Experimentally, it is observed that within a specific phase of matter, the temperature change of a substance increases proportionally with the energy supplied. The proportionality constant $C$ is called the heat capacity. This leads to the following definition:
Heat Capacity
If a body with the heat capacity $C$ is heated up by the temperature difference $\Delta T$, the energy change can be calculated according to
$$
\Delta Q = C \Delta T
$$
This formula implies a direct relationship between energy and temperature which will be discussed in more detail later for gases.
The heat capacity of a body depends primarily on the composition of its material. If it is made of a homogeneous material, a specific heat capacity $c$ can be defined, interpreted as the heat capacity per unit mass. This leads to the formula:
$$
\Delta Q = cm\Delta T
$$
The unit of $c$ is usually given in the literature as kJ/(kg·K). If the specific heat capacity is multiplied by the mass of the body, the total heat capacity $C$ is obtained, with the unit kJ/K.
A small overview of specific heat capacities, melting heats, and vaporization heats for various materials is shown in the table below.
| Substance | Heat Capacity $c$ [kJ/(kg·K)] | Melting Heat $c_\mathrm{s}$ [kJ/kg] | Vaporization Heat $c_\mathrm{v}$ [kJ/kg] |
|---|---|---|---|
| Water (l) | 4.18 | 334 | 2257 |
| Ethanol (l) | 2.43 | 105 | 854 |
| Helium (g) | 5.19 | 6 | 21 |
| Iron (s) | 0.45 | 227 | 6340 |
| Mercury (l) | 0.14 | 11.3 | 295 |
Energy input can occur in various ways. For example, friction losses or electric currents can heat a body, liquid, or gas. If an object falls from a certain height into a water basin, its potential energy is completely converted into thermal energy, resulting in a small, albeit hardly measurable, temperature increase of the water.
In the case of electrical heating, the power output must be multiplied by the duration $\Delta t$ to calculate the energy input $\Delta Q$. The required time for a heating process at constant power $P$ can then be determined as:
$$
\Delta t = \frac{cm\Delta T}{P}
$$
It is important to note that this formula assumes constant power delivery, which is usually approximately valid for electric devices like water kettles.