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Química Geral · Reactions in water and gases

Misturas gasosas, movimento molecular e gases reais

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O que você vai aprender

  • Use Dalton's law and mole fractions to find partial and total pressures of ideal gas mixtures.
  • Correct the pressure of a gas collected over water for water vapour and use it to find moles, masses and molar masses.
  • State the postulates of kinetic molecular theory and relate average kinetic energy to absolute temperature.
  • Calculate root-mean-square speeds and use Graham's law to compare effusion and diffusion rates and to find molar masses.
  • Explain when and why real gases deviate from ideal behaviour and calculate a pressure with the van der Waals equation.

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Dalton's law and mole fractions

In a mixture of ideal gases each gas behaves as if it were alone in the container. Its partial pressure is Pᵢ = nᵢRT/V, and the total pressure is the sum of the partial pressures: P_total = P₁ + P₂ + … (Dalton's law). Dividing one equation by the other gives Pᵢ = χᵢ × P_total, where the mole fraction χᵢ = nᵢ/n_total has no units and all the mole fractions add up to 1. Example: 0.200 mol N₂ and 0.050 mol O₂ at a total pressure of 1.25 atm. χ(O₂) = 0.050/0.250 = 0.200, so P(O₂) = 0.200 × 1.25 atm = 0.250 atm and P(N₂) = 1.00 atm. Partial pressures depend on moles, not masses: equal masses of two gases have partial pressures in the inverse ratio of their molar masses. Adding another gas to a rigid flask at constant temperature raises the total pressure but leaves every existing partial pressure unchanged, because nᵢ, V and T of each original gas are unchanged; only the mole fractions fall. When two flasks are joined, each gas expands into the total volume, so its new partial pressure is P × V_initial/V_total (Boyle's law), and the new partial pressures are then added.

Collecting a gas over water

A gas made in a reaction (H₂ from a metal and acid, O₂ from decomposing KClO₃) is often collected by displacing water from an inverted bottle. The collected gas is saturated with water vapour, so P_total = P_gas + P(H₂O), where P(H₂O) is the vapour pressure of water at the temperature of the water (about 17.5 torr at 20 °C and 23.8 torr at 25 °C). Raising or lowering the bottle until the water levels inside and outside match makes the total pressure inside equal to the barometric pressure. Example: H₂ collected at 25 °C with the barometer reading 760.0 torr has P(H₂) = 760.0 − 23.8 = 736.2 torr. Only this dry-gas pressure goes into n = PV/RT; using the barometric pressure overestimates the moles of gas and, in a molar-mass experiment, underestimates the molar mass.

Kinetic molecular theory

The ideal-gas model rests on a few postulates. A gas is made of particles in constant, random, straight-line motion. The particles' own volume is negligible compared with the container volume. The particles neither attract nor repel one another. Collisions with each other and with the walls are elastic, so no kinetic energy is lost overall. The average translational kinetic energy is proportional to the absolute temperature: per molecule it is (3/2)k_B·T, and per mole it is (3/2)RT. Gases at the same temperature therefore have the same average kinetic energy whatever their molar mass. With R = 8.314 J mol⁻¹ K⁻¹, at 500 K the average kinetic energy is 1.5 × 8.314 × 500 = 6.24 × 10³ J/mol = 6.24 kJ/mol. Always use kelvin: going from 50 °C to 100 °C raises the average kinetic energy by the factor 373.15/323.15 = 1.15, not 2. Pressure comes from wall collisions. At constant volume, heating makes the molecules hit the walls both more often and harder, and each effect scales with the speed, so P ∝ T. The speeds follow the Maxwell–Boltzmann distribution: on heating the peak shifts to higher speed and the curve becomes lower and broader, while the area under it (the total number of molecules) stays the same.

