The model, not the gas in the room

An ideal gas is a theoretical gas that follows a simplified particle model: its particles move randomly, take up no significant space, exert no intermolecular forces on one another, and collide without losing total kinetic energy. No gas is perfectly ideal. Even so, many gases behave close enough to this model to make useful predictions.

A scientific model is a simplified representation of a system that keeps the features needed for prediction and leaves out features judged to be less important. That last part matters. A model works only if we remember what it leaves out.

Assumptions you should be able to recognize

The ideal gas model depends on a short list of assumptions:

• Gas particles are in continuous random motion. They travel in straight lines between collisions.
• The particles are treated as points: their own volume is negligible compared with the volume of the container.
• There are no intermolecular forces between particles. In this model, particles neither attract nor repel one another except during collisions.
• Collisions between particles, and between particles and the container walls, are perfectly elastic.
• The average kinetic energy of the particles depends on absolute temperature, so hotter gas particles move faster on average.

An elastic collision is a collision in which the total kinetic energy of the colliding particles is unchanged. A particle may speed up or slow down after a collision, but the total for the particles involved is conserved.

An intermolecular force is an attractive or repulsive interaction between molecules or particles. The ideal gas model deliberately ignores these forces; real gases do not.

How pressure arises in the model

Gas pressure is caused by particles hitting the walls of the container. When collisions happen more often, or with greater force, the pressure is higher. In the ideal model, these collisions are clean rebounds: no energy is drained into attractions between particles or into deformation of the particles themselves.

That is what makes the assumptions useful. If particles have negligible volume and no attractions, we can link the measurable properties of a gas — pressure, volume, temperature and amount — using simple mathematical relationships.

Why real gases are not perfectly ideal

A real gas is a gas whose particles have finite volume and may experience intermolecular forces. Many real gases behave almost ideally, especially when their particles are far apart and moving quickly, but the ideal gas model starts to break down under certain conditions.

Keep two main limitations in mind.

First, real particles take up space. At high pressure, the gas is squeezed into a smaller volume, so the volume of the particles becomes a noticeable fraction of the total volume. The “negligible volume” assumption is no longer safe.

Second, real particles attract one another. At low temperature, particles have lower kinetic energy, so intermolecular attractions affect their motion and collisions more strongly. The “no intermolecular forces” assumption becomes weak, and collisions can no longer be treated as perfectly ideal rebounds.

Conditions for greatest deviation

Real gases deviate most from ideal behaviour at low temperature and high pressure.

At low temperature, particles move more slowly on average. Attractions between particles can noticeably change their paths and reduce the force or frequency of wall collisions compared with the ideal prediction.

At high pressure, particles are pushed close together. With less empty space between them, particle size and intermolecular attractions both matter more.

The opposite conditions — high temperature and low pressure — favour ideal behaviour. The particles are far apart, moving rapidly, and have little time to interact significantly.

Why some gases deviate more than others

Under comparable conditions, gases with stronger intermolecular forces deviate more from ideal behaviour. Polar molecules, molecules capable of hydrogen bonding, and molecules with large electron clouds tend to have stronger attractions than small non-polar molecules.

Larger gas particles also deviate more because their actual volume is less negligible. This links neatly to bonding and structure: the type and strength of intermolecular forces studied in Structure 2.2 help explain why different gases are not equally ideal.

No mathematical treatment of real-gas corrections is required here. Focus on the explanation: identify which ideal assumption breaks down and connect it to temperature, pressure, particle size or intermolecular forces.

Avogadro’s law and molar volume

Avogadro’s law says that equal volumes of gases, at the same temperature and pressure, contain equal numbers of particles. For an ideal gas, the gas identity doesn’t matter. Hydrogen, oxygen and carbon dioxide would each contain the same number of particles if their volumes, temperatures and pressures matched.

The main quantities used in gas measurements are $P$, where $P$ is pressure (Pa), $V$, where $V$ is volume ($\mathrm{m^3}$), $T$, where $T$ is absolute temperature (K), and $n$, where $n$ is amount of substance (mol). In this topic, use kelvin for temperature in gas calculations, because gas volume and pressure depend on absolute temperature rather than the Celsius scale.

A molar volume is the volume occupied by one mole of a substance under specified conditions of temperature and pressure. For a gas, we write it as

The data booklet gives the molar volume of an ideal gas at standard temperature and pressure.

