Boyle's Law states:

At constant temperature, the volume of a gas varies inversely with absolute pressure, while the density of a gas varies directly with absolute pressure.

Boyle's Law is mathematically stated as:

Boyle's Law is important to divers because it relates changes in pressure i.e., depth, to changes in the volume of a gas and defines the relationship between pressure and volume in breathing gas supplies.

Example #1: Suppose you had a balloon containing 1 cubic foot of air at the surface of the water. This balloon is under 1 Atmosphere-Absolute (ATA) or 14.7 psi or 1 Bar (approx) of pressure. If we take the balloon underwater to a depth of 33 feet (10m), it is now under 29.4 psi (2 Bar) of pressure. Boyle's Law then tells us that since we have twice the pressure, the volume of the balloon will be decreased to one half. It follows then, that taking the balloon to 66 feet (20m), the pressure would compress the balloon to one third its original size, 99 feet (30m) would make it 1/4, etc.

If we bring the balloon in the previous example back up to the surface, it would increase in size due to the lessening pressure until it reached the surface and returned to its original size, 1 cubic foot. This is because the air in the balloon is compressed from the pressure when submerged, but returns to its normal size and pressure when it returns to the surface.

Here is a Calculator for ATA and volume fractions.

Along with the volume of air in the balloon, the surrounding pressure will affect the density of the air as well. Density, simply stated, is how close the air molecules are packed together. The air in the balloon or container at the surface is at its standard density, but when we descend to the 33 feet (10m) where its volume is reduced to one half, the density has doubled. At 66 feet (20m), the density has tripled. This is because the pressure has pushed the air molecules closer together. The reverse also happens, suppose we inflate a balloon at 99 feet (30m) We know the air at this depth is 4 times denser than at the surface. As the balloon ascends, the external pressure lessens and  the balloon will expand, eventually bursting.

Example #2: This is a much tougher problem, but has practical applications. You want to make another dive, but your buddy does not have a full tank to use. What do you do? You have a full tank but she does not. You also have an equalization hose. You want to equalize both tanks so you can dive. So what pressure are we able to put into the 50 CF tank from a full 100 CF tank? Tank equalization takes time in order for Boyles law to apply. Pressure and fill calculation are governed by Boyle's law as long temperature does not have to be considered. To calculate a very fast fill (seconds) we would need to use Charles Law and some calculus since it would then be a time based integration problem due to fast temperature changes. We are going to hook up the hoses, open the valves slowly, and wait 15-20 min for the temperatures to also equalize so we can use Boyles Law and make things much simpler. This is of particularly useful in filling your own pony bottles from a full primary tank. So we will work in Cubic feet and psi. We will assume that the 100 CF tank is a low pressure steel tank with 2500 psi in it. We also assume that the 50 CF tank has 500 psi left in it from a previous dive. So: P1 = 2500 psi (gauge pressure in 100 CF tank) Pb = 500 psi (gauge pressure left in 50 CF tank) V1 = 100 Cubic Feet V2 = 150 Cubic Feet (100 CF tank + 50 CF tank) Arranging Boyles equation for this example we get: P2 = (P1 - Pb + 14.7)*V1/V2 + Pb - 14.7 P2 = (2500 - 500 + 14.7) * 100 / 150 + 500 - 14.7 = 1828 psi Now we have 1828 psi in both tanks, since the pressures in both tanks must be equal (in a nominal time allowing for temperature equalization also). We can now make another dive.

Here is a Calculator for you to do Tank Equalizations. Tank #1 is the higher pressure tank

In these examples of Boyle's Law, the temperature of the gas was considered a constant value. However, temperature significantly affects the pressure and volume of a gas; it is therefore essential to have a method of including this effect in calculations of pressure and volume. To a diver, knowing the effect of temperature is essential, because the temperature of deep water is often significantly different from the temperature of the air at the surface. The gas law that describes the physical effects of temperature on pressure and volume is "Charles' Law"