P10: Density and States of Matter
The particle model, density calculations, the density practical, internal energy, changes of state and heating/cooling curves.
The Particle Model
The particle model explains the behaviour of matter in its three states: solid, liquid and gas. In this model, matter is made up of tiny particles that are always moving.
Properties of Solids
- Particles are arranged in a regular, fixed pattern
- Particles vibrate about fixed positions but cannot move freely
- Solids have a fixed shape and fixed volume
- Solids are dense and cannot be compressed
- Strong forces of attraction between particles hold them in place
Properties of Liquids
- Particles are close together but in a random arrangement
- Particles can move past each other freely
- Liquids have a fixed volume but take the shape of their container
- Liquids are dense and cannot be compressed easily
- Weaker forces of attraction between particles than in solids
Properties of Gases
- Particles are far apart in a random arrangement
- Particles move very quickly in all directions
- Gases have no fixed shape and no fixed volume — they fill any container
- Gases have low density and are easily compressed
- Very weak forces of attraction between particles
| Property | Solid | Liquid | Gas |
|---|---|---|---|
| Arrangement of particles | Regular, close together | Random, close together | Random, far apart |
| Movement of particles | Vibrate about fixed positions | Move past each other | Move rapidly in all directions |
| Forces between particles | Strong | Weaker than solids | Very weak |
| Shape | Fixed | Takes shape of container | Fills container |
| Volume | Fixed | Fixed | Variable |
| Compressibility | Cannot be compressed | Cannot be compressed | Easily compressed |
| Density | High | High | Low |
Density
Density is a measure of how much mass is packed into a given volume. Different materials have different densities because their atoms are arranged differently.
Density Equation
ρ = m / V
ρ = density in kilograms per metre cubed (kg/m³)
m = mass in kilograms (kg)
V = volume in metres cubed (m³)
Key Facts About Density
- Density depends on the material, not the size or shape of the object
- Objects with a density less than water (1000 kg/m³) will float
- Metals are generally very dense because their atoms are closely packed
- Gases have very low densities because the particles are far apart
Worked Example
A block of aluminium has a mass of 5.4 kg and a volume of 0.002 m³. Calculate its density.
ρ = m / V = 5.4 / 0.002 = 2700 kg/m³
Worked Example
The density of iron is 7900 kg/m³. Calculate the mass of an iron block with a volume of 0.05 m³.
m = ρ × V = 7900 × 0.05 = 395 kg
Worked Example
A rock has a mass of 240 g and a volume of 80 cm³. Calculate its density in kg/m³.
ρ = m / V = 240 / 80 = 3 g/cm³
To convert: 3 g/cm³ = 3 × 1000 = 3000 kg/m³
The Density Practical
You need to be able to measure the density of regular and irregular objects experimentally.
Finding the Density of a Regular Solid
- Measure the mass using an electronic balance
- Measure the dimensions using a ruler, vernier callipers or micrometer
- Calculate the volume using the appropriate formula (e.g. V = l×w×h for a cuboid)
- Calculate density using ρ = m / V
Finding the Density of an Irregular Solid
- Measure the mass using an electronic balance
- Fill a measuring cylinder with enough water to submerge the object and record the volume
- Lower the object into the water on a thread
- Record the new water level
- Volume of object = final volume − initial volume
- Calculate density using ρ = m / V
Finding the Density of a Liquid
- Place an empty measuring cylinder on an electronic balance and record the mass
- Add a known volume of liquid to the cylinder
- Record the new mass
- Mass of liquid = new mass − mass of empty cylinder
- Calculate density using ρ = m / V
When describing the density practical, always state the measuring instrument and the resolution. For example, use a ruler with 1 mm resolution, a measuring cylinder with 1 cm³ resolution, and a balance with 0.01 g resolution.
Internal Energy
All matter contains internal energy due to the kinetic and potential energy of its particles.
What is Internal Energy?
