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Lecture notes for CHEM 1000 The Hydrogen Atom Explaining the periodic table by understanding the hydrogen atom Presented by Dr. Maggie Austen Monday, May 9th, 2005 @ York University Understanding atoms by understanding electrons Some history of atomic theory… In the mid 1800’s Faraday discovered cathode rays. In 1897 Thomson determined that the rays were composed of small particles; electrons. In 1911 Rutherford proposed the nuclear model of atoms. He discovered protons in 1919. Understanding electrons by studying atomic spectra • Each atom has a characteristic spectrum of emission and absorption. • The wavelengths of the hydrogen atom are described by a simple mathematical formula. 1 ~ 1 1 = R 2 − 2 λ n1 n2 The Bohr model of the atom, 1913 • Assumed that electrons orbit the nucleus with discrete (‘quantized’) values of angular momentum, l = nh/2π. • Explained how electrons could occupy space around the nucleus without crashing into it. • Reproduced the observed energy spectrum for hydrogen atoms. • Did not reproduce the energy spectra for multielectron atoms. • Did not explain the effects of magnetic fields on atomic spectra. • Gave no reason why the angular momentum should be quantized. • One other problem… Electrons as matter waves • In 1905, to explain the photoelectric effect, Einstein said that light can be particle-like. • De Broglie suggested in 1924 that matter might also be wave-like, λp = h. • Confirmed by electron diffraction patterns in 1927. • Standing waves with l = nh/2π needed to avoid destructive interference. Explains quantization, but … Matter waves and wavefunctions The probability, or electron density, is the square of the wavefunction. • The wave-like properties of electrons are described by their wavefunction. • A wavefunction is a mathematical function of 3D space (x,y,z) that contains all the measurable information about the system, such as: – energy – momentum – spatial probability distribution. At the nodes the probability is zero Why do we need to talk about probabilities? • Schrodinger’s equation H Ψ = EΨ relates the energy of the particle to its wavefunction. • For a one-dimensional box-like potential well the wavefunctions are sine waves with wavelength = 2L/n and E = n2h2/8mL2. Particle in a box • As the box gets smaller, the wavelengths get shorter and the energy increases. En = n2h2 / 8mL2 = p2/2m • As the particle gets more localized its momentum, p = h/λ, increases. • If there are no walls, we can localize an electron by summing many sine waves of different wavelengths, i.e. many different momenta. See also Heisenberg’s Uncertainty Principle Probability distributions • A single sine wave extends to infinite distance. • Particles of known energy (momentum) cannot be localized. • Electrons in atoms are therefore not limited to circular orbits. • The electron is distributed over the (infinite) volume of the atom. • The electron density is determined by the potential energy field, centered at the nucleus. Orbitals of the hydrogen atom • Solutions to Schrodinger’s equation for the H atom reproduce the observed energy spectrum. • There are n2 different wavefunctions with each energy, En. A wavefunction for one electron is called an orbital. Shells, Subshells & Orbitals For atoms with many electrons we often talk about shells and subshells. E.g. the ‘valence shell’ contains the chemically active electrons. • This figure shows 3 energy levels, or shells, n. • Each shell has n subshells, or ‘types’ of orbitals, l, and a total of n2 orbitals. • Each subshell consists of 2l+1 orbitals. Increasing nodes • Like the particle in a box wavefunctions, the number of nodes increases as the energy and principal quantum number, n, increase. • Other quantum numbers describe other properties of the wavefunction. Radial nodes Spherical s orbitals • The simplest orbitals are the spherical ‘s’ orbitals. • The highest probability (or electron density) is at the nucleus, then it falls off exponentially. • The density falls off more slowly as n increases Î larger ‘size’ of orbital; 1s < 2s < 3s Ψ2(r) ρ(r) 90 % Summary of Some Quantum Numbers of Electrons in Atoms Name Symbol Permitted Values Property principal n positive integers (1,2,3,…) orbital energy & size total nodes = n-1 angular momentum l integers from 0 to n-1 orbital shape planar nodes = l (The l values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.) magnetic ml integers from -l to 0 to +l orbital orientation (The ml values do not directly