Boyle's Law is named after Robert Boyle (1627-1691). Edme Mariotte (1620-1684), a French physicist, discovered the law independently at roughly the same time, so this law is often known as Mariotte's or Boyle Mariotte's law. During scuba diving classes the law is usually called Boyle's Law. Boyle's law relates the volume and pressure of a gas held at a constant temperature. Boyle's law is:
PV = k
Where P is the pressure of gas,
V is the volume of gas
and k is a constant. The constant k does not need to be known to understand the relationship between P and V.
Boyle's law basically says this:
When you increase the pressure the volume decreases.
When you increase the volume the pressure decreases.
This Law is often applied to demonstrate what happens during ascent and descent and how it compensates the pressure in the BCD, lungs, mask and anywhere else that air is contained. As you descend, pressure increases. As pressure increases volume decreases, so the same amount of air takes up less space. Because of this, as you descend you will notice that your BCD "deflates". It is not losing air, it is compressing the same air into a smaller volume. You may also notice other effects of airspaces being compressed - wet suits fit less snugly (if you are wearing a thick neoprene suit this is even more apparent) and the airspace behind ears becomes decompressed. When equalizing you need to let more air into your Eustachian tubes to compensate for the reduction in volume.
Likewise, as you ascend pressure decreases and volume increases. A full BCD at depth will become fuller as you ascend - that is why you must release air from your BCD as you ascend. More importantly, it is important to exhale from your lungs as you ascend. If you hold your breath while ascending the air in your lungs expands beyond capacity, which can cause painful internal injuries.
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