The Density of materials and the particle model of matter

Doc Brown's Physics Revision Notes

Suitable for GCSE/IGCSE Physics/Science courses or their equivalent

 A simple experiment to determine the density of a material is described,

 and how to use the formula for density to calculate density.

This section also looks at density from the point of view of a particle model of matter.

 In everyday language, dense objects are described as 'heavy' and less dense objects as 'light', but this is not a correct scientific description of the object or the nature of the material.

 What is density? and what is the formula for density?

Density is a measure of how compact a material is - it indicates how much space or volume a given mass occupies.

The greater the mass of material in a given volume, the greater the density of the material.

The density of a material depends on what it is made up of (atoms and their arrangement) and its physical state.

The density for a given material is the same whatever its shape or size for a given physical state.

The scientific symbol for density is the Greek letter rho  (ρ)

The formula for density is: ρ = m ÷ v

DENSITY (kg/m3) = MASS (kg) ÷ VOLUME (m3)

Density units in physics are usually kg/m3.

However, in chemistry, density data is often quoted in g/cm3 because most quantitative measurements in a school/college chemistry laboratory are usually quoted in grams (g) and ml (cm3), so I've sometimes quoted both sets of units,

so don't get them muddled! and note that:   g/cm3 = kg/m3 ÷ 1000  (can you work out why?)

Its advisable to be able to convert mass and volume units e.g.

mass: 1 kilogram = 1000 grams, so:  g ÷ 1000 = kg   and   kg x 1000 = g

volume: ml = cm3, 1 cubic metre = 1 million cm3,  so:  cm3 ÷ 106 = m3   and   m3 x 106 = cm3


Density is very important property to know about a material

e.g. if the density of an object is less than that of water (~1000 kg/m3) it floats

if the density of an object is more than that of water it sinks!

In general: if the object has a density < fluid it floats and if density of object is > fluid it sinks.

However, although shape doesn't affect density, shape does, otherwise, how can a steel ship float on water!?

All is explained in the section on 'floating and sinking'


Experiments to determine the density of a material

All the apparatus needed for the different methods for measuring density is described below and illustrated in one big diagram via parts (1a), (1b), (2a) and (2b).

By some means you need to know a specific mass and volume of a material to calculate its density.

You need to relate the methods described to the picture diagram, which I've repeated a few times.

Simple experiments to measure the density of a solid material

(i) An irregularly shaped object

Any lump of a solid material can be accurately weighed on an electronic balance (2b) (usually grams, g).

Don't forget to tare the mass balance to zero before placing the object on it.

If the solid is a lump with an irregular shape (and insoluble in water!) you can use a Eureka can (displacement can) to measure its volume.

The eureka can (1a) is filled with water above the spout and any excess drains off into the measuring cylinder so that the water level is just under the spout.

Then empty the collection measuring cylinder and place it under the spout again.

You then carefully lower the object into the water attached to a very fine thread.

As the object enters the water, the water level rises and a volume of water equal to the volume of the object is displaced from the eureka can down the spout and measured on collection in the measuring cylinder (1b).

Only measure the volume when the spout has stopped dripping, otherwise you won't measure the correct volume!

You cannot use this method if the object has a density <1.0 g/cm3 (1000 kg/m3), because it floats on water and only partially displaces the water. Repeat several times and calculate the average volume measured.

Then apply the density formula: ρ = m ÷ v


(ii) A regular shaped object

Again the block of material is weighed on the electronic balance (2b).

If the material is a perfectly shaped cube or rectangular block (solid cuboid), you can then accurately measure the length, breadth and height and calculate the volume: V = l x b x h.

If it is a regular solid shaped cylinder you need to work out the surface area of the end and multiply this by the length or height of the cylinder. Area of a circle = πr2, where π = pi = 3.142 and r = radius of the cylinder.

Therefore volume of cylinder: V = πr2l

Then apply the density formula:  ρ = m ÷ v


Calculation of density

Q1 Irregular shaped solid object

A stone weighing 27.2 g displaced 8.5 cm3 of water (8.5 ml), calculate its density.

 density of solid ρ = m ÷ v, ρ = 27.2 ÷ 8.5 = 3.2 g/cm3

However, you may have to calculate the density in kg/m3 and this is arithmetically a bit more awkward!

