Quantum theory

1. Quantum theory Electron Clouds and Probability

2. Bohr’s model of the atom is unable to describe electron (e-) behavior in an atom • Problem: multiple spectral lines closely spaced • deBroglie Hypothesis • Believed e- had a Dual-nature • Acted as particles with mass and waves of energy with no mass, simultaneously • Combined Einstein’s Relativity equation (E=mc2) with Plank’s quantum equation (E= hv) (Plank’s constant, frequency of a wave) • mc2 = hv substitute c with v ( general velocity) • v(frequency of wave)= v/l velocity/wavelength)

3. Final equation l = h mv • Enabled de Broglie to predict the wavelength of a particle of mass m and velocity v • Showed that an e- stream acted as a group of particles and and in some ways as a ray of light • Waves can act as particles and particles can act as waves • Wave-particle duality of nature

4. Momentum (p) is the product of mass and velocity (speed and direction of motion) l = h/p • Wavelength inversely proportional to momentum • Only worth studying for particles of small mass • Quantum mechanic: small particles traveling near speed of light

5. Schrodenger • Studied e- as waves • Found amplitude of wave was related to distance or point in space an e- was from the nucleus • Developed an equation using e- energies and amplitude along with quantum levels to describe wave function • Max Born found that the square of the amplitude gave the probability of finding the e- at that point in space for which the equation is solved

6. Since the e- is traveling at the speed of light and appearing at all these positions, the e- appears to be everywhere • The area the e- occupies appears to be a region of negative charge with a specific shape • This is referred to as an Electron cloud

7. Heisenberg studied e- as particles • Noted it was impossible to determine both the exact position and exact momentum of an e- at the same time • Due to the fact that you interacted with the particle to “see” it and changed one of the two properties • There is always uncertainty • Proposed Heisenberg Uncertainty Principle

8. Impossible to know the exact position and momentum of an e- at the same time • Scientists are unable to describe the exact structure of an atom due to this • But it can be determined with probability • Can determine with high probability where an e- is most likely to be found in the energy levels of an atom at any one given time

9. Probability • Is the ratio between the number of times the e- is in that current position and the total number of times it is at all positions • The higher the probability, the more likely the e- will be found in a given position • The probability plots give a three dimensional shape to a region of space an e- is most likely to be found

10. How do e- “jump” to higher energy levels? • Change amplitude: it’s a wave function not a particle function • Extend into higher energy levels only if unfilled • Explains gaps or lack of some colors (cannot fit into filled levels) • So what is importance of particle nature? • Only particles carry charge • Keep e- from cancelling each other out • Makes e- occupy specific regions and evenly spreads them out

11. So do e- actually jump? Not really. Change the amplitude which can extend the reach of the energy of the e- into a higher energy state (level) Goes from ground state to excited state (notice word state). • e- can extend into lower states or levels, but the repulsion of e- in the lower levels force e- to maintain their region of space based on the average energy of the e- amplitude from its wave energy. Wave energy function explains why e- seem to go near to nucleus and amplitude also explains why e- appear to be in a region and are found at probability locations, specific distances from nucleus.

12. If a higher region, energy level or state, is completely filled (occupied) by e-, an e- that gains enough energy to extend its amplitude into that region is unable, because there is not enough energy for that e- to penetrate that level (e- act as a barrier). However, if the e- gains enough energy to “jump” to a higher unfilled level, it will pass through the lower energy level (now has more energy) and go to the higher energy state. This can explain why some colors or gaps of energy are skipped when looking at a spectrum. But there is still the problem of multiple lines of the same color. Jumps between energy levels next to each other can explain some but not all. There has to be another reason and that is the next part of the energy levels that needs to be explained.