- Internal energy is the total energy stored by the particles that make up a substance
- It is the sum of the kinetic energy (due to particle motion) and potential energy (due to the forces between particles and their positions)
- Heating a substance increases its internal energy
- The temperature of a substance is related to the average kinetic energy of its particles
Heating and Internal Energy
- When a substance is heated, energy is transferred to the particles
- If the substance is not changing state, the kinetic energy increases — the particles move faster and the temperature rises
- If the substance is changing state, the potential energy increases — the particles move further apart but the temperature stays constant
Changes of State
When a substance changes state, its particles rearrange but the substance itself does not change — it is still the same material.
Changes of State
- Melting — solid to liquid
- Freezing — liquid to solid
- Boiling / evaporation — liquid to gas
- Condensation — gas to liquid
- Sublimation — solid directly to gas
Key Points About Changes of State
- During a change of state, the temperature stays constant — energy is used to overcome or form bonds between particles
- The mass is conserved — the number of particles does not change
- Changing state is a physical change, not a chemical change — the substance is still the same material
- The energy needed to change state is called latent heat
Do not confuse melting with dissolving. Melting is a change of state where a solid becomes a liquid due to heating. Dissolving is when a substance mixes into a solvent to form a solution.
Heating and Cooling Curves
A heating or cooling curve shows how the temperature of a substance changes as it is heated or cooled over time.
Reading a Heating Curve
- Diagonal sections show the substance is heating up — energy is increasing the kinetic energy of particles
- Horizontal flat sections show a change of state — the temperature stays constant while energy is used to overcome bonds
- The first flat section is the melting point (solid to liquid)
- The second flat section is the boiling point (liquid to gas)
- The longer the flat section, the more energy is needed for the change of state
Reading a Cooling Curve
- Diagonal sections show the substance is cooling down
- Horizontal flat sections show the substance is changing state — bonds are forming and energy is released
- The first flat section is the condensation point (gas to liquid)
- The second flat section is the freezing point (liquid to solid)
- Some substances show supercooling — the temperature drops below the freezing point before the flat section appears
Interpreting a Heating Curve
A sample of ice at −10 °C is heated. The temperature rises to 0 °C (diagonal line). It then stays at 0 °C while the ice melts (horizontal line). Once fully melted, the water temperature rises to 100 °C (diagonal line). It stays at 100 °C while the water boils (horizontal line). After boiling, the steam temperature rises above 100 °C (diagonal line).
Anomalous Expansion of Water
Water behaves unusually near its freezing point — it expands when it freezes, which means ice is less dense than liquid water.
Why Ice Floats
- Most substances are denser as solids than as liquids
- Water is most dense at 4 °C
- Below 4 °C, water expands as it cools towards freezing
- When water freezes, the molecules form an open crystalline structure with gaps, making ice less dense than water
- This is why ice floats on water and ponds freeze from the top down, allowing aquatic life to survive underneath
Practice Questions
1. A metal sphere has a mass of 3.2 kg and a volume of 400 cm³. Calculate its density in kg/m³.
V = 400 cm³ = 400 × 10−6 m³ = 4 × 10−4 m³
ρ = m / V = 3.2 / (4 × 10−4) = 8000 kg/m³
2. Describe the arrangement and motion of particles in a gas.
In a gas, particles are far apart in a random arrangement. They move very rapidly in all directions. Forces between particles are very weak. Gases have no fixed shape or volume and fill their container.
3. Explain why the temperature stays constant while a substance is melting.
During melting, the energy supplied is used to overcome the bonds between particles rather than increase their kinetic energy. Since temperature depends on average kinetic energy, the temperature stays constant while the substance changes state.
4. Describe how you would find the density of an irregular stone.
Measure the mass using an electronic balance. Fill a measuring cylinder with water and record the initial volume. Lower the stone into the water on a thread and record the new volume. The volume of the stone equals the increase in water level. Calculate density using ρ = m / V.
