What density means

Definition

Density

Density is the mass per unit volume of a substance.

Density compares the mass of a substance with the volume that mass occupies.
A denser material fits more mass into the same volume, so a $cm31\ \text{cm}^3$ cube of lead has a far greater mass than a $cm31\ \text{cm}^3$ cube of aluminium.
Density is a property of the material rather than of the object, so a small steel bolt and a large steel girder have the same density.
Two objects with equal volumes have different masses when their materials have different densities.
Water has a density of about $kg/m31000\ \text{kg/m}^3$ , aluminium about $kg/m32700\ \text{kg/m}^3$ , lead about $kg/m311\,300\ \text{kg/m}^3$ and air about $kg/m31.2\ \text{kg/m}^3$ .
An object floats in a fluid when the density of the object is lower than the density of the fluid, so a heavy steel ship floats while a small steel nail sinks.

The density equation

The equation is $ρ=mV\rho=\dfrac{m}{V}$ , where $ρ\rho$ is density, $m$ is mass and $V$ is volume.
In SI units, $ρ\rho$ is in kilograms per cubic metre, $kg/m3\text{kg/m}^3$ , $m$ is in kilograms, $kg\text{kg}$ , and $V$ is in cubic metres, $m3\text{m}^3$ .
The symbol $ρ\rho$ is the Greek letter rho, and it is used for density so that $d$ can be kept for distance.
Rearranging for mass gives $m=ρVm=\rho V$ .
Rearranging for volume gives $V=mρV=\dfrac{m}{\rho}$ .
The volume of a cuboid is $V=l×w×hV=l\times w\times h$ , and the volume of a cylinder is $V=πr2hV=\pi r^{2}h$ .

Units and conversions

Mass and volume units must match, so kilograms with cubic metres give $kg/m3\text{kg/m}^3$ and grams with cubic centimetres give $g/cm3\text{g/cm}^3$ .
Converting between the two density units uses $kg/m31\ \text{g/cm}^3=1000\ \text{kg/m}^3$ .
Converting mass uses $kg1\ \text{g}=0.001\ \text{kg}$ , so $kg250\ \text{g}=0.250\ \text{kg}$ .
Converting volume uses $m31\ \text{cm}^3=1\times10^{-6}\ \text{m}^3$ , because there are $100$ centimetres in a metre and $100^{3}=1\,000\,000$ .
Liquid volumes are often measured in millilitres, and $cm31\ \text{ml}=1\ \text{cm}^3$ .
Read the question to find which density unit is wanted, then convert the mass and the volume before substituting into the equation.

Example

Density of a metal block

A rectangular block measures $cm8.0\ \text{cm}$ by $cm5.0\ \text{cm}$ by $cm2.0\ \text{cm}$ and has a mass of $g216\ \text{g}$ .
The volume is $cm3V=8.0\times5.0\times2.0=80\ \text{cm}^3$ .
Using $ρ=mV\rho=\dfrac{m}{V}$ gives $g/cm3\rho=\dfrac{216}{80}=2.7\ \text{g/cm}^3$ .
Converting gives $kg/m3\rho=2.7\times1000=2700\ \text{kg/m}^3$ .
The density of the block is $kg/m32700\ \text{kg/m}^3$ , which matches aluminium.

Example

Finding a mass from a volume

A tank holds $m30.25\ \text{m}^3$ of water, and the density of water is $kg/m31000\ \text{kg/m}^3$ .
Rearranging $ρ=mV\rho=\dfrac{m}{V}$ gives $m=ρVm=\rho V$ .
Substituting gives $kgm=1000\times0.25=250\ \text{kg}$ .
The mass of water in the tank is $kg250\ \text{kg}$ .

Example

Density from displaced water

A stone of mass $g96\ \text{g}$ is lowered into a measuring cylinder, and the water level rises from $cm345\ \text{cm}^3$ to $cm385\ \text{cm}^3$ .
The volume of the stone is the increase in the reading, $cm3V=85-45=40\ \text{cm}^3$ .
Using $ρ=mV\rho=\dfrac{m}{V}$ gives $g/cm3\rho=\dfrac{96}{40}=2.4\ \text{g/cm}^3$ .
In SI units this is $kg/m3\rho=2400\ \text{kg/m}^3$ .

