What is the relationship between volume and pressure?

The pressure (P) of gas is inversely proportional to the volume (V) of gas. This means that as we hold temperature (T) and amount (n) of gas constant (same), as the pressure of gas molecules increase, the volume of gas molecules decrease. Mathematically, we can write this relationship as:

P ∝1/V

If we remove the proportionality sign and introduce an equal sign and a proportionality constant, we will get:

P = K/V

If we multiply both sides of the equation by V, we will get the expression: PV = K.

Notice that we have held the temperature (T) and amount of gas (n) constant.

Since P times (x) V is equal to a constant: K, then it follows that different conditions of pressure and volume for the same mass of gas at constant temperature can be expressed as:

P1V1 =K= P2V2, thus, P1V1= P2V2,   equation  (1)

where;

• P1 = old pressure
• V1 = old volume
• P2 = new pressure
• V2 = new volume

Basically, if you know three of the four variables in equation 1, you can solve for the fourth variable. However, you must make sure your units agree with each other. That is V1 & V2 should be in the same units and P1 and P2 should also be in the same units.

How to use equation 1 to calculate gas volume or pressure

At constant temperature of 35 ºC, a sample of gas occupies a volume of 5.0 L and has a pressure of 2 atm. If the volume of the gas decreased to 2.0 L, calculate its new pressure.

Answer

Let’s use the following models to make sense of the problem

Clearly, from the model you can see that as the volume of gas decreased, its molecules become more crowded and collide more often with the walls of the container. As a result, the new pressure inside the container should be higher than the old pressure.

Therefore,

V1, old volume = 5.0 L

P1, old pressure = 2 atm

V2, new volume = 2.0 L

P2, new pressure = ?

Since P1V1 = P2V2, then it follows that;

5.0 L x 2 atm = 2.0 L x P2

If you isolate P2 by dividing the left and right sides of the equal sign, you will get:

P2 = (5.0 L x 2.0 atm)/2.0 L = 5.0 atm.

Hence, the new pressure is 5.0 atm. This new pressure is higher than the old one and agrees with our initial prediction.

How do Graphs of pressure versus volume look?

The relationship between pressure and volume (P-V relationship) is usually called Boyle’s law in honor of Robert Boyle, who was first to uncover the relationship.

If you plot pressure versus volume or pressure versus inverse of volume (1/v), you will get graphs like the ones in this model.

Where can we apply Boyle’s law?

Anytime we breathe, we apply Boyle’s law. To breathe in, we must expand our rib cage to increase its volume so that the pressure inside our lungs can decrease. Once the pressure inside our lungs becomes lower than the atmospheric pressure, air molecules are able to rush in through our nostrils. Similarly, to breathe out, we must contract our rib cage to decrease its volume so that the pressure inside our lungs can become greater. Once the pressure inside our lungs becomes greater than the atmospheric pressure, air molecules are able to rush out through our nostrils.

Now, let’s use the piston-cylinder model to illustrate Boyle’s law

How can you demonstrate Boyle’s law at home?

You can use a syringe to demonstrate the pressure-volume relationship. Draw some air into syringe, then use one hand to close its mouth, while you use the other hand to apply force on the plunger. You will notice that as you apply more force the volume of gas gets smaller and smaller.

To learn more about the relationship between temperature and volume of gas, click here.

To learn more about the properties of gases, click here.

.

All the particles (atoms and molecules) of a substance are continually moving and so possess kinetic energy. In gases the movement of the particles is highly energetic and this is the reason why gases form, the particles have enough energy to overcome the attractive forces holding the particles together. In gases the particles are moving very quickly and freely in a random manner constantly bumping into each other and their surroundings. It is these collisions between the particles of the gas and the walls of the container it is confined to that creates gas pressure. The gas pressure is the overall force of all these collisions divided by the area of the walls of the container it is confined in.

The relationship of a gas with pressure and volume was developed by the scientist Robert Boyle at around 1660 and is known as Boyle’s Law.

Boyle’s law states:

"For a fixed mass of gas, at a constant temperature, the product (pressure x volume) is a constant."

Pressure x Volume = constant

p x V = constant

The animation below gives and explanation of Boyle's law:

A sealed cylinder with no leaks contains a fixed mass of a gas kept at a constant temperature. The gas pressure is created by the collision of the moving gas particles with each other and against the walls of the cylinder.

The above set up is used to investigate the relationship between pressure and volume for a gas. A force is exerted on the piston to compress the gas. The corresponding pressure and volume values are recorded for different applied forces.

By plotting the recorded values of pressure (p) against volume (V) a curve is produced. We can see from the values that when the pressure is doubled the volume is halved. If the pressure was to increase by 3 the volume would decrease to a third. Thus, the volume is inversely proportional to the pressure. By plotting pressure (p) against the reciprocal of the volume (1/V) a straight line is obtained the gradient of which is the constant in Boyle’s Law.

A decrease in volume increases the number of gas particles per unit volume. This results in an increase in the number of gas particles close to the cylinder walls and therefore an increase in the number of collisions with the wall. As the number of collisions per unit area increases so does the force per unit area thereby giving an increase in pressure.

Using the example of the sealed cylinder above, the volume of gas at the start is 50 cm3 with a pressure of 1.2 x 105 Pascals. The piston is pushed slowly into the syringe until the pressure on the gauge reads 2.0 x 105 Pascals. What is the volume of gas?

Solution:

We know

p x V = constant

therefore,

p1 x V1 = p2 x V2

p1= 1.2 x 105 Pascals
V1 = 50 cm3
p2 = 2.0 x 105 Pascals
V2 = ?

p1 x V1 = p2 x V2

V2 = 30 cm3