Kinetic theory and gas laws

Kinetic theory and gas laws

A model of particles in matter

Matter can be described as many particles in continual motion. In solids particles vibrate about comparatively fixed positions; in liquids they remain close but move past one another; in gases they have much greater average spacing and move throughout the available volume. Interactions and motion explain differences in shape, density and compressibility. Heating generally adds microscopic energy, but a phase change can use it to change arrangement rather than temperature. A picture showing a few large circles is a representation, not a scale drawing of every atom. Keep measured macroscopic properties separate from the molecular interpretation offered by the model.

Brownian motion and diffusion

Visible suspended smoke or pollen particles move irregularly because surrounding much smaller fluid molecules collide unevenly with them. That is Brownian motion; the visible objects are not individual gas molecules. Diffusion is net spreading from higher concentration toward lower concentration through random motion when other transport does not dominate. It is generally faster in gases than liquids under comparable conditions because particles move more freely. Warming can speed diffusion, but convection currents can also carry material. A good demonstration distinguishes these effects rather than interpreting every spreading coloured plume as evidence of molecular diffusion alone. Repeated observation and controlled conditions strengthen the explanation.

How collisions create pressure

Molecules striking a wall change momentum and exert force. Pressure is the wall-normal force per area. At fixed volume and amount, higher temperature increases average translational kinetic energy and can increase pressure. At constant temperature, compression increases number density and wall collision rate, also raising pressure. A sealed gas sample does not lose particles just because it cools. The ideal model treats particle volume and attractions as negligible compared with the available volume and kinetic effects. It is most suitable at relatively low density and away from condensation. Actual gases can depart significantly from this approximation near liquefaction or at high pressures.

Boyle's law and its assumptions

For a fixed amount at constant temperature, pV is constant, giving p₁V₁. Pressure and volume are inversely proportional, so doubled pressure means half volume within the model. State both constant temperature and fixed amount; neither condition can be omitted. A p-against-V graph is curved, whereas p-against-1/V is linear for an ideal offset-free model. Use absolute pressure when gauge pressure and atmospheric pressure are relevant. Rapid compression can warm the gas, so an experiment needs suitable waiting for thermal equilibrium before interpreting its readings as isothermal. 'Pressure increases when volume falls' is not a complete quantitative law unless the constraints are clear.

Charles's law and pressure law

At constant pressure and amount, V/T is constant. At constant volume and amount, p/T is constant. T must be absolute temperature. A gas at27°C is near300 K; at127°C it is near400 K. Its constant-pressure volume rises by400/300, not127/27. Equal Celsius and kelvin intervals do not make their temperature ratios equal, because their zero points differ. Extrapolating ideal straight-line behaviour motivates absolute zero, but ordinary gases condense before a simple laboratory sample reaches that endpoint. Do not describe zero kelvin as a practically measured point reached by continuing an unchanged gas law indefinitely.

Combined law and ideal equation

For fixed amount pV/T is constant. Including amount gives, with n in moles. With, use pascals, cubic metres and kelvin. One litre or dm³ is10⁻³ m³. A particle-count form uses instead, with N total particles and Boltzmann constant kB. The forms agree because and. Do not substitute a mole amount into the particle-count equation or use the wrong constant. Listing known values with units before rearranging is a reliable way to avoid an apparently plausible but badly scaled result.

Molar volume, mass and density

Equal ideal-gas volumes at equal temperature and pressure contain equal particle numbers. Molar volume is RT/p and is therefore condition-dependent, not a universal22.4 or24 dm³/mol. Use the given value and stated conditions. Substituting into the gas equation gives densityρ =pM/(RT), with molar mass M in kg/mol for SI use. At fixed pressure a warmed gas becomes less dense as its volume grows. Heating a sealed rigid vessel does not change mass density merely from temperature increase because mass and volume remain fixed. Separate a material's pressure response from its conserved mass account.

Mixtures and partial pressure

For a nonreacting ideal mixture, total pressure is the sum of component partial pressures. A component's partial pressure is its mole fraction times total pressure, equivalent to what it would exert alone in the same volume and temperature. A20% mole fraction at100 kPa gives20 kPa partial pressure. Gas collected above water can include water vapour, so subtract its partial pressure to find dry-gas pressure when supplied with appropriate data. Adding an inert gas at fixed temperature and volume raises total pressure without changing existing ideal components' partial pressures. At fixed total pressure the volume can change, giving a different comparison.

Real gases and molecular energy

For a simple monatomic ideal gas, mean translational energy per particle is3kBT/2. Doubling T doubles that energy, while characteristic speed changes as√T at fixed molecular mass. Different gases at equal T have equal mean translational energy but not equal characteristic speeds. A one-mole van der Waals model, introduces corrections for attractions and finite effective particle volume. The constants are material-dependent. This is an improved model within a range, not exact at every phase condition. Explain which ideal assumptions fail rather than merely saying real gases are imperfect.

Self-check and laboratory reasoning

A gas at120 cm³ and100 kPa becomes80 cm³ at150 kPa if temperature and amount stay fixed. A sealed fixed-volume sample at300 K and100 kPa reaches150 kPa at450 K. For a combined-law problem write both sets of p,V,T before rearranging; convert temperatures to kelvin and pressures to appropriate absolute values. Ratio equations can retain the same volume unit on both sides, while with SI R requires m³. Pressure experiments need rated apparatus and supervision, so heating a rigid sealed container is not a casual demonstration. A successful calculation should agree with the expected direction of volume or pressure change.