Physics
Cambridge IGCSE

TOPIC 2: THERMAL PHYSICS

# 2.1b Gas Pressure

The science behind steam power was the key to the industrial revolution. Understanding gases became a 'hot' topic. Scientists used particle models to help them understand how gases such as steam behaved.

The animation here shows gas particles moving freely in a box. The gas particles bounce off the sides and collide with each other. They are in constant random motion.

In this section, we will consider how these particles behave and what should happen when we change the temperature of the gas.

Figure 1. Particles in an ideal gas

Some particles are coloured red to see their motion more clearly.

### Pressure

Pressure is a measurement of the force acting on a surface, as covered in section 1.8. The international unit of pressure is the pascal (Pa). The particles in the box shown above (figure 1) hit the sides of the container at very high speeds. These collisions produce a force on the sides of the container. Therefore, the movement of particles make a pressure on the container.

This is a very common question in exams - you should be able to clearly explain the cause of this pressure.

Once scientists realised that these collisions caused the pressure in a trapped gas, they began to consider the effect of changing the conditions, and asking some simple questions using this model. What would happen to the pressure if we change variables such as the volume of the container or the temperature of the gas?

## 1. Volume and Pressure

What would happen if the volume of the box (shown above in figure 1) is increased? In this case, there would be fewer collisions on the sides in a certain time, as the particles would have a greater distance to travel between each collision. Therefore, an increase in volume leads to a decrease in pressure.
A bike tyre has air pumped into it, and so it is compressed into a small volume. This makes the pressure in the tyre high as explained above, and keeps them inflated for a smooth ride.

Figure 2. Pressure in bike tyres

### Calculating pressure when the volume changes

The relationship between volume and pressure led scientists to try to calculate and predict the pressure in a gas for a certain volume. A scientist called Robert Boyle realised that P x V was a constant value. If the volume is doubled, the pressure changes to half the original value, and so P x V remains constant.

Here's the formula used to show Boyle's Law for pressure and volume:

pressure x volume = constant

P V = constant

[Pa] x [m3] = [Pa m3]

Learn this formula!

To make this formula easier to use, it is common to use the following argument:
If the pressure x volume is constant for different volumes, then after any change of volume, P x V is the same at the start and at the end. Then we can write this as follows:

initial pressure x initial volume = final pressure x final volume

P1 V1 = P2 V2

[Pa] x [m3] = [Pa] x [m3]

We suggest you learn this formula - it makes it much easier to calculate values!

Notes:

• This formula is only valid if the mass and temperature are fixed, as shown in figure 3 below. The gas needs to be compressed slowly to ensure the temperature does not change.
• The units for volume can be in m3 or cm3, as long as you use the same units on both sides of the equation.

## Boyle's Law

It is easy to repeat Boyle's experiment and test how a smaller volume increases the pressure. A syringe or similar container with a piston / plunger is often used to measure the gas volume, with a pressure sensor attached. This is shown in the animation in figure 3.

Notice the shape of this famous graph showing the pressure increasing as the volume decreases. The line is curved. This experiment can be used to show that, for any fixed mass of gas at a constant temperature:

P V = constant

Figure 3: Pressure changing with volume of a gas
(By NASA's Glenn Research Center - Public Domain)

The graph shown above in figure 3 shows that pressure is inversely proportional to volume, and you are expected to be able to show this in a graphical form.

Example:
A bubble of gas is underwater, and has a volume of 30 cm3 when at a pressure of 200 kPa. As it rises, the pressure decreases to 150 kPa. Calculate the volume of the bubble of gas at 150 kPa.

For this question, we can label the starting pressure and volume:
P1 = 200 kPa = 200 000 Pa
V1 = 30 cm3 (as long as we use cm3 for V1 and V2 on both sides of this formula, we do not need to do a conversion.)
We know the final pressure P2 is 150 kPa = 150 000 Pa. Now we can substitute into the formula above:

P1 x V1 = P2 x V2
200 000 x 30 = 150 000 x V2
So V2 = 6 000 000 / 150 000 = 40 cm3

Questions:

1. A diver is deep under the sea. When divers breathe out, they release bubbles of gas which rises to the surface. Explain what happens to:

• a) The pressure in the water surrounding a bubble as it rises to the surface.
• b) The volume of a bubble as it rises to the surface.

a) The pressure in water depends on the height (h) of water above that point due to the formula P = ρ g h.
Therefore as a bubble rises, h decreases, and so the pressure decreases.

b) If the pressure of the water decreases as a bubble rises, then the pressure on the gas bubble also decreases. If the pressure decreases, then as PV = constant, then the volume increases.

