We can define the energy as a function of n, Z, and R_HĮnergy ≔ n, Z, R → − Z 2 ⋅ R n 2 Įnergy ≔ n, Z, R ↦ − Z 2 R `/` n 2 H SI ≔ evalf Constant ' h ', units R SI ≔ evalf Constant ' h ', units ⋅ Constant ' c ', units ⋅ Constant ' R infinity ', units a0 SI ≔ evalf Constant ' a 0 ', units We define additional scientific constants that we will need in this parts of the lesson: Planck's constant ( h ), Rydberg's constant ( R H ), and the Bohr radius ( a 0 ). Where a_0 is a constant known as the Bohr radius The radius of each orbit can be computed from the following formula Where the positive integer n is the quantum number, Z is the number of protons, and R_H is the Rydberg constant. The energy of each orbit can be computed from the following formula The orbits are labeled by positive integers n known as the principal quantum numbers.įigure 2: Sketch of the Bohr model's circular orbits with emission of radiation (Author: JabberWok ( CC BY-SA 3.0 no changes)) The predicted frequencies of emission matched the measured emission spectra of hydrogen gases. Each orbit has an energy with transitions between orbits corresponding to the emission and absorption of radiation. In 1913 Niels Bohr proposed a model for the hydrogen atom in which electrons rotate around the positively charge nucleus in discrete circular orbits. (d) Changing the numbers in (b), what is the fraction of the velocity in (c) to the speed of light? (c) Changing the numbers in (a), determine the velocity when a voltage of 120 V (Volts) is applied to the cathode ray tube. (Note that we are not considering relativistic effects). The velocity of the electron is only 3/1000 of the speed of light. (b) What is the fraction of this velocity to the speed of light?Īnswer: We can compute the ratio of the velocity to the speed of light constant
V SI ≔ subs m = m SI, q = e SI, V = 2 ⋅ Unit ' volt ', sqrt 2.0 ⋅ q ⋅ V m (a) What is the velocity when a voltage of 2 V (Volts) is applied to the cathode ray tube?Īnswer: We can substitute a voltage of 2 V as well as the charge and the mass into the expression for the velocity
With ScientificConstants : e SI ≔ evalf Constant ' e ', units m SI ≔ evalf Constant ' m e ', units c SI ≔ evalf Constant ' c ', units The values of these constants are built into the Maple engine. We also load the ScientificConstants package and define the following scientific constants that we will need in this part of the lesson where the subscript SI indicates that we are using Standard International (SI) units: electron charge (e), electron mass (m), and the speed of light (c). We load the Units package with Maple's with command Using this formula, we can compute the velocity of the electrons for a given applied voltage. If the forces from the electric and magnetic fields are chosen to balance, preventing the deflection of the cathode ray, then the charge-to-mass ratio of the particles can be computed from the formulaīecause the kinetic energy of the electron must be equal to the charge of the electron multiplied by the applied voltage, we have The applied voltage causes a stream (cathode ray) of negatively charged particles (electrons) to flow through the tube. Thomson in his third cathode-ray-tube experiment used electric and magnetic fields to measure the charge-to-mass ratio of the electron.įigure 1: Schematic of a cathode ray tube with deflection from an applied electric field (Author: Kurzon (Public Domain)) Thomson, Bohr, and Schrödinger would receive Nobel Prizes for their work in 1906, 1922, and 1933 respectively. (3) Erwin Schrödinger's equation for quantum atoms and molecules (1925) (2) Niels Bohr's model for one-electron atoms (1913) Thomson's third cathode-ray experiment (1897) The structure of the modern atom is explored through three key advances: