Energy Quantization Objective: To calculate the energy of a hydrogen atom; to describe the Bohr model; to describe the photon model of light; to know the range of energies for visible light; to describe what produces an absorption spectrum and an emission spectrum; to apply conservation of energy to determine the energy of a photon emitted or a photon absorbed by a hydrogen atom.

Photon model of light There are two models of light that are useful to explain various experiments. One model of light describes it as an electromagnetic wave made up of a propagating wave made up of an oscillating electric field and oscillating magnetic field (in a plane perpendicular to the electric field). The color of light depends on its wavelength (or frequency where λf = c in a vacuum). Light can be made up of many electromagnetic waves of various colors giving you what is called a spectrum. White light is made up of equal amounts of all of the colors of the visible spectrum. Another model of light describes it as a collection of particles called photons. These photons have no rest mass but do have energy (kinetic energy). The energy of a photon is E = hf where h is Planck’s constant and f is the frequency of the light. Thus the “color” of a photon depends on its energy. Visible light is a small region of the entire range of possible energies of photons. The entire range is called the electromagnetic spectrum which ranges from gamma rays with very high energy to radio with very low energy. The range of energies of the spectrum is shown on pg. 214 of your textbook. The visible range of the spectrum is from 1.8 eV (red) to 3.1 eV (violet). The unit of energy most often used for light is the electron-Volt. Note that 1 eV= 1.6 × 10−19 J.

Bohr model of hydrogen The Bohr model is a semi-classical model of the hydrogen atom. It is wrong! However, it had mild success in predicting the ground state energy of atomic hydrogen, and it’s an easy model to understand. In the Bohr model, just as planets attracted to a star orbit the star, an electron attracted to a proton orbits the proton. However, unlike planet-star systems, the electron-proton system is only allowed to have certain orbits, i.e. at certain distances with certain energies. A better model of a hydrogen is the “electron cloud” model. An electron can be found, with a certain probability, anywhere inside a certain cloud. The shape, size, and configuration of this cloud depends on the energy of the electron. Though this model is more accurate, our simple minds prefer simple models based on things we already understand. Thus, we’ll often appeal to the Bohr model as we think about energy states of a hydrogen atom. In the Bohr model, an electron orbits the proton in a circular orbit at a certain distance away. Only certain orbits are allowed. Those orbits are the ones where the kinetic energy of the electron (the kinetic energy of the proton is negligible because it is so much more massive than the electron) and the electric potential energy of the electron-proton system are −13.6 eV E = K + U = N = 1, 2, 3, ... (1) N 2 N is an integer that represents the energy state (i.e. the particular orbit in the Bohr model) of the atom. The lowest energy of the atom is called the ground state, N=1. Note that we are not including their rest energies in this equation for the “energy” of the atom. When an electron moves from a lower energy to a higher energy, the electron-proton system (i.e. atom) gains energy. How can this be? There are a number of possibilities including: (1) the atom absorbed a photon of exactly the right amount of energy; (2) an incoming external electron collided with the electron imparting energy to the atomic electron. Note that only photons equal to the change in the energy between two states can be absorbed. When an electron moves from a higher energy to a lower energy, the electron-proton system loses energy. In this case, a photon is emitted.

Application 1. Suppose a photon is absorbed by a hydrogen atom causing the electron to be “excited” from the N=1 state to the N=5 state. What “kind” (i.e. x-ray, ultraviolet, visible, infrared, etc.) of photon was this? 2. What color photon would be emitted if an electron made a transition from N=3 to N=2? 3. A beam of photons, each with energy, 12.09 eV is incident on a container of atomic hydrogen. To what state will a particular atom be excited? 4. A beam of photons, each with energy, 11 eV is incident on a container of atomic hydrogen. To what state will a particular atom be excited? 5. What is the speed of an electron in hydrogen when in the ground state?