correspond to x,y, and z.) Each electron is described by a set of quantum numbers, (n, l, ml). Each quantum number is restricted to certain values, depending on the previous quantum number. Orbital shapes, described by l. l = # of planar nodes s Visualizing p orbitals l = 1; ml = +1, 0, -1 0 R2 Ψ2 = Y2R2 p 1 d 2 Y Visualizing d orbitals l = 2; ml = +2, +1, 0, -1, -2 • Only exist for n > 2; 3d, 4d, etc. • There are 2l + 1 = 5 possible values of ml, so there are 5 different orientations for the d orbitals. • The dz2 orbital has cone-shaped nodes, rather than nodal planes. Y2 Orbits versus Orbitals • We have modified the Bohr model to account for waveparticle duality and the uncertainty principle. • An additional quantum number, ms, is required to explain magnetic field effects. • Schrödinger’s equation can be written for any electron system, but cannot be solved exactly for more than one electron. • We must make some assumptions / approximations. • Start by modifying the H-atom orbitals to fit the multielectron atom picture. How does this help with explaining the periodic table? • The periodic table has rows of 2, 8, 18 & 32 {= 2n2} elements. • It has blocks of 2, 6, 10 & 14 {= 2(2l+1)} elements. • This suggests that orbitals are being filled sequentially with 2 electrons per orbital. Questions remaining: The spin quantum number • The effects of a magnetic field can be explained by postulating the electrons have intrinsic magnetic moments, associated with the charged particle ‘spinning’ about an axis. • Pauli was able to use the idea of a spin quantum number, ms, having two possible values: ±½, to explain why each orbital can accept no more than 2 electrons. The Pauli Exclusion Principle Why 2 electrons / orbital? Why s before p before d? • Why 4s before 3d? Energy level splitting in multielectron atoms There must be something about the electron-electron repulsion in multi-electron atoms that makes s orbitals lower in energy than p & d orbitals. If this effect become very extreme, the 4s might be lower in energy than the 3d orbitals. These effects are best understood by looking at radial distribution functions. Radial Probability Distributions • What is the probability of finding the electron in a given ring, or rather on a given sphere? • P(r, dr) = N / Ntot P(r , dr ) = ρ (r ) ⋅ V (r , dr ) ρ (r ) ⋅ 4π ⋅ r 2 dr = N tot N tot P(r , dr ) ∝ r 2 ⋅ R 2 (r )dr Radial distribution functions When the hydrogen atom is in the n = 1 state (1s orbital), the most probable distance from the nucleus is the Bohr radius, ao. R2(r) The most probable distance increases with n but decreases with l. Note also that the radial nodes decrease with l. Notice that the 2s orbital has a small region of electron density in close to the nucleus, while the 2p orbital does not. r2R2(r) r2R2(r) Radial distribution functions The 2s orbital penetrates the n = 1 shell more than the 2p does. Thus the 2p orbital is more shielded from the nuclear charge. The surface area of a sphere is 4πr2. The probability of finding the electron at a given distance, summed (integrated) over all angles is 4πr2R2(r). There are more points at larger distances; no points at r = 0. Similarly, the 3s orbital penetrates more and is less shielded than 3p, which is less shielded than 3d. Therefore 3s is least repelled by the core (1s, 2s, 2p) electrons, thus most attracted to the nucleus, and lowest in energy. Summary of H-atom orbitals Summary of Electron Quantum Numbers in Hydrogen Atoms Name principal angular momentum magnetic Symbol n l ml spin s spin ms Permitted Values Property positive integers(1,2,3,…) orbital energy & size integers from 0 to n -1 integers from -l to 0 to +l ‘type’ l min. n s 0 1 # ml planar nodes # ‘lobes’ 1 0 1 1 2 2 4 3 8 l 2l 0 orbital shape p 1 2 orbital orientation d e- ½ for all electrons magnitude of +½ or -½ direction of e- spin 2 3 5 0, ±1, ±2 spin For multi-electron atoms, the other quantum numbers also affect the energy, and thus the order of orbital filling. 3 -1, 0, +1 f 3 4 7 0, ±1, ±2, ±3 summary l l +1 2l +1 0, ±1… ± l shape Uncertainty Principle, 1920s • Particles interact with photons of a similar wavelength. • Measurements are affected by the measuring device. • We cannot know the exact location and momentum simultaneously. • Heisenberg found that ∆x∆p > h/4π Dmitri Mendeleev 1871 The hydrogen atom orbitals, YR The Schrödinger Equation HΨ = EΨ Ψ = Yl,m(θ,φ) Rn,l (r) h2 d 2 d2 d 2 Ze 2 Hˆ = − 2 2 + 2 + 2 − 8π m dx dy dz r Mosley, 1913: X-ray spectra & Bohr model reveal atomic numbers