Ideal gas molecules

What you'll learn

How random molecular motion produces gas pressure.
Why absolute zero is −273 °C and why the Kelvin scale starts there.
How temperature affects the average kinetic energy and speed of gas molecules.
How to use the gas relationships $p1T1=p2T2\frac{p_1}{T_1}=\frac{p_2}{T_2}$ and $p_1V_1=p_2V_2$ .

1. The particle picture of a gas

A gas is made of particles, usually molecules, that are far apart compared with their size. A molecule is a small particle made from atoms joined together.

Gas molecules are in random motion. This means they move in many different directions, with no fixed pattern. They continually collide with each other and with the walls of their container.

When a gas molecule hits a wall, it changes direction. During the collision, it exerts a force on the wall. One molecule gives a tiny force, but a gas contains huge numbers of molecules, so the combined effect of many collisions each second produces a measurable pressure.

Definition

Gas pressure

Pressure is force per unit area on a surface. In a gas, pressure is caused by molecules colliding with the walls of the container. The SI unit of pressure is the pascal, Pa; kPa is also commonly used.

The diagram links the particle model to the two gas relationships you need for this topic.

Gas molecules moving randomly, producing pressure, with pressure-volume and pressure-temperature graphs

Example

Explaining pressure in a tyre

A bicycle tyre is pumped up with more air at roughly the same temperature. Explain why the pressure increases.

Adding air increases the number of gas molecules in the fixed volume of the tyre.
With more molecules moving randomly, more molecules strike each square metre of tyre wall each second.
More collisions per second means a larger total force per unit area on the wall, so the gas pressure increases.

2. Temperature and molecular motion

A moving object has kinetic energy, which is energy due to motion. Gas molecules have kinetic energy because they are constantly moving.

Temperature is linked to the motion of molecules. If you heat a gas, energy is transferred to its molecules. Their average kinetic energy increases, so their average speed increases.

Key Idea

Heating a gas

An increase in temperature means the gas molecules have greater average kinetic energy and therefore a greater average speed.

The word average matters. Not every molecule in a gas moves at the same speed. Some are faster and some are slower, but increasing the temperature raises the average.

Common Mistake

Not every molecule speeds up equally

Do not write that “all molecules move at the same speed”. In a gas, molecules have a range of speeds; temperature tells you about their average kinetic energy.

3. Absolute zero

If cooling a gas reduces the average kinetic energy of its molecules, there must be a lowest possible temperature. In the IGCSE particle model, this is where the molecules have the minimum possible kinetic energy.

Definition

Absolute zero

Absolute zero is the lowest possible temperature. It is −273 °C, which is 0 K on the Kelvin scale.

There cannot be a temperature below absolute zero because average kinetic energy cannot be negative.

4. The Kelvin scale

The Kelvin scale is the temperature scale used in gas calculations. It starts at absolute zero:

0 K = −273 °C
273 K = 0 °C
373 K = 100 °C

A temperature change of 1 K is the same size as a temperature change of 1 °C, but Kelvin temperatures start from a different zero point.

Tip

Writing Kelvin units

Write Kelvin temperatures as 300 K, not 300 °K. Kelvin does not use a degree symbol.

To convert between Celsius and Kelvin:

Kelvin temperature = Celsius temperature + 273

TK=TC+273T_\text{K}=T_\text{C}+273

Celsius temperature = Kelvin temperature − 273

TC=TK−273T_\text{C}=T_\text{K}-273

Example

Converting between Celsius and Kelvin

Convert 27 °C to K, and convert 250 K to °C.

For Celsius to Kelvin, add 273: $KT_\text{K}=27+273=300\ \text{K}$ .
For Kelvin to Celsius, subtract 273: $TC=250−273=−23∘CT_\text{C}=250-273=-23^\circ\text{C}$ .

5. Kelvin temperature and average kinetic energy

For a gas, the Kelvin temperature is proportional to the average kinetic energy of its molecules.

