A fixed temperature

Definition

Gas pressure

The force per unit area that gas molecules exert on the walls of their container when they collide with them.

Picture a fixed amount of gas trapped in a cylinder by a movable piston, with the temperature of the gas held constant.
A constant temperature means the molecules keep the same average speed and the same average kinetic energy throughout.
Each collision with a wall is therefore just as hard as before, whatever happens to the volume.
The pressure can still change, because the volume decides how often the molecules reach the walls.

Increasing the volume

Pulling the piston out gives the same number of molecules a larger volume to move around in.
The molecules are now more spread out, so there are fewer molecules in each cubic centimetre of gas.
Each molecule has further to travel between one collision with a wall and the next.
The walls also have a larger area, so the collisions are shared out over more surface.
As a result, each square metre of wall is hit less often every second.
Each collision exerts the same force as before, because the molecules are moving at the same speed.
Fewer collisions per second on each square metre give a smaller force per unit area, so the pressure decreases.
Pushing the piston in has the opposite effect: the molecules are crowded into a smaller volume, hit each square metre of wall more often, and the pressure increases.

Pressure and volume

Definition

Inversely proportional

A relationship between two quantities in which their product stays constant, so doubling one of them halves the other.

For a fixed mass of gas at constant temperature, pressure and volume are inversely proportional.
Doubling the volume halves the pressure, and halving the volume doubles the pressure.
The relationship is $pressure×volume=constant\text{pressure} \times \text{volume} = \text{constant}$ $pressure \times volume = constant$ , or $pV=constantpV = \text{constant}$ $p V = constant$ .
1. $p$ is the pressure in pascals, $Pa\text{Pa}$ .
2. $V$ is the volume in cubic metres, $m3\text{m}^3$ .
If the gas starts at pressure $p_1$ and volume $V_1$ and changes to pressure $p_2$ and volume $V_2$ , the product stays the same, so $p_1 V_1 = p_2 V_2$ .
Any units can be used in $p_1 V_1 = p_2 V_2$ , such as $cm3\text{cm}^3$ or $kPa\text{kPa}$ , as long as both volumes are in the same unit and both pressures are in the same unit.
You do not need to memorise this equation, but you must be able to choose it from a list of equations and use it.
The equation only works if the temperature stays constant and no gas leaks in or out.
A graph of pressure against volume is a curve that falls steeply at first and then levels off, getting closer to the axes without ever touching them.
A graph of pressure against $1V\dfrac{1}{V}$ is a straight line through the origin, which is the clearest test that pressure and volume are inversely proportional.
Compressing a gas quickly makes it warm up, so in an experiment the volume must be changed slowly to keep the temperature constant.

Example

Compressing air in a cylinder

A cylinder holds $m30.060\ \text{m}^3$ of air at a pressure of $Pa100\,000\ \text{Pa}$ .
The piston is pushed in slowly until the volume is $m30.020\ \text{m}^3$ , and the temperature does not change.

p_1 V_1 = p_2 V_2

100\,000 \times 0.060 = p_2 \times 0.020

p_2 = \dfrac{100\,000 \times 0.060}{0.020} = 300\,000\ \text{Pa}

The volume has been divided by $3$ , so the pressure has been multiplied by $3$ , which matches the inverse relationship.

Example

Finding a new volume

A syringe contains $cm340\ \text{cm}^3$ of air at $kPa100\ \text{kPa}$ .
The plunger is pulled out slowly until the pressure falls to $kPa80\ \text{kPa}$ , at constant temperature.
Rearrange $p_1 V_1 = p_2 V_2$ to make $V_2$ the subject.

V_2 = \dfrac{p_1 V_1}{p_2}

V_2 = \dfrac{100 \times 40}{80} = 50\ \text{cm}^3

Both pressures are in kilopascals and the volume is in $cm3\text{cm}^3$ , so the answer comes out in $cm3\text{cm}^3$ with no conversion needed.

