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Atomic Structures & Period Properties. Electromagnetic Radiation and Radiation Energy Photoelectric Effect and Its Frequency Dependence Atomic Spectrum of Hydrogen Gas The Bohr’s Model of H-atom Quantum Mechanic Model for Electrons in Atoms Atomic Orbitals and Quantum Numbers
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Atomic Structures & Period Properties • Electromagnetic Radiation and Radiation Energy • Photoelectric Effect and Its Frequency Dependence • Atomic Spectrum of Hydrogen Gas • The Bohr’s Model of H-atom • Quantum Mechanic Model for Electrons in Atoms • Atomic Orbitals and Quantum Numbers • Electron Spin and Pauli Exclusion Principle • Electron Configurations for Atoms with many Electrons • Periodic Trends and Atomic Properties
Thomson’s Atomic Model* (1904) sphere of uniform positive charge negatively charged “corpuscle” Equal angular intervals d ~ “atomic dimensions” * Joseph J. Thomson, “On the Structure of the Atom” Philosophical Magazine and Journal of Science, Series 6, Vol. 7, No. 39, pp. 237-265
Atomic Modeling in theEarly 20th Century: 1904-1913 Charles Baily University of Colorado, Boulder Oct 12, 2008
Key Themes to Atomic Modeling Stability of the atom Dynamics of its parts Chemical/spectral properties
Electromagnetic Radiation • Electromagnetic Radiation = Light: • radiation energy that propagates through space in wave form • The speed of light is constant in a given medium • The speed of light through space is c = 2.998 x 108 m/s • c = ln(where l = wavelength, n = frequency) • Light with longer wavelength has lower frequency, and one with higher frequency has shorter wavelength. • According to Quantum Theory: Radiation Energy depends only on frequency: En = hn • where the Planck constant and h = 6.626 x 10-34 J.s
Photoelectric Effect • Photoelectric current
Photoelectric Effect • When light with energy greater than the minimum value strikes a metal plate (the cathode), electrons are ejected • A potential gradient is created and electrons flow in the circuit and photoelectric current is produced. • Different metals require different minimum energy to produce photoelectric effect. This is called the work function. • If light with energy lower than the minimum value is used, no photoelectric effect is produced. • The minimum energy needed to produce photoelectric effect corresponds to the binding energy of electrons on the metal surface.
Photoelectric Effect • Light with minimum frequency needed to eject electrons
Photoelectric Voltage & Current • The energy and speed of ejected electrons depends on the frequency (n) of incident light, which must be greater than the threshold (minimum) value for the metal used. Ee = En – Eo (Eo = minimum energy) Ee = h(ni – no) (no = minimum frequency) = hc(1/li - 1/lo) (lo = longest wavelength) • Speed of electron: ve = (2Ee/me)½ • The photoelectric voltage is directly related to the energy of ejected electrons, which depends on the frequency of light. • The photoelectric current (or the Amps) depends on the intensity of incident light – higher light intensity produces more current.
Einstein’s Explanation of PhotoelectricEffect • Light is composed of energy particles called photon • Energy of each photon is dependent only on the frequency of light emitting the photon: Ep = hn; • Total energy of electromagnetic radiation (light) = Nhn, • where N being the number of photon. • When light strikes on the metal, the photon is absorbed by an electrons on the metal surface, such that one electron absorbs only a photon (a quantum of energy) and the electron becomes excited. • If the photon carries energy greater than the binding energy of the metal, that electron will be ejected from the metal surface. The excess energy becomes the kinetic energy of electron. • Light is considered to have both wave and particle properties
E = mc2 & E = hc/l l = h/mc • Portrait of Albert Einstein:
Continuous Spectrum • White light produces a continuous spectrum
Atomic Spectrum • Spectrum produced by hydrogen gas discharge contains discrete lines:
Hydrogen Spectrum • Balmer’s equation for hydrogen spectrum in the visible region: 1/l = 1.097 x 107 m-1(1/22 – 1/n2); (n > 2) 1/l = 1.097 x 10-2 nm-1(1/22 – 1/n2); (n > 2) • If n = 3, 1/l = 1.097 x 10-2 nm-1(1/22 – 1/32) = 1.524 x 10-3 nm-1 l = 656.3 nm If n = 4, 1/l = 1.097 x 10-2 nm-1(1/22 – 1/42) = 2.057 x 10-3 nm-1 l = 486.2 nm
General equations for hydrogen spectrum: 1/l = 1.097 x 107 m-1(1/n12 – 1/n22); (n1 > 0, n2 > n1 ) n = 3.289 x 1015 s-1(1/n12 – 1/n22); (n1 > 0, n2 > n1 )
Spectral Series of Hydrogen Spectrum • Recurring patterns of line spectra for hydrogen were observed in different spectral regions, such as in ultraviolet region, visible region, infrared region, etc. • Spectral lines in ultravioletregion are called the Lyman series, which are due to electronic transitions from higher energy levels to level n = 1; • Spectral lines observed in the visible region, called the Balmer series, are due to electronic transitions from upper energy levels to level n = 2; • Spectral lines that appear in infrared region, called the Paschen series, are due to electronic transitions from upper energy levels to level n = 3.