27.2 g = 27.2/1000 = 0.0272 kg   and   8.5 cm3 = 8.5 / 106 = 8.5 x 10-6 m3    (1 m3 = 106 cm3)

 ρ = m ÷ v, ρ = 0.0272 ÷ (8.5 x 10-6) = 0.0272 ÷ 0.0000085 = 3200 kg/m3

It might be handy to know that kg/m3 is 1000 x g/cm3 !!! 3.2 x 1000 = 3200 !!!


Q2 A regular solid block

A block of iron had dimensions of 3.0 cm x 5.0 cm x 12.0 cm and weighed 1.420 kg, calculate the density of iron.

Volume of block = 3 x 5 x 12 = 180 cm3,  volume = 180/106 = 1.8 x 10-4 m3   (0.00018)

 density of solid ρ = m ÷ v, ρ = 1.42 ÷ 0.00018 = 7889 kg/m3


Q3 A regular solid cylinder of an alloy has a diameter of 3.0 cm, a length of 12.0 cm and a mass of 750 g.

Calculate the density in kg/m3

radius of cylinder = diameter/2 = 3.0/2 = 1.5 cm,

cross-section area = πr2 = 3.142 x 1.52 = 7.070 cm2

volume of cylinder = 7.070 x 12.0 = 84.83 cm3, 84.83/106 = 8.483 x 10-5 m3   (remember 1 m3 = 106 cm3)

mass of cylinder = 750/1000 = 0.750 kg  (1 kg = 1000 g)

 density of solid ρ = m ÷ v, ρ = 0.750 ÷ 8.483 x 10-5 = 8841 = 8840 kg/m3  (3 sf)


Simple experiments to determine the density of a liquid

To measure the density of liquid you need to weighed an accurately measured volume of the liquid.

(i) Using a measuring cylinder (2b)

If you are only using 10 ml of liquid, you should use a 10 ml measuring cylinder , a 50 ml measuring cylinder would not be accurate enough.

A clean empty measuring cylinder is weighed on the electronic balance (m1).

The liquid under investigation is poured to a convenient volume v eg 50 cm3 (in a 50 ml measuring cylinder for best accuracy)

Make sure the bottom of the liquid's meniscus rests exactly on the 50 cm3 mark.

The measuring cylinder and liquid are then reweighed (m2).

The difference between the weights gives you the mass m of the liquid (m = m2 - m1)

density of liquid ρ = m ÷ v


(ii) Using a burette or pipette (both more accurate than a measuring cylinder) (2a) and (2b)

Weigh a suitable container, eg a conical flask or beaker, on the electronic balance (m1).

Carefully measure into the container an accurately known volume v of the liquid under investigation.

You can use a burette (for any volume from 10.00 to 50 cm3) or a 25.00 ml (25 cm3) pipette.

In either case make sure the bottom of the liquid's meniscus rests exactly on the calibration mark for selected volume.

The container and liquid are reweighed (m2) and the difference in weight is the mass m of the liquid (m = m2 - m1)

density of liquid ρ = m ÷ v


To calculate the density of liquid is no different from the calculations for a solid so I don't feel the need to add any more density calculations.

DENSITY and the particle model

(c) doc b (c) doc b (c) doc b

The density of a material depends on the nature of the material eg air, water, wood or iron AND the physical state of the material, which is how the particles are arranged. We can use the particle model of matter to partly explain the differences in density between different materials, and in particular the difference in density between gaseous, liquid and solid state of specific substance. When explaining different density values, you must consider both the kinetic energy of the particles and the arrangement of the particles.

Applying the particle model to the different densities of the states of matter.

(c) doc bGASES: The particles have more kinetic energy than in liquids or solids and can move around at random quite freely. This enables the particles to spread out and fill all the available space giving a material a very low density compared to liquids and solids. With very weak forces of attraction between the particles there is no constraint on their movement - they can't club together to form a liquid or solid.