At standard temperature and pressure (STP) for IB gas calculations, the temperature is 273.15 K and the pressure is 100 kPa. The molar volume of an ideal gas at STP is $22.7\ \mathrm{dm^3\ mol^{-1}}$, the same as $2.27\ \times\ 10^{-2}\ \mathrm{m^3\ mol^{-1}}$.

This idea can feel less obvious than it is: one mole of helium atoms and one mole of chlorine molecules take up the same volume at the same temperature and pressure, even though their particles have very different masses. Same $n$, same $T$, same $P$ gives same $V$ for an ideal gas.

Investigating gas relationships

For a fixed mass of ideal gas, $n$ stays constant. You can then test how two variables relate while keeping the others controlled.

• If $T$ and $n$ are constant, $P$ is inversely proportional to $V$. Compressing the gas increases the frequency of wall collisions per unit area.
• If $P$ and $n$ are constant, $V$ is directly proportional to $T$. Heating the gas makes particles move faster, so the gas must expand to keep pressure constant.
• If $V$ and $n$ are constant, $P$ is directly proportional to $T$. Heating the gas increases the force and frequency of collisions with the container walls.

You do not need to learn the names of the individual gas laws for assessment. What matters is recognising the relationships and reading their graphs correctly.

Ideal-gas graph relationships for a fixed amount of gas.

A good investigation controls variables on purpose. For example, if you use a simulation to vary temperature and record pressure, temperature is the independent variable, pressure is the dependent variable, and volume and amount of gas are controlled variables. A spreadsheet helps because you can calculate extra columns, such as reciprocal volume, and plot graphs quickly.

Sketches versus plotted data

A sketch graph helps when you want to show the shape of a relationship clearly, such as direct proportionality, inverse proportionality, or a straight line through the origin. The drawback is that it does not show experimental scatter, uncertainty or the quality of the data.

An accurately plotted graph uses measured or simulated data points. It can show scatter, anomalies and the strength of a relationship, but a poor scale or too narrow a data range can hide the simple model. In practice, we often use both: a plotted graph to analyse evidence, and a sketch to communicate the ideal relationship.

The ideal gas equation

The ideal gas equation puts the four measurable gas variables into a single relationship:

You’ll find the value of $R$ in the data booklet. With SI units, use pressure in Pa, volume in $m^3$, temperature in K and amount in mol. The pascal is the SI unit of pressure: 1 Pa is 1 N m$^{-2}$. Kilopascals are an SI multiple. Watch the volume units, though. If a question gives volume in $dm^3$, convert it to $m^3$ unless you are deliberately using a consistent equivalent unit set allowed by the data booklet.

Most calculation questions become straightforward once you rearrange the ideal gas equation:

$n = PV/(RT)$

$V = nRT/P$

$P = nRT/V$

$T = PV/(nR)$

Temperature has to be in K before you substitute it. Add 273.15 to the Celsius value.

The combined gas law

If the amount of gas stays fixed, you can compare the same sample before and after a change with the combined gas law:

Use this equation only when no gas is added or removed and you’re comparing two states of the same gas sample. Pressure and volume units must match on both sides; temperatures must be in kelvin.

For a fixed amount of ideal gas, $PV/T$ stays constant. If temperature increases, the gas responds through pressure, volume, or both, depending on what the container allows.

Calculating molar mass from experimental gas data

You can use the ideal gas equation to find the molar mass of a gas when you know the sample’s mass and collect a known volume at a known temperature and pressure.

A molar mass is the mass of one mole of a substance. Calculate it using

The practical steps are:

1. Measure the mass of gas collected, often from the mass lost by a gas source.
2. Measure the gas volume, temperature and pressure.
3. Use $n = PV/(RT)$ to calculate the amount of gas.
4. Use $M = m/n$ to calculate molar mass.

When gas is collected over water, the pressure inside the measuring cylinder is not just the pressure of the dry gas. Water vapour also adds to the total pressure. For more accurate processing, subtract the water vapour pressure at the measured temperature from the total pressure before using the ideal gas equation.

Experimental judgement matters here too. A suitable method uses glassware or apparatus that measures volume precisely enough for the gas sample, gives a reliable temperature measurement, and includes either a barometer reading or trustworthy atmospheric pressure data. Repeats help separate random scatter from a systematic error such as gas escaping before collection.