Balmer’s Equation: 1/l = RH(1/22 – 1/n2) • Portrait of Johann Balmer:
Electronic Transitions in Hydrogen Discharge • Electronic transitions that produce different sets of line spectra
Bohr’s Model for Hydrogen • Electron orbits the nucleus in the manner Earth orbits the Sun • Only a particular set of orbits is allowed – each orbit must satisfy the condition that the angular momentum: mever = nh/2p (r = orbit radius) • While in a particular orbit, electron neither gains nor loses energy each orbit is called stationary state • Electronic energy in a given orbit is given by the expression: En = -2.18 x 10-18 J(1/n2) (n = 1, 2, 3,….) • Electron gains energy when it jumps from an inner orbit to the outer orbit, and loses energy when it jumps from an outer orbit to an inner one, such that, • DE = -2.18 x 10-18 J (1/nf2 - 1/ni2); • (n = 1, 2, 3, …)
Energy in Hydrogen Atom: En = -B(Z2/n2) • Portrait of Niels Bohr:
“Electrons occupy discrete orbits of constant energy. These orbits are described using the ordinary mechanics, while the passing of the system between different stationary states cannot be treated on this basis” * Niels Bohr, “On the Constitution of Atoms and Molecules” Philosophical Magazine and Journal of Science, Series 6, Vol. 26, No. 151, pp. 1-25
“In making a transition between stationary states, a single photon will be radiated…” * Niels Bohr, “On the Constitution of Atoms and Molecules” Philosophical Magazine and Journal of Science, Series 6, Vol. 26, No. 151, pp. 1-25
Applying Bohr’s Model to Hydrogen Atom • Consider an electron jumps from energy levels n = 3 to n = 2: • Ei = E3 = -2.178 x 10-18 J(1/32) = -2.420 x 10-19 J • Ef = E2 = -2.178 x 10-18 J(1/22) = -5.445 x 10-19 J DE = E2 – E3 = -2.178 x 10-18 J(1/22 - 1/32) = -3.025 x x 10-19 J • Energy lost by electron is emitted as radiation energy, En = hc/l l = hc/En = (6.626 x 10-34 J.s)(2.998 x 108 m/s)/(3.025 x 10-19 J) • = 6.567 x 10-7 m = 656.7 nm • Calculated wavelength agrees with observed values of alpha (red) line in hydrogen spectrum. • When electron jumps from levels n = 4 to n = 2, emitted photon calculated wavelength agrees with the beta (blue) line in H-spectrum with l = 486.4 nm.
Limitation of Bohr’s Model • Bohr’s model works only for hydrogen atom and other one-electron (hydrogen-like) ionic species, such as He+, Li2+, etc. • For H-atom, Bohr’s energy given by: En = -2.178 x 10-18 J(1/n2) • For other one-electron particle energy given by: En = -2.178 x 10-18 J(Z2/n2) • Bohr’s model cannot explain atomic spectra of atoms having more than one electron;
Traveling and Standing Waves • Light waves are traveling waves – values of wavelengths and frequencies are infinite • Waves on plucked strings (guitar, violin, cello, etc.) are standing waves – their motions limited within a boundary • The wavelengths of a standing wave is limited by the length of the string – that is, • = 2L/n L = distance the wave has to travel within a boundary and n = 1, 2, 3,…(integer) • Standing waves are quantized – the wavelength has certain fixed values (not arbitrary values) that are limited by 2L/n.
