Core quantities and constants

• Pressure: $P=\dfrac{F}{A}$, where force acts perpendicular to the surface.

• Amount of substance: $n=\dfrac{N}{N_A}$.

• Avogadro constant: $N_A=6.02\times10^{23}\ \mathrm{mol^{-1}}$.

• Use kelvin in all gas-law equations: $T(\mathrm{K})=,^{\circ}\mathrm{C}+273$.

• Know the two forms of the ideal gas equation: $PV=nRT$ and $PV=Nk_BT$.

• Universal gas constant: $R=8.31\ \mathrm{J,mol^{-1},K^{-1}}$.

• Boltzmann constant: $k_B=1.38\times10^{-23}\ \mathrm{J,K^{-1}}$.

Empirical gas laws and the combined law

• Boyle’s law: for constant temperature, $PV=\text{constant}$, so $P\propto \dfrac{1}{V}$.

• Charles’s law: for constant pressure, $\dfrac{V}{T}=\text{constant}$, so $V\propto T$.

• Pressure law / Gay-Lussac’s law: for constant volume, $\dfrac{P}{T}=\text{constant}$, so $P\propto T$.

• Combining the three gives $\dfrac{PV}{T}=\text{constant}$ for a fixed amount of gas.

• For the same sample of gas between two states: $\dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2}$.

• In exam questions, first identify which variable is constant before choosing the equation.

• On a $P$–$V$ diagram, an isothermal change follows a curved inverse relation; a constant-volume change is vertical; a constant-pressure change is horizontal.

This diagram links Boyle’s law, Charles’s law, Gay-Lussac’s law, Avogadro’s law, the combined gas law, and the ideal gas law. It is useful for seeing which quantities are held constant in each relationship. It works well as a quick revision map before exam questions. Source

This graph shows the inverse relationship between pressure and volume for Boyle’s law. It is useful for linking the algebraic statement $PV=\text{constant}$ to the curved shape seen on graphs. Students should connect this to isothermal changes of a fixed mass of gas. Source

Ideal gas law: what it means

• Ideal gas = a model used to approximate real gases.

• The model treats gas particles as tiny point particles in random motion.

• Particles have negligible volume compared with the container volume.

• There are no intermolecular forces except during collisions.

• Collisions between particles and with walls are perfectly elastic.

• The ideal gas law can be written as $PV=nRT$ or $PV=Nk_BT$.

• These two forms are equivalent because $R=N_Ak_B$ and $N=nN_A$.

• Use $PV=nRT$ when the question gives moles; use $PV=Nk_BT$ when it gives number of molecules.

This figure shows gas particles widely separated in a container, helping explain why an ideal gas can be modelled as particles with negligible size and negligible intermolecular forces. It supports the microscopic assumptions behind the ideal gas law. It is especially helpful when comparing ideal and real gas behaviour. Source

Pressure from molecular collisions

• Gas pressure arises from the change in momentum of particles colliding with container walls.

• More frequent collisions or harder collisions give a greater pressure.

• The kinetic-theory result is $P=\dfrac{1}{3}\rho\overline{c^2}$, often written as $P=\dfrac{1}{3}\rho v^2$.

• Here $\rho$ is the gas density and $v^2$ means the mean square translational speed.

• This shows pressure depends on both how much mass is packed into the volume and how fast the molecules are moving.

• If temperature increases, average molecular kinetic energy rises, so collisions become more forceful.

• At constant volume, this leads to higher pressure.

This diagram shows a gas molecule colliding with a wall and reversing its momentum component perpendicular to the surface. It helps explain how repeated molecular collisions produce pressure. This is the key microscopic picture behind the kinetic-theory expression for pressure. Source

Temperature and internal energy

• For an ideal gas, temperature is linked to the average translational kinetic energy of molecules.

• For a monatomic ideal gas, the internal energy is only the random kinetic energy of the molecules.

• Required equations: $U=\dfrac{3}{2}Nk_BT$ and $U=\dfrac{3}{2}nRT$.

• So for a monatomic ideal gas, internal energy depends only on temperature.

• If temperature stays constant, internal energy stays constant.

• If the number of molecules doubles at the same temperature, internal energy doubles.

• Do not apply these $U$ equations to gases unless the question specifies ideal monatomic gas.

Ideal gas vs real gas

• A real gas differs from an ideal gas because molecules have finite volume and can exert intermolecular forces on each other.

• The ideal gas approximation works best at high temperature, low pressure, and low density.

• Under these conditions, particles are far apart and interactions are less important.

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

• In explanations, connect this to particles being closer together, so their size and intermolecular attraction matter more.

• If the question asks when the model is valid, state the conditions and the particle-level reason.

This image shows several ideal-gas isotherms on a $P$–$V$ diagram. It helps students recognise how pressure changes with volume for different constant temperatures. It is useful for interpreting qualitative graph questions and linking them to Boyle’s law. Source

Exam traps and strategy

• Always convert temperature to kelvin before substituting into gas equations.

• Check whether the gas amount is given as $N$ or $n$ before choosing $k_B$ or $R$.

• Distinguish carefully between microscopic explanations (molecules, collisions, momentum) and macroscopic quantities ($P,V,T$).

• In proportionality questions, decide first which variables are constant.

• In “explain” questions, mention collision frequency, momentum change, and molecular kinetic energy where relevant.

• For graph questions, describe both the shape and the constant variable.

• Remember that the syllabus limits gas laws to constant volume, constant temperature, constant pressure, and the ideal gas law.