Internal energy; Temperature changes in a system and specific heat capacity

SPEC 4.3.2.1 & 4.3.2.2 INTERNAL ENERGY & SHC

Internal Energy & SHC

The particle-model angle on energy. What heating actually does at the particle level, and the rule that catches students every series: Δθ is the change, not the final temperature.

Why This Lesson Earns Marks

You met SHC in Lesson 1.03. This is the particle-model revisit: what does heating actually do to the particles? The key new idea is internal energy as the sum of two parts — and the rule that heating does either one thing or the other, never both at once.

KE + PE

Internal Energy

ΔE = mcΔθ

On Equation Sheet

Δθ

Change Not Final T

By The End

Define internal energy as the spec does. State the rule: heating either raises the temperature OR causes a change of state. Apply ΔE = mcΔθ with the right Δθ (the change, not the final value).

Internal Energy — Definition

This is the new concept for Module 3. It's the umbrella term that captures all the energy stored inside a system at the particle level.

Definition [DEF] — spec wording

Internal energy is the total kinetic energy and potential energy of all the particles (atoms and molecules) that make up a system.

Two Parts

Kinetic energy (KE): from the motion of the particles — vibrating, sliding, or flying around.

Potential energy (PE): stored in the bonds between the particles — like the energy in a stretched spring, but at the molecular scale.

Internal energy is the sum of these two contributions across every particle in the substance.

"Thermal Energy" vs "Internal Energy"

AQA uses both terms. They're related, not opposing: "thermal energy" is the name for the energy transferred to a substance by heating (it's the ΔE in the SHC equation). "Internal energy" is the energy stored within the substance — the total KE + PE of all particles. When heating raises temperature with no state change, the thermal energy supplied becomes an increase in internal energy (all as KE).

KE + PE — At The Particle Level

Both parts are present in every state of matter — just in different proportions.

Mark-Scheme Phrasing

"Internal energy is the sum of the kinetic energies of all the particles (their motion) and the potential energies stored in the bonds (or forces) between them."

What Heating Does — The XOR Rule

Heat goes into a substance → internal energy increases. The crucial spec point: the energy goes into one or the other, never both at the same instant.

The Rule

Heating either raises the temperature of the system or produces a change of state — not both at once. While ice is melting at 0°C, the temperature stays at 0°C even as energy keeps going in. Only after all the ice has melted does the water start to warm up.

Specific Heat Capacity — Recap

You met this equation in Lesson 1.03. Here we link it to the particle picture: the energy you put in becomes KE of the particles, and that shows up as a temperature rise.

Definition [DEF] — spec wording

The specific heat capacity of a substance is the amount of energy required to raise the temperature of one kilogram of the substance by one degree Celsius.

ΔE = m × c × Δθ

On The Equation Sheet

ΔE = mcΔθ is provided in the exam — you don't have to memorise it. But you do need to recognise when to use it and substitute correctly.

Symbols & Units

Strict SI units everywhere. Get a single unit wrong and you'll be out by a factor of 1000.

SymbolQuantitySI Unit

ΔEChange in thermal energyjoules (J)

mMasskilograms (kg)

cSpecific heat capacityjoules per kg per °C (J/kg°C)

ΔθChange in temperaturedegrees Celsius (°C)

Conversion Trap

If the question gives mass in grams, convert to kg first (divide by 1000). If it gives energy in kilojoules, convert to J (multiply by 1000). Mixing units gives an answer off by a factor of 1000 — spotted instantly by examiners.

The Δθ Trap — Change, Not Final

The single most common SHC mistake. The Greek letter delta (Δ) means "change in" — not "the value of".

ScenarioWrongRight

Water heated from 20°C to 100°CΔθ = 100Δθ = 100 − 20 = 80°C

Iron block heated from 15°C to 65°CΔθ = 65Δθ = 65 − 15 = 50°C

Liquid cools from 80°C to 30°CΔθ = 30Δθ = 80 − 30 = 50°C

Why It Doesn't Matter If You Use °C or K

For a change in temperature, Δθ in °C and Δθ in kelvin are numerically identical (the scales differ only by a constant offset, which cancels in a difference). A 20°C change is the same as a 20 K change. You'll see both notations in mark schemes.

ESUA — Worked Example

A heater supplies 26 000 J of energy to a 2.0 kg iron block. The temperature rises from 15°C to 45°C. Calculate the specific heat capacity of iron. 4 marks

E

Equation: ΔE = mcΔθ, rearranged: c = ΔE / (m Δθ). (M1)

S

Substitute: first find Δθ.

Δθ = 45 − 15 = 30°C(M1 — the Δθ mark)

c = 26 000 / (2.0 × 30) (M1)

U

Unit: J/kg°C

A

Answer: c = 26 000 / 60 = 433 J/kg°C (3 sig fig) (A1)

Sense check: published SHC of iron is ~450 J/kg°C. ✔

Reference SHCs — And Why Water Wins

The bigger the SHC, the more energy is needed per kg per °C. Water's value is unusually high — with real-world consequences.

SubstanceSHC (J/kg°C)Why It Matters

Water4200Used in central heating — stores lots of energy per kg

Aluminium~900Saucepans heat quickly with little energy

Iron / steel~450Common in calculation questions

Copper~390Hot-water pipes warm fast, lose heat fast

Real-World Implication

Water's high SHC is why coastal climates are milder than inland ones: the ocean takes a lot of energy to warm up and a lot of energy to cool down, so it buffers temperature swings. Same principle: hot-water bottles stay warm for hours.

