Density and Its Units

Introduction

You have already met density as a property that tells you how tightly mass is packed into space. This page looks at density from the point of view of units. Density is a perfect example of a derived unit: it is built from the kilogram and the metre by division. Once you see exactly how its unit is made, you can write it in any form an examiner uses and convert between the SI unit and the laboratory unit without guessing.

Core explanation

Units follow the equation. Density is defined as

ρ = m ÷ V

Whatever units you use for mass and volume, the unit of density is simply "mass unit ÷ volume unit". If mass is in kilograms and volume in cubic metres, the unit is kg ÷ m³. If mass is in grams and volume in cubic centimetres, the unit is g ÷ cm³.

Two ways of writing the same unit. "Per" can be written with a slash or with a negative index:

- kg/m³ is the same as kg m⁻³ - g/cm³ is the same as g cm⁻³

The negative index comes from the rule that 1/x³ = x⁻³. Many exam boards and data books prefer the index form because it avoids stacking several slashes in more complicated units.

The SI unit. Because the kilogram and the metre are SI base units, kg m⁻³ is the SI unit of density. Water has a density of about 1000 kg m⁻³ at room temperature, air about 1.2 kg m⁻³ and iron about 7870 kg m⁻³.

The laboratory unit. Laboratory samples are measured in grams and cm³, so chemists usually quote g cm⁻³ . Water is then about 1.00 g cm⁻³, which is a convenient number to remember. For liquids you may also see g mL⁻¹, which is identical to g cm⁻³ because 1 mL = 1 cm³.

Converting between them. One gram is 1/1000 of a kilogram, and one cubic centimetre is 1/1 000 000 of a cubic metre. So

1 g cm⁻³ = (0.001 kg) ÷ (0.000 001 m³) = 1000 kg m⁻³

To go from g cm⁻³ to kg m⁻³, multiply by 1000; to go back, divide by 1000. A useful sanity check: the numbers in kg m⁻³ are always the larger ones. A third unit, g dm⁻³ (grams per litre), is numerically equal to kg m⁻³ and is often used for gases and for the mass concentration of solutions.

Formulae

ρ = m ÷ V

1 g cm⁻³ = 1000 kg m⁻³ = 1000 g dm⁻³

Step-by-step reasoning

To calculate a density and give it in SI units:

1. Write down the mass and volume with their units. 2. Divide mass by volume and write the unit as mass unit ÷ volume unit. 3. If the result is in g cm⁻³, multiply the number by 1000 to obtain kg m⁻³. 4. Check that the answer is reasonable: most solids and liquids lie between about 500 and 20 000 kg m⁻³.

Visual explanation

Draw a 1 cm cube of water and label it "1 g". Next to it draw a 1 m cube and label it "1 000 000 cubes of 1 cm³ = 1 000 000 g = 1000 kg". The drawing shows why the same substance has density 1 g cm⁻³ and 1000 kg m⁻³: both describe one packing, counted in different-sized boxes.

Real-world analogy

Quoting a price as "pence per gram" or "pounds per kilogram" describes the same cheese at the same shop; only the units change, so the number changes. Density works the same way: g cm⁻³ and kg m⁻³ describe the same material, with a factor of 1000 between the numbers.

Real-world example

Engineering tables usually list densities in kg m⁻³ because engineers calculate the masses of large structures in kilograms and cubic metres. A concrete bridge deck, at roughly 2400 kg m⁻³, lets an engineer multiply directly by a volume in m³ to find a mass in kg without extra conversions.

Why?

Why is the conversion factor 1000 and not 1 000 000? Changing grams to kilograms divides the top by 1000, while changing cm³ to m³ divides the bottom by 1 000 000. Dividing the denominator by the larger number makes the value bigger overall, and the net effect is 1 000 000 ÷ 1000 = 1000.

Common misconception

"Because 1 m = 100 cm, you convert density by multiplying by 100." This ignores the cube. Volume units are cubed, so 1 m³ = 100³ cm³ = 1 000 000 cm³, and mass units change as well. The correct factor between g cm⁻³ and kg m⁻³ is 1000.

Worked example

Question: A 25.0 cm³ sample of olive oil has a mass of 22.9 g. Find its density in g cm⁻³ and in kg m⁻³.

Reasoning: ρ = m ÷ V = 22.9 g ÷ 25.0 cm³ = 0.916 g cm⁻³. Multiply by 1000 to convert to SI: 0.916 × 1000 = 916 kg m⁻³. This is less than water's 1000 kg m⁻³, which fits the fact that oil floats on water.

Answer: 0.916 g cm⁻³, which is 916 kg m⁻³.

Quick check

1. Write the unit "grams per cubic centimetre" using negative-index notation. Answer: g cm⁻³.

Exam focus

Be ready to express density units as kg m⁻³ or g cm⁻³ and to convert with the factor 1000. When a question gives mass in kg but volume in cm³, convert one of them before dividing, otherwise the unit is mixed and the number is meaningless.

Advanced insight

In chemistry you will later meet mass concentration , measured in g dm⁻³, which has the same form as density but describes the mass of a solute dissolved per unit volume of solution rather than the mass of the whole material. Recognising that two quantities share a unit does not mean they are the same quantity; always read what the mass and volume refer to.

Summary

Density is mass ÷ volume, so its unit is any mass unit divided by any volume unit. The SI unit is kg m⁻³; the laboratory unit is g cm⁻³ (identical to g mL⁻¹). The slash and negative-index forms mean the same thing. To convert, 1 g cm⁻³ = 1000 kg m⁻³, a factor that arises from both the mass and volume conversions.

Practice questions

1. Explain why kg m⁻³ is the SI unit of density. Answer: Density is mass divided by volume; the SI unit of mass is the kilogram and the SI unit of volume is the cubic metre, so the SI unit is kg ÷ m³ = kg m⁻³. 2. Convert the density of aluminium, 2.70 g cm⁻³, into kg m⁻³. Answer: 2.70 × 1000 = 2700 kg m⁻³. 3. Convert the density of air, 1.2 kg m⁻³, into g cm⁻³. Answer: 1.2 ÷ 1000 = 0.0012 g cm⁻³. 4. A student divides a mass of 0.050 kg by a volume of 20 cm³ and writes "0.0025". Explain what is wrong and give a correct density in g cm⁻³. Answer: The units are mixed (kg and cm³) and no unit is given. Converting the mass to 50 g gives 50 ÷ 20 = 2.5 g cm⁻³.