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Química Geral · Atoms, electrons and the periodic table

Luz, fótons e o átomo de Bohr

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O que você vai aprender

  • Interconvert the wavelength, frequency and photon energy of light, per photon and per mole of photons, and place radiation in the electromagnetic spectrum.
  • Use the work function to explain the threshold of the photoelectric effect and to calculate the threshold wavelength and the kinetic energy of ejected electrons.
  • Explain line spectra with quantized energy levels and use the Bohr energies or the Rydberg equation for hydrogen emission, absorption and ionization.
  • Calculate de Broglie wavelengths and state what the Heisenberg uncertainty principle means for the motion of an electron in an atom.

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Ler a aula gratuita (7 partes)

Light as a wave

Light is an electromagnetic wave. Its wavelength λ (the distance between successive crests) and frequency ν (crests passing a point per second, in Hz = s⁻¹) are linked by the speed of light: c = λν, with c = 2.998 × 10⁸ m/s in a vacuum for every colour. Wavelength must be in metres before you divide: 1 nm = 10⁻⁹ m, 1 pm = 10⁻¹² m. For orange light of 600. nm, ν = 2.998 × 10⁸ m/s ÷ 6.00 × 10⁻⁷ m = 5.00 × 10¹⁴ Hz. Longer wavelength means lower frequency. In order of increasing frequency (and decreasing wavelength) the electromagnetic spectrum runs radio waves, microwaves, infrared, visible (about 400 nm violet to 700 nm red), ultraviolet, X-rays (roughly 0.01–10 nm) and gamma rays.

Photons: energy comes in packets

Light is absorbed and emitted in packets called photons. One photon carries E = hν = hc/λ, with h = 6.626 × 10⁻³⁴ J s, so shorter wavelength means more energy per photon. A 600. nm photon carries 6.626 × 10⁻³⁴ × 2.998 × 10⁸ ÷ 6.00 × 10⁻⁷ = 3.31 × 10⁻¹⁹ J; a mole of them (× 6.022 × 10²³ mol⁻¹) carries 199 kJ/mol. Comparing a molar photon energy with a bond energy tells you whether one photon can break one bond: the Br–Br bond (Br₂, drawn) needs 193 kJ/mol, so λ = hcNA/E = 0.11963 J m mol⁻¹ ÷ 193 000 J/mol = 6.20 × 10⁻⁷ m, and any photon shorter than about 620 nm can split it. Brightness (intensity) sets how many photons arrive each second, not how much energy each one carries: a dim ultraviolet lamp delivers photons that can break bonds, while a bright infrared heat lamp only warms. Total energy = number of photons × energy per photon.

The photoelectric effect

Light shining on a clean metal ejects electrons only if each photon has at least the work function Φ, the minimum energy needed to remove one electron from that metal. One photon is absorbed by one electron, so below the threshold frequency ν₀ = Φ/h (threshold wavelength λ₀ = hc/Φ) no electrons leave, however intense the light or however long it shines. Above the threshold the surplus becomes kinetic energy: KEmax = hν − Φ. Raising the intensity at a fixed wavelength ejects more electrons per second but does not make them faster; raising the frequency makes them faster. A plot of KEmax against ν is a straight line of slope h for every metal; it meets the frequency axis at ν₀. Example: for Φ = 4.00 × 10⁻¹⁹ J, λ₀ = 1.9865 × 10⁻²⁵ J m ÷ 4.00 × 10⁻¹⁹ J = 497 nm. Light of 300. nm gives photons of 6.62 × 10⁻¹⁹ J, so KEmax = 6.62 × 10⁻¹⁹ − 4.00 × 10⁻¹⁹ = 2.62 × 10⁻¹⁹ J, and the electron speed follows from KE = ½mv² with mₑ = 9.109 × 10⁻³¹ kg. Work functions are often quoted in electronvolts: 1 eV = 1.602 × 10⁻¹⁹ J.

Line spectra

A hot solid glows with a continuous spectrum, but a gas of excited atoms, such as the neon (drawn) in a sign or hydrogen in a discharge tube, emits only a few sharp wavelengths: a line spectrum. Each element has its own set of lines, the same in emission and in absorption. Sharp lines mean the atom can hold only certain energies. An electron that drops from a higher level to a lower one emits one photon whose energy equals the gap, Ephoton = Eupper − Elower; absorbing a photon of exactly that energy lifts it back up. Light of any other energy is not absorbed. Atoms excited to level n can cascade down by every possible pair of levels, so up to n(n − 1)/2 different lines can appear (6 from n = 4).