13. Now lets put this into the Bohr Model • Electrons are assumed to have a circular path and to always be found at a specific distance from the nucleus dependent on their P.E.(ground state) • But there is the probability of any e- at trillions if not more points in space • Many of these points have high probability • Connect all these points together and you obtain a 3D shape • The most probable place to find the e- is on the surface of this shape

14. Now lets put this into the Bohr Model • Electrons are assumed to have a circular path and to always be found at a specific distance from the nucleus dependent on their P.E.(ground state) • But there is the probability of any e- at trillions if not more points in space • Many of these points have high probability • Connect all these points together and you obtain a 3D shape • The most probable place to find the e- is on the surface of this shape

15. Remember that e- move near speed of light • The e- random movement causes it to appear as a cloud • The e- occupies all the volume of this cloud • Does not normally go beyond the outer volume area(ground state)

17. Now in order to describe an e- behavior we need to represent different energy states • Do this by use of quantum numbers • The differences correspond to the lines observed in the spectrum of atoms • Easily described using H • When an e- moves from ground to excited state, energy emitted as a form of light • Represented by a line in the H spectrum

18. Atoms with more than one e- have problems • Interactions of the other e- cause problems • as well as the increased nuclear pull • It is assumed that the various e- in a multi electron atom occupy the same energy states without affecting each other • To describe an e- in an atom, four quantum numbers are required • Quantum numbers are ID’d by the letters n, l, m, s • Each e- has its own unique set of these four numbers

19. Any one e- can occupy only a specific energy level based on its total and P.E. • These energy levels are represented by whole integers starting with 1 • The number of the energy level, represented by the letter n, is called the Principle Quantum Number 1,2,3….n • Electrons can be found in each energy level of an atom • Greatest number of e- in a level is 2n2

20. Second Quantum number is l Azimuthal quantum number • Represents the energy sublevels and orbitals • Each energy level is a group of energy states • Represented by the number of spectral lines we saw of the same color • These are closely grouped together • States called sublevels • # sublevels in an energy level is equal to the principle quantum number

21. Principle quantum level 1 has 1 sublevel • Principle quantum level 2 has 2 sublevels • Principle quantum level 3 has 3 sublevels • And so on… • The lowest sublevel of energy in any principle energy is always designated by the letter s • 2nd sublevel is p so 2nd level has an s and p • 3rd sublevel isd so 3rd has a s,p,d • 4th sublevel is f so 4th has s,p,d,f

22. Each sublevel holds a maximum number of e- • Every s can contain 2e- (one pair) • Every p can contain 6e- (three pair) • Every d can contain 10e- (five pair) • Every f can contain 14e- (seven pair)

23. Each pair in a sublevel has a different place in space • Due to the interactions of the e- within a sublevel on each other • Try to be as far from each other as possible due to the fact they are all same charge • The space occupied by one pair of e- is called an orbital • Designated by quantum number ml magnetic quantum number • Defines each orbital by indicating its direction in space

24. Ex. Sublevel s is simply spherical in nature • Sublevel p with three orbitals (6e-, 3 pairs) has e- in 3D along the x,y,z axis • Orbitals of the same sublevel are alike in size and shape and have same energy

26. Electrons in the same orbital must coexist together • How if they are repulsive • Fourth quantum number is spin ms spinmagnetic quantum number • Electrons in the same orbital spin in opposite directions • Sets up opposite magnetic fields, so e- become slightly attractive to each other • Up and down arrows E used to show spin direction • Pauli Exclusion Principle: no two e- in the same atom can have the exact same four quantum values

27. So lets see how e- would start to occupy positions in an atom • Aufbau principle states that e- will always occupy lowest available energy levels first • So lets see how this might look:

28. As more e- are added to the atom and occupy higher energy levels, the interactions become greater between e- of different energy levels and sublevels • Also remember that the nucleus is also gaining protons and its overall charge is increasing causing it to pull harder • All these interactions force sublevels to begin to overlap each other • It changes the filling pattern of e- in atoms

31. Note: starting with energy level three, 4s fills before 3d • Thus have a new filling pattern for e- • Can use an ARROW DIAGRAM to determine the pattern • ECN Electron Configuration Notation • Short cut

32. One more e- filling fact • Hund’s Rule: • The most stable atoms are those which have the most parallel (same direction) spinning e- • Designated by using boxes and the arrows for e- spin • EON Electron Orbital Notation • EDD Electron Dot Diagrams

33. Quantum Numbers • n,l,m,s • O • Na • Cu