What density means
Definition
Density
The mass per unit volume of a material.
Definition
Mass
The amount of matter in an object, measured in kilograms.
Definition
Volume
The amount of space that an object or substance takes up, measured in cubic metres.
- Density measures how much mass is packed into each unit of volume of a material.
- A material with a high density has a large mass in a small volume, while a material with a low density has a small mass spread through a large volume.
- A 1 kg1\ \text{kg}1 kg bag of sugar and a 1 kg1\ \text{kg}1 kg bag of crisps have the same mass, but the crisps fill a much larger volume, so they have a much lower density.
- Density is a property of the material, not of the object, so a steel paper clip and a steel girder have the same density even though their masses are very different.
- Density is found using the equation ρ=mV\rho = \dfrac{m}{V}ρ=Vm, where ρ\rhoρ is density, mmm is mass and VVV is volume.
- In words, the equation is density=massvolume\text{density} = \dfrac{\text{mass}}{\text{volume}}density=volumemass, and it must be recalled rather than looked up.
- The equation can be rearranged to m=ρVm = \rho Vm=ρV to find a mass, or to V=mρV = \dfrac{m}{\rho}V=ρm to find a volume.
Units of density
- The SI unit of density is the kilogram per cubic metre, written kg/m3\text{kg/m}^3kg/m3, which comes from mass in kg\text{kg}kg divided by volume in m3\text{m}^3m3.
- In the laboratory, density is often given in grams per cubic centimetre, written g/cm3\text{g/cm}^3g/cm3, which comes from mass in g\text{g}g divided by volume in cm3\text{cm}^3cm3.
- A cube of side 1 m1\ \text{m}1 m contains 100×100×100=1 000 000100 \times 100 \times 100 = 1\,000\,000100×100×100=1000000 cubes of side 1 cm1\ \text{cm}1 cm, so 1 m3=1×106 cm31\ \text{m}^3 = 1 \times 10^{6}\ \text{cm}^31 m3=1×106 cm3.
- A density of 1 g/cm31\ \text{g/cm}^31 g/cm3 is equal to 1000 kg/m31000\ \text{kg/m}^31000 kg/m3, so a value in g/cm3\text{g/cm}^3g/cm3 is multiplied by 100010001000 to give it in kg/m3\text{kg/m}^3kg/m3.
- Pure water has a density of about 1000 kg/m31000\ \text{kg/m}^31000 kg/m3, aluminium about 2700 kg/m32700\ \text{kg/m}^32700 kg/m3, steel about 7900 kg/m37900\ \text{kg/m}^37900 kg/m3 and air about 1.2 kg/m31.2\ \text{kg/m}^31.2 kg/m3.
- The mass and the volume must be in matching units before dividing, so a mass in grams is never divided by a volume in cubic metres.
Example
Density of a stone
- A stone has a mass of 0.60 kg0.60\ \text{kg}0.60 kg and a volume of 2.4×10−4 m32.4 \times 10^{-4}\ \text{m}^32.4×10−4 m3.
- Write the equation: ρ=mV\rho = \dfrac{m}{V}ρ=Vm.
- Substitute the values: ρ=0.602.4×10−4\rho = \dfrac{0.60}{2.4 \times 10^{-4}}ρ=2.4×10−40.60.
- Calculate the result: ρ=2500 kg/m3\rho = 2500\ \text{kg/m}^3ρ=2500 kg/m3.
Example
Mass of air in a classroom
- A classroom measures 8.0 m8.0\ \text{m}8.0 m by 6.0 m6.0\ \text{m}6.0 m by 3.0 m3.0\ \text{m}3.0 m, and air has a density of 1.2 kg/m31.2\ \text{kg/m}^31.2 kg/m3.
- Find the volume: V=8.0×6.0×3.0=144 m3V = 8.0 \times 6.0 \times 3.0 = 144\ \text{m}^3V=8.0×6.0×3.0=144 m3.
- Rearrange the density equation: m=ρVm = \rho Vm=ρV.
- Substitute the values: m=1.2×144=172.8 kgm = 1.2 \times 144 = 172.8\ \text{kg}m=1.2×144=172.8 kg.
- To two significant figures the air in the room has a mass of 170 kg170\ \text{kg}170 kg, which is more than the mass of two adults.
Measuring mass and volume
Definition
Water displacement
A method of finding the volume of an object from the volume of water it pushes aside when fully submerged.
Definition
Meniscus
The curved surface of a liquid in a narrow container such as a measuring cylinder.
Definition
Upthrust
The upward force that a fluid exerts on an object that is partly or fully submerged in it.
- Mass is measured with a top-pan balance, which should be set to zero before the object is placed on it.
- The volume of a regular solid, such as a cube, cuboid or cylinder, is calculated from its measured dimensions.
- The volume of an irregular solid, such as a stone or a key, is found by water displacement, because a fully submerged object pushes aside a volume of water equal to its own volume.
- The volume of a liquid is read directly from a measuring cylinder, looking at the bottom of the meniscus with the eye level with the liquid surface.
- Reading the scale from above or below eye level causes a parallax error, which makes every reading too high or too low.
- When an object is lowered into water, the water pushes up on it with a force called upthrust, which is why objects feel lighter when submerged.
Practical
Determining the density of solids and liquids
- Aim: to determine the densities of a regular solid, an irregular solid and a liquid by measuring their mass and volume.
