Specific Heat Capacity and Calorimetry

Specific heat capacity is the energy needed to raise the temperature of 1 kg of a substance by 1 K (or 1 °C), measured in J/(kg·K), and the thermal energy transferred is Q = mcΔT; calorimetry uses this relation to measure heat transfers in an insulated container. In brief: water has a high specific heat capacity, 4,180 J/(kg·K), so it heats up and cools down slowly. In an insulated system, heat lost by hot objects equals heat gained by cold objects, which allows the final temperature or an unknown c to be calculated.

Sand on a beach becomes hot in the sun much faster than the sea, and water is used in heating systems to carry energy. These are consequences of specific heat capacity. This guide defines specific heat capacity, explains the formula and the method of mixtures, compares materials in a table, works through a heating and a calorimetry problem, and corrects common mistakes.

What is specific heat capacity?

When a substance is heated, the energy supplied needed for a certain temperature rise depends on the mass, the temperature change and the material. This is summarized in

Q = mcΔT

where Q is the thermal energy transferred (J), m the mass (kg), c the specific heat capacity (J/kg·K) and ΔT the temperature change (K or °C, the change is the same in both). A large c means that much energy is needed to change the temperature. The reason for the high value in water is that energy is also stored in the bonds and hydrogen bonding between the molecules, not only in their motion. The heat capacity of an object is C = mc (J/K), the energy per kelvin for the whole object.

What is calorimetry?

Calorimetry is the measurement of heat transfers. A calorimeter is an insulated container, often of polystyrene or copper with insulation, in which a process happens, and the temperature change is measured. Two common methods:

• Method of mixtures: a hot object of known mass and temperature is put in water of known mass and temperature. By conservation of energy: heat lost by the hot object = heat gained by the water (and the calorimeter).
• Electrical heating: a heater of power P is run for time t, and Q = Pt is compared with mcΔT, to find c.

Heat losses to the surroundings make the measured temperature rise smaller than ideal, and cause errors, so lagging and a lid are used.

Specific heat capacities compared

Worked example: heating water and mixing metal with water

1. Heating water. How much energy is needed to heat 0.50 kg of water from 20 °C to 100 °C? Q = mcΔT = 0.50 × 4,180 × 80 = 167,200 J.
2. Time with a kettle. A 2,000 W heater (ignoring losses) needs t = Q ÷ P = 167,200 ÷ 2,000 = 83.6 s.
3. Mixing. A 0.20 kg copper block (c = 385 J/kg·K) at 100 °C is dropped into 0.30 kg of water at 20 °C in an insulated container. Let T be the final temperature.
   • Heat lost by copper = 0.20 × 385 × (100 − T) = 77 × (100 − T).
   • Heat gained by water = 0.30 × 4,180 × (T − 20) = 1,254 × (T − 20).
   • Set them equal: 7,700 − 77T = 1,254T − 25,080, so 32,780 = 1,331T and T = 24.6 °C.
4. Interpretation: the water temperature rises only by 4.6 °C, even though the copper cools by 75 °C, because the water has a large mass and a large specific heat capacity.

How was it discovered?

Joseph Black introduced the idea of specific heat in the 1760s while observing that equal masses of different substances need different amounts of heat for the same temperature change. Antoine Lavoisier and Pierre-Simon Laplace built an ice calorimeter in 1782 to 1783. In 1819, Pierre Dulong and Alexis Petit noticed that, for many solid elements, the molar heat capacity is nearly constant, about 25 J/(mol·K).

Where is it used?

Water in central heating systems and car radiators carries large amounts of energy. The high c of oceans moderates the climate of coastal regions, and nutrition scientists measure the energy in food by burning it in a bomb calorimeter.

Which mistakes are common?

Mistake 1: Using the final temperature instead of the change

ΔT = T_final − T_initial.

Mistake 2: Mixing units of mass

c in J/(kg·K) needs the mass in kilograms. Convert grams first.

Mistake 3: Using Q = mcΔT during a change of state

While a substance melts or boils, the temperature does not change, and latent heat is used instead.

Mistake 4: Ignoring the container

In accurate work, the calorimeter also absorbs heat, with its own heat capacity.