Practical

Investigating the densities of solids and liquids

Aim: to determine the density of a regular solid, an irregular solid and a liquid by measuring mass and volume and using $ρ=mV\rho=\dfrac{m}{V}$ .
Apparatus: digital balance, $cm30\ \text{cm}$ ruler, vernier calipers, regular solid such as a metal cuboid, irregular solid such as a stone, measuring cylinder, displacement can, beaker, water, the liquid under test, paper towel and a heatproof tray to catch spills.
Variables: the material being tested is the independent variable, the calculated density is the dependent variable, and the temperature of the samples, the measuring instruments used and the method of reading the scales are controlled.
Method, regular solid:
- Check that the balance reads zero with nothing on the pan, then place the solid on it and record the mass.
- Measure the length, width and height with a ruler, or with calipers for a small dimension, taking each measurement in three different places.
- Calculate a mean for each dimension, then calculate the volume from the appropriate geometric formula.
- Calculate the density from $ρ=mV\rho=\dfrac{m}{V}$ , keeping the mass and volume units consistent.
Method, irregular solid:
- Measure and record the mass of the dry solid on the balance.
- Part fill a measuring cylinder with water and record the initial volume $V_{1}$ , reading the bottom of the meniscus with your eye level with the scale.
- Lower the solid gently on a thread until it is fully submerged, avoiding splashes and shaking out any trapped air bubbles.
- Record the new volume $V_{2}$ , so the volume of the solid is $V=V_{2}-V_{1}$ .
- Use a displacement can instead if the solid will not fit in the cylinder: fill the can until it stops dripping, submerge the solid, collect the overflow in a measuring cylinder and read its volume.
- Dry the solid before repeating any mass measurement, then calculate the density.
Method, liquid:
- Measure the mass $m_{1}$ of an empty, dry measuring cylinder.
- Pour in a measured volume of the liquid and record that volume.
- Measure the combined mass $m_{2}$ of the cylinder and liquid, so the mass of liquid is $m=m_{2}-m_{1}$ .
- Repeat for at least five different volumes of the same liquid, recording mass and volume each time.
Results: each material gives a value of density that stays the same whatever size of sample is used, which is what makes density useful for identifying a material.
Maths: plot mass on the vertical axis against volume on the horizontal axis for the liquid, draw a line of best fit through the origin, and take the gradient as the density.
Watch out: parallax when reading the meniscus, trapped air bubbles on the solid, water splashed out of the cylinder, a porous solid that soaks up water, and a balance that has not been zeroed all shift the result.
Improvements: read every scale at eye level, use a larger sample so the percentage uncertainty in each reading falls, use calipers for small dimensions, repeat each measurement and take a mean, and hold a floating solid under the surface with a sinker whose own volume is then subtracted.
Safety: stand glass measuring cylinders on a tray so they cannot topple, lower dense solids gently rather than dropping them, mop up spills at once to prevent slips, and wear eye protection if the liquid is anything other than water.

Exam technique

Density calculation method

Write $ρ=mV\rho=\dfrac{m}{V}$ before substituting, because the equation itself often carries a mark.
Show the volume calculation as a separate line when the volume has to be worked out from measured dimensions.
Give the unit with the final answer, since a correct number with no unit loses the last mark.
Name both measuring instruments when a question asks how density was found, then say which quantity each one measures.
State the subtraction explicitly for a displacement method, writing $V=V_{2}-V_{1}$ rather than describing it in words alone.
Name the change and its effect when asked for an improvement, such as reading at eye level so the meniscus is not misread.

Common Mistake

Do not treat density as the same thing as mass, because a large object made of a low density material can still be heavier than a small dense one.
Do not use the final measuring cylinder reading as the volume of an irregular solid, because the volume is the increase in the reading.
Do not include the mass of the container in the mass of a liquid, because the empty container must be weighed and subtracted.
Do not mix units, because grams divided by cubic metres gives a value that is wrong by a factor of a million.

Self review

State the density equation and give the SI unit of each quantity in it.
Calculate the density of a $kg0.50\ \text{kg}$ sample that occupies $m32.0\times10^{-4}\ \text{m}^3$ .
Convert $g/cm37.8\ \text{g/cm}^3$ into kilograms per cubic metre.
Describe how to measure the volume of an irregular solid that is too large for a measuring cylinder.
Explain how the mass of a liquid is obtained in the density practical.
Explain what the gradient of a mass against volume graph represents.

What density means

Definition

Density

Density is the mass per unit volume of a substance.

Density compares the mass of a substance with the volume that mass occupies.
A denser material fits more mass into the same volume, so a $cm31\ \text{cm}^3$ cube of lead has a far greater mass than a $cm31\ \text{cm}^3$ cube of aluminium.
Density is a property of the material rather than of the object, so a small steel bolt and a large steel girder have the same density.
Two objects with equal volumes have different masses when their materials have different densities.
Water has a density of about $kg/m31000\ \text{kg/m}^3$ , aluminium about $kg/m32700\ \text{kg/m}^3$ , lead about $kg/m311\,300\ \text{kg/m}^3$ and air about $kg/m31.2\ \text{kg/m}^3$ .
An object floats in a fluid when the density of the object is lower than the density of the fluid, so a heavy steel ship floats while a small steel nail sinks.