Particle-Wave Duality • According to Einstein, light can be both particle and wave. • Louis de Broglie proposed that other particles too can have both particulate and wave properties. • He proposed that a particle with mass m traveling at a speed v will exhibit a wavelength given by the following formula: • l = h/mv (h is Planck constant) • For example, an electron (me = 9.11 x 10-31 kg) traveling at 3.00 x 107 m/s acquires a wave characteristic such that, • l = (6.626 x 10-34 J.s)/{(9.11 x 10-31 kg)(3.00 x 108 m/s)} • = 2.42 x 10-11 m = 24.2 pm
De Broglie’s Equation:l = h/mv • Portrait of Louis de Broglie:
Heisenberg Uncertainty Principle • It is impossible to know simultaneously both the exact location and the energy of an electron in a given atom. • If the momentum or energy of an electron is determined accurately, then the knowledge of its location becomes less precise. • Heisenberg’s uncertaintyprinciple can be expressed as: • Dx.(mDv) >h/4p; • where Dx represent uncertainty in determining the location and andDv represents the uncertainty in the speed • (Such uncertainty is insignificant in macroscopic objects, but becomes very dominant when applied to a subatomic system.) • According to Heisenberg Uncertainty principle, it is not appropriate to assume that electrons are moving around the nucleus in a well-defined orbit, as stated in the Bohr’s model.
Heisenberg Uncertainty:Dx.Dp>h/4p • Portrait of Werner Heisenberg:
Quantum Mechanical Model • Also called wave mechanics – treating all motions of particles as wave-like; • Louis de Broglie originated the idea that, like light, all particulate motions have wave characteristics; • a new mathematical formula that incorporates both particulate and wave characteristics was needed. • Heisenberg uncertainty principle implies that we cannot know the position and energy of an electron in atom at the same time with some degree of certainty. • If we determine precisely the energy of electrons in atoms, we can only approximate their where about • Erwin Shrödinger derived a mathematical model for hydrogen that assumed electron to behave a standing wave.
Schrödinger’s Wave Function, y(x,y,z) • (h2/8p2me)[(d2/dx2) + (d2/dy2) + (d2/dz2)] – (Zq1q2/r)y = Ey • The equation is a bit complicated and Schrödinger wasn’t even sure if it works • We’ll try to understand the meaning of this equation • The wave functiony(x,y,z) has no physical meaning, but • [y(x,y,z)]2 implies probability • The square of the wave function yields a probability about finding an electron having a particular energy at a given location in the atom – just probability, not a definite location. • The sum of the squares of these wave functions yields a probability space called orbital.
Orbitals • Orbital • It is a probability space inside the atom where the chances of finding an electron with particular energy value is greater than 90% • Each orbital is described by a set of three quantum numbers: n, l, and ml ; • The number of orbitals in a subshell is equal to (2l + 1) and the number of orbitals in a shell is equal to n2; • As a consequence of the Pauli exclusion principle, each orbital can accommodate a maximum of two electrons, which must have opposite spins
Quantum Numbers • A set of numbers that describe an orbital or an electron • The principal quantum number (n)has the integral values: 1, 2, 3,…, ∞. It is related to the size and energy of the orbital • The angular momentum quantum number (l) has the integral values: 0, 1, 2,…,(n – 1). It is related to the shape of atomic orbitals. Each value of l is designated a letter symbol, which is summarized below: • Values of l: 0 1 2 3 • Letter symbols: spdf • The magnetic quantum number (ml) is related to the orientation of the orbital in the Cartesian coordinates x, y, and z. • ml has values from – l to +l (including 0)
Other Meanings of The Quantum Numbers • The principal quantum number (n) also describes the primary electronic shell or main energy level • The angular momentum quantum number (l) also implies the sub-shell or energy sub-level • The number of sub-shell in a given energy shell is equal to n: • Shelln = 1 has one subshell - the 1s-subshell; • shell n = 2 has two subshells - the 2s- and 2p-subshells; • shell n = 3, has three subshells - the 3s-, 3p-, and 3d-subshells, and so on,… • The number of orbitals in a given subshell is determined by the possible values that ml can have, which is equal to (2l + 1): • subshell l = 0 has one orbital; l = 1 has three orbitals; l = 2 has five orbitals; l = 3 has seven orbitals, and so on…
Quantum Numbersand Orbital Designations • The combination of quantum numbers: n, l, and ml, describes a particular orbital in the atom. • n = 1, l = 0, and ml = 0, orbital 1s; • n = 2, l = 0, and ml = 0, orbital 2s; • n = 3, l = 0, and ml = 0, orbital 3s; • n = 2, l = 1, and ml = 0, orbital 2p; • n = 3, l = 1, and ml = 0, orbital 3p; • n = 3, l = 2, and ml = 0, orbital 3d; • All orbitals with l = 0 have spherical shape, but the size becomes larger as the value of n increases; • Each orbital-p has two lobes, like a dumb-bell, with a nodal plane
Experiment by Stern & Gerlac Led to The Concept of Electron Spins
