Mathematically, this can be represented as:
Volume = Constant/Pressure
Pressure x Volume = Constant
Substituting in variables, the formula is:
Because the formula is equal to a constant, it is possible to solve for a change in volume or pressure using a proportion:
Explanation and Discussion:
Boyle's law explains an inverse relationship between the volume and pressure of a gas. For example, if the volume of a gas doubled, the pressure it exerted would be cut by half. If the volume tripled, the pressure would be a third of what it was before.
Why does this change? First, let's consider some of the properties of gases. Molecules in a gas are in constant motion. They have kinetic energy, or energy of motion. The average energy of a gas is its temperature. This energy causes the gas molecules, which are (ideally) not held together very tightly, to move in a straight line through space until they impact something. They can impact other molecules or the side of their container. Ideally, these impacts are totally elastic; no energy would be lost and the temperature would remain constant. These impacts cause a force on the surface of the container, pressure. (For Boyle's law, we assume that the temperature does not change).
The motion of gases also causes them to expand and fill their container, giving them a volume equal to that of their container. If they did not strike the sides of the container, they would continue on in a straight path.
Why are pressure and volume related then? Aren't they just two unique properties of gases? The link comes in how pressure is defined, and how volume affects the pressure.
Pressure is a derived unit. Pressure is force divided by a two-dimensional surface measurement. Force is often measured in newtons (N), a unit derived from a kilogram-meter (kg-m).Surface area is often measured in square meters or square centimeters. A pressure unit would then be a newton per square meter (N/m2), the Pascal (Pa). Because a Pascal is relatively small, force is often measured more conveniently in kiloPascals (kPa).
With a gas, the pressure is exerted on the sides of the container. If there is a greater surface area, the force will remain the same, so the pressure will go down. For example, if a ten newton force is exerted over ten square meters, the pressure is 1 kPa. If the surface area increases to twenty square meters, the pressure is reduced to 0.5 kPa. If the surface area decreases to five square meters, the pressure is increased to 2 kPa.
The volume of the container dictates its internal surface area for the gas. If the volume of a gas decreases, because a gas expands to fill its container, the container's volume must have decreased. Therefore, there is a smaller surface area, and the pressure increases. If the volume of the gas expands, meaning a larger container, the pressure would go down. This type of relationship is called an inverse relationship.
That's most of Boyle's law! Mathematically, this is represented as PV. Boyle's law produces a constant, K, so extended, the formula is PV=K. However, because the constant stays the same, additional pressures and volumes can be equated and solutions can be found for unknowns. This equation can be a powerful tool in solving for or converting pressures and volumes.
Let's try an example with our new equation. If the volume of an gas is 0.312 liters at 822 kPa, how would we find a new volume at 948 kPa?
0.312 L x 822 kPa = 948 kPa x V1
(0.312 L x 822 kPa) / 948 kPa = V1
V1 = (0.312 L x 822 kPa) / 948 kPa
V1 = 0.2705 L
V1 = 0.271 L (to the correct number of significant digits)
Checking the answer: Because the pressure increased, we would expect the volume to decrease. Because the pressure increased by 115% (115/100), we would expect the volume to decrease by 87% (100/115). Dividing 0.271 by 0.312 get .868, or 87%. Our answer is correct.