Gas Laws (AQA A Level Physics): Revision Note

Ideal gas laws

• The ideal gas laws are the experimental relationships between pressure ($p$), volume ($V$) and temperature ($T$) of an ideal gas

• An explanation of the following three ideal gas laws is required:

  • Boyle's law

  • Charles's law

  • Pressure law

• The mass and the number of molecules of the gas is assumed to be constant for all of these

Boyle’s law

• If the temperature $T$ of an ideal gas is constant, then Boyle’s law is given by:

$p\propto \frac{1}{V}$

• This means the pressure is inversely proportional to the volume of a gas

• The relationship between the pressure and volume for a fixed mass of gas at constant temperature can also be written as:

$p_1{V}_1=p_2{V}_2$

• Where:

  • $p_1$ = initial pressure (Pa)

  • $p_2$ = final pressure (Pa)

  • $V_1$ = initial volume (m³)

  • $V_2$ = final volume (m³)

• If the temperature is higher but still constant, then the graph has the same shape but is shifted further above the origin

Charles's law

• If the pressure $p$ of an ideal gas is constant, then Charles’s law is given by:

$V\propto T$

• This means the volume is directly proportional to the temperature of a gas

• The relationship between the volume and thermodynamic temperature for a fixed mass of gas at constant pressure can also be written as:

$\frac{V_1}{T_1}=\frac{V_2}{T_2}$

• Where:

  • $V_1$ = initial volume (m³)

  • $V_2$ = final volume (m³)

  • $T_1$ = initial temperature (K)

  • $T_2$ = final temperature (K)

• A straight line through the origin shows direct proportionality

Pressure law

• If the volume $V$ of an ideal gas is constant, the Pressure law is given by:

$p\propto T$

• This means the pressure is proportional to the temperature

• The relationship between the pressure and thermodynamic temperature for a fixed mass of gas at constant volume can also be written as:

$\frac{p_1}{T_1}=\frac{p_2}{T_2}$

• Where:

  • $p_1$ = initial pressure (Pa)

  • $p_2$ = final pressure (Pa)

  • $T_1$ = initial temperature (K)

  • $T_2$ = final temperature (K)

• A straight line through the origin shows direct proportionality between pressure and temperature in kelvin

Relationships between pressure, volume & temperature

• An ideal gas is one that obeys the relation:

$pV\propto T$

• Where:

  • $p$ = pressure of the gas (Pa)

  • $V$ = volume of the gas (m³)

  • $T$ = thermodynamic temperature (K)

• The molecules in a gas move around randomly at high speeds, colliding with surfaces and exerting pressure upon them

• Imagine molecules of gas free to move around in a box

• The temperature of a gas is related to the average speed of the molecules:

  • The hotter the gas, the faster the molecules move

  • Hence the molecules collide with the surface of the walls more frequently

• Since force is the rate of change of momentum:

  • each collision applies a force across the surface area of the walls

  • the faster the molecules hit the walls, the greater the force on them

• Since pressure is the force per unit area:

  • Higher temperature leads to higher pressure

• If the volume $V$ of the box decreases, and the temperature $T$ stays constant:

  • there will be a smaller surface area of the walls and hence more collisions

  • this also creates more pressure

• Since this equates to a greater force per unit area, pressure in an ideal gas is therefore defined by:

  The frequency of collisions of the gas molecules per unit area of a container