Boyle's Law: Pressure-Volume Relationship

Robert Boyle investigated, experimentally, how the volume of a fixed amount of gas responds to a change in pressure at constant temperature. His apparatus was a J-shaped glass tube, closed at the short end (trapping a fixed column of air) and open at the long end, into which mercury could be poured (Figure 6.1).

With the mercury level equal in both arms (Figure 6.1a), the trapped air is at atmospheric pressure, 1 atm, and occupies some volume V. Adding mercury until the trapped air is compressed to half its original volume (Figure 6.1b) creates a height difference of 760 mm between the two mercury columns -- and since a 760 mm column of mercury is by definition 1 atm, the pressure difference caused by the added mercury is 1 atm, so the gas pressure is now $1+1=2$ atm. Compressing further to a third of the original volume (Figure 6.1c) raises the height difference to 1520 mm (2 atm of added pressure), so the gas pressure becomes $1+2=3$ atm... except the book's own reading records it more simply: pressure has doubled when volume halved, and by the same pattern quadruples when volume is cut to a quarter. This experiment led Boyle to his conclusion: at a given temperature, the volume occupied by a fixed mass of gas is inversely proportional to its pressure.

Mathematically, with T and n (temperature and number of moles) held fixed,

where k is a proportionality constant. Rearranging (6.2),

Boyle's law holds for any gas regardless of its chemical identity, provided the pressure is not too high. So for a given mass of gas under two different sets of conditions, at the same temperature,

Why compressing a gas raises its pressure. Gas pressure arises from the constant bombardment of the container walls by gas molecules. If a fixed amount of gas is compressed to half its volume, the number density of molecules doubles, so twice as many molecules strike a given patch of wall in a given time -- and the pressure doubles as a direct consequence.

Consequence: pressure and density. Since density $d=\dfrac{\text{mass}}{\text{volume}}=\dfrac{m}{V}$, i.e. $V=\dfrac{m}{d}$, substituting into Boyle's law $P_1V_1=P_2V_2$ gives $P_1\dfrac{m}{d_1}=P_2\dfrac{m}{d_2}$, so

In other words, at constant temperature the density of a fixed mass of gas is directly proportional to its pressure -- squeeze a gas into a smaller volume and it gets denser in direct proportion to how much harder you squeeze it. …