Density: how much matter fills each unit of volume
Definition
Density
Mass per unit volume.
ρ=mV \rho = \frac{m}{V} ρ=Vmwhere ρ\rhoρ (rho) is the density, mmm is the mass and VVV is the volume; the unit is g/cm3\text{g/cm}^3g/cm3 or kg/m3\text{kg/m}^3kg/m3.
- Recall the equation and its rearrangements, m=ρVm = \rho Vm=ρV and V=m/ρV = m/\rhoV=m/ρ, because there is no formula sheet.
- Density is a property of the material: a paperclip and a girder of the same steel have the same density, because doubling the volume of a material also doubles its mass.
- The two units are g/cm3\text{g/cm}^3g/cm3 (convenient in the laboratory) and kg/m3\text{kg/m}^3kg/m3 (SI), with 1 g/cm3=1000 kg/m31\ \text{g/cm}^3 = 1000\ \text{kg/m}^31 g/cm3=1000 kg/m3.
- Water has a density of about 1.0 g/cm31.0\ \text{g/cm}^31.0 g/cm3, the same as 1000 kg/m31000\ \text{kg/m}^31000 kg/m3.
Practical
Investigation: measuring the density of liquids and solids
- Aim: to measure the density of a liquid, a regular solid and an irregular solid, using ρ=m/V\rho = m/Vρ=m/V.
- Apparatus: eye protection, balance, measuring cylinder, ruler (or vernier calipers), a regular block or cylinder, an irregular solid that sinks, thread, and water.
- Method:
- Regular solid: measure the mass on a balance and the dimensions with a ruler, then calculate the volume from the shape.
- Liquid: mass the empty measuring cylinder, pour in the liquid and read its volume at the bottom of the meniscus, re-mass, and subtract to get the liquid's mass.
- Irregular solid: measure its mass, then find its volume by displacement, the rise in level when it is lowered fully into water in a measuring cylinder.
- Calculate ρ=m/V\rho = m/Vρ=m/V for each.
- Measure:
- Measure the mass with the balance and the volume from the dimensions or from displacement.
- Tabulate the results and calculate each density to 2 significant figures.
- Result: each material has its own characteristic density, given by ρ=m/V\rho = m/Vρ=m/V.
- Watch out:
- Read the meniscus at eye level to avoid parallax.
- Measure a sphere's diameter by trapping it between two blocks, and submerge the solid fully when using displacement.
- Quote sensible significant figures for each density.
- Safety: wear eye protection.
Measuring the density of a liquid
- The mass of a liquid cannot be measured without a container, so the container is subtracted.
- Find the mass of an empty, dry measuring cylinder on a balance.
- Pour in the liquid and record its volume, reading the scale at eye level and at the bottom of the meniscus.
- Find the mass of the cylinder plus liquid, then take mass of liquid === (cylinder +++ liquid) −-− (empty cylinder).
- Calculate ρ=m/V\rho = m/Vρ=m/V.
Measuring the density of a regular solid
- For a regular shape the volume can be calculated from measured lengths, so no water is needed.
- Measure the mass on a balance.
- Measure the dimensions with a ruler (or vernier calipers or a micrometer for small dimensions), taking a mean of two or three readings since blocks are rarely perfectly uniform.
- Calculate the volume: for a rectangular block V=l×w×hV = l \times w \times hV=l×w×h; for a cylinder V=πr2hV = \pi r^2 hV=πr2h.
- Calculate ρ=m/V\rho = m/Vρ=m/V.
Measuring the density of an irregular solid by displacement
- An irregular solid that sinks pushes aside its own volume of liquid, so the liquid measures the volume for you.
- Measure the mass of the dry solid on a balance.
- Part-fill a measuring cylinder with water and record the initial volume V1V_1V1.
- Lower the solid in gently on a thread until it is completely submerged, avoiding splashes, and record the new volume V2V_2V2.
- The volume of the solid is V=V2−V1V = V_2 - V_1V=V2−V1; then calculate ρ=m/V\rho = m/Vρ=m/V.
- A displacement (eureka) can does the same job: fill it to the spout, lower the solid in, and catch the overflow in a measuring cylinder; the volume collected equals the volume of the solid.
Common Mistake
- Do not confuse density with mass. "Lead is heavier than water" says nothing until the volume is fixed; lead is denser, so every cubic centimetre has more mass.
- Get the conversion right: 1 g/cm3=1000 kg/m31\ \text{g/cm}^3 = 1000\ \text{kg/m}^31 g/cm3=1000 kg/m3, not 100100100 and not 101010.
- Subtract the container for a liquid: the mass of the measuring cylinder is not part of the liquid.
- Read the bottom of the meniscus, eye level with it, not the top and not from above.
- Submerge the solid fully for displacement; a solid left part-floating gives the wrong volume.
- Keep the unit: a density without g/cm3\text{g/cm}^3g/cm3 or kg/m3\text{kg/m}^3kg/m3 loses the accuracy mark.
Exam technique
- Describe how to determine is worth a mark per distinct step, so write numbered steps, each naming the instrument and the quantity it measures ("measure the mass of the stone with a balance").
- Make the last step state ρ=m/V\rho = m/Vρ=m/V: a method without the calculation leaves a mark behind.
- In calculations, substitute with consistent units (all grams and cubic centimetres, or all kilograms and cubic metres), then give the answer to sensible significant figures with its unit.
Self review
- Define density, give the equation and its two common units.
- How do you find the density of a liquid using a balance and a measuring cylinder?
- How do you find the volume of an irregular solid that sinks?
- Convert 2.6 g/cm32.6\ \text{g/cm}^32.6 g/cm3 into kg/m3\text{kg/m}^3kg/m3.
- Why does a large block of a material have the same density as a small piece of it?