Definition

Proportional

Two quantities are proportional if multiplying one by a certain factor multiplies the other by the same factor. For an ideal gas, average kinetic energy is proportional to Kelvin temperature.

So if the Kelvin temperature doubles, the average kinetic energy doubles. This is why you must use Kelvin, not Celsius, when comparing gas temperatures.

Example

Comparing average kinetic energies

A gas warms from 20 °C to 40 °C. By what factor does the average kinetic energy increase?

Convert both temperatures to Kelvin because average kinetic energy is proportional to Kelvin temperature: 20 °C is 293 K, and 40 °C is 313 K.
Compare the Kelvin temperatures: factor $K=1.07=\frac{313\ \text{K}}{293\ \text{K}}=1.07$ .
The average kinetic energy increases by a factor of 1.07, so it is about 7% greater, not twice as large.

Common Mistake

Using Celsius as if it starts at zero energy

A gas at 40 °C does not have twice the average kinetic energy of a gas at 20 °C. You must convert to Kelvin before comparing temperatures in gas questions.

6. Gas relationships for a fixed mass

A fixed mass of gas means the amount of gas stays the same: no gas enters or leaves. The same idea is often described as a fixed amount of gas.

You need two ideal gas relationships in this topic:

pressure and volume at constant temperature
pressure and Kelvin temperature at constant volume

Common Mistake

When the simple gas laws apply

These relationships assume a fixed mass of gas. If gas is added or removed, or if the gas changes state into a liquid, the simple relationships no longer apply.

7. Pressure and volume at constant temperature

At constant temperature, the average kinetic energy and average speed of the gas molecules stay the same.

If the volume decreases, the molecules have less space to move around in. They hit the walls more often. Since the molecules are not slower, the increased collision rate causes a higher pressure.

So, for a fixed mass of gas at constant temperature:

smaller volume means larger pressure
larger volume means smaller pressure
pressure and volume have an inverse relationship

The equation is:

pressure 1 multiplied by volume 1 = pressure 2 multiplied by volume 2

p_1V_1=p_2V_2

where:

$p_1$ is the initial pressure
$V_1$ is the initial volume
$p_2$ is the final pressure
$V_2$ is the final volume

Useful rearrangements are:

p2=p1V1V2V2=p1V1p2\begin{aligned} p_2 &= \frac{p_1V_1}{V_2}\\ V_2 &= \frac{p_1V_1}{p_2} \end{aligned}

Example

Compressing a gas at constant temperature

A sealed syringe contains gas at a pressure of 100 kPa and a volume of 0.060 m³. It is compressed slowly at constant temperature to a volume of 0.020 m³. Calculate the final pressure.

Choose $p_1V_1=p_2V_2$ because the mass of gas is fixed and the temperature is constant.
Rearrange for final pressure: $p2=p1V1V2p_2=\frac{p_1V_1}{V_2}$ .
Substitute with units: $kPap_2=\frac{100\ \text{kPa}\times 0.060\ \text{m}^3}{0.020\ \text{m}^3}=300\ \text{kPa}$ .
The volume has become one-third of its original value, so the pressure becoming three times larger matches the inverse relationship.

8. Pressure and Kelvin temperature at constant volume

At constant volume, the container size does not change. If the gas is heated, its molecules have greater average kinetic energy and move faster on average.

Faster molecules collide with the walls more often and with a greater effect each time. This increases the pressure.

So, for a fixed mass of gas at constant volume:

higher Kelvin temperature means higher pressure
lower Kelvin temperature means lower pressure
pressure is directly proportional to Kelvin temperature

The equation is:

pressure 1 divided by Kelvin temperature 1 = pressure 2 divided by Kelvin temperature 2

p1T1=p2T2\frac{p_1}{T_1}=\frac{p_2}{T_2}

where temperature must be in K.

Useful rearrangements are:

p2=p1T2T1T2=p2T1p1\begin{aligned} p_2 &= \frac{p_1T_2}{T_1}\\ T_2 &= \frac{p_2T_1}{p_1} \end{aligned}

Example

Heating a gas at constant volume

A sealed metal cylinder contains gas at 100 kPa at 27 °C. It is heated to 87 °C. Calculate the new pressure.