The Bohr model and the Rydberg equation

Bohr gave the hydrogen electron fixed energies Eₙ = −2.179 × 10⁻¹⁸ J/n², n = 1, 2, 3 … The zero of energy is the electron removed completely (n = ∞), so every bound level is negative, and the levels crowd together as n grows. For a transition, ΔE = 2.179 × 10⁻¹⁸ J × (1/n_low² − 1/n_high²), which is the energy of the photon emitted (falling) or absorbed (rising). Dividing by hc gives the Rydberg equation, 1/λ = R_H(1/n_low² − 1/n_high²) with R_H = 1.097 × 10⁷ m⁻¹. Example: n = 4 → n = 2 releases 2.179 × 10⁻¹⁸ × (1/4 − 1/16) = 4.09 × 10⁻¹⁹ J, a 486 nm blue-green photon. Lines ending on n = 1 (Lyman series) are ultraviolet, those ending on n = 2 (Balmer series: 656, 486, 434 and 410 nm) are visible, and those ending on n = 3 (Paschen) are infrared. Ionization takes the electron from its level to n = ∞: from n = 1 that costs 2.179 × 10⁻¹⁸ J per atom, or 1312 kJ/mol, the ionization energy of hydrogen. The model works for hydrogen and other one-electron species but fails for atoms with two or more electrons, and its sharp circular orbits are not real.

Matter waves and uncertainty

De Broglie proposed that every moving particle has a wavelength λ = h/(mv) = h/p. An electron (mₑ = 9.109 × 10⁻³¹ kg) at 2.00 × 10⁶ m/s has λ = 6.626 × 10⁻³⁴ ÷ (9.109 × 10⁻³¹ × 2.00 × 10⁶) = 3.64 × 10⁻¹⁰ m = 364 pm, comparable to the spacing of atoms in a crystal, so electron beams diffract. If the kinetic energy is given instead, use p = √(2m·KE). A thrown ball has a wavelength around 10⁻³⁴ m, far too small to detect, so everyday objects show no wave behaviour. The Heisenberg uncertainty principle states that a particle's position and momentum cannot both be known exactly at the same time: Δx·Δp ≥ h/4π. The more tightly an electron is confined, the larger the minimum uncertainty in its momentum. An orbit would fix the electron's position and momentum together at every instant, so quantum mechanics replaces Bohr's orbits with orbitals, regions that give the probability of finding the electron, while the energy levels stay quantized.

What to carry forward

Convert with c = λν and E = hν = hc/λ, keeping wavelengths in metres and multiplying by NA for a mole of photons. One photon acts on one electron: it ejects a photoelectron only if hν ≥ Φ, with KEmax = hν − Φ, and it is emitted or absorbed by an atom only if its energy matches a gap between levels. For hydrogen, Eₙ = −2.179 × 10⁻¹⁸ J/n² and 1/λ = R_H(1/n_low² − 1/n_high²). Every particle has λ = h/mv, and position and momentum cannot both be known exactly, which is why electrons occupy orbitals rather than orbits.

Problemas de prática (40)

Praticar é grátis com uma conta.

Preveja o produto 2

The photoelectric effect

  • Doubling the intensity of the lightPredict how the intensity of light above the threshold changes the number and the kinetic energy of photoelectrons.

Matter waves and uncertainty

  • Squeezing an electron into a smaller spacePredict how confining an electron more tightly changes the minimum uncertainty in its momentum.

Classificar 2

Light and photons

  • Which region is 1.0 × 10¹⁸ Hz?Assign radiation of a given frequency to its region of the electromagnetic spectrum by converting it to a wavelength.

Line spectra and the Bohr model

  • Which transition emits light?Classify hydrogen transitions as emission or absorption from the direction of the change in n.

Comparar 5

Matter waves and uncertainty

  • Rank particles by de Broglie wavelengthRank moving particles by de Broglie wavelength by comparing their momenta.

Line spectra and the Bohr model

  • Rank hydrogen emissions by photon energyRank hydrogen emission transitions by the energy of the emitted photon using the Bohr energy expression.

Light and photons

  • Red and violet photonsCompare the frequency, speed and photon energy of red and violet light.
  • Rank radiation by photon energyOrder the regions of the electromagnetic spectrum by the energy of their photons.

The photoelectric effect

  • One light source, three metalsCompare the photoelectric response of several metals to the same light using their work functions.