The density equation

The equation is $ρ=mV\rho=\dfrac{m}{V}$ , where $ρ\rho$ is density, $m$ is mass and $V$ is volume.
In SI units, $ρ\rho$ is in kilograms per cubic metre, $kg/m3\text{kg/m}^3$ , $m$ is in kilograms, $kg\text{kg}$ , and $V$ is in cubic metres, $m3\text{m}^3$ .
The symbol $ρ\rho$ is the Greek letter rho, and it is used for density so that $d$ can be kept for distance.
Rearranging for mass gives $m=ρVm=\rho V$ .
Rearranging for volume gives $V=mρV=\dfrac{m}{\rho}$ .
The volume of a cuboid is $V=l×w×hV=l\times w\times h$ , and the volume of a cylinder is $V=πr2hV=\pi r^{2}h$ .

Units and conversions

Mass and volume units must match, so kilograms with cubic metres give $kg/m3\text{kg/m}^3$ and grams with cubic centimetres give $g/cm3\text{g/cm}^3$ .
Converting between the two density units uses $kg/m31\ \text{g/cm}^3=1000\ \text{kg/m}^3$ .
Converting mass uses $kg1\ \text{g}=0.001\ \text{kg}$ , so $kg250\ \text{g}=0.250\ \text{kg}$ .
Converting volume uses $m31\ \text{cm}^3=1\times10^{-6}\ \text{m}^3$ , because there are $100$ centimetres in a metre and $100^{3}=1\,000\,000$ .
Liquid volumes are often measured in millilitres, and $cm31\ \text{ml}=1\ \text{cm}^3$ .
Read the question to find which density unit is wanted, then convert the mass and the volume before substituting into the equation.

Example

Density of a metal block

A rectangular block measures $cm8.0\ \text{cm}$ by $cm5.0\ \text{cm}$ by $cm2.0\ \text{cm}$ and has a mass of $g216\ \text{g}$ .
The volume is $cm3V=8.0\times5.0\times2.0=80\ \text{cm}^3$ .
Using $ρ=mV\rho=\dfrac{m}{V}$ gives $g/cm3\rho=\dfrac{216}{80}=2.7\ \text{g/cm}^3$ .
Converting gives $kg/m3\rho=2.7\times1000=2700\ \text{kg/m}^3$ .
The density of the block is $kg/m32700\ \text{kg/m}^3$ , which matches aluminium.

Example

Finding a mass from a volume

A tank holds $m30.25\ \text{m}^3$ of water, and the density of water is $kg/m31000\ \text{kg/m}^3$ .
Rearranging $ρ=mV\rho=\dfrac{m}{V}$ gives $m=ρVm=\rho V$ .
Substituting gives $kgm=1000\times0.25=250\ \text{kg}$ .
The mass of water in the tank is $kg250\ \text{kg}$ .

Example

Density from displaced water

A stone of mass $g96\ \text{g}$ is lowered into a measuring cylinder, and the water level rises from $cm345\ \text{cm}^3$ to $cm385\ \text{cm}^3$ .
The volume of the stone is the increase in the reading, $cm3V=85-45=40\ \text{cm}^3$ .
Using $ρ=mV\rho=\dfrac{m}{V}$ gives $g/cm3\rho=\dfrac{96}{40}=2.4\ \text{g/cm}^3$ .
In SI units this is $kg/m3\rho=2400\ \text{kg/m}^3$ .

