Density Calculation and Measurement
Density is a fundamental property of matter that quantifies how much mass is packed into a given volume. ESAT questions frequently test your ability to calculate density, handle unit conversions, and apply the concept to experimental scenarios without a calculator.
Part of the ESAT Physics syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Density is an intrinsic property of a substance; a small piece of iron has the same density as a large block of iron.
- The state of matter significantly affects density, with most substances being densest as solids, slightly less dense as liquids, and far less dense as gases.
- Water is a crucial exception: solid ice is less dense than liquid water, which is why it floats.
- The density of an object with a complex shape or made of multiple materials is found by dividing its total mass by its total volume.
- Experimental determination of volume for an irregular object often uses water displacement, measuring the volume of water the object pushes aside.
- The enormous density difference between gases and condensed states (solids/liquids) is due to the vast empty space between gas particles.
› Why does this happen?
Why does density change with the state of matter?
It's all about how the particles are arranged. In a solid, particles are tightly packed in a fixed, regular pattern, vibrating on the spot. In a liquid, particles are still packed closely but can move past each other, so the spacing between them increases slightly. In a gas, particles are very far apart with lots of empty space between them. For the same amount of mass (the same number of particles), a gas takes up a much larger volume than a liquid or solid. Since density = mass / volume, gases are far less dense.
Why is ice less dense than liquid water?
Water is a rare exception to the normal solid-liquid density rule. As liquid water cools and freezes, its molecules arrange themselves into a fixed, open, crystal structure. This rigid structure holds the molecules further apart on average than when they are in the liquid state, where they can move around and pack more closely together. Since the same mass of water molecules takes up more volume as solid ice, its density is lower, which is why icebergs float.
Formulae
ρ = m / V Use this to find density (ρ), mass (m), or volume (V) when the other two quantities are known. Ensure units are consistent before calculating.
Definitions
- Density (ρ)
- The mass of a substance per unit of its volume. It is a measure of how compact the matter in a substance is.
Worked example
A solid metal cube has a side length of 20 cm and a mass of 6.4 kg. A cylindrical hole with a radius of 5 cm is drilled through the center of the cube, from one face to the opposite face. What is the density of the metal in g/cm³? (Use π = 3).
- 1
First, calculate the initial volume of the solid cube before drilling:
Vcube = (side)3 = (20 cm)3 = 8000 cm³ - 2
Next, calculate the volume of the cylindrical hole that was removed.
The cylinder's height is the cube's side length (20 cm):
Vhole = π × r2 × h = 3 × (5 cm)2 × 20 cm = 3 × 25 × 20 = 1500 cm³ - 3
Find the actual volume of the metal by subtracting the hole's volume from the cube's volume:
Vmetal = Vcube - Vhole = 8000 cm³ - 1500 cm³ = 6500 cm³ - 4
Convert the mass from kilograms to grams:
m = 6.4 kg × 1000 g/kg = 6400 g - 5
Finally, calculate the density using the mass and volume of the metal:
ρ = m / V = 6400 g / 6500 cm³ = 64/65 g/cm³
Answer: 64/65 g/cm³
Common mistakes
- ×Unit Conversion Errors: A frequent mistake is converting g/cm³ to kg/m³ incorrectly. Remember that 1 m³ = (100 cm)³ = 1,000,000 cm³, so 1 g/cm³ is equal to 1000 kg/m³, not 100.
- ×Inconsistent Units: Performing calculations with mixed units, such as mass in kg and volume in cm³. Always convert all quantities to a consistent system (e.g., grams and cubic centimetres) before applying the formula.
- ×Volume Misinterpretation: In water displacement experiments, using the final water level as the object's volume instead of the *change* in water level (Vfinal - Vinitial).
- ×Average vs Material Density: Confusing the average density of a hollow object with the density of the material it's made from. You must use the volume of the material only (Total Volume - Hollow Volume).
No-calculator tips
- ✓Simplify the conversion from g/cm³ to kg/m³: just multiply by 1000. For example, the density of water is 1 g/cm³, which is 1000 kg/m³.
- ✓When dividing in the density formula, simplify the fraction by cancelling zeros or common factors before performing the long division. E.g., 5400 / 90 becomes 540 / 9 = 60.
- ✓Approximate to check your answer. If you calculate a density for a common metal like steel to be 0.8 g/cm³, you should be suspicious, as it's less dense than water and should be much higher (around 8 g/cm³).