 T_{C} + 32$ . A temperature change is different. A change of $10.0 ° C$ is a change of $10.0 K$ , because the two degrees are the same size.

A property factor holds only where its measurement holds. A density belongs to one substance at one temperature, and $1.20 g / mL$ for one liquid says nothing about another.

Label what each unit counts when two things share a unit. In a mixture, grams of salt and grams of solution do not cancel each other, even though both are grams.

Correct units are necessary but not enough. A chain with the right units still gives a wrong answer if a number in it is wrong, such as $100 cm$ in a meter written as $10 cm$ .

T_{K} = T_{C} + 273.15

Check your answer

Read the units. Cancel them on paper before you touch a calculator. If what is left is not the unit the question asked for, a factor is flipped or missing.

Ask which way the number should move. Converting to a smaller unit gives a bigger number, and converting to a larger unit gives a smaller one. $250.0 mL$ is a quarter of a liter, so $0.2500 L$ is right and $250, 000$ is not.

Go back the other way. Multiply your answer by the factors turned over. You should land on the number you started with.

In a Chemistry box, write the whole chain on one row, with a unit on every number, as $250.0 mL \times 1 L / 1000 mL$ $= 0.2500 L$ . The Checker names each factor it knows, as in “1 L / 1000 mL: exact by definition.” It reports the units that are left, as in “Units cancel to L.” Then it checks the arithmetic.

If a flipped factor leaves units that cannot be the answer’s, the row reads ✗. The Checker names the factor to turn over, as in “Check the factor 1000 mL / 1 L: turned over, it would cancel mL.”

If you write the right final value anyway, the row reads ✓, because the quantity really is unchanged. Look for “Units cancel to” rather than “Units convert by definition” to be sure every unit cancelled.

A density inside a chain reads ✓ when the density is on a row above it, such as $ρ = 1.20 g / mL$ , with the note “the density ρ given above.” A density written only in the chain reads ? with the note “not a ratio Chalk can derive, so its value is taken as given.” A value that is not the $ρ$ above reads ? too. Another way to have it checked is to give $ρ$ and $V$ on their own rows, write $m = ρ V$ , and substitute. That row reads ✓.

Write a dose, such as $15 mg$ per kilogram of body mass, on a row of its own, dose $= 15 mg / kg$ . The chain below it, $68.0 kg \times 15 mg / 1 kg$ $= 1.0 \times 10^{3} mg$ , then reads ?, with the dose taken as given. Without the dose row, Chalk reads $15 mg / 1 kg$ as a unit conversion, and the row reads ✗.

Dimensions: checking a formula, not only a conversion

Units come in kinds, called dimensions. Length, mass, time and amount of substance are four of them. Meters and miles are different units of the same dimension, and only quantities of the same dimension can be added or set equal.

That gives a check on any formula. Kinetic energy, $\frac{1}{2} m v^{2}$ , has units $kg \cdot m^{2} / s^{2}$ , which is exactly a joule. A formula that came out as $kg \cdot m / s$ could not be an energy, whatever its numbers.

The check has limits. It catches a missing factor of mass or a time in the wrong place. It cannot see a pure number, so it would not notice a missing $\frac{1}{2}$ . A quantity inside a logarithm or an exponent must have no units at all, which is why $ln K$ and $e^{- E_{a} / R T}$ use ratios such as $E_{a} / R T$ .

[\frac{1}{2} m v^{2}] = kg \cdot \frac{m^{2}}{s^{2}} = J

Common mistakes

Flipping a factor. $375 mL \times 1000 mL / 1 L$ gives $375, 000 {mL}^{2} / L$ , not liters. The milliliters must sit on the bottom to cancel, so the right chain is $375 mL \times 1 L / 1000 mL$ $= 0.375 L$ .
Putting a prefix’s power on the wrong side. $1 nm = 10^{9} m$ would make a nanometer enormous. A nanometer is tiny, so many of them make one meter, and the equality is $1 m = 10^{9} nm$ .
Converting a squared unit with an unsquared factor. $1 m^{2}$ is not $100 {cm}^{2}$ . Each side of a square meter is $100 cm$ , so its area is $100 \times 100 = 10^{4} {cm}^{2}$ .
Counting the digits of an exact factor. $250.0 mL \times 1 L / 1000 mL$ is $0.2500 L$ , with four figures, not one. The $1$ and the $1000$ are a definition, so only $250.0$ limits the answer.
Rounding partway through a chain. $65.0 mi / h$ is $104.6 km / h$ , and rounding that to $105$ before converting to $m / s$ gives $29.2 m / s$ instead of $29.1$ . Keep every digit until the last step.
Converting a temperature with a factor. $25.0 ° C$ is $298.2 K$ , found by adding $273.15$ . A factor such as $1 K / 1 ° C$ would leave $25.0 K$ , colder than liquid nitrogen.

Key terms

Dimensional analysis: Solving a problem by multiplying by conversion factors, which are ratios equal to $1$ such as $60 \min \div 1 h$ , so the unwanted units cancel. An example is grams → moles → grams. If the units don’t come out right, the setup has a mistake.
Unit conversion: Expressing the same quantity in another unit using a known relation between the units. Most conversions multiply by a conversion factor, a ratio equal to $1$ , but converting between temperature scales also needs an offset.
Exact number: A number with no uncertainty: a count, a defined relationship such as $1 kg = 1000 g$ , or a coefficient in a balanced equation. Exact numbers do not limit the significant figures of a result.
Density: Mass per unit volume, such as $g / mL$ . It converts between the mass and the volume of a sample, and it changes with temperature.
Physical quantity: A measurable property expressed as a numerical value together with a unit or reference. The same quantity can have different numerical values in different units.
Significant figures: The digits in a measurement that are known reliably, plus one final estimated digit. Exact counts and defined conversions, such as $1 m = 100 cm$ , never limit the significant figures of an answer.

Work through an example

A sample occupies $25.0 mL$ and has density $1.20 g / mL$ . What is its mass?

Find mass from volume and density →

Convert a speed from $mi / h$ to $m / s$ →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

After reading

Study it, then try it

Practice the key terms as flashcards, then open the example in its tool, change a value and keep a useful result on your board.

Study the 6 key terms Build the factor chain Check the mass in a Chemistry box Open worked example on a board Conversion factors

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