The first law deals with the relationship between the pressure and the volume of an ideal gas. Robert Boyle and Edme Mariotte discovered this relationship independently of each other.
To understand this law, imagine the ideal gas contained in a vessel that is sealed airtight with a movable piston. The piston is weighted, so that a defined pressure is established in the gas, which can be calculated from the weight force and the cross-sectional area of the piston.
It is important that during the investigation the temperature is kept constant at all times, i.e. an isothermal process takes place. Furthermore, the container must be so well sealed that no exchange of particles with the environment can occur; otherwise, it would no longer be a closed system.
If the number of weights is increased and the piston is thus pushed further into the vessel, an initial rise in temperature is observed.
Therefore, the measurement can only be taken once the temperature has returned to room temperature.
It is then observed that the height of the piston, and thus the volume, decreases in inverse proportion to the pressure, i.e.:
$$
V \propto \frac{1}{p}
$$
In other words: the product of pressure and volume is always constant for an ideal gas:
$$
pV = \text{const}
$$
This experimental result makes sense, since we have already linked the product \(pV\) with energy, which must always be conserved in a closed system.
From this relationship, it immediately follows that in a change of state, the product of \(p\) and \(V\) must be equal before and after:
$$
p_1 V_1 = p_2 V_2
$$
All three forms are equivalent to one another and are used in the literature to represent Boyle–Mariotte’s law.