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Quantum Numbers A Short Tutorial Bohr Model of Hydrogen Atom • an e-’s is found in specific energy levels. • These levels represent a fixed distance from the nucleus. • A quantum of energy is the amount of energy needed to move one electron. • Neils Bohr was also a GK for the Danish National Team ! 5.1 The Bohr Model • Like the rungs of this ladder, the energy levels in an atom are not equally spaced. • The higher the energy level of an e-, the less energy it takes to move the e- to the next higher energy level. • Moving requires a quantum leap. Modern View of Atoms 1. Energy of e-’s is quantized. 2. e-’s sometimes act like waves. 3. Position of e- is based on probability. • Modern view is the quantum-mechanical model of the atom. Location of Electrons Based on Probability • e-’s are mathematically predicted to exist in electron clouds. • 90% chance that e-’s exist in this shape. • e- clouds have high electron density. Schrodinger: • An Atom contains 4 quantum numbers • Schrodinger was always very serious about Quantum Numbers. Principal Quantum Number • Symbol is n. • Distance from nucleus. • Values are n = 1, 2, 3, 4, 5, 6, 7, …. Clearly, the further the distance from the nucleus, the bigger the volume of the energy level is. n=1 n=2 n=3 Orbital Quantum Number Symbol is l. It corresponds to the type of orbital, either s, p, d or f. It has values of 0, 1, 2, 3, …. Check the next several slides for a look at the different orbitals. The s - orbital The s-orbital is shaped like a sphere with its center at the origin of a 3 dimensional grid. The axes on the grid are the x-axis, the yaxis, and the z-axis. The p - orbital The p – orbitals are dumbbell shaped and lie across each axis, so they are 90° from each other. There are always 3 p – orbitals for each energy level. The p - orbital Another look at p – orbitals and how they orientate themselves to each other. 5.1 The d - orbitals The d – orbitals are also dumbbell shaped and generally lie between the axes. There are always 5 d – orbitals. The f - orbitals Here is a list of the 7 f – orbitals. fxyz, fx3, fy3, fz3, fz(x2-y2), fy(z2-x2), fx(z2-y2) They have complex shapes. Magnetic Quantum Number Symbol is m though sometimes lm. It corresponds to the orientation of the orbital around the axis. It has values of - l, … 0, …. + l You have seen these on earlier slides. Check the next slide in the presentation to look at the p – orbitals again. Magnetic Quantum Numbers • Notice the x, y, and z part of the orbital notation below ? • This is telling us how the orbital is orientated around the axes. • This is the magnetic quantum number. Magnetic Quantum Number • There are as many orientations as there are orbitals. • Check the table: Orbital type s p d f # of orbitals Magnetic QN 1 0 3 -1 0 + 1 5 -2 -1 0 +1 +2 7 -3 -2 -1 0 +1 +2 +3 Magnetic Quantum Number • Now let’s apply this to the p – orbitals. • Recall: There are three p – orbitals and they are the px , py , and pz . • From the slide above we see that the magnetic quantum numbers for the p – orbitals are as follows: -1 , 0, +1. • Therefore the px corresponds to -1, the py corresponds to 0, and the pz corresponds to + 1. • The other orbitals work in the same way – there are as many orbitals as magnetic quantum numbers. Spin Quantum Number (ms) • Symbol is s, though sometimes ms • It tells us that the two electrons in the orbital have opposite spins. • It’s values are either + ½ or – ½ . • Remember that there can be no more than 2 electrons per orbital. Is there a pattern here? for n = 1 : 1orbital for n = 2 : 4 orbitals (one s and three p orbitals) for n = 3 : 9 orbitals (one s, three p, and five d orbitals) For a given energy level n, there are n2 orbitals Since there can be 2 electrons per orbital, the number of electrons in any energy level is described as 2 n2 Lets put all the info together Lets put all the info together Lets put all the info together Summary • Remember that the quantum numbers are the solutions to the Schrodinger equations. • They are actually numbers but it would be confusing to have 4 numbers right next to each other so some are given letter designations to make it easier to read. • For example: For orbital quantum numbers the s – orbital = 0. The p – orbital = 1. The d – orbital = 2 etc. Summary • Let’s look at a certain electron in an Iron atom. We will look at the last electron to be filled in its’ electron configuration which would be 3d6 • • • • Principal QN = 3 It is in the 3rd energy level. Orbital QN = 2 It is a d – orbital Magnetic QN = -2 It is filling up the d – orbital Spin QN = - ½ It has an opposite spin from the electron already there.