2. A syringe contains 10 cm3 of air at a pressure of 100 kPa.

• a) Calculate the new pressure in the syringe when the gas is compressed to a volume of 4 cm3.
• b) State one condition required for the calculation above to be true.

a) We know that:
P1 x V1 = P2 x V2
Substituting in values for the initial volume (10 cm3) and pressure (100 000 Pa), and the final volume (4 cm3) gives:
100 000 x 10 = P2 x 4
rearranging this gives:
 P2 = 1 000 000 4

So the final pressure P2 = 250 000 Pa (250 kPa)

b) This formula only applies if the temperature or mass of gas remains constant.

3. A diver descends to a depth of 40 m, breathing air that is at a pressure of 500 kPa. The diver's lungs have a total volume of 0.006 m3.

a) Calculate the volume the air in the diver's lungs would occupy at the surface, where the pressure is only 100 kPa.

b) Suggest why it is dangerous for divers to hold their breath whilst swimming upwards to the surface.

a) Using the formula:
P1 x V1 = P2 x V2
Then substituting in values for the initial and final values gives:
500 000 x 0.006 = 100 000 x V2
rearranging this gives:

 V2 = 500 000 x 0.006 100 000

So the final volume of air at the surface will be V2 = 0.03 m3.

b) This volume is 5 times larger than the volume of the lungs. The diver is in danger of causing damage to his /her lungs or even rupturing them, as the air expands outwards.

## 2. Temperature and Pressure

If we increase the temperature, the gas particles move faster. Scientists now know that the temperature of a substance is directly related to the average kinetic energy of the particles. If we increase the temperature of a gas, then two things will happen:

• They will hit the sides harder.
• They will hit the sides more often, as they will take less time to travel between the sides of the container.

An increase in temperature leads to a pressure increase. This assumes that the volume of the container is kept constant.

Now have a go at these questions to check you understand this section:

Questions:

4. A basketball is left outside in strong sunlight. As the ball heats up, it appears to be harder to compress and bounces higher. This is because the pressure inside the ball has increased.
Using the idea of particles in a gas, explain why the gas pressure inside the ball has increased.

As the temperature increases, the particles move faster. This means they hit the sides of the container harder, and also more often. This produces a larger force on the sides and therefore a larger pressure.

5. An aerosol can contains a fluid under pressure. Explain why it is dangerous to expose the can to high temperatures.

If you increase the temperature, the pressure inside the can increases further. This could cause the can to break open or even explode in high temperatures.

## The Kelvin scale

Is there a limit to how hot a gas or any substance can be? We know the surface of the Sun is at about 6000 °C from observations. Scientists estimate the temperature in the core of the Sun to be about 15 million °C! These particles could still speed up to even higher temperatures. There is no upper limit. However, there is a lower limit to temperature.

If you keep cooling a gas, the particles move slower and slower. Eventually, at - 273 °C, the particles stop moving all together and cannot be cooled further. This is the lowest temperature possible, and is hence called absolute zero.

If this is the true 'zero' of temperature, it seems sensible to start measuring temperature from this zero point, and increase from there. This new scale is called the Kelvin scale. Absolute zero at - 273 °C is equal to zero Kelvin (0 K). The melting point of ice, is 0 °C, equal to +273 K, and so on. This is shown in figure 4 below:

Figure 3: Pluto's surface
A very cold 44 K or -229 °C

Figure 4. An imaginary thermometer showing the Kelvin scale.

As you can see from this figure, the Kelvin scale is just the centigrade (also known as the Celsius) scale shifted down by 273 units to the left on this diagram. To calculate the temperature in Kelvin is really easy as shown by this formula:

Temperature (in Kelvin) = Temperature (in Celsius/centigrade) + 273

[K] = [°C] + 273

Learn this formula!

Now have a go at these questions to check you understand the concept of absolute zero and the Kelvin scale:

Questions:

6. Scientists are studying the conditions found on the planet Mars.

• a) One block of frozen CO2 is at a temperature of 150 K. What is this temperature in degrees Celsius?
• b) Surface rocks can get as warm as 20 °C in the day time. What is this temperature in Kelvin?

a) The temperature will be 150 K - 273 = -123 °C

b) The temperature will be 20 °C +273 = 293 K

### Temperature and K.E.

As a substance is heated above absolute zero, the particles move faster. This means they increase in kinetic energy. Any two substances that are at the same temperature have particles with the same average kinetic energy. If the kelvin temperature is doubled, the average kinetic energy doubles. So next time you use a thermometer to measure air temperature, remember you are actually measuring the kinetic energy of the gas molecules around you.

The total thermal energy stored in a substance is called the internal energy as explained in section 1.7a. Therefore a temperature increase will lead to a kinetic energy increase of all the particles in a substance, and therefore an internal energy increase.

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