Molecular speeds and Graham's law

Setting (1/2)M·u² equal to (3/2)RT per mole gives the root-mean-square speed u_rms = √(3RT/M). With R = 8.314 J mol⁻¹ K⁻¹, M must be in kg/mol to give m/s. For O₂ at 300 K: u_rms = √(3 × 8.314 × 300 / 0.03200) = 484 m/s. At the same temperature lighter molecules move faster, in proportion to 1/√M. Effusion is escape through a pinhole into a vacuum; Graham's law says rate₁/rate₂ = √(M₂/M₁) at the same temperature and pressure. Helium effuses √(32.00/4.003) = 2.83 times as fast as O₂. Because time is inversely proportional to rate, the time for the same amount to effuse is proportional to √M, so M_unknown = M_known × (t_unknown/t_known)². The same square-root ratio gives an approximate guide to diffusion: in a tube with NH₃ at one end and HCl at the other, the white NH₄Cl ring forms nearer the HCl end because NH₃ (17.03 g/mol) moves faster than HCl (36.46 g/mol). Repeated effusion of UF₆ separates ²³⁵U from ²³⁸U because the lighter molecule escapes slightly faster.

Real gases and the van der Waals equation

Real gases deviate from PV = nRT most at high pressure and low temperature, where molecules are close together and move slowly; they behave most ideally at low pressure and high temperature. The compressibility factor Z = PV/(nRT) equals 1 for an ideal gas. Attractions pull molecules away from the walls and lower the pressure, so Z < 1; at very high pressure the molecules' own volume dominates and Z > 1. The van der Waals equation, (P + an²/V²)(V − nb) = nRT, corrects for both: a measures intermolecular attraction (large for polar or polarizable molecules: He 0.0341, Ar 1.34, CO₂ 3.59, H₂O 5.46 L² atm mol⁻²) and b measures the volume excluded by the molecules (in L/mol). Solved for pressure, P = nRT/(V − nb) − an²/V². Example: 1.00 mol N₂ (a = 1.39 L² atm mol⁻², b = 0.0391 L/mol) in 1.00 L at 273.15 K gives 22.41/(1.00 − 0.0391) − 1.39 = 23.33 − 1.39 = 21.9 atm, below the ideal 22.4 atm because attractions dominate. For a gas with a tiny a, such as H₂ or He, the b term wins and the van der Waals pressure is above the ideal value. Attractions also explain why real gases condense on cooling, which an ideal gas never would.

What to carry forward

In a mixture, each gas has Pᵢ = χᵢP_total and the partial pressures add; subtract the vapour pressure of water from a gas collected over water. Average kinetic energy depends only on absolute temperature, (3/2)RT per mole, while speed also depends on mass: u_rms = √(3RT/M) with M in kg/mol, and effusion rates scale as 1/√M. Real gases depart from ideality at high pressure and low temperature; the van der Waals a term (attraction) lowers the pressure and the b term (molecular volume) raises it.

Problemas de prática (40)

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Preveja o produto 4

Partial pressures and mole fractions

  • Adding helium to a flask of oxygenPredict how adding a second gas at constant volume and temperature changes partial pressure, mole fraction and total pressure.

Molecular speeds and effusion

  • Where the NH₄Cl ring formsUse Graham's law to predict where two gases diffusing toward each other first meet.

Kinetic molecular theory

  • Doubling the volume at constant temperaturePredict how average speed and wall-collision frequency respond when a gas expands at constant temperature.

Real gases and van der Waals

  • When is a real gas most nearly ideal?Predict the conditions of temperature and pressure under which a real gas behaves most ideally.

Classificar 3

Kinetic molecular theory

  • Which is not a postulate of kinetic molecular theory?Distinguish the postulates of kinetic molecular theory from a common misstatement about molecular speeds.

Partial pressures and mole fractions

  • Which relation holds for an ideal mixture?Identify the relation between mole fraction, partial pressure and total pressure for an ideal gas mixture.

Real gases and van der Waals

  • What do a and b correct for?Assign each van der Waals constant to the non-ideal effect it corrects for.

Comparar 5

Kinetic molecular theory

  • Helium at 300 K or CO₂ at 400 KSeparate the temperature dependence of average kinetic energy from the mass dependence of molecular speed.

Real gases and van der Waals

  • Rank van der Waals compressibility factorsRank gases by van der Waals compressibility factor from their a and b constants at the same molar volume and temperature.
  • Rank noble gases by the van der Waals a constantRank the noble gases by the strength of their intermolecular attractions as reflected in the van der Waals a constant.

Partial pressures and mole fractions

  • Rank partial pressures in an equal-mass mixtureRank the partial pressures of gases present in equal masses using their molar masses.

Molecular speeds and effusion

  • Rank rms speeds at one temperatureRank gases at the same temperature by root-mean-square speed using molar mass.