Practical

Pressure and volume of trapped air

Aim: to measure how the pressure of a fixed mass of air changes as its volume is changed at constant temperature.
Apparatus: a $cm360\ \text{cm}^3$ plastic syringe, a short length of thick-walled rubber tubing, a pressure sensor connected to a data logger or a pressure gauge reading in $kPa\text{kPa}$ , a clamp stand with a boss and clamp, and a thermometer.
Method:
- Set the plunger at the $cm350\ \text{cm}^3$ mark and connect the nozzle to the pressure sensor with the shortest possible length of tubing.
- Clamp the syringe so that its scale can be read at eye level, then record the starting volume, pressure and room temperature.
- Push the plunger in slowly to $cm345\ \text{cm}^3$ and hold it still for about $30$ seconds so that the air returns to room temperature.
- Record the pressure, then repeat in steps of $cm35\ \text{cm}^3$ down to about $cm325\ \text{cm}^3$ .
- Pull the plunger back out through the same volumes, recording the pressure again at each step, and work out a mean pressure for each volume.
- Multiply each mean pressure by its volume, then plot one graph of pressure against volume and another of pressure against $1V\dfrac{1}{V}$ .
Variables: the independent variable is the volume, the dependent variable is the pressure, and the control variables are the temperature and the mass of trapped air.
Results: the product $p V$ comes out roughly the same for every reading, the graph of $p$ against $V$ is a falling curve, and the graph of $p$ against $1V\dfrac{1}{V}$ is a straight line through the origin.
Watch out: the air inside the tubing is not shown on the syringe scale, so every volume reading is slightly too small, which is a systematic error that short tubing keeps small.
Watch out: pushing the plunger too quickly warms the air and raises the pressure above its true value, so always wait before taking a reading.
Watch out: a leak lets air escape at high pressure and makes $p V$ drift downwards, so check that every joint is tight before you start.
Safety: do not compress the air beyond the range of the sensor or gauge, and keep your face away from the joints in case the tubing slips off.

Common Mistake

Do not say the pressure falls because the molecules slow down, because the temperature is constant and their average speed does not change.
Do not say the extra space fills up with air or anything else, because the same molecules simply spread out through a larger volume.
Do not use $p_1 V_1 = p_2 V_2$ if the temperature changes or if gas escapes.
Do not mix units, because if $V_1$ is in $cm3\text{cm}^3$ then $V_2$ must be in $cm3\text{cm}^3$ too.

Exam technique

Explaining a pressure change at constant temperature

Say that the molecules are more spread out or travel further between collisions, then link this to fewer collisions with the walls each second.
State that the speed of the molecules does not change because the temperature is constant, since this is the step that weaker answers leave out.
In calculations, write $p_1 V_1 = p_2 V_2$ , substitute both pairs of values, and only then rearrange, so that each step can earn credit.
Check that the answer makes sense, because a smaller volume must always give a larger pressure.

Self review

Explain why increasing the volume of a gas at constant temperature decreases its pressure.
State what happens to the speed of the molecules when a gas is compressed slowly at constant temperature.
Write the relationship between pressure and volume for a fixed mass of gas at constant temperature.
A gas at $kPa200\ \text{kPa}$ occupies $cm330\ \text{cm}^3$ ; calculate its pressure when its volume is increased to $cm360\ \text{cm}^3$ at the same temperature.
Describe the shape of a graph of pressure against volume for a gas at constant temperature.

A fixed temperature

Definition

Gas pressure

The force per unit area that gas molecules exert on the walls of their container when they collide with them.

Picture a fixed amount of gas trapped in a cylinder by a movable piston, with the temperature of the gas held constant.
A constant temperature means the molecules keep the same average speed and the same average kinetic energy throughout.
Each collision with a wall is therefore just as hard as before, whatever happens to the volume.
The pressure can still change, because the volume decides how often the molecules reach the walls.

