Day 04

in 1913, Niels Bohr postulated that for a hydrogen atom

1. the electron moves about the nucleus with speed u in one of a fixed set of circular orbits; as long as the electron remains in a given orbit, its energy is constant and no energy is emitted thus, each orbit is characterized by a fixed radius, r, and a fixed energy, E 2. the electron's angular momentum, l = mur, is an integer multiple of (h/2π), with n = 1, 2, 3,... 3. an electron emits energy as a photon when the electron falls from an orbit of higher energy and larger radius to an orbit of lower energy and smaller radius

two things the Bohr theory allows us to calculate

1. the radii if the allowed orbits in a hydrogen atom r = n^2a0 where n = 1, 2, 3.. and a0 = 53 pm 2. the energies of these orbits (only for hydrogen) E = -Rh/n^2 where Rh (Rydberg constant) = 2.17868 X 10^(-18)J

equation 1 of the Bohr model

E = -Rh/n^2 where Rh (Rydberg constant) = 2.17868 X 10^(-18)J the energy of the hydrogen atom is quantized all the allowed energy values are negative by convention, the energy of the electron is defined to be zero when it is free of the nucleus (i.e. n = ∞) equation can also be used to explain certain features of the emission spectrum of the hydrogen atom

wave-particle duality

Einstein suggested particle-like properties of light could explain the photoelectric effect diffraction patterns suggest photons are wave-like

four series of the hydrogen atom

Lyman series (n = 1; UV region) Balmer series (n = 2; Visible region) Paschen series (n = 3; Infrared region) Bracket series (n = 4)

one view of atomic structure in early 20th century was that

an electron traveled about the nucleus in an orbit

two assumptions of atomic model in classical physics

any orbit should be possible and so is any energy but orbiting electrons should be constantly accelerating and should radiate energy end result should be destruction

Bohr model of the hydrogen atom and the equation for the change of energy

assumption: the electron moves in a circular orbit of radius r about the nucleus the electron has only a fixed set of allowed orbits

emission spectroscopy

bright lines are observed on a dark background of the photographic plate

energy of electrons

by convention, a free electron has 0 energy if a free electron has 0 energy, a bound electron (which is more stable) should have negative energy

absorption spectroscopy

dark lines are observed on a bright background of the photographic plate

spectroscopy

emission spectroscopy absorption spectroscopy observe which ν of light the atoms absorb (appear as dark lines on the film)

inadequacies of the Bohr model

experimentally, the Bohr theory cannot explain the emission spectra of atoms and ions with more than one electron ex) can only explain H, He+, Li2+; doesn't count the repulsion between electrons fundamentally, the postulate of quantized angular momentum forcing an electron into a fixed orbit has no basis but still has importance as the conceptual bridge

ionization energy of hydrogen and hydrogen-like atoms

for single electron systems ionization: electron is removed from the orbit to n → ∞ He+ (Z = 2), Li2+ (Z = 3), Be3+ (Z = 4)

Balmer's equation

in 1885, Johann Balmer, deduced a formula for the wavelengths of hydrogen's spectral lines ν = 3.2881 X 10^(15) s^(-1) (1/2^2 - 1/n^2) where ν is the frequency of the spectral line, and n must be an integer > 2 for λ = constant (m or nm) X (1/2^2 - 1/n^2) for E = constant (J) X (1/2^2 - 1/n^2)

Bohr model of the hydrogen atom

n = 1, 2, 3, and so on excitation of the atom raises the e- to higher-numbered orbits light is emitted when the e- falls to a lower-numbered orbit two transitions in the Balmer series of the hydrogen are shown

for the Balmer series of the hydrogen atom, the final orbit is always

n = 2

equation 2 from Bohr model

only applies to hydrogen comes from equation 1 positive for absorption negative for emission |ΔE| = |Ef - Ei| = Ephoton = hν

what is the significance of having only a limited number of well-defined wavelength lines for the hydrogen atomic spectrum?

only certain energies are allowed for the electron in the hydrogen atom energy of the electron in the hydrogen atom is quantized (fixed)

de Broglie relation

p and mu gets huge for macroscopic objects, so cannot observe wave properties de Broglie called the waves associated with material particles "matter waves" in one equation de Broglie summarized the concepts of waves and particles (the momentum, mv, is a particle property, whereas λ is a wave property), with noticeable effects if the objects are small

de Broglie, 1924

small particles of matter may at times display wave-like properties matter also has dual properties

energy-level diagram for the hydrogen atom

the e- in a hydrogen atom normally stays in the orbit closest to the nucleus (n = 1) this lowest allowed energy is called the ground state when the e- gains a quantum of energy, it moves to a higher level (n = 2, 3,...) and the atom is in an excited state when the e- drops from a higher to a lower level, a unique quantity of energy is emitted

the hydrogen spectrum

the most well-studied atomic spectrum the Balmer series (correspond to the visible region) series for hydrogen atoms discontinuous spectra just a part of the spectrum

how to show small particles do have wave properties?

wave properties of electrons demonstrated λ of X-rays is ~1Å

electrons can be wave-like

λ applies to a wave p applies to particles matter has both wave and particle properties

equation relating νphoton to the Energy of electrons

νphoton = (Ei - Ef)/h