Ideal gases (AQA A-Level Physics): Revision Notes

Ideal gases

Gas laws

The gas laws show how pressure (p), volume (V), and temperature (T) are related for a fixed mass of gas. These laws are based on experimental observations rather than theoretical predictions. There are three gas laws you need to understand:

Boyle's law

When temperature is held constant, pressure and volume are inversely proportional. This means that as one increases, the other decreases proportionally.

The equation for Boyle's law is:

$pV = k$

where $k$ is a constant value for that particular gas sample at that temperature.

Charles' law

When pressure is held constant, volume is directly proportional to absolute temperature. This means that as temperature increases, volume increases proportionally.

The equation for Charles' law is:

$\frac{V}{T} = k$

where $k$ is a constant value for that particular gas sample at that pressure.

The pressure law

When volume is held constant, pressure is directly proportional to absolute temperature. This means that as temperature increases, pressure increases proportionally.

The equation for the pressure law is:

$\frac{p}{T} = k$

where $k$ is a constant value for that particular gas sample at that volume.

Absolute temperature and the kelvin scale

All thermal physics equations require temperature to be measured in kelvin (K), which is the absolute scale of temperature. The relationship between temperature changes is straightforward: a change of 1 K equals a change of 1°C.

To convert between Celsius and kelvin, use:

$K = C + 273$

where $K$ is the temperature in kelvin and $C$ is the temperature in Celsius.

Absolute zero

Absolute zero is the lowest temperature possible in the universe. It occurs at -273°C, which equals 0 K on the kelvin scale. At absolute zero, particles possess no kinetic energy whatsoever, and both the volume and pressure of a gas theoretically reach zero.

The ideal gas equation

The three gas laws can be combined into a single relationship. When you consider all three laws together, you get:

$\frac{pV}{T} = k$

However, the constant $k$ depends on the amount of gas present, which is measured in moles ($n$). This leads to a more useful form:

$\frac{pV}{T} = nR$

where $n$ is the number of moles of gas, and $R$ is the molar gas constant.

The molar gas constant has the value:

$R = 8.31 \text{ J mol}^{-1} \text{ K}^{-1}$

Rearranging this relationship gives the ideal gas equation:

$pV = nRT$

This equation allows you to calculate any one variable (pressure, volume, number of moles, or temperature) if you know the other three.

Moles and molecules

One mole of any substance contains exactly $6.02 \times 10^{23}$ atoms or molecules. This number is called the Avogadro constant ($N_A$):

$N_A = 6.02 \times 10^{23} \text{ mol}^{-1}$

You can convert between the number of moles ($n$) and the number of molecules ($N$) using:

$N = n \times N_A$

or rearranged:

$n = \frac{N}{N_A}$

Alternative form of the ideal gas equation

By substituting the relationship between molecules and moles into the ideal gas equation, you can express it in terms of the number of molecules rather than moles:

$pV = \frac{NRT}{N_A}$

This can be simplified further using the Boltzmann constant ($k$), which is defined as:

$k = \frac{R}{N_A}$

This leads to the alternative form:

$pV = NkT$

This form is particularly useful when working with individual molecules rather than bulk quantities measured in moles.

Molar mass

Molar mass is the mass (in grams) of one mole of a substance. You can determine the molar mass by finding the relative molecular mass, which equals approximately the sum of all nucleons (protons and neutrons) in one molecule of the substance.

Work done by gases

When a gas changes volume at constant pressure, work is done. This typically occurs through the transfer of thermal energy to or from the gas. The work done can be calculated using:

$W = p\Delta V$

where $p$ is the pressure and $\Delta V$ is the change in volume.