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Specific Heat Capacity and Heat Balance

Different materials need different amounts of energy to warm up. The specific heat capacity c of a material is the energy needed to raise the temperature of 1 kg of it by 1 °C (or 1 K). Its unit is J/(kg·°C) or J/(kg·K). The energy needed to heat any amount is Q = m × c × ΔT. Water has a very high c (about 4200 J/(kg·°C)), so it heats and cools slowly. When a hot and a cold body touch and nothing is lost, the heat given by the hot body equals the heat taken by the cold body (heat balance). A calorimeter uses this idea to measure c.

🎬 Step-by-step story

  1. Two beakers sit on identical heaters. One has 1 kg of water, the other 1 kg of cooking oil. Both start at 20 °C.
  2. We switch both heaters on for the same time, so both get the same energy. The oil gets hot fast. The water warms up slowly.
  3. The energy needed to warm 1 kg by 1 °C is called specific heat capacity, c. Water needs about 4200 J. Oil needs about 2000 J. Count the cubes: 4 kJ against 2 kJ.
  4. Energy needed = mass × c × temperature rise, or Q = mcΔT. Heat 1 kg of water by 10 °C: 42 cubes. Heat 2 kg: 84 cubes. Double the mass, double the energy.
  5. Now mix hot water and cold water. The hot water cools and gives heat. The cold water warms and takes heat. Heat given = heat taken. They meet at 50 °C.
  6. Free play: pick a material, change the mass and the temperature rise, and count the energy cubes.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Both beakers got the same energy. Why did the oil get hotter?

Oil needs only about 2000 J to warm 1 kg by 1 °C, while water needs about 4200 J. The same energy gives oil about twice the temperature rise.

What does 4200 J/(kg·°C) actually mean?

To make 1 kg of water 1 °C warmer you must give it 4200 joules. Watch the 4 cubes (about 4 kJ) for water and 2 cubes for oil.

Why does doubling the mass double the energy?

2 kg is just two 1 kg pieces side by side. Each piece needs the same energy, so the total doubles. The cube count goes from 42 to 84.

Is the final temperature in mixing always the average?

Only when the masses and materials are equal, as in the 3D (0.5 kg + 0.5 kg of water gives 50 °C). With unequal masses the result is closer to the bigger mass's temperature.

Is a large c good or bad?

It depends. For a cooking pan you want a small c so it heats fast. For a hot-water bag or a radiator coolant you want a large c so it stores lots of heat. Try both in free play.

Heat, temperature and internal energy

Temperature tells how hot something is. It is measured in °C or K. Heat is energy that moves from a hotter body to a colder body. It is measured in joules (J).

The particles in a body always move. The total energy of these moving particles is its internal energy. Heating a body gives it more internal energy, so its particles move faster and its temperature goes up.

A big bucket of warm water can hold more heat than a small cup of boiling water. So temperature and heat are not the same thing.

What is specific heat capacity?

Specific heat capacity (c) is the energy needed to raise the temperature of 1 kg of a material by 1 °C (1 K is the same step).

Unit: J/(kg·°C) or J/(kg·K).

Typical values

A large c means the material needs lots of energy to warm up, and it gives out lots of energy as it cools. A small c means it heats up and cools down quickly. Metal pans have a small c, so they heat fast.

Heat capacity (without "specific") is for a whole object: C = m × c, in J/°C.

The quantity of heat: Q = mcΔT

The heat Q needed (or given out) depends on three things:

  1. the mass m (more stuff needs more energy),
  2. the material, through c,
  3. the temperature change ΔT = final − initial.

Q = m × c × ΔT

Use m in kg, c in J/(kg·°C) and ΔT in °C to get Q in J. If ΔT is negative, the body gives out heat as it cools.

Rearranged: c = Q ÷ (mΔT), m = Q ÷ (cΔT), ΔT = Q ÷ (mc).

Heat balance and calorimetry

When a hot body and a cold body touch, heat flows from hot to cold until both reach the same temperature (thermal equilibrium). If no heat escapes to the air:

Heat given by the hot body = heat taken by the cold body
m₁c₁(T₁ − T) = m₂c₂(T − T₂), where T is the final temperature.

This is just conservation of energy. A calorimeter is an insulated cup (often copper or aluminium, with a lid, a stirrer and a thermometer) used to do such mixing with very little heat loss.

Measuring c with a heater

Put an electric heater of known power P into a block of mass m for time t. The energy is Q = P × t. Measure the temperature rise and use c = Pt ÷ (mΔT). Real results come out a bit high because some heat escapes; insulation reduces this error.

Where a high c helps

Water in car radiators and in home heating, hot-water bottles, sea breezes and the mild climate near oceans all depend on water's large c.

Key formulas and definitions

Worked examples

1. How much energy heats 2 kg of water from 20 °C to 30 °C? (c = 4200 J/(kg·°C))

ΔT = 30 − 20 = 10 °C. Q = mcΔT = 2 × 4200 × 10 = 84 000 J = 84 kJ.

2. A 0.5 kg iron block (c = 450 J/(kg·°C)) cools from 100 °C to 40 °C. How much heat does it give out?

ΔT = 60 °C. Q = 0.5 × 450 × 60 = 13 500 J = 13.5 kJ given out.

3. 18 000 J warms 2 kg of a metal by 10 °C. Find c.

c = Q ÷ (mΔT) = 18 000 ÷ (2 × 10) = 900 J/(kg·°C). This is close to aluminium.

4. A 1 kW kettle runs for 84 s and heats 0.5 kg of water. Find the temperature rise (no heat loss).

Q = Pt = 1000 × 84 = 84 000 J. ΔT = Q ÷ (mc) = 84 000 ÷ (0.5 × 4200) = 40 °C.

5. 0.2 kg of water at 80 °C is mixed with 0.3 kg of water at 20 °C. Find the final temperature.

Heat lost = heat gained: 0.2 × 4200 × (80 − T) = 0.3 × 4200 × (T − 20). c cancels: 16 − 0.2T = 0.3T − 6, so 0.5T = 22, T = 44 °C.

6. A 0.1 kg copper ball (c = 390) at 200 °C is dropped into 0.39 kg of water at 20 °C. Find the final temperature (ignore the cup).

Heat lost by copper = 0.1 × 390 × (200 − T) = 39(200 − T). Heat gained by water = 0.39 × 4200 × (T − 20) = 1638(T − 20). So 7800 − 39T = 1638T − 32 760, 1677T = 40 560, T ≈ 24.2 °C. The water hardly warms because its c is so large.

Common mistakes

Practice quiz

1. Specific heat capacity is the energy needed to warm:
2. The SI unit of specific heat capacity is:
3. Equal masses get equal energy. Which warms the most?
4. Q = mcΔT. If m doubles and ΔT stays the same, Q:
5. In a mixing experiment with no heat loss:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is specific heat capacity in simple words?

It is how much energy 1 kg of a material needs to get 1 °C hotter. Water needs a lot (about 4200 J); metals need little.

Why is the specific heat capacity of water so high?

Water molecules hold on to each other strongly with hydrogen bonds. Much of the energy goes into these bonds, so less goes into raising the temperature.

What is the difference between heat capacity and specific heat capacity?

Heat capacity is for a whole object (J/°C). Specific heat capacity is for 1 kg of the material (J/(kg·°C)). Heat capacity = mass × specific heat capacity.

Where this is taught

RomaniaClasa a VIII-aThermal phenomena
Ukraine8 класThermal phenomena
Russia8 классThermal phenomena
Russia8 классThermal phenomena
China九年级（初三）Ch.13 Internal energy

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