Convert to Kelvin: $KT_1=27+273=300\ \text{K}$ and $KT_2=87+273=360\ \text{K}$ .
Choose $p1T1=p2T2\frac{p_1}{T_1}=\frac{p_2}{T_2}$ because the gas mass and volume stay fixed.
Rearrange for final pressure: $p2=p1T2T1p_2=\frac{p_1T_2}{T_1}$ .
Substitute with units: $kPap_2=\frac{100\ \text{kPa}\times 360\ \text{K}}{300\ \text{K}}=120\ \text{kPa}$ .

Tip

Choosing the right gas relationship

If volume changes but temperature is constant, use $p_1V_1=p_2V_2$ . If temperature changes but volume is constant, use $p1T1=p2T2\frac{p_1}{T_1}=\frac{p_2}{T_2}$ . In both cases, the mass of gas must be fixed.

Exam technique

In the exam

Always convert °C to K before using any pressure-temperature gas equation.
Check what is being kept constant: constant temperature points to $p_1V_1=p_2V_2$ , while constant volume points to $p1T1=p2T2\frac{p_1}{T_1}=\frac{p_2}{T_2}$ .
Keep pressure units consistent, such as using kPa throughout or Pa throughout, and carry the units through your substitution.

Self review

Check yourself

Why do gas molecules create pressure on the walls of a container?
What is 35 °C in Kelvin, and why is Kelvin needed in gas calculations?
A fixed mass of gas is compressed at constant temperature. What happens to its pressure, and why?

What you'll learn

How random molecular motion produces gas pressure.
Why absolute zero is −273 °C and why the Kelvin scale starts there.
How temperature affects the average kinetic energy and speed of gas molecules.
How to use the gas relationships $p1T1=p2T2\frac{p_1}{T_1}=\frac{p_2}{T_2}$ and $p_1V_1=p_2V_2$ .

1. The particle picture of a gas

A gas is made of particles, usually molecules, that are far apart compared with their size. A molecule is a small particle made from atoms joined together.

Gas molecules are in random motion. This means they move in many different directions, with no fixed pattern. They continually collide with each other and with the walls of their container.

Definition

Gas pressure

The diagram links the particle model to the two gas relationships you need for this topic.

Gas molecules moving randomly, producing pressure, with pressure-volume and pressure-temperature graphs

Example

Explaining pressure in a tyre

A bicycle tyre is pumped up with more air at roughly the same temperature. Explain why the pressure increases.

Adding air increases the number of gas molecules in the fixed volume of the tyre.
With more molecules moving randomly, more molecules strike each square metre of tyre wall each second.
More collisions per second means a larger total force per unit area on the wall, so the gas pressure increases.

2. Temperature and molecular motion

A moving object has kinetic energy, which is energy due to motion. Gas molecules have kinetic energy because they are constantly moving.

Temperature is linked to the motion of molecules. If you heat a gas, energy is transferred to its molecules. Their average kinetic energy increases, so their average speed increases.

Key Idea

Heating a gas

An increase in temperature means the gas molecules have greater average kinetic energy and therefore a greater average speed.

The word average matters. Not every molecule in a gas moves at the same speed. Some are faster and some are slower, but increasing the temperature raises the average.

Common Mistake

Not every molecule speeds up equally

Do not write that “all molecules move at the same speed”. In a gas, molecules have a range of speeds; temperature tells you about their average kinetic energy.

3. Absolute zero

Definition

Absolute zero

Absolute zero is the lowest possible temperature. It is −273 °C, which is 0 K on the Kelvin scale.

There cannot be a temperature below absolute zero because average kinetic energy cannot be negative.

4. The Kelvin scale

The Kelvin scale is the temperature scale used in gas calculations. It starts at absolute zero:

0 K = −273 °C
273 K = 0 °C
373 K = 100 °C

A temperature change of 1 K is the same size as a temperature change of 1 °C, but Kelvin temperatures start from a different zero point.

