The Atom What was know about the atom in 1900? First, the existence of atoms was not universally accepted at this time, but for those who did think atoms existed, they knew: 1. Atoms are small, but they are stable. 2. Atoms are electrically neutral, but contain negative electrons. 3. Atoms emit and absorb electromagnetic radiation (i.e. light) For example: the emission spectrum of Hydrogen: Hydrogen gas light Diffraction grating (or prism) Screen Different wavelengths have maxima at different locations heat or excite Result for Hydrogen
Location of Hydrogen Spectral Lines In 1884, a Swiss school teacher, Johann Balmer, found a formula by trial and error that reproduces the locations of the Hydrogen spectral lines:
The Problem of the Atom Possibly the central question (the Holy Grail) of physics in the early 20 th century was this: What internal structure of the atom would yield the observed patterns of spectral lines? The efforts to answer this question would ultimately lead to a new physics, Quantum Mechanics The Thomson Plum Pudding Model... a first try: Pudding Uniformly Charged Positive sphere Atom Plums The electrons vibrate back and forth, thus emitting and absorbing radiation. However, the calculated spectra aren t anywhere close to the observations. PhET H-Atom
Rutherford Scattering In 1910, Ernst Rutherford (and his assistants Geiger and Marsden) fired high energy alpha particles (Helium nuclei) at a thin sheet of gold foil. They didn t expect much to happen. alpha particles gold foil Flash of light Screen If the gold atoms were like Thomson s plum pudding model, there would be very little chance of the alpha particles scattering at high angles. So Rutherford was quite surprised when Geiger and Marsden reported that they had observed alpha particles scatter at high angles up to 180 o! Rutherford s reaction: It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. PhET Ruth The Conclusion: This can only happen if the positive charge in an atom is concentrated in a small volume The Nucleus. PhET Ruth
The Basics of the Electric Force (we ll do a lot more with this in PHY192) Recall what Newton said for the Gravitational Force between two masses: In 1785, Charles Coulomb proposed something similar for the Electric Force between two electric charges where charge q is measured in Coulombs (C): The Electric Force: (Coulomb s Law) The Electric Potential Energy: Note for the electric potential energy, the sign is determined by the sign of the charges.
Whiteboard Problem QM-4: Rutherford Scattering An alpha particle with kinetic energy 8 MeV rebounds at 180 degrees from a Gold Nucleus. What is the alpha s closest approach distance to the nucleus? (LC) Hint: use the electric potential energy and energy conservation. You may assume the mass of the nucleus is much much larger than the mass of the alpha, so the nucleus does not recoil. Look up adequate estimates of the charges. The alpha particle is composed of 2 protons and 2 neutrons and has a mass ~ 4 m p and a charge = +2e. The Gold nucleus has a charge of +79e. (see constants sheet for value of m p, e, and k) This closest approach distance puts an upper limit on the size of the nucleus.
A Quick Review up to this point In 1900, Max Planck was trying to explain blackbody radiation. He modelled the blackbody as a cavity radiator and found that he could successfully reproduce the continuous spectrum only if assumed that the walls of the cavity emitted and absorbed radiation in integer multiples of discrete quanta with energies given by E = hf where f is the radiation frequency and h is a constant. In 1905, Albert Einstein extended Planck s idea of the quantum to explain the photoelectric effect. He proposed that the radiation itself is composed of discrete quanta with energy E = hf. These light quanta were later named photons, particles of light. In 1910, Ernst Rutherford was probing the structure of the atom with high energy alpha particles. He found that the positive charge and most of the mass is concentrated in a small volume the atomic nucleus.
Rutherford s Model of the Nuclear Atom Based on the conclusions of his scattering experiments which showed that the positive charge and most of the mass in an atom is concentrated in a small volume at the center, Rutherford proposed a model of the atom that looks like a miniature solar system: But there s a big problem with this: The electrons have uniform circular motion that requires continuous acceleration. Accelerated charges emit electromagnetic waves (radiation) and lose energy. The electron quickly spirals into the nucleus i.e. the atom is not stable! PhET H-atom
The Bohr Atom In 1913, a young Danish physicist, Niels Bohr, was working in Rutherford s lab. He wasn t having much success as an experimentalist, but in the last few months of his stay in England, he put together the first working model of the atom. Bohr started with Rutherford s model and borrowed and extended Einstein s idea of the quantum to include not just the radiation, but the matter itself. Bohr s starting point was three assumptions (Postulates): Bohr s work is amazing in it s simplicity a true back-of the envelope calculation; for the first time, the mysterious spectrum of Hydrogen could be understood.
Whiteboard QM-5: Bohr Energy Levels of Hydrogen Here, you will derive the energy levels of an electron in a Hydrogen atom the same way that Bohr did it in 1913. (Note: the derivation in your text uses de Broglie standing waves that s not the way Bohr did it; he just used what he learned in PHY191 and PHY192.) The Model: Electron Nucleus (proton): (to a good approximation, the proton doesn t move) Step 1: Use what you know about uniform circular motion to find an expression for the square of the speed of the electron in terms of its radius and other physical constants. (LC)
Whiteboard QM-5: Bohr Energy Levels of Hydrogen Step 2: Use Bohr s angular momentum postulate to obtain another expression for the speed of the electron. (LC) Step 3: Combine the expressions from steps 1 and 2 and find an expression for the allowed orbital radii of the electron. (LC) (the Bohr radius) Step 4: Calculate the Bohr radius in nm (LC), and sketch the n = 1, 2, & 3 orbits to scale.
Whiteboard QM-5: Bohr Energy Levels of Hydrogen Step 5: Write the total energy of the electron as the sum of its kinetic and potential energies. Use what you have from the previous steps to express the total energy, E, in terms of n and physical constants. (LC) (the Bohr Energy ) Step 6: Calculate E B the Bohr Energy (in ev) (LC); note, -E B is the ground state (n = 1) energy of the electron. Cass Bohr
The Bohr Model of Hydrogen, a Summary Allowed electron orbital radii and speeds: Allowed electron energy levels:
Whiteboard QM-6: What does Bohr s Model predict for the Hydrogen Spectrum? Consider the emission of a photon: Energy Level Diagram: photon Find an expression for the wavelength of the emitted photon in terms of the initial and final quantum numbers. How does your expression compare with Balmer s formula? Exactly the same as Balmer s PhET H-atom