5. Explain why ice floats on water in terms of density and particle arrangement.
When water freezes, the molecules form an open crystalline structure with gaps between them. This makes the same mass of water occupy a larger volume in the solid state, so ice has a lower density than liquid water and floats.
🔬 Required Practical
Required Practical: Measuring Density of Regular and Irregular Objects
Aim: To determine the density of regular solid objects, an irregular solid object, and a liquid.
Method for a regular solid (e.g. a cuboid): Measure the mass of the object using an electronic balance. Measure the length, width, and height using a ruler, vernier callipers, or micrometer. Calculate the volume using the appropriate formula (V = l × w × h for a cuboid, V = πr²h for a cylinder). Calculate the density using ρ = m/V.
Method for an irregular solid (e.g. a stone): Measure the mass using an electronic balance. Fill a measuring cylinder with enough water to submerge the object and record the initial volume. Carefully lower the object into the water on a thread. Record the new water level. Volume of object = final volume − initial volume. Calculate density using ρ = m/V.
Method for a liquid: Place an empty measuring cylinder on a balance and record the mass. Add a known volume of liquid using the measuring cylinder. Record the new mass. Mass of liquid = new mass − mass of empty cylinder. Calculate density using ρ = m/V.
Variables: Independent: the material or object being measured. Dependent: mass and volume (used to calculate density). Control: the measuring instruments used, the temperature of the water (for displacement method).
Analysis: Compare the calculated density values with known reference values to identify the material. Objects with density less than 1000 kg/m³ will float in water. Uncertainty can be assessed by considering the resolution of each measuring instrument and calculating percentage uncertainty.
Common exam questions: "Why should you read the measuring cylinder at eye level?" — To avoid parallax error, which could give an incorrect volume reading. "Why should the object be fully submerged?" — The displacement method only measures the volume of the part underwater; if part is above water, the measured volume will be too low. "Why measure the mass before the volume for the irregular solid?" — If the object is wet from the displacement method, the measured mass would be too high, giving an incorrect density value.
🔢 Maths Skills
Mathematical Skills for this Topic
Calculating density using ρ = m/V: Rearrange to find mass: m = ρ × V. Rearrange to find volume: V = m / ρ. Always ensure units are consistent. The standard units are kg and m³, giving density in kg/m³. If you use g and cm³, density is in g/cm³. To convert g/cm³ to kg/m³, multiply by 1000. For example, 2.7 g/cm³ = 2700 kg/m³.
Calculating volume of regular shapes: Cuboid: V = l × w × h. Cylinder: V = πr²h. Sphere: V = (4/3)πr³. Always use the radius (not the diameter) in the cylinder and sphere formulas. If dimensions are given in cm, the volume will be in cm³. Convert to m³ by dividing by 1 000 000 (1 m³ = 1 000 000 cm³).
Unit conversion: cm³ to m³: To convert cm³ to m³, divide by 1 000 000 (or multiply by 10−6). To convert m³ to cm³, multiply by 1 000 000. This is a common source of error. For example, 150 cm³ = 150 × 10−6 m³ = 1.5 × 10−4 m³. A quick check: 1 m = 100 cm, so 1 m³ = 100 × 100 × 100 = 1 000 000 cm³.
Percentage uncertainty: Uncertainty in a measurement is approximately half the resolution of the instrument. Percentage uncertainty = (uncertainty / measured value) × 100%. For a ruler with 1 mm resolution, uncertainty = 0.5 mm = 0.0005 m. If measuring a length of 50 mm, percentage uncertainty = (0.5/50) × 100% = 1%. Multiple measurements increase overall uncertainty.
⚠️ Common Misconceptions
Watch Out!
Students often think that heavy objects are always dense. Wrong: Heavy objects are always dense — if something weighs a lot, it must have a high density. Correct: Density depends on both mass AND volume (ρ = m/V). A heavy object can have a low density if it is very large. For example, a large log is heavy but has low density because wood is less dense than water. A small gold ring is light but has very high density (19 300 kg/m³). Weight alone tells you nothing about density without knowing the volume.
