Heat Capacity and Specific Heat Capacity

Heat capacity is the amount of heat needed to raise the temperature of an object by 1 K, and specific heat capacity is the heat needed to raise the temperature of 1 g of a substance by 1 K, related by q = mcΔT. In brief: water has a specific heat of 4.184 J/(g·K), so warming 250 g of water from 20 °C to 80 °C takes 250 × 4.184 × 60 = 62,760 J, or about 62.8 kJ.

Specific heat explains why sand on a beach becomes hot while the sea stays cool, why a metal spoon feels cold and why water is used for cooling. It is also the key quantity in all calorimetry calculations. This article defines the terms, shows the formula in use, gives data for common substances and explains how specific heat can be used to identify a metal.

What are heat capacity and specific heat?

• Heat capacity (C) is the heat required to raise the temperature of a whole object by 1 K. Its unit is J/K. It depends on the amount of material, so a large pot has a larger heat capacity than a small one.
• Specific heat capacity (c) is the heat required to raise the temperature of 1 g (or 1 kg) of a substance by 1 K. The unit is J/(g·K), which is the same as J/(g·°C). It is a property of the material.
• Molar heat capacity (C_m) is the heat required to raise the temperature of 1 mol by 1 K, in J/(mol·K). C_m = c × M.

The two are connected by C = m × c. The specific heat of a gas depends on whether it is measured at constant pressure (c_p) or constant volume (c_v), and c_p is larger because some of the heat is used for expansion work.

What is the formula for heat transfer?

q = m × c × ΔT

Here q is the heat absorbed (positive) or released (negative), m is the mass, c the specific heat, and ΔT = T_final − T_initial. Since a change of 1 K is the same as a change of 1 °C, ΔT can be calculated directly from Celsius temperatures. The formula is valid only when there is no phase change; during melting or boiling, heat is absorbed without a temperature change.

Specific heats of common substances

Water has one of the highest specific heats among common substances. Metals have low values, so they heat up and cool down quickly.

Molar heat capacities of metals

The molar heat capacities of the solid metals are all close to 3R = 24.9 J/(mol·K), a result known as the Dulong–Petit law. It means that each atom, no matter what it is, stores about the same amount of energy for a given temperature rise.

Worked examples

Example 1: heating water

How much heat is needed to warm 250 g of water from 20.0 °C to 80.0 °C? q = 250 × 4.184 × (80.0 − 20.0) = 62,760 J = 62.8 kJ.

Example 2: finding the specific heat of a metal

A 50.0 g piece of metal is heated to 100.0 °C and placed in 100.0 g of water at 20.0 °C in an insulated cup. The final temperature is 24.0 °C. Identify the metal.

1. Heat gained by the water: q = 100.0 × 4.184 × (24.0 − 20.0) = 1673.6 J.
2. The metal lost the same amount of heat: q_metal = −1673.6 J.
3. ΔT_metal = 24.0 − 100.0 = −76.0 °C.
4. c = q / (mΔT) = −1673.6 / (50.0 × −76.0) = 0.440 J/(g·K).

This is close to 0.449 J/(g·K), the value for iron, so the metal is probably iron. The small difference arises from the heat absorbed by the cup and lost to the air.

Example 3: how long to boil a kettle

An electric kettle with a power of 2000 W heats 1.00 kg of water from 20 °C to 100 °C. The heat needed is 1000 × 4.184 × 80 = 334,720 J. Time = 334,720 J / 2000 J/s = 167 s, or about 2.8 minutes. In practice it takes a little longer, because the kettle itself is heated and some heat is lost.

Example 4: heat capacity of a calorimeter

A calorimeter absorbs 840 J of heat when its temperature rises by 2.0 °C. Its heat capacity is C = 840 / 2.0 = 420 J/°C.

Why is the specific heat of water so high?

Heating water increases the motion of the molecules, but part of the energy goes into stretching and breaking hydrogen bonds, and into the rotation and vibration of the molecules. This extra energy sink means that more energy is needed per degree. The consequence is important in nature: oceans store large amounts of heat in summer and release it in winter, which moderates the climate of coastal regions. The human body, which is about 60 % water, also has a high heat capacity, which stabilizes body temperature.

Who measured specific heats?

Joseph Black introduced the idea of specific heat in the 1760s, based on his measurements of the heat exchanged when equal masses of different substances were mixed. Pierre Dulong and Alexis Petit published their law in 1819, which was an important help in determining atomic masses. Deviations from the law at low temperature, which were unexplained for decades, were explained by Einstein in 1907 with a quantum theory of solids, and this was an early success of quantum theory.

Which mistakes are common?

• Using q = mcΔT during a phase change: the temperature is constant, but heat is absorbed.
• Wrong sign of ΔT: ΔT is final minus initial.
• Confusing C and c: C is for an object and c is per gram.
• Using the specific heat of water for another liquid: use the correct value.
• Forgetting the calorimeter: the container also absorbs heat.