## Boyle’s law: pressure and volume of an enclosed gas

Denition; Boyle’s Law The pressure of a ﬁxed quantity of gas is inversely proportional to the volume it occupies so long as the temperature remains constant.

P α 1/V

P=K/V

P1V1 = P2V2

Can you use the kinetic theory of gases to explain this inverse relationship between the pressure and volume of a gas?

Let’s think about it. If you decrease the volume of a gas, this means that the same number of gas particles are now going to come into contact with each other and with the sides of the container much more often.

You may remember from earlier that we said that pressure is a measure of the frequency of collisions of gas particles with each other and with the sides of the container they are in.

So, if the volume decreases, the pressure will naturally increase. The opposite is true if the volume of the gas is increased.

Now, the gas particles collide less frequently and the pressure will decrease.

It was an Englishman named Robert Boyle who was able to take very accurate measurements of gas pressures and volumes using excellent vacuum pumps.

He discovered the startlingly simple fact that the pressure and volume of a gas are not just vaguely inversely related, but are exactly inversely proportional.

This can be seen when a graph of pressure against the inverse of volume is plotted.

When the values are plotted, the graph is a straight line. This relationship is shown in the graph below.

The graph of pressure plotted against the inverse of volume, produces a straight line. This shows that pressure and volume are exactly inversely proportional. Note

In this tutorial, the terms directly proportional and inversely proportional will be used a lot, and it is important that you understand their meaning.

We will look at two examples to show the diﬀerence between directly proportional and inversely proportional.

Directly proportional; two variables are said to be directly proportional when both values are increasing or decreasing.:

Y ∝ x

Y=kx

Inversely proportional; Two variables are inversely proportional if one of the variables is increasing and the other is decreasing or vice versa

We can write this relationship symbolically as

p∝1/ V

This equation can also be written as follows:

p = k V

where k is a proportionality constant. If we rearrange this equation, we can say that:

pV = k

This equation means that, assuming the temperature is constant, multiplying any pressure and volume values for a ﬁxed amount of gas will always give the same value.

So, for example,

p1V1 = k and p2V2 = k,

From this, we can then say that:

p1V1 = p2V2

P1 =initial pressure  V1=initial volume

P2 = final pressure    V2 = final volume

In the gas equations, k is a ”variable constant”. This means that k is constant in a particular set of situations, but in two diﬀerent sets of situations it has diﬀerent constant values.

calculations

Question 1

: A sample of helium occupies a volume of 160 cm3 at 100 kPa and 25 ◦C. What volume will it occupy if the pressure is adjusted to 80 kPa and if the temperature remains unchanged?

Answer Step 1 : Write down all the information that you know about the gas.

V1 = 160 cm3 and V2 = ?

p1 = 100 kPa and p2 = 80 kPa

Step 2 : Use an appropriate gas law equation to calculate the unknown variable. Because the temperature of the gas stays the same, the following equation can be used: p1V1 = p2V2

If the equation is rearranged, then

V2 = (p1V1) /p2

Step 3 : Substitute the known values into the equation, making sure that the units for each variable are the same. Calculate the unknown variable.

V2 =100×160/ 80

V2 = 200cm3

The volume occupied by the gas at a pressure of 80kPa, is 200 cm3

Question 2

The pressure on a 2.5 l volume of gas is increased from 695 Pa to 755 Pa while a constant temperature is maintained. What is the volume of the gas under these pressure conditions?

Answer Step 1 : Write down all the information that you know about the gas.

V1 = 2.5 l and V2 = ?

p1 = 695 Pa and p2 = 755 Pa

Step 2 : Choose a relevant gas law equation to calculate the unknown variable. At constant temperature,

p1V1 = p2V2

Therefore,

V2 =p1V1 / p2

Step 3 : Substitute the known values into the equation, making sure that the units for each variable are the same.

Calculate the unknown variable.

V2 =695×2.5 /755

V2 = 2.3l

Exercise:

1. An unknown gas has an initial pressure of 150 kPa and a volume of 1 L. If the volume is increased to 1.5 L, what will the pressure now be?
2. A bicycle pump contains 250 cm3 of air at a pressure of 90 kPa. If the air is compressed, the volume is reduced to 200 cm3. What is the pressure of the air inside the pump?
3. The air inside a syringe occupies a volume of 10 cm3 and exerts a pressure of 100 kPa. If the end of the syringe is sealed and the plunger is pushed down, the pressure increases to 120 kPa. What is the volume of the air in the syringe?