Tip

Writing Kelvin units

Write Kelvin temperatures as 300 K, not 300 °K. Kelvin does not use a degree symbol.

To convert between Celsius and Kelvin:

Kelvin temperature = Celsius temperature + 273

TK=TC+273T_\text{K}=T_\text{C}+273

Celsius temperature = Kelvin temperature − 273

TC=TK−273T_\text{C}=T_\text{K}-273

Example

Converting between Celsius and Kelvin

Convert 27 °C to K, and convert 250 K to °C.

For Celsius to Kelvin, add 273: $KT_\text{K}=27+273=300\ \text{K}$ .
For Kelvin to Celsius, subtract 273: $TC=250−273=−23∘CT_\text{C}=250-273=-23^\circ\text{C}$ .

5. Kelvin temperature and average kinetic energy

For a gas, the Kelvin temperature is proportional to the average kinetic energy of its molecules.

Definition

Proportional

Two quantities are proportional if multiplying one by a certain factor multiplies the other by the same factor. For an ideal gas, average kinetic energy is proportional to Kelvin temperature.

So if the Kelvin temperature doubles, the average kinetic energy doubles. This is why you must use Kelvin, not Celsius, when comparing gas temperatures.

Example

Comparing average kinetic energies

A gas warms from 20 °C to 40 °C. By what factor does the average kinetic energy increase?

Convert both temperatures to Kelvin because average kinetic energy is proportional to Kelvin temperature: 20 °C is 293 K, and 40 °C is 313 K.
Compare the Kelvin temperatures: factor $K=1.07=\frac{313\ \text{K}}{293\ \text{K}}=1.07$ .
The average kinetic energy increases by a factor of 1.07, so it is about 7% greater, not twice as large.

Common Mistake

Using Celsius as if it starts at zero energy

A gas at 40 °C does not have twice the average kinetic energy of a gas at 20 °C. You must convert to Kelvin before comparing temperatures in gas questions.

6. Gas relationships for a fixed mass

A fixed mass of gas means the amount of gas stays the same: no gas enters or leaves. The same idea is often described as a fixed amount of gas.

You need two ideal gas relationships in this topic:

pressure and volume at constant temperature
pressure and Kelvin temperature at constant volume

Common Mistake

When the simple gas laws apply

These relationships assume a fixed mass of gas. If gas is added or removed, or if the gas changes state into a liquid, the simple relationships no longer apply.

7. Pressure and volume at constant temperature

At constant temperature, the average kinetic energy and average speed of the gas molecules stay the same.

If the volume decreases, the molecules have less space to move around in. They hit the walls more often. Since the molecules are not slower, the increased collision rate causes a higher pressure.

So, for a fixed mass of gas at constant temperature:

smaller volume means larger pressure
larger volume means smaller pressure
pressure and volume have an inverse relationship

The equation is:

pressure 1 multiplied by volume 1 = pressure 2 multiplied by volume 2

p_1V_1=p_2V_2

where:

$p_1$ is the initial pressure
$V_1$ is the initial volume
$p_2$ is the final pressure
$V_2$ is the final volume

Useful rearrangements are:

p2=p1V1V2V2=p1V1p2\begin{aligned} p_2 &= \frac{p_1V_1}{V_2}\\ V_2 &= \frac{p_1V_1}{p_2} \end{aligned}

Example

Compressing a gas at constant temperature

A sealed syringe contains gas at a pressure of 100 kPa and a volume of 0.060 m³. It is compressed slowly at constant temperature to a volume of 0.020 m³. Calculate the final pressure.

Choose $p_1V_1=p_2V_2$ because the mass of gas is fixed and the temperature is constant.
Rearrange for final pressure: $p2=p1V1V2p_2=\frac{p_1V_1}{V_2}$ .
Substitute with units: $kPap_2=\frac{100\ \text{kPa}\times 0.060\ \text{m}^3}{0.020\ \text{m}^3}=300\ \text{kPa}$ .
The volume has become one-third of its original value, so the pressure becoming three times larger matches the inverse relationship.

