Density

Key points

What density means

• Density is the mass per unit volume of a substance. It is the mass of each cubic metre (or each cubic centimetre) of the substance.
• ρ = m / V. The symbol ρ is the Greek letter rho.
• The unit of density is kg/m³ or g/cm³.
• 1 g/cm³ = 1000 kg/m³. Use matching units: mass in g with volume in cm³, or mass in kg with volume in m³.
• Pure water has a density of about 1000 kg/m³ (1.0 g/cm³).
• A large block and a small block of the same material have the same density. The large block has more mass, but it also has more volume.

Measuring density

• In every method, you measure the mass with a balance and find the volume. Then calculate ρ = m / V.
• A liquid:
  1. Measure the mass of an empty measuring cylinder on a balance.
  2. Pour in some liquid. Read the volume at the bottom of the meniscus, with your eye level with the surface.
  3. Measure the mass of the cylinder and the liquid together.
  4. Mass of the liquid = (mass of cylinder + liquid) − (mass of empty cylinder).
  5. Density = mass of liquid ÷ volume of liquid.
• A regularly shaped solid (for example, a cuboid):
  1. Measure the mass on a balance.
  2. Measure the length, width and height with a ruler.
  3. Volume = length × width × height.
  4. Density = mass ÷ volume.
• For other regular shapes, use the right volume formula. For example, the volume of a cylinder is πr²h, where r is the radius and h is the height.
• An irregularly shaped solid that sinks (volume by displacement):
  1. Measure the mass on a balance.
  2. Part-fill a measuring cylinder with water. Read the volume, V₁.
  3. Tie the object to a thread. Lower it gently into the water until it is completely under the surface. Lowering it gently stops water splashing out.
  4. Read the new volume, V₂.
  5. Volume of the object = V₂ − V₁. The object pushes aside (displaces) its own volume of water.
  6. Density = mass ÷ (V₂ − V₁).
• The object must sink. An object that floats is not completely under the water, so it displaces less water than its own volume.
• Check that no air bubbles stick to the object. Bubbles add to the volume reading, so the density would come out too small.
• If the object is too large for a measuring cylinder, use a displacement can (overflow can). Fill the can with water up to the spout. Lower the object in. Collect the water that overflows in a measuring cylinder. Its volume is the volume of the object.

Floating and sinking

• An object floats in a liquid if its density is less than the density of the liquid.
• An object sinks in a liquid if its density is greater than the density of the liquid.
• To decide, compare the two densities in the same units.
• For an object made of more than one material, use its average density: its total mass ÷ its total volume.
• A steel ferry floats although steel is denser than water. The hull encloses a large volume of air. So the average density of the whole ferry is less than the density of the seawater.
• Extra detail: an object with exactly the same density as the liquid neither rises nor sinks. It stays where it is put.
• Two liquids that do not mix: the liquid with the lower density floats on top of the liquid with the higher density. Extended
• Cooking oil is less dense than water and does not mix with it. So oil floats on water. Extended
• With several liquids that do not mix, they form layers. The densest liquid is at the bottom and the least dense is at the top. Extended

Common mistakes

• Students mix units, for example a mass in g with a volume in m³. / The mark scheme wants matching units: g with cm³ (answer in g/cm³), or kg with m³ (answer in kg/m³).
• Students use the reading V₂ as the volume of the stone. / The mark scheme wants the difference: volume of the object = V₂ − V₁.
• Students use the mass of the cylinder and liquid together. / The mark scheme wants: subtract the mass of the empty cylinder first.
• Students write that heavy objects sink and light objects float. / The mark scheme wants a comparison of densities: the object floats if its density is less than the liquid’s density.
• Students write that an object floats on a liquid because it is “less dense” without saying less dense than what. / The mark scheme wants both densities compared, using the values given.
• Students say the denser liquid floats on top. / The mark scheme wants: the less dense liquid floats on top, if the liquids do not mix. Extended

Exam tips

• Define density in a few words: “mass per unit volume”.
• Describe a method as numbered steps: what you measure, with which instrument, and how you calculate. Name the measuring cylinder and the balance.
• For the displacement method, say “lower the object gently until it is fully submerged” and “volume = new reading − first reading”.
• Give at least one precaution: read the bottom of the meniscus at eye level; make sure no air bubbles stick to the object.
• Calculate: show ρ = m / V, the substitution and the unit. If you change units, show the change (for example 2.7 g/cm³ = 2700 kg/m³).
• Determine whether it floats: compare the two density values in the same units, then say “floats” or “sinks”.
• Determine which liquid is on top: compare the densities and state that the less dense liquid floats on top, because the liquids do not mix. Extended
• A typical 1-mark answer: “Density is the mass per unit volume.”
• A typical 2-mark answer to “Will the block float in oil of density 900 kg/m³? The block’s density is 1200 kg/m³.”: “No, it sinks (1). The density of the block is greater than the density of the oil (1).”
• A typical 3-mark answer to “Describe how to find the volume of a small stone”: “Read the volume of water in a measuring cylinder (1). Lower the stone in gently until it is fully submerged and read the new volume (1). The volume of the stone is the difference between the two readings (1).”