Plan for Fri, 31 Oct 08 Lecture –Emission spectrum of atomic hydrogen (7.3) –The Bohr model of hydrogen (7.4) –Quantum mechanical model (7.5) Quiz 5

Photon Emission Relaxation from one energy level to another by emitting a photon. With  E = hc/ If = 440 nm,  = 4.5 x J Emission

Atomic Emission When we heat a sample of an element, the atoms become excited. When the atom relaxes it emits visible light. The color of the light depends on the element. Li Na KCaSr H Li Ba When the light emitted from excited atoms is passed through a prism, we see discrete bands of color at specific wavelengths.

Emission spectrum of H (cont.) Light Bulb Hydrogen Lamp Quantized, not continuous

Emission spectrum of H (cont.) We can use the emission spectrum to determine the energy levels for the hydrogen atom.

Balmer Model Joseph Balmer (1885) first noticed that the frequency of visible lines in the H atom spectrum could be reproduced by: n = 3, 4, 5, ….. The above equation predicts that as n increases, the frequencies become more closely spaced.

Rydberg Model Johann Rydberg extends the Balmer model by finding more emission lines outside the visible region of the spectrum: n 1 = 1, 2, 3, ….. This suggests that the energy levels of the H atom are proportional to 1/n 2 n 2 = n 1 +1, n 1 +2, … R y = 3.29 x /s

The Bohr Model Niels Bohr uses the emission spectrum of hydrogen to develop a quantum model for H. Central idea: electron circles the “nucleus” in only certain allowed circular orbitals. Bohr postulates that there is Coulombic attraction between e- and nucleus. However, classical physics is unable to explain why an H atom doesn’t simply collapse.

The Bohr Model of the atom Principle Quantum number: n An “index” of the energy levels available to the electron.

The Bohr Model (cont.) Bohr model for the H atom is capable of reproducing the energy levels given by the empirical formulas of Balmer and Rydberg. Z = atomic number (1 for H) n = integer (1, 2, ….) R y x h = x J (!)

The Bohr Model (cont.) Energy levels get closer together as n increases at n = infinity, E = 0

Identify the energy level diagram that best represents hydrogen:

The Bohr Model (cont.) We can use the Bohr model to predict what  E is for any two energy levels

The Bohr Model (cont.) Example: At what wavelength will emission from n = 4 to n = 1 for the H atom be observed? 14

The Bohr Model (cont.) Example: What is the longest wavelength of light that will result in removal of the e - from H?  1

The n = 4 to n = 1 transition in hydrogen corresponds to a wavelength of 97.4 nm. At what wavelength is the n = 4 to n = 2 transition expected? A.    B.    C.    D.  

Extension to Higher Z The Bohr model can be extended to any single electron system….must keep track of Z (atomic number). Examples: He + (Z = 2), Li +2 (Z = 3), etc. Z = atomic number n = integer (1, 2, ….)

Extension to Higher Z (cont.) Example: At what wavelength will emission from n = 4 to n = 1 for the He + atom be observed? 2 14

So what happened to the Bohr Model? Although it successfully described the line spectrum of hydrogen and other one-electron systems, it failed to accurately describe the spectra of multi-electron atoms. The Bohr model was soon scrapped in favor of the Quantum Mechanical model, although the vocabulary of the Bohr model persists. However, Bohr pioneered the idea of quantized electronic energy levels in atoms, so we owe him big. Thanks Niels Bohr!

Quantum Concepts The Bohr model was capable of describing the discrete or “quantized” emission spectrum of H. But the failure of the model for multielectron systems combined with other issues (the ultraviolet catastrophe, workfunctions of metals, etc.) suggested that a new description of atomic matter was needed.

Quantum Concepts (cont.) This new description was known as wave mechanics or quantum mechanics. Recall, photons and electrons readily demonstrate wave-particle duality. The idea behind wave mechanics was that the existence of the electron in fixed energy levels could be though of as a “standing wave”.

Quantum Concepts (cont.) What is a standing wave? A standing wave is a motion in which translation of the wave does not occur. In the guitar string analogy (illustrated), note that standing waves involve nodes in which no motion of the string occurs. Note also that integer and half- integer values of the wavelength correspond to standing waves.

Quantum Concepts (cont.) Louis de Broglie suggests that for the e - orbits envisioned by Bohr, only certain orbits are allowed since they satisfy the standing wave condition. not allowed

Quantum Concepts (cont.) Erwin Schrodinger developed a mathematical formalism that incorporates the wave nature of matter: H, the “Hamiltonian,” is a special kind of function that gives the energy of a quantum state, which is described by the wavefunction, .

Quantum Concepts (cont.) What is a wavefunction? = a probability amplitude Probability of finding a particle in space: Probability = With the wavefunction, we can describe spatial distributions. The Probability Distribution for the Hydrogen 1s Orbital in Three- Dimensional Space (b) The Probability of Find the Electron at Points Along a Line Drawn From the Nucleus Outward in Any Direction for the Hydrogen 1s Orbital

Hydrogen’s Electron Cross Section of the Hydrogen 1s Orbital Probability Distribution Divided into Successive Thin Spherical Shells (b) The Radial Probability Distribution Two Representations of the Hydrogen 1s, 2s, and 3s Orbitals (a) The Electron Probability Distribution (b) The Surface Contains 90% of the Total Electron Probability (the Size of the Oribital, by Definition)

Quantum Concepts (cont.) Another limitation of the Bohr model was that it assumed we could know both the position and momentum of an electron exactly. Werner Heisenberg development of quantum mechanics leads him to the observation that there is a fundamental limit to how well one can know both the position and momentum of a particle. Uncertainty in position Uncertainty in momentum where…

Quantum Concepts (cont.) Example: What is the uncertainty in velocity for an electron in a 1 Å radius orbital in which the positional uncertainty is 1% of the radius. (1 Å = m)  x = (1 Å)(0.01) = 1 x m huge

Quantum Concepts (cont.) Example (you’re quantum as well): What is the uncertainty in position for a 80 kg student walking across campus at 1.3 m/s with an uncertainty in velocity of 1%.  p = m  v = (80kg)(0.013 m/s) = 1.04 kg.m/s Very small……we know where you are.