Students often think that particles themselves expand when a substance is heated. Wrong: When a substance is heated, the particles expand and get bigger, which is why the substance expands. Correct: Particles themselves do not change size when heated. When a substance is heated, the particles gain kinetic energy and move faster, pushing further apart from each other. It is the gaps between the particles that increase, not the size of the particles. In a solid, the particles vibrate more and take up more space; in a liquid, they move faster and slightly further apart; in a gas, they move much faster and spread out much more.
✍️ 6-Mark Question
Extended Answer Question
6 marks: Describe how you would determine the density of an irregular solid object and a liquid. Explain the sources of uncertainty in each method.
For an irregular solid, first measure the mass using an electronic balance. Then use the water displacement method to find the volume: fill a measuring cylinder with enough water to submerge the object and record the initial volume. Carefully lower the object into the water on a thin thread and record the new volume. The volume of the object equals the increase in water level (final volume − initial volume). Calculate density using ρ = m/V.
For a liquid, first place an empty measuring cylinder on an electronic balance and record the mass (this is the mass of the cylinder). Pour a known volume of the liquid into the cylinder using the measuring cylinder scale. Record the new mass. The mass of the liquid equals the new mass minus the mass of the empty cylinder. Calculate density using ρ = m/V.
There are several sources of uncertainty. For the irregular solid, the measuring cylinder has limited resolution (typically 1 cm³ or 0.5 cm³), so the volume reading has an uncertainty of ±0.5 cm³ at each end of the scale, giving ±1 cm³ for the volume difference. The object may trap air bubbles when submerged, making the measured volume too large. If the object is porous, it may absorb water, also giving an incorrect volume. The mass should be measured before the volume to avoid measuring a wet object, which would give a falsely high mass.
For the liquid, the measuring cylinder may not be perfectly dry when first weighed, and some liquid may stick to the walls above the meniscus, giving a small error in the volume reading. Reading the meniscus at the wrong angle (parallax error) is another source of uncertainty. Taking repeated measurements and calculating an average can reduce the effect of random errors.
Mark scheme: 2 marks for describing the irregular solid method, 1 mark for describing the liquid method, 2 marks for explaining sources of uncertainty (resolution, parallax, air bubbles, wet mass), 1 mark for suggesting improvements (repeats, average, measure mass first)
📊 AO3: Analyse & Evaluate
Analysis and Evaluation
A student measured the density of four metal samples and recorded the following results:
| Sample | Mass (g) | Volume (cm³) | Density (g/cm³) |
|---|---|---|---|
| A | 27.0 | 10.0 | 2.70 |
| B | 89.0 | 10.0 | 8.90 |
| C | 78.6 | 10.0 | 7.86 |
| D | 47.4 | 6.0 | 7.90 |
Known densities: aluminium = 2.70 g/cm³, copper = 8.96 g/cm³, iron = 7.87 g/cm³, lead = 11.3 g/cm³, silver = 10.5 g/cm³.
Identify each sample and evaluate the reliability of the measurements for sample D, given that the measuring cylinder has a resolution of 1 cm³.
Answer: Sample A = aluminium (2.70 g/cm³ matches exactly). Sample B = copper (8.90 is close to 8.96; small difference may be due to impurities or measurement uncertainty). Sample C = iron (7.86 is close to 7.87). Sample D = iron (7.90 is close to 7.87). However, sample D's volume is only 6.0 cm³, measured with a 1 cm³ resolution cylinder. The uncertainty in volume is ±0.5 cm³ at each reading, giving ±1 cm³ for the displacement. Percentage uncertainty in volume = (1/6.0) × 100% = 16.7%. This means the density could range from (47.4/7.0) = 6.77 g/cm³ to (47.4/5.0) = 9.48 g/cm³. This large range overlaps with several metals, making the identification of sample D unreliable. A larger object or a more precise measuring cylinder (e.g. 0.5 cm³ resolution) would reduce the percentage uncertainty and improve reliability.
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