Increasing the volume

Pulling the piston out gives the same number of molecules a larger volume to move around in.
The molecules are now more spread out, so there are fewer molecules in each cubic centimetre of gas.
Each molecule has further to travel between one collision with a wall and the next.
The walls also have a larger area, so the collisions are shared out over more surface.
As a result, each square metre of wall is hit less often every second.
Each collision exerts the same force as before, because the molecules are moving at the same speed.
Fewer collisions per second on each square metre give a smaller force per unit area, so the pressure decreases.
Pushing the piston in has the opposite effect: the molecules are crowded into a smaller volume, hit each square metre of wall more often, and the pressure increases.

Pressure and volume

Definition

Inversely proportional

A relationship between two quantities in which their product stays constant, so doubling one of them halves the other.

For a fixed mass of gas at constant temperature, pressure and volume are inversely proportional.
Doubling the volume halves the pressure, and halving the volume doubles the pressure.
The relationship is $pressure×volume=constant\text{pressure} \times \text{volume} = \text{constant}$ $pressure \times volume = constant$ , or $pV=constantpV = \text{constant}$ $p V = constant$ .
1. $p$ is the pressure in pascals, $Pa\text{Pa}$ .
2. $V$ is the volume in cubic metres, $m3\text{m}^3$ .
If the gas starts at pressure $p_1$ and volume $V_1$ and changes to pressure $p_2$ and volume $V_2$ , the product stays the same, so $p_1 V_1 = p_2 V_2$ .
Any units can be used in $p_1 V_1 = p_2 V_2$ , such as $cm3\text{cm}^3$ or $kPa\text{kPa}$ , as long as both volumes are in the same unit and both pressures are in the same unit.
You do not need to memorise this equation, but you must be able to choose it from a list of equations and use it.
The equation only works if the temperature stays constant and no gas leaks in or out.
A graph of pressure against volume is a curve that falls steeply at first and then levels off, getting closer to the axes without ever touching them.
A graph of pressure against $1V\dfrac{1}{V}$ is a straight line through the origin, which is the clearest test that pressure and volume are inversely proportional.
Compressing a gas quickly makes it warm up, so in an experiment the volume must be changed slowly to keep the temperature constant.

Example

Compressing air in a cylinder

A cylinder holds $m30.060\ \text{m}^3$ of air at a pressure of $Pa100\,000\ \text{Pa}$ .
The piston is pushed in slowly until the volume is $m30.020\ \text{m}^3$ , and the temperature does not change.

p_1 V_1 = p_2 V_2

100\,000 \times 0.060 = p_2 \times 0.020

p_2 = \dfrac{100\,000 \times 0.060}{0.020} = 300\,000\ \text{Pa}

The volume has been divided by $3$ , so the pressure has been multiplied by $3$ , which matches the inverse relationship.

Example

Finding a new volume

A syringe contains $cm340\ \text{cm}^3$ of air at $kPa100\ \text{kPa}$ .
The plunger is pulled out slowly until the pressure falls to $kPa80\ \text{kPa}$ , at constant temperature.
Rearrange $p_1 V_1 = p_2 V_2$ to make $V_2$ the subject.

V_2 = \dfrac{p_1 V_1}{p_2}

V_2 = \dfrac{100 \times 40}{80} = 50\ \text{cm}^3

Both pressures are in kilopascals and the volume is in $cm3\text{cm}^3$ , so the answer comes out in $cm3\text{cm}^3$ with no conversion needed.