8. Pressure and Kelvin temperature at constant volume

At constant volume, the container size does not change. If the gas is heated, its molecules have greater average kinetic energy and move faster on average.

Faster molecules collide with the walls more often and with a greater effect each time. This increases the pressure.

So, for a fixed mass of gas at constant volume:

higher Kelvin temperature means higher pressure
lower Kelvin temperature means lower pressure
pressure is directly proportional to Kelvin temperature

The equation is:

pressure 1 divided by Kelvin temperature 1 = pressure 2 divided by Kelvin temperature 2

p1T1=p2T2\frac{p_1}{T_1}=\frac{p_2}{T_2}

where temperature must be in K.

Useful rearrangements are:

p2=p1T2T1T2=p2T1p1\begin{aligned} p_2 &= \frac{p_1T_2}{T_1}\\ T_2 &= \frac{p_2T_1}{p_1} \end{aligned}

Example

Heating a gas at constant volume

A sealed metal cylinder contains gas at 100 kPa at 27 °C. It is heated to 87 °C. Calculate the new pressure.

Convert to Kelvin: $KT_1=27+273=300\ \text{K}$ and $KT_2=87+273=360\ \text{K}$ .
Choose $p1T1=p2T2\frac{p_1}{T_1}=\frac{p_2}{T_2}$ because the gas mass and volume stay fixed.
Rearrange for final pressure: $p2=p1T2T1p_2=\frac{p_1T_2}{T_1}$ .
Substitute with units: $kPap_2=\frac{100\ \text{kPa}\times 360\ \text{K}}{300\ \text{K}}=120\ \text{kPa}$ .

Tip

Choosing the right gas relationship

Exam technique

In the exam

Always convert °C to K before using any pressure-temperature gas equation.
Check what is being kept constant: constant temperature points to $p_1V_1=p_2V_2$ , while constant volume points to $p1T1=p2T2\frac{p_1}{T_1}=\frac{p_2}{T_2}$ .
Keep pressure units consistent, such as using kPa throughout or Pa throughout, and carry the units through your substitution.

Self review

Check yourself

Why do gas molecules create pressure on the walls of a container?
What is 35 °C in Kelvin, and why is Kelvin needed in gas calculations?
A fixed mass of gas is compressed at constant temperature. What happens to its pressure, and why?

What you'll learn

1. The particle picture of a gas

2. Temperature and molecular motion

3. Absolute zero

4. The Kelvin scale

5. Kelvin temperature and average kinetic energy

6. Gas relationships for a fixed mass

7. Pressure and volume at constant temperature

8. Pressure and Kelvin temperature at constant volume

Review Ideal gas molecules by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Ideal gas molecules

What you'll learn

1. The particle picture of a gas

2. Temperature and molecular motion

3. Absolute zero

4. The Kelvin scale

5. Kelvin temperature and average kinetic energy

6. Gas relationships for a fixed mass

7. Pressure and volume at constant temperature

8. Pressure and Kelvin temperature at constant volume

Review Ideal gas molecules by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Ideal gas molecules

What you'll learn

1. The particle picture of a gas

2. Temperature and molecular motion

3. Absolute zero

4. The Kelvin scale

5. Kelvin temperature and average kinetic energy

6. Gas relationships for a fixed mass

7. Pressure and volume at constant temperature

8. Pressure and Kelvin temperature at constant volume

Review Ideal gas molecules by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Ideal gas molecules Revision Guide

Ideal gas molecules

What you'll learn

1. The particle picture of a gas

2. Temperature and molecular motion

3. Absolute zero

4. The Kelvin scale

5. Kelvin temperature and average kinetic energy

6. Gas relationships for a fixed mass

7. Pressure and volume at constant temperature

8. Pressure and Kelvin temperature at constant volume

Review Ideal gas molecules by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Ideal gas molecules Revision Guide