Heat and temperature

Why this matters

Heat is energy on the move, and almost every engineered system has to manage it: engines and power plants turn heat into work, electronics must shed heat to survive, buildings are insulated against it, and chemical reactors are held at precise temperatures. Chemists use heat measurements (calorimetry) to find reaction energies, and materials scientists heat-treat metals to change their properties. This lesson gives you the core calculation, Q = m·c·ΔT, plus the three ways heat travels.

Temperature and heat are different

• Temperature measures how energetic the random motion of a material's particles is, on average. It is measured in degrees Celsius (°C) or kelvin (K).
• Heat Q is energy transferred because of a temperature difference, always flowing from hotter to colder. It is measured in joules.

A bathtub of warm water and a cup of boiling water: the cup is hotter (higher temperature), but the bathtub holds far more thermal energy and can melt far more ice.

The kelvin scale starts at absolute zero:

T(K) = T(°C) + 273.15

A change of 1 °C is exactly the same as a change of 1 K, so in ΔT you can use either. Use kelvin whenever a formula uses the temperature itself (as in the gas laws).

Specific heat capacity

Different materials need different amounts of energy to warm up. The specific heat capacity c is the energy needed to raise 1 kg by 1 K:

Q = m · c · ΔT

Water's unusually high c is why oceans moderate climate and why water is used as a coolant in engines and power plants.

Worked example 1: boiling water in a kettle

How much heat raises 1.5 kg of water (1.5 litres) from 20 °C to 100 °C?

Q = 1.5 × 4186 × (100 − 20) = 502,320 J ≈ 502 kJ

A 2,000 W kettle supplies 2,000 J per second. If all of it went into the water:

t = Q / P = 502,320 / 2,000 = 251 s ≈ 4.2 minutes

Real kettles take a bit longer because some heat warms the kettle itself and escapes to the room. The ratio of useful heat to energy supplied is the efficiency.

Worked example 2: mixing (calorimetry)

Pour 0.20 kg of water at 80 °C into 0.30 kg of water at 20 °C in an insulated cup. If no heat escapes, the heat lost by the hot water equals the heat gained by the cold water:

0.20 × 4186 × (80 − T) = 0.30 × 4186 × (T − 20)
16 − 0.20·T = 0.30·T − 6
T = 22 / 0.50 = 44 °C

With the same material on both sides, c cancels and the answer is a mass-weighted average. With different materials, keep each c in the equation.

Phase changes

While a substance melts or boils, heat goes into breaking bonds instead of raising the temperature. The energy per kilogram is the latent heat L:

Q = m · L

For water, the latent heat of fusion (melting) is about 334 kJ/kg and of vaporisation (boiling) about 2,260 kJ/kg. Boiling 1.5 kg of water away completely would take another 3.4 MJ, almost seven times as much as heating it from 20 °C to 100 °C. That huge number is why steam burns are so severe and why evaporative cooling works.

How heat travels

1. Conduction through a solid, particle to particle. For a flat slab:

P = k · A · ΔT / d

k is the thermal conductivity in W/(m·K), A the area, d the thickness. A 10 m² wall section with 0.10 m of insulation (k ≈ 0.04 W/(m·K)) and a 20 K difference loses about 0.04 × 10 × 20 / 0.10 = 80 W. Copper (k ≈ 400) conducts ten thousand times better, which is why heat sinks are metal.

2. Convection: a moving fluid carries heat with it. Warm air rising from a radiator, a fan cooling a CPU.

3. Radiation: every object emits electromagnetic radiation, mostly infrared at everyday temperatures. It needs no medium, which is how sunlight reaches Earth.

Real building and equipment calculations combine all three and use code-approved methods and material data; this lesson gives the physics, not a design procedure.

In code

def mix_temperature(m1, t1, m2, t2):
    """Final temperature when two masses of water mix with no heat loss."""
    return (m1 * t1 + m2 * t2) / (m1 + m2)

print(mix_temperature(0.20, 80, 0.30, 20))   # 44.0

Common mistakes

• Using grams with c in J/(kg·K). 250 g is 0.250 kg.
• Confusing heat and temperature. More mass at the same temperature holds more energy.
• Forgetting phase changes. Heating ice at −10 °C to water at 20 °C has three steps: warm the ice, melt it, warm the water.
• Using °C in a formula that needs kelvin. ΔT is fine in °C; absolute temperatures in gas laws are not.

Quick recap

• Heat is energy transferred by a temperature difference; temperature is not heat.
• Q = m·c·ΔT; water's c ≈ 4,186 J/(kg·K).
• Phase changes: Q = m·L, with no temperature change.
• Heat travels by conduction, convection and radiation.

Try it

In the Do tab, write the heat equation and use it to work out how long a kettle takes to boil.