Practical

Investigating the densities of solids and liquids

Aim: to determine the density of a regular solid, an irregular solid and a liquid by measuring mass and volume and using $ρ=mV\rho=\dfrac{m}{V}$ .
Apparatus: digital balance, $cm30\ \text{cm}$ ruler, vernier calipers, regular solid such as a metal cuboid, irregular solid such as a stone, measuring cylinder, displacement can, beaker, water, the liquid under test, paper towel and a heatproof tray to catch spills.
Variables: the material being tested is the independent variable, the calculated density is the dependent variable, and the temperature of the samples, the measuring instruments used and the method of reading the scales are controlled.
Method, regular solid:
- Check that the balance reads zero with nothing on the pan, then place the solid on it and record the mass.
- Measure the length, width and height with a ruler, or with calipers for a small dimension, taking each measurement in three different places.
- Calculate a mean for each dimension, then calculate the volume from the appropriate geometric formula.
- Calculate the density from $ρ=mV\rho=\dfrac{m}{V}$ , keeping the mass and volume units consistent.
Method, irregular solid:
- Measure and record the mass of the dry solid on the balance.
- Part fill a measuring cylinder with water and record the initial volume $V_{1}$ , reading the bottom of the meniscus with your eye level with the scale.
- Lower the solid gently on a thread until it is fully submerged, avoiding splashes and shaking out any trapped air bubbles.
- Record the new volume $V_{2}$ , so the volume of the solid is $V=V_{2}-V_{1}$ .
- Use a displacement can instead if the solid will not fit in the cylinder: fill the can until it stops dripping, submerge the solid, collect the overflow in a measuring cylinder and read its volume.
- Dry the solid before repeating any mass measurement, then calculate the density.
Method, liquid:
- Measure the mass $m_{1}$ of an empty, dry measuring cylinder.
- Pour in a measured volume of the liquid and record that volume.
- Measure the combined mass $m_{2}$ of the cylinder and liquid, so the mass of liquid is $m=m_{2}-m_{1}$ .
- Repeat for at least five different volumes of the same liquid, recording mass and volume each time.
Results: each material gives a value of density that stays the same whatever size of sample is used, which is what makes density useful for identifying a material.
Maths: plot mass on the vertical axis against volume on the horizontal axis for the liquid, draw a line of best fit through the origin, and take the gradient as the density.
Watch out: parallax when reading the meniscus, trapped air bubbles on the solid, water splashed out of the cylinder, a porous solid that soaks up water, and a balance that has not been zeroed all shift the result.
Improvements: read every scale at eye level, use a larger sample so the percentage uncertainty in each reading falls, use calipers for small dimensions, repeat each measurement and take a mean, and hold a floating solid under the surface with a sinker whose own volume is then subtracted.
Safety: stand glass measuring cylinders on a tray so they cannot topple, lower dense solids gently rather than dropping them, mop up spills at once to prevent slips, and wear eye protection if the liquid is anything other than water.

Exam technique

Density calculation method

Write $ρ=mV\rho=\dfrac{m}{V}$ before substituting, because the equation itself often carries a mark.
Show the volume calculation as a separate line when the volume has to be worked out from measured dimensions.
Give the unit with the final answer, since a correct number with no unit loses the last mark.
Name both measuring instruments when a question asks how density was found, then say which quantity each one measures.
State the subtraction explicitly for a displacement method, writing $V=V_{2}-V_{1}$ rather than describing it in words alone.
Name the change and its effect when asked for an improvement, such as reading at eye level so the meniscus is not misread.

Common Mistake

Do not treat density as the same thing as mass, because a large object made of a low density material can still be heavier than a small dense one.
Do not use the final measuring cylinder reading as the volume of an irregular solid, because the volume is the increase in the reading.
Do not include the mass of the container in the mass of a liquid, because the empty container must be weighed and subtracted.
Do not mix units, because grams divided by cubic metres gives a value that is wrong by a factor of a million.

Self review

State the density equation and give the SI unit of each quantity in it.
Calculate the density of a $kg0.50\ \text{kg}$ sample that occupies $m32.0\times10^{-4}\ \text{m}^3$ .
Convert $g/cm37.8\ \text{g/cm}^3$ into kilograms per cubic metre.
Describe how to measure the volume of an irregular solid that is too large for a measuring cylinder.
Explain how the mass of a liquid is obtained in the density practical.
Explain what the gradient of a mass against volume graph represents.

14.1.2 Density

What density means

The density equation

Units and conversions

Density of a metal block

Finding a mass from a volume

Density from displaced water

Investigating the densities of solids and liquids

Density calculation method

Review 14.1.2 Density by teaching Genie

14.1.2 Density

What density means

The density equation

Units and conversions

Density of a metal block

Finding a mass from a volume

Density from displaced water

Investigating the densities of solids and liquids

Density calculation method

Review 14.1.2 Density by teaching Genie

14.1.2 Density

What density means

The density equation

Units and conversions

Density of a metal block

Finding a mass from a volume

Density from displaced water

Investigating the densities of solids and liquids

Density calculation method

Review 14.1.2 Density by teaching Genie

14.1.2 Density Revision Guide

14.1.2 Density

What density means

The density equation

Units and conversions

Density of a metal block

Finding a mass from a volume

Density from displaced water

Investigating the densities of solids and liquids

Density calculation method

Review 14.1.2 Density by teaching Genie

14.1.2 Density Revision Guide