Calcular 16

Gas collected over water

  • Pressure of O₂ collected over waterSubtract the vapour pressure of water to find the pressure of a dry gas collected over water.
  • Moles of hydrogen collected over waterFind the amount of a gas collected over water from its volume, temperature and the corrected pressure.
  • Mass of magnesium from the hydrogen it releasesCombine a water-vapour correction with reaction stoichiometry to find the mass of metal that reacted.

Partial pressures and mole fractions

  • Equal masses of helium and argonConvert masses to moles to find a partial pressure in a mixture of gases with very different molar masses.
  • Opening the valve between two flasksCombine Boyle's law and Dalton's law to find the pressure after two gas samples are joined.
  • Mole fraction of neon from partial pressuresFind a mole fraction as the ratio of a partial pressure to the total pressure.
  • Mole fraction of NO₂ after reactionUse partial pressures as reacting amounts in a rigid vessel to find a mole fraction after a gas-phase reaction.
  • Partial pressure of O₂ in a three-gas mixtureCalculate a partial pressure from the moles of each gas and the total pressure.

Molecular speeds and effusion

  • Helium content of the first gas to escapeCalculate the composition of the gas that first effuses from an equimolar mixture of two gases.
  • How much faster does H₂ effuse than CO₂?Apply Graham's law to find the ratio of effusion rates of two gases.
  • rms speed of N₂ at room temperatureCalculate the root-mean-square speed of a gas with consistent SI units.
  • Enrichment factor for uranium hexafluorideCalculate the ideal single-stage effusion separation factor for two isotopic molecules.

Kinetic molecular theory

  • Heating a gas from 27 °C to 327 °CCalculate the factor by which average kinetic energy changes with temperature using kelvin.
  • Translational kinetic energy of a mole of gasCalculate the average translational kinetic energy of one mole of an ideal gas from its temperature.

Real gases and van der Waals

  • Van der Waals pressure of CO₂Calculate the pressure of a real gas with the van der Waals equation.
  • Van der Waals pressure of compressed H₂Calculate a van der Waals pressure for a gas whose excluded-volume correction outweighs its attraction correction.

Encontre o erro 2

Kinetic molecular theory

  • Does 50 °C to 100 °C double the energy?Diagnose a reasoning error that uses Celsius temperature in the kinetic-energy relation and estimate the correct change.

Gas collected over water

  • A low molar mass for butaneDiagnose how omitting the water-vapour correction biases a molar mass found from a gas collected over water.

Interprete as evidências 4

Gas collected over water

  • Molar mass of lighter gas from dataCalculate the molar mass of a gas from mass, volume and water-corrected pressure data.

Real gases and van der Waals

  • A compressibility factor of 0.85Interpret a compressibility factor below 1 in terms of intermolecular attractions and molar volume.

Molecular speeds and effusion

  • Molar mass from effusion timesFind the molar mass of an unknown gas from effusion times measured against a reference gas.

Kinetic molecular theory

  • The speed distribution after heatingInterpret how the Maxwell–Boltzmann speed distribution of a gas changes when the temperature rises.

Trabalhe de trás para frente 2

Partial pressures and mole fractions

  • Composition of heliox from its densityWork back from the density of a two-gas mixture to its mole fractions through the average molar mass.

Molecular speeds and effusion

  • How hot must O₂ be to match helium's speed?Work back from a required rms speed to the temperature at which a heavier gas matches a lighter one.

Planeje uma síntese 1

  • Oxygen content of a deep-diving gasChoose the mole percent of a gas in a breathing mixture that gives a target partial pressure at a higher total pressure.

Explique por quê 3

Gas collected over water

  • Why match the water levels?Explain why the water levels are equalized before reading the volume of a gas collected over water.

Kinetic molecular theory

  • Why a heated sealed can builds pressureExplain the increase of gas pressure with temperature at constant volume in terms of molecular collisions.

Real gases and van der Waals

  • Why ammonia liquefies but an ideal gas never wouldExplain why real gases condense on cooling and compression in terms of the failure of an ideal-gas postulate.

Travou em misturas gasosas, movimento molecular e gases reais? Escaneie o problema.

Tire uma foto da reação da sua lista ou do livro e desenhe o mecanismo com cada seta corrigida. 1 scan grátis agora, sem cadastro; mais 1 com uma conta gratuita.

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