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.
Density is also mentioned in
Pressure & upthrust in liquids, density and why do
objects float or sink?
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 more spread out the particles, the
lower the material's density  which is why gases have a very low density.
The more closely the particles are packed
together, the greater the density  which is why solids have the highest
density.
See the
particle model of the states of
matter
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/m^{3})
= MASS (kg)
÷
VOLUME (m^{3})
Density units in physics are usually kg/m^{3}.
However, in chemistry, density data is often
quoted in g/cm^{3} because most quantitative measurements in a
school/college chemistry laboratory are usually quoted in grams (g) and ml (cm^{3}),
so I've sometimes quoted both sets of units,
so don't get them muddled! and note that:
g/cm^{3} = kg/m^{3}
÷ 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 = cm^{3}, 1
cubic metre = 1 million cm^{3}, so: cm^{3} ÷
10^{6} = m^{3} and m^{3}
x 10^{6} = cm^{3}
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/m^{3}) 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) Measuring the density of 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,
attached to a very fine thread, into
the water.
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/cm^{3} (1000 kg/m^{3}), 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) Measuring the density of an 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 = πr^{2}, where π
= pi = 3.142 and r = radius of the cylinder.
Therefore volume of cylinder: V = πr^{2}l
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 cm^{3}
of water (8.5 ml), calculate its density.
density of solid
ρ
= m ÷ v,
ρ = 27.2 ÷ 8.5 =
3.2 g/cm^{3}
However, you may have to calculate the
density in kg/m^{3} and this is arithmetically a bit more awkward!
27.2 g = 27.2/1000 = 0.0272 kg
and 8.5 cm^{3} = 8.5 / 10^{6} = 8.5 x 10^{6}
m^{3} (1 m^{3} = 10^{6} cm^{3})
ρ
= m ÷ v,
ρ = 0.0272 ÷
(8.5 x 10^{6}) = 0.0272 ÷ 0.0000085 =
3200 kg/m^{3}
It might be handy to know that kg/m^{3}
is 1000 x g/cm^{3} !!! 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 cm^{3},
volume = 180/10^{6} = 1.8 x 10^{4} m^{3}
(0.00018)
density of solid
ρ
= m ÷ v,
ρ = 1.42 ÷
0.00018 = 7889
kg/m^{3}
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/m^{3}
radius of cylinder = diameter/2 = 3.0/2 = 1.5 cm,
crosssection area = πr^{2} =
3.142 x 1.5^{2} = 7.070 cm^{2}
volume of cylinder = 7.070 x 12.0 = 84.83
cm^{3}, 84.83/10^{6} = 8.483 x 10^{5} m^{3}
(remember 1 m^{3} = 10^{6} cm^{3})
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/m^{3} (3 sf)
Q4
A cube of material has a side length of 2.5 cm.
If the material has a density of 5000
kg/m^{3}, what is the mass of the cube in g and kg?
l = 2.5/100 = 0.025 m, therefore volume =
l^{3} = 0.025^{3} = 1.5625 x 10^{5} m^{3}
ρ
= m ÷ v, m =
ρ x v = 5000 x 1.5625 x 10^{5}
=
0.078 kg (x 1000 =
78 g)
Q5
?
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) Measuring the density of a liquid 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 cm^{3} (in a 50 ml measuring
cylinder for best accuracy)
Make sure the bottom of the liquid's
meniscus rests exactly on the 50 cm^{3} 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
This is not a very accurate method,
the measuring cylinder is not calibrated to a high standard.
(ii) Measuring the density of a liquid 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 to 50 cm^{3}) or a 25 ml (25 cm^{3}) 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
This is a more accurate method than
(ii), a pipette or burette are calibrated to a higher standard than a
measuring cylinder.
(iii) Measuring the relative density of a
liquid using a density bottle
The liquid must not be too viscous (too
sticky to run freely).
A density bottle is a glass chamber with
a ground neck entrance. Into the neck goes a precisely fitting stopper with
a capillary tube running down its central axis  see diagrams above. Its
volume can be as little as 1020 cm^{3} (10 ml).
Procedure
You need to do the experiment at a
constant temperature because liquids expand/contract if heated/cooled i.e.
density is temperature dependent.
(3a)
The density bottle and stopper must be completely empty and dried in an
oven before use  obviously must be allowed to cool down to room
temperature at ~25^{o}C.
The density bottle and stopper
are then weighed on a mass balance, preferably to
± 0.01 g (m1)
(3b)
The density bottle is then completely filled with pure water and the
stopper pushed down to expel any excess water.
