The mathematical expression of Boyle’s law

Introduction

Mariotte’s law is a different name from Boyle’s law. It expresses the relationship between gas compression and expansion at a constant temperature. In 1662, Robert Boyle, a scientist, explained the relationship between pressure and volume. According to him, the volume of a given gas is inversely proportional to its pressure at a constant temperature. The link between pressure and volume was discovered in 1676 by Edme Mariotte, a French physicist.

Under the assumption of an ideal gas, the law can be inferred from gas kinetic theory. Real gases satisfy Boyle’s law at sufficiently low pressures, while the product PV decreases slightly as the gas deviates from perfect behaviour at higher pressures.

The mathematical expression of Boyle’s law

The mathematical expression of Boyle’s law states the relationship between pressure and volume. This Boyle’s law is used in many applications nowadays. The mathematical expression is generated so that we can easily understand the relationship between pressure and volume.

Now,P is the pressure applied

V is the volumes of the gas

T is the constant temperature,

K is constant of proportionality

At fixed temperature and a fixed amount of gas,

According to Boyle’s law

This can also be expressed as

                                           V∝1/P

                                            V=K/P

                                            PV=K

From the equation, we can see that pressure and volume are inversely proportional. This equation can also be derived from the ideal gas law equation.

Applications of Boyle’s law

As stated above, Boyle’s law is used or applied on a day-to-day basis. For example, while we were breathing etc.

Respiration: Every minute, all humans are required to use Boyle’s law. Because we will be using Boyle’s law while breathing. When we inhale, the volume increases as our body is filled with air and the pressure lowers. As you exhale. The volume shrinks while the pressure rises.

Soda Bottle

This is the best illustration of Boyle’s law. The soda bottle is usually half-filled with carbon dioxide and half-filled with water. The air molecules are suffocated when the bottle is closed, and they have no room to move. The air molecules in the bottle evaporate when the bottle is opened, lowering the pressure.

Syringe: Syringes are mostly used in the medical field. It is used to insert or add fluids, among other things. When the plunger is pushed down, the number of liquid increases, resulting in a drop in pressure. When the plunger is pushed up, however, the volume decreases and the pressure rises.

Adding air to flat tyres

Everyone owns a vehicle. Let’s look at Cycles to get a better understanding of it. When we add air in cycles with an air pumper, the number of air molecules in the tyre increases, indicating that the volume has grown and the pressure has reduced.

Mathematical expression of boyle’s law

To understand it more clearly,we can say  that at constant temperature and quantity, any change taken place in the volume of the gas effects the total pressure exerted.This can be mathematically expressed as 

P1V1 = P2V2

The above expression tells that,The product of initial volume and the initial pressure is equal to the volume of final pressure and final volume.

The terms can be defined as

P1 is initial pressure exerted by the gas

P2 is final pressure exerted by the gas

V1 is initial volume occupied by the gas

V2 is the final volume occupied by the gas

The above mathematical expression is by the known relationship between pressure and volume. We know that  PV = k

The initial pressure and the initial volume are kept constant, Hence,

P1V1  = k

The final pressure and final volume are kept constant

 Also  P2V2  = k

 Therefore, when we equalise both we will get the mathematical expression as

P1V1 = P2V2

Boyle’s law graph:

To understand the relationship or the expression, Look at the graph below

The graph shows the hyperbola shapes which states that as the pressure increases, the volume decreases.

Hyperbola has two asymptotes, horizontal and vertical. This explains that even though pressure applied is larger or infinity, the volume will never be zero. Likewise, even though the volume is increased to infinity or larger, the pressure will never be decreased to zero.This explains that both pressure and volume are inversely proportional to each other.

Example for Boyle’s law mathematical expression 

Let us take a problem to understand the expression of Boyle’s law

A fixed amount of gas takes up 2L of space and exerts a pressure of 800 kPa on the container’s walls. What pressure would the gas exert if it were totally moved into a new container with a volume of 6 litres (provided the temperature and quantity of gas remained constant)?

Solution:

The initial volume of gas occupied is 2 litres           

The initial  pressure of the gas exerted is 800kpa          

The final volume of gas occupied is 6 litres

The final pressure exerted by the gas is?                                                     

we know the expression 

As per Boyle’s law                          

P1V1 = P2V2 

Conclusion

Boyle’s law explains how gases behave. And it is stated as the pressure and volume are reciprocal to each other at a fixed amount of gas and temperature. This is used in many day-to-day life applications.