In a substance like air, the particles are widely spread out in the atmosphere giving air a very low density.

Density of air in the atmosphere is 1.2 kg/m3. Density of steam = 0.6 kg/m3.

Hydrogen and helium are the 'lightest', lowest density gases, ρ (H2) = 0.10 kg/m3, ρ (He) = 0.17 kg/m3

Carbon dioxide sinks in air because it is 'heavier', it has a higher density than air, density = 1.9 kg/m3 (air is 1.2).

If you compress a gas, you force the molecules closer together, same mass in smaller volume, so the density increases.

(c) doc bLIQUIDS: In liquids, the particles are close together, usually giving densities a bit less than the solid, but much greater than the density of gases. The forces between liquid particles are greater than those between gaseous particles and are strong enough so they are attracted close together leaving little free space. However, liquid particles still have enough kinetic energy to move around at random and create a little space and on average are spaced out just that little bit more than in solids, hence their slightly lower density than the solid.

Water has a density of 1000 kg/m3, more than 1600 times more dense than steam!

Density of liquid air is 974 kg/m3, see this sharply contrasts with gaseous air, which is over 800 times less dense at 1.2 kg/m3!

Hydrocarbon petroleum oil, typically has a density of 820 kg/m3, less dense than water, so it floats on it!

Mercury atoms have a much greater atomic mass than hydrogen, oxygen or carbon so liquid mercury has a density of 13 584 kg/m3, that is more than 13 times more dense than water!

If you heat a liquid, increasing its temperature, the particles gain kinetic energy and the collisions become more frequent and energetic and this enables the particles on average to spread out a bit more, increasing the liquid volume and lowering the density.

(c) doc bSOLIDS: The strongest interparticle forces of attraction occur in solids where particles are attracted and compacted as much as is possible. The particles can only vibrate around fixed positions in the structure and do not have sufficient kinetic energy to break free and move around creating a little space like in liquids. The result is the highest density for the state of a specific material. Although liquid densities for a specific material are just a bit less than those of the solid, both the solid and liquid states have much greater densities than the gaseous or vapour state.

Generally speaking the solid state exhibits the highest density.

In a very dense material like iron, the particles (iron atoms) are not only heavy, but very close together giving the high density of 7870 kg/m3.

Because of the particular crystal structure of ice (ρ = 931 kg/m3), solid water is less dense than liquid water (ρ = 1000 kg/m3), so ice floats on water! Very unusual! In the solid the water molecules form a very open crystal structure in which the water molecules are actually slightly further apart on average compared to their compactness in liquid water.

When dealing with insulating materials, beware!, their densities are much lower than the 'bulk' solid because very low density gases like air or carbon dioxide are trapped in them, considerably lowering the overall density of the original solid material.

If you heat a solid, increasing its temperature, the particles gain kinetic energy and the vibrations become more energetic and this enables the particles on average to spread out a bit more ('pushing' each other apart), increasing the solid volume and lowering the density.

OCR A gcse 9-1 physics P1.1d Be able to define density. From measurements of length, volume and mass be able to calculate density.  See also the investigation of Archimedes’ Principal using eureka cans. density (kg/m3) = mass (kg) / volume (m3)    d = m/v P1.1e Be able to explain the differences in density between the different states of matter in terms of the arrangements of the atoms and molecules. P1.1f Be able to apply the relationship between density, mass and volume to changes where mass is conserved. You should be familiar with the structure of matter and the similarities and differences between solids, liquids and gases. You should have a simple idea of the particle model and be able to use it to model changes in particle behaviour during changes of state. You should be aware of the effect of temperature in the motion and spacing of particles and an understanding that energy can be stored internally by materials. P1.2a Be able to describe how mass is conserved when substances melt, freeze, evaporate, condense or sublimate.  Use of a data logger to record change in state and mass at different temperatures.  Demonstration of distillation to show that mass is conserved during evaporation and condensation. P1.2b Be able to describe that these physical changes differ from chemical changes because the material recovers its original properties if the change is reversed.

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