Calcular 18

Matter waves and uncertainty

  • Wavelength of a thrown baseballCalculate the de Broglie wavelength of a macroscopic object and judge whether it could be observed.
  • Wavelength of a moving electronCalculate the de Broglie wavelength of an electron from its mass and speed.
  • Electron and proton with the same energyCompare the de Broglie wavelengths of two particles with the same kinetic energy using p = √(2m·KE).

Light and photons

  • Energy of one blue photonCalculate the energy of a single photon from its wavelength with E = hc/λ.
  • Longest wavelength that splits Cl₂Find the longest wavelength whose photons can each break one Cl–Cl bond, given the molar bond energy.
  • Frequency of a green laserConvert a visible wavelength in nanometres into a frequency with c = λν.
  • Photons from a laser pointerCount the photons emitted per second by a laser from its power and wavelength.
  • Energy of a mole of 254 nm photonsCalculate the energy carried by one mole of ultraviolet photons from their wavelength.
  • Microwave photons to heat waterCombine q = mcΔT with E = hν to count the microwave photons needed to heat a sample of water.

The photoelectric effect

  • Threshold wavelength for cesiumCalculate the longest wavelength that can eject electrons from a metal from its work function.
  • Speed of the fastest photoelectronsCalculate the maximum speed of photoelectrons from the photon wavelength, the work function and KE = ½mv².
  • Kinetic energy of photoelectrons from sodiumCalculate the maximum kinetic energy of photoelectrons from the photon energy and the work function.

Line spectra and the Bohr model

  • Wavelength of the red hydrogen lineUse the Rydberg equation to calculate the wavelength emitted in the n = 3 → n = 2 transition of hydrogen.
  • Ionization energy of hydrogenCalculate the molar ionization energy of ground-state hydrogen from the Bohr energy of n = 1.
  • Energy of hydrogen's n = 3 levelCalculate the energy of a Bohr level of hydrogen with Eₙ = −2.179 × 10⁻¹⁸ J/n².
  • Ionizing hydrogen from an excited stateCalculate the energy needed to ionize hydrogen atoms that are already in the n = 2 level.
  • Longest wavelength that ionizes hydrogenCalculate the longest photon wavelength able to ionize a ground-state hydrogen atom.
  • How many lines from n = 4?Count the distinct emission lines possible when excited hydrogen atoms cascade from a given level to the ground state.

Encontre o erro 2

Light and photons

  • Find the error in a frequency calculationDiagnose a unit-conversion error in a wavelength-to-frequency calculation from an implausible result.

Line spectra and the Bohr model

  • Does a higher start mean a bigger photon?Diagnose the error in assuming that transitions between higher hydrogen levels emit more energetic photons.

Interprete as evidências 2

Line spectra and the Bohr model

  • What do hydrogen's visible lines share?Interpret hydrogen's visible emission lines as one series ending on a common lower level.

The photoelectric effect

  • Reading a photoelectric plotInterpret the slope and frequency-axis intercept of plots of photoelectron kinetic energy against light frequency for two metals.

Trabalhe de trás para frente 3

Matter waves and uncertainty

  • Electron energy for a 100 pm wavelengthWork back from a target de Broglie wavelength to the kinetic energy an electron must have.

Line spectra and the Bohr model

  • Which level did the electron start from?Work back from an observed hydrogen emission wavelength and its lower level to the upper level with the Rydberg equation.

The photoelectric effect

  • Work function from photoelectron energyWork back from the measured maximum kinetic energy of photoelectrons to the work function of the metal.

Planeje uma síntese 1

  • Getting photoelectrons from sodiumChoose a light source that will produce photoelectrons by comparing its wavelength with the threshold wavelength.

Explique por quê 5

Line spectra and the Bohr model

  • Where the Bohr model failsIdentify which observation lies beyond the Bohr model and explain why.
  • Why a neon sign glows at only certain coloursExplain a line emission spectrum in terms of quantized atomic energy levels.

The photoelectric effect

  • Why bright red light ejects no electronsExplain the threshold frequency of the photoelectric effect in terms of single photons and the work function.

Light and photons

  • Why UV damages and infrared only warmsExplain why photon energy, not total light energy, decides whether light can break chemical bonds.

Matter waves and uncertainty

  • Why orbits became orbitalsExplain why the uncertainty principle rules out Bohr's orbits while keeping quantized energy levels.

Travou em luz, fótons e o átomo de Bohr? 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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