- Apparatus: top-pan balance reading to 0.01 g0.01\ \text{g}0.01 g, ruler with a millimetre scale, vernier callipers, a metal cuboid, a small irregular stone, a eureka can, a 100 cm3100\ \text{cm}^3100 cm3 measuring cylinder, a 250 cm3250\ \text{cm}^3250 cm3 beaker, thread, a supply of water, the liquid to be tested and paper towels.
- Method, regular solid:
- Zero the balance, place the metal cuboid in the centre of the pan and record its mass in grams.
- Measure the length, width and height of the cuboid with vernier callipers, closing the jaws gently so that they touch the metal without squeezing it.
- Measure each dimension in three different places and calculate a mean, because the block may not be perfectly even.
- Calculate the volume in cm3\text{cm}^3cm3 using V=length×width×heightV = \text{length} \times \text{width} \times \text{height}V=length×width×height, then calculate the density using ρ=mV\rho = \dfrac{m}{V}ρ=Vm.
- Method, irregular solid:
- Zero the balance, dry the stone with a paper towel and record its mass.
- Stand the eureka can on a level bench and fill it with water until water starts to run out of the spout.
- Wait until the dripping has completely stopped, so that the water surface sits exactly level with the bottom of the spout.
- Place the empty measuring cylinder under the spout.
- Tie the thread around the stone and lower it slowly into the eureka can until it is fully submerged, without splashing and without letting it touch the sides.
- Wait until water has stopped dripping from the spout, then read the volume of water in the measuring cylinder at eye level from the bottom of the meniscus.
- Record this volume as the volume of the stone, then calculate its density using ρ=mV\rho = \dfrac{m}{V}ρ=Vm.
- Method, liquid:
- Place the empty, dry measuring cylinder on the balance and record its mass.
- Pour about 50 cm350\ \text{cm}^350 cm3 of the liquid into the cylinder and read its exact volume at eye level from the bottom of the meniscus.
- Put the cylinder back on the balance and record the combined mass of the cylinder and the liquid.
- Subtract the mass of the empty cylinder from the combined mass to find the mass of the liquid.
- Add further 10 cm310\ \text{cm}^310 cm3 portions of liquid, recording the total volume and combined mass each time, and calculate the density for every reading.
- Results: record every measurement in a table with the quantity and its unit in each column heading, and give each density to a consistent number of significant figures.
- Maths: for the liquid, plot mass on the yyy-axis against volume on the xxx-axis, and the gradient of the straight line through the origin gives the density of the liquid.
- Watch out: air bubbles clinging to the stone make the displaced volume too large, water left in the spout from filling makes it too small, and a wet stone gives a mass that is too large.
- Safety: mop up spilled water straight away so no one slips, keep the balance away from the water, and wear eye protection if the test liquid is anything other than water.
Practical
Investigating Archimedes' principle
- Aim: to compare the upthrust on a submerged object with the weight of the water it displaces.
- Apparatus: a 000 to 10 N10\ \text{N}10 N newton meter, a metal mass of about 500 g500\ \text{g}500 g with a hook, a eureka can, a small beaker, a top-pan balance, a clamp stand and a supply of water.
- Method:
- Zero the newton meter, hang the metal mass from it and record the weight of the mass in air.
- Find the mass of the empty, dry beaker on the balance and record it.
- Fill the eureka can until water runs from the spout, wait for the dripping to stop, then place the beaker under the spout.
- Lower the mass on the newton meter slowly into the eureka can until it is completely under water but not touching the bottom or sides.
- Record the new reading on the newton meter while the mass is submerged and still.
- When the dripping stops, find the mass of the beaker and the displaced water, then subtract the mass of the empty beaker.
- Convert the mass of displaced water into kilograms and multiply by 10 N/kg10\ \text{N/kg}10 N/kg to find its weight.
- Results: the upthrust is the weight in air minus the reading when submerged, and this value should be very close to the weight of the displaced water.
- Maths: if the reading falls from 5.0 N5.0\ \text{N}5.0 N to 4.4 N4.4\ \text{N}4.4 N, the upthrust is 0.6 N0.6\ \text{N}0.6 N, and the displaced water should have a mass of about 60 g60\ \text{g}60 g.
- Watch out: the newton meter must be read at eye level while the mass is still, and any water lost by splashing makes the displaced weight too small.
- Safety: clamp the stand firmly to the bench so that the heavy mass cannot fall on your feet, and keep water away from any electrical sockets.
Exam technique
Describing a density method
- Name the measuring instrument for each quantity, such as a balance for mass and a measuring cylinder or eureka can for volume.
- State clearly that the volume of water displaced is equal to the volume of the object.
- Finish every method with the calculation step, writing out ρ=mV\rho = \dfrac{m}{V}ρ=Vm.
- Give an improvement with its reason, for example repeating each reading and calculating a mean to reduce the effect of random errors.
- In a calculation, show the equation, the substituted numbers and the answer with its unit, because each stage can earn credit.
Common Mistake
- Do not confuse mass with density, because a large wooden log can have a greater mass than a steel bolt while having a much lower density.
- Do not divide a mass in grams by a volume in cubic metres, because the units must match before the calculation.
- Do not start collecting water from a eureka can until it has stopped dripping after filling, or the measured volume will be too large.
- Do not forget that 1 m31\ \text{m}^31 m3 is one million cubic centimetres, not one hundred.
Self review
- Define density.
- Write the equation linking density, mass and volume, and rearrange it to make volume the subject.
- Convert a density of 2.7 g/cm32.7\ \text{g/cm}^32.7 g/cm3 into kg/m3\text{kg/m}^3kg/m3.
- Describe how the volume of an irregular solid is measured using a eureka can.
- Explain why the bottom of the meniscus should be read at eye level.