With a tissue of filter paper,
any excess fluid on the top of the stopper should be carefully
removed.
You should also make sure there
is no spilt liquid on the outside of the bottle, if so, this should
be carefully wiped off too.
In other words, all the exterior
of the density bottle and stopper must be dry.
Also make sure you don't absorb
any liquid from the capillary tube of the stopper.
(3c)
The filled density bottle of water is then reweighed on the mass balance
(m2).
(3d)
The steps from 3a to 3c are then repeated and the final mass of the
bottle/stopper filled is measured (m3).
Results (fiction!)
m1 = mass of empty bottle +
stopper = 35.51 g
m2 = mass of empty bottle +
stopper + water = 50.62 g
m3 = mass of empty bottle +
stopper + liquid X = 48.42 g
Calculation
You can get the accurate density of pure
water from a data table on the internet e.g.
temperature 
10^{o}C 
20^{o}C 
30^{o}C 
density g/cm^{3} 
0.9997 
0.9982 
0.9957 
density kg/m^{3} 
999.7 
998.2 
995.7 
(Note: g/cm^{3} x 1000 = kg/m^{3})
However, as you can see, by assuming the
density of water is 1.00 g/cm^{3} (1000 kg/m^{3}), at 20^{o}C
you only introduce a 0.2% error.
{((1.00  0.998)/1.00) x 100},
for a school/college laboratory,
that's a pretty good accuracy for doing any real experiment.
Assuming constant temperature and
therefore constant volume, the calculation is as follows.
m2  m1 = mass of water = m4 = 50.62
 35.51 = 15.11 g of water
m3  m1 = mass of liquid X = m5 =
48.42  35.51 = 12.91 g of liquid X
Density of liquid ρ_{water}
= m_{water}
÷ v_{water},
v = m ÷
ρ
Therefore
volume of bottle = volume of water = mass of water
÷ density of water
= 15.11 ÷ 1.00 =
15.11 cm^{3}
We can now calculate the density
of liquid X.
Density of liquid
X ρ_{X}
= m_{X} ÷ v_{X}
= mass of liquid X ÷ volume of liquid X
= 12.91 ÷ 15.11 =
0.854 g/cm^{3}
(x 1000 =
854 kg/m^{3})
Source of errors
Not ensuring there is no excess
liquid on the outside of the density bottle or on the top of the
stopper.
Not ensuring the bottle is
completely dry and empty before use.
Not ensuring all measurements are
done at the same recorded temperature.
Again, this
again is much more accurate method than (ii) and more probably more
accurate than (iii) too, because you are calibrating the density bottle to a
high standard.
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
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.
In a closed system, when a
substance changes state, it is a physical change (NOT
chemical), there is no change in mass (same number of
particles), but there is a change in particle arrangement which
leads to a change in density.
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
GASES:
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
very widely spread out in the atmosphere giving air a very low density.
Density of air in the atmosphere is 1.2
kg/m^{3}. Density of steam = 0.6 kg/m^{3}.
Hydrogen and helium are the 'lightest',
lowest density gases, ρ (H_{2})
= 0.10 kg/m^{3}, ρ (He)
= 0.17 kg/m^{3}
Carbon dioxide sinks in air because it is
'heavier', it has a higher density than air, density = 1.9 kg/m^{3}
(air is 1.2).
If you compress a gas, you force the
molecules closer together, same mass in smaller volume, so the density
increases.
If you heat a gas and it can expand, the
particles are further apart and the density will decrease.
LIQUIDS:
In liquids, the particles are close together, usually giving
high 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/m^{3},
more than 1600 times more dense than steam!
Density of liquid air is 974 kg/m^{3},
see this sharply contrasts with gaseous air, which is over 800 times less
dense at 1.2 kg/m^{3}!
Hydrocarbon petroleum oil, typically has
a density of 820 kg/m^{3}, 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/m^{3}, 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.
This effect is used in liquid
thermometers e.g. those containing a capillary tube of mercury or
coloured alcohol.
The thermal expansion is proportional
to temperature so you can fit a calibrated linear scale next to the
capillary tube.
SOLIDS:
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/m^{3}.
Because of the particular crystal
structure of ice (ρ
= 931 kg/m^{3}), solid water is less dense than liquid water (ρ
= 1000 kg/m^{3}), 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.
Density is also mentioned in
Pressure & upthrust in liquids, density and why do
objects float or sink?
OCR A gcse 91
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/m^{3}) = mass (kg)
/ volume (m^{3}) 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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