Practical

Pressure and volume of trapped air

Aim: to measure how the pressure of a fixed mass of air changes as its volume is changed at constant temperature.
Apparatus: a $cm360\ \text{cm}^3$ plastic syringe, a short length of thick-walled rubber tubing, a pressure sensor connected to a data logger or a pressure gauge reading in $kPa\text{kPa}$ , a clamp stand with a boss and clamp, and a thermometer.
Method:
- Set the plunger at the $cm350\ \text{cm}^3$ mark and connect the nozzle to the pressure sensor with the shortest possible length of tubing.
- Clamp the syringe so that its scale can be read at eye level, then record the starting volume, pressure and room temperature.
- Push the plunger in slowly to $cm345\ \text{cm}^3$ and hold it still for about $30$ seconds so that the air returns to room temperature.
- Record the pressure, then repeat in steps of $cm35\ \text{cm}^3$ down to about $cm325\ \text{cm}^3$ .
- Pull the plunger back out through the same volumes, recording the pressure again at each step, and work out a mean pressure for each volume.
- Multiply each mean pressure by its volume, then plot one graph of pressure against volume and another of pressure against $1V\dfrac{1}{V}$ .
Variables: the independent variable is the volume, the dependent variable is the pressure, and the control variables are the temperature and the mass of trapped air.
Results: the product $p V$ comes out roughly the same for every reading, the graph of $p$ against $V$ is a falling curve, and the graph of $p$ against $1V\dfrac{1}{V}$ is a straight line through the origin.
Watch out: the air inside the tubing is not shown on the syringe scale, so every volume reading is slightly too small, which is a systematic error that short tubing keeps small.
Watch out: pushing the plunger too quickly warms the air and raises the pressure above its true value, so always wait before taking a reading.
Watch out: a leak lets air escape at high pressure and makes $p V$ drift downwards, so check that every joint is tight before you start.
Safety: do not compress the air beyond the range of the sensor or gauge, and keep your face away from the joints in case the tubing slips off.

Common Mistake

Do not say the pressure falls because the molecules slow down, because the temperature is constant and their average speed does not change.
Do not say the extra space fills up with air or anything else, because the same molecules simply spread out through a larger volume.
Do not use $p_1 V_1 = p_2 V_2$ if the temperature changes or if gas escapes.
Do not mix units, because if $V_1$ is in $cm3\text{cm}^3$ then $V_2$ must be in $cm3\text{cm}^3$ too.

Exam technique

Explaining a pressure change at constant temperature

Say that the molecules are more spread out or travel further between collisions, then link this to fewer collisions with the walls each second.
State that the speed of the molecules does not change because the temperature is constant, since this is the step that weaker answers leave out.
In calculations, write $p_1 V_1 = p_2 V_2$ , substitute both pairs of values, and only then rearrange, so that each step can earn credit.
Check that the answer makes sense, because a smaller volume must always give a larger pressure.

Self review

Explain why increasing the volume of a gas at constant temperature decreases its pressure.
State what happens to the speed of the molecules when a gas is compressed slowly at constant temperature.
Write the relationship between pressure and volume for a fixed mass of gas at constant temperature.
A gas at $kPa200\ \text{kPa}$ occupies $cm330\ \text{cm}^3$ ; calculate its pressure when its volume is increased to $cm360\ \text{cm}^3$ at the same temperature.
Describe the shape of a graph of pressure against volume for a gas at constant temperature.

1.3.4 Gas volume and pressure at constant temperature

A fixed temperature

Increasing the volume

Pressure and volume

Compressing air in a cylinder

Finding a new volume

Pressure and volume of trapped air

Explaining a pressure change at constant temperature

Review 1.3.4 Gas volume and pressure at constant temperature by teaching Genie

1.3.4 Gas volume and pressure at constant temperature

A fixed temperature

Increasing the volume

Pressure and volume

Compressing air in a cylinder

Finding a new volume

Pressure and volume of trapped air

Explaining a pressure change at constant temperature

Review 1.3.4 Gas volume and pressure at constant temperature by teaching Genie

1.3.4 Gas volume and pressure at constant temperature

A fixed temperature

Increasing the volume

Pressure and volume

Compressing air in a cylinder

Finding a new volume

Pressure and volume of trapped air

Explaining a pressure change at constant temperature

Review 1.3.4 Gas volume and pressure at constant temperature by teaching Genie

1.3.4 Gas volume and pressure at constant temperature Revision Guide

1.3.4 Gas volume and pressure at constant temperature

A fixed temperature

Increasing the volume

Pressure and volume

Compressing air in a cylinder

Finding a new volume

Pressure and volume of trapped air

Explaining a pressure change at constant temperature

Review 1.3.4 Gas volume and pressure at constant temperature by teaching Genie

1.3.4 Gas volume and pressure at constant temperature Revision Guide