Why Water's High SHC Matters

Water has an unusually high SHC compared to most other substances — about 10× that of copper per kg per °C. The mechanism is beyond GCSE, but the consequences are heavily exam-relevant.

What High SHC Means

Water needs lots of energy to warm up by 1°C.

Water releases lots of energy when it cools by 1°C.

Water resists temperature change — once hot, it stays hot longer; once cold, it warms slowly.

Real-World Applications

Central heating systems use water to carry thermal energy from boiler to radiators — large amounts of energy stored per kg.

Hot-water bottles stay warm for hours, releasing energy slowly.

Coastal climate buffering: the ocean takes lots of energy to warm up or cool down, so coastal regions have milder seasonal swings than inland regions.

The human body is ~60% water — its high SHC helps keep core temperature stable.

Examiner Alert

The Δθ Substitution Error

The single most-cited error in SHC mark schemes: students substitute the final temperature instead of the change. Always work out final − initial first — even if it's a "simple" calculation. Mark schemes typically award an explicit mark just for finding Δθ correctly.

When A Question Says "Internal Energy"

If a question explicitly asks about internal energy, mention both KE and PE of the particles (the full definition). When the question is about a calculation, "thermal energy" (the ΔE from the equation) is what you compute — that's the energy transferred, which equals the increase in internal energy when no state change happens.

Heating Curve Hint

If you ever see a graph of temperature against time with flat sections, those flat sections are where state changes are happening — energy still going in, but going into PE (breaking bonds), not KE. That's the link to specific latent heat in Lesson 3.04.

Synoptic 6-Mark Frame

A recurring 6-mark question (Jun 2023 P1 was one example) asks you to combine particle description with internal energy reasoning for a substance being heated through a state change. Mark scheme has nine indicative points spread across three Levels.

Question stem (typical): "Ice in a beaker is heated steadily. Describe how the arrangement and motion of the particles change, and explain in terms of internal energy what happens as the ice melts and the resulting water then warms." 6 marks

Indicative Points — Cover All Three Phases

Particle picture (state comparison): in the solid, particles are in a regular pattern and vibrate about fixed positions; in the liquid, particles are in a random arrangement and move freely past each other.

During melting (state change): the temperature stays constant at the melting point because the supplied energy increases the potential energy of the particles — bonds between particles are being weakened/broken — not the kinetic energy.

As the resulting water warms: the supplied energy increases the kinetic energy of the particles — they move faster — so the temperature rises.

Synoptic line: internal energy is the total kinetic and potential energy of all the particles. Both phases above represent increases in internal energy — just stored differently.

Levels Breakdown

Level 1 (1–2 marks): isolated points, no logical linking. Level 2 (3–4 marks): particle picture + one of (PE during melting OR KE during warming). Level 3 (5–6 marks): all three phases linked, with explicit mention of internal energy as the sum.

Weak vs Strong — The Δθ Mark

Same calculation. Same numbers. Different mark allocations.

Weak — 2/4

"ΔE = mcΔθ

ΔE = 0.5 × 4200 × 100

ΔE = 210 000 J"

Why it loses: substituted Δθ = 100 (the final temperature) instead of the change. M1 awarded for equation, but no Δθ mark and no answer mark.

Strong — 4/4

"ΔE = mcΔθ

Δθ = 100 − 20 = 80°C

ΔE = 0.5 × 4200 × 80

ΔE = 168 000 J = 1.68 × 10⁵ J"

Why it scores: equation, explicit Δθ line, correct substitution, correct answer with unit.

Practice 1 — SHC Calculation

A 1500 W kettle heats 0.40 kg of water from 18°C to 98°C. SHC of water = 4200 J/kg°C. Calculate the energy transferred to the water and, assuming all the kettle's electrical power becomes thermal energy in the water, the time taken. 5 marks

E

Equations: ΔE = mcΔθ for the energy. Then P = E/t → t = E/P for the time.

S

Substitute (Step 1 — energy):

Δθ = 98 − 18 = 80°C

ΔE = 0.40 × 4200 × 80 = 134 400 J

Step 2 — time:

t = E/P = 134 400 / 1500

U

Units: ΔE in J; t in seconds.

A

Answers:

ΔE = 134 400 J ≈ 1.34 × 10⁵ J (134 kJ, 3 sig fig)

t = 134 400 / 1500 = 89.6 s (~90 s — realistic for kettle)

Practice 2 — PEE Answer

A teacher heats a beaker of water at a steady rate. At first the temperature rises steadily, then stays at 100°C for several minutes while the water boils. The heater is still on the whole time. Using the particle model, explain where the energy is going during the period at 100°C. 3 marks

P

Point: The energy is increasing the potential energy of the water particles, not their kinetic energy.

E

Evidence: The water is changing state from liquid to gas (boiling). Energy is needed to break the bonds between the molecules so they can escape as a gas.

E

Explain: Because temperature reflects the average kinetic energy of particles, and the supplied energy is going into PE (bond-breaking) rather than KE, the temperature stays constant. Heating raises the temperature OR causes a state change — not both at once.

Checkpoint — You Can Now

Knowledge

Define internal energy as the total KE + PE of all particles in a system (spec wording).

Define specific heat capacity as energy to raise 1 kg by 1°C (spec wording).

State the rule: heating either raises temperature or causes a change of state — not both at once.

Skills

Apply ΔE = mcΔθ with the right Δθ (the change , not the final value).

Convert g → kg and kJ → J before substituting.

Rearrange the equation to solve for any of E, m, c, or Δθ.

Write a PEE answer linking energy supply to KE (temperature) or PE (state change).