Equations we use in Chem 132 - Chemistry

3) Heisenberg uncertainty principle:∆p∆x ≥ h/4πThis equation states that a particle’s momentum p and position x cannot be knownsimultaneously to better than a certain precision. ∆p stands for the uncertainty inmomentum (p = mv) and ∆x for the uncertainty in position. Once again, h is Planck’sconstant. You can think of this in two ways. One interpretation is that the act ofmeasuring one of these quantities changes the other so that it cannot be measured withgreater than a certain precision. Another interpretation is that these quantities really donot have precise meaning simultaneously so that asking about a particle’s exact positionand velocity is just not a meaningful question.4) Schrödinger wave equationHψ = Eψ :(-h 2 /8π 2 m)d 2 ψ/dx 2 + V(x)ψ = EψHere H is a mathematical operator called the Hamiltonian. It has units of energy and ismeant to describe the total energy (kinetic plus potential) of a particle or set of particles.Again, h stands for Planck’s constant, m is the particle mass and x the particle position(this would be r in three dimensions). V(x) is the potential energy that the particle feels.For us in Chem 132, the most important potential is the Coulomb potential (see below)describing the attraction of electrons to nuclei and interelectron repulsion. We have alsoalluded to the harmonic potential used for springs (Hooke’s Law) describing bondvibrations of molecules. Typically, the input to the Schrodinger equation is V(x) and wesolve for the allowed energies E that the particle can have and the correspondingwavefunctions ψ(x) that the particle can have. In general, ψ(x) is a complex number andits squared modulus |ψ(x)| 2 is proportional to the probability of finding the particle at thelocation x. I would like to explain to you why the Schrödinger equation works and why|ψ(x)| 2 describes its location in a probabilistic way at a more fundamental level that isintuitive but I can’t. I suggest treating it as a brute fact that it works and explainsexperimental data exquisitely well.5) Bohr formulaE n = -Z 2 me 4 /(8n 2 h 2 ε 0 2 ) = 2.178 x 10 -18 (Z 2 /n 2 ) (in Joules)When we solve the Schrödinger equation with the Coulomb potential V = -Ze 2 /4πε 0 r, theallowed energies for an electron bound to a proton that we get are described by the aboveformula. We were able to “derive” this formula by using Bohr’s idea that electrons travelin circular orbits and that the radii are selected by the criterion that the DeBrogliewavelength of the electron folds back onto itself. That assumption plus some classicalmechanics gave us the same result. This formula applies onto to one electron atoms likeH, He + , Li 2+ , etc. Z stands for the atomic number (number of protons in the nucleus) andn is an integer (1,2,3, …) we call the principal quantum number. Remember that the case