Behaviour of Perfect Gases and Kinetic Theory of Gases

The One Thing

Gas laws describe the macroscopic relationships among pressure, volume, temperature, and amount of gas, while kinetic theory explains these relationships through the random motion, collisions, and temperature-dependent kinetic energy of molecules. Perfect-gas behaviour is most closely approximated at low pressure and high temperature.

Reactions, Processes and Experiments

Key Terms

• Perfect or ideal gas: A hypothetical gas whose molecules occupy negligible volume, exert no intermolecular forces except during collisions, and obey the ideal-gas equation under all conditions.
• Pressure: Normal force exerted per unit area by gas molecules striking the walls of their container.
• Volume: The space occupied by a gas, usually equal to the volume of its container.
• Absolute temperature: Temperature measured from absolute zero on the Kelvin scale; $T(\mathrm{K})=t{(}^{\circ }\mathrm{C})+273.15$.
• Amount of substance: The quantity of gas measured in moles; one mole contains approximately $6.022\times 10^{23}$ particles.
• Boyle’s law: For a fixed mass of gas at constant temperature, pressure is inversely proportional to volume: $PV=\text{constant}$.
• Charles’s law: For a fixed mass of gas at constant pressure, volume is directly proportional to absolute temperature: $V/T=\text{constant}$.
• Pressure law: For a fixed mass of gas at constant volume, pressure is directly proportional to absolute temperature: $P/T=\text{constant}$.
• Avogadro’s law: At constant pressure and temperature, gas volume is directly proportional to the number of molecules or moles: $V\propto n$.
• Combined gas law: For a fixed mass of gas, $PV/T$ remains constant when pressure, volume, and temperature change.
• Ideal-gas equation: $PV=nRT$, where $P$ is pressure, $V$ is volume, $n$ is number of moles, $R$ is the universal gas constant, and $T$ is absolute temperature.
• Gas constant: The universal constant $R$, with value $8.314{\mathrm{J}\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$, or approximately $0.0821{\mathrm{L}\mathrm{atm}\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$.
• Kinetic theory: A microscopic model that explains gas properties using molecular motion, collisions, and kinetic energy.
• Random molecular motion: Continuous molecular motion in all directions, with no preferred direction when averaged over a large number of molecules.
• Elastic collision: A collision in which total kinetic energy and total momentum are conserved.
• Mean free path: The average distance travelled by a molecule between two successive collisions.
• Root-mean-square speed: A representative molecular speed defined by $c_{\text{rms}}=\sqrt{\text{average of }{c}^2}$, given for an ideal gas by $c_{\text{rms}}=\sqrt{3RT/M}$ or $\sqrt{3kT/m}$.
• Boltzmann constant: The energy constant $k=1.38\times 10^{-23}{\mathrm{J}\mathrm{K}}^{-1}$, related to $R$ by $R=N_{A}k$.
• Degrees of freedom: The independent ways in which a molecule can possess energy, including translational, rotational, and vibrational motion.
• Law of equipartition of energy: At thermal equilibrium, each quadratic degree of freedom contributes an average energy of one-half $kT$ per molecule.
• Internal energy of an ideal gas: The total microscopic kinetic energy of the molecules; for an ideal gas, it depends only on temperature.
• Molar mass: The mass of one mole of gas molecules, represented by $M$ in the root-mean-square speed and density equations.
• Specific heat at constant volume: The heat capacity $C_V$ associated with heating an ideal gas while its volume remains constant.
• Specific heat at constant pressure: The heat capacity $C_P$ associated with heating an ideal gas while its pressure remains constant.
• Heat-capacity ratio: $\gamma =C_P/C_V$, whose value depends on the active degrees of freedom.

Easily Confused

• Boyle’s law and Charles’s law: Boyle’s law holds temperature constant and relates pressure inversely to volume; Charles’s law holds pressure constant and relates volume directly to absolute temperature.
• Charles’s law and the pressure law: Charles’s law describes $V$ against $T$ at constant pressure; the pressure law describes $P$ against $T$ at constant volume.
• Pressure law and the combined gas law: The pressure law applies at constant volume for a fixed mass; the combined gas law allows pressure, volume, and temperature all to change.
• Temperature and average translational kinetic energy: Temperature measures average translational kinetic energy, not the kinetic energy of every individual molecule.
• $R$ and $k$: $R$ applies to molar quantities, whereas $k$ applies to individual molecules; they are related by $R=N_{A}k$.
• Molecular speed and average translational kinetic energy: At the same temperature, different gases can have different root-mean-square speeds, but their average translational kinetic energy per molecule is the same.
• Perfect gases and real gases: Perfect gases are modelled as having negligible molecular volume and no intermolecular forces; real gases deviate especially at high pressure and low temperature.
• $C_V$ and $C_P$: $C_V$ is the heat capacity at constant volume, whereas $C_P$ is the heat capacity at constant pressure and is greater by $R$ for an ideal gas.
• Mean free path and molecular speed: Mean free path is an average distance between collisions; molecular speed describes molecular motion and is temperature-dependent.

What Gets Asked

• Apply Boyle’s, Charles’s, or the pressure law to two states of a gas. Marks are lost by using the wrong law or failing to keep the stated constant variable fixed.
• Use the combined gas law, $P_1{V}_1/T_1=P_2{V}_2/T_2$. The specific error is using Celsius temperatures instead of kelvin.
• Calculate an unknown quantity with $PV=nRT=NkT$. Marks are lost by confusing the number of moles $n$ with the number of molecules $N$, or by using an inconsistent value and unit system for $R$.
• Derive or apply the kinetic-theory pressure relation. The relevant equation is $P=(1/3)\rho c_{\text{rms}}^2=(1/3)(Nm/V)c_{\text{rms}}^2$, or equivalently $PV=(1/3)Nm{c}_{\text{rms}}^2$; the common slip is omitting the factor $1/3$.
• Calculate root-mean-square speed. Use $c_{\text{rms}}=\sqrt{3kT/m}=\sqrt{3RT/M}$, with $M$ in $\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$ when using SI units.
• Explain internal energy, degrees of freedom, or specific heats. Marks depend on distinguishing $U=(f/2)nRT$, $C_V=(f/2)R$, $C_P=C_V+R$, and $\gamma =1+2/f$, and on recognising that vibrational modes are inactive in the stated approximation for a diatomic gas at ordinary temperatures.