What is the mathematical formula for boyle's law?

Answer 1

PV = const
for pressure Pand volume V, at constant temperature T, PV = constant. This is Boyle's Law.

The Law of Atmospheres: An Ocean of Water

First, a quick reminder of how we found the pressure variation with depth in an ocean of water at rest. We imagine isolating a small cylinder of water, with its axis vertical, and construct a free body diagram:

The pressure forces from the surrounding water acting on the curved sides obviously all cancel each other. So the only forces that count are the weight of the cylinder of water, and the pressure forces on the top and the bottom-that on the bottom being greater, since it must balance the pressure on the top plus the weight, since the cylinder is at rest.

Taking the cylinder to have cross-section area A, height , and the water to have density , the cylinder has volume , mass , and therefore weight

The pressure P is a function of height h above the bottom,

We've measured h here from the bottom of the ocean, because in the next section, we'll apply the same analysis to the atmosphere, where we do live at the bottom of the "sea".

The pressure on top of the cylinder exerts a downward force equal to

the bottom feels an upward pressure , so since the total force must be zero,

.

This equation can be rearranged to:

.

Recalling that the differential is defined by , we see that this pressure equation in the limit becomes:

.

Since is a constant, the solution is simple:

where we've written the constant of integration in the form. Notice the pressure in this ocean drops to zero at height h= h0 - obviously the surface! This means our formula describes water pressure in an ocean of depth h0, and is just a different way of writing that the pressure is times the depth below the surface. (We are subtracting off the atmospheric pressure acting down on the ocean's surface from the air above it-we're just considering the extra pressure from the weight of the water itself as we descend. Remember air pressure is the same as approximately thirty feet of water, so is a small correction in a real ocean)

An Ocean of Air

We now go through exactly the same argument for an "ocean of air", drawing the same free body diagram for a small vertical cylinder, and arriving at the same differential equation,

But it doesn't have the same solution! The reason is that , which we took to be constant for water (an excellent approximation), is obviously not constant for air. It is well known that the air thins out with increasing altitude.

The key to solving this equation is Boyle's Law: for a given quantity of gas, it has the form , but notice that means that if the pressure of the gas is doubled, the gas is compressed into half the space, so its density is also doubled.

So an alternative way to state Boyles law is

where C is a constant (assuming constant temperature). Putting this in the differential equation:

.

This equation can be solved (if this is news to you, see the footnote at end of this section):

.

The air density decreases exponentially with height: this equation is the Law of Atmospheres.

This density decrease doesn't happen with water because water is practically incompressible. One analogy is to imagine the water to be like a tower of bricks, one on top of the other, and the air a tower of brick-shaped sponges, so the sponges at the bottom are squashed into much greater density-but this isn't quite accurate, because at the top of the atmosphere, the air gets thinner and thinner without limit, unlike the sponges.

Footnote: Solving the Differential Equation

The equation is the same as , where a is a constant. If you are already familiar with the exponential function, and know that , you can see the equation is solved by the exponential function. Otherwise, the equation can be rearranged to , then integrated using to give , with c a constant of integration. Finally, taking the exponential of each side, using , gives , where .

Answer 2

Boyle's law states that pressure and volume have an inverse relationship. In an equation form: P1V1=P2V2. Starting pressure times starting volume equals ending pressure times ending volume.