The Bohr Model; Wave Mechanics and Orbitals

Attempt to explain H line emission spectrum Why lines? Why the particular pattern of lines? Emission lines suggest quantized E states… Bohr’s Quantum Model of the Atom

 e - occupies only certain quantized energy states  e- orbits the nucleus in a fixed radius circular path  E e- in the n th state  depends on Coulombic attraction of nucleus(+) and e-(-)  always negative Bohr’s Model of the H Atom E n = x J ( ) 1 n2n2 n = 1,2,3,… nucleus

First Four e- Energy Levels in Bohr Model n=1 n=2 n=3 n=4 nucleus n=3 n=2 n=1 E ground state excited states n=4 E Levels are spaced increasingly closer together as n

E n = x J ( ) 1 n2n2 n = 1,2,3,… Energy of H atom e- in n=1 state? In J/atom: E n=1 = x J/(1 2 ) = x J/atom In J/mole: E n=1 = x J/atom(6.02 x atoms/mol)(1kJ/1000J) = -1310kJ/mol

n=1 n=2 n=3 n=4 n=3 n=2 n=1 E x J/atom x J/atom x J/atom n= x J/atom First Four e- Energy Levels in Bohr Model the more -, the lower the E n

n=1 n=2 n=3 n=4 n=3 n=2 n=1 E x J/atom x J/atom x J/atom n= x J/atom What is  E for e- transition from n=4 to n=1? (Problem 1)  E = E n=1 - E n=4 = x J/atom - (-1.36 x J/atom) = x J/atom

What is of photon released when e- moves from n=4 to n=1? (Problem 1)   E photon = |  E| = hc/ 2.04 x J/atom = (6.63 x Js/photon)(3.00 x 10 8 m/s)/   = 9.75 x m or 97.5 nm A line at 97.5 nm (UV region) is observed in H emission spectrum.

Bohr Model Explains H Emission Spectrum  E n calculated by Bohr’s eqn predicts all ’s (lines). Quantum theory explains the behavior of e- in H. But, the model fails when applied to any multielectron atom or ion.

Wave Mechanics Quantum, Part II

Wave Mechanics Incorporates Planck’s quantum theory But very different from Bohr Model Important ideas Wave-particle duality Heisenberg’s uncertainty principle

Wave-Particle Duality e- can have both particle and wave properties Particle: e- has mass Wave: e- can be diffracted like light waves e- or light wave wave split into pattern slit

 h/mu u = velocity m = mass Wave-Particle Duality Mathematical expression (deBroglie) Any particle has a but wavelike properties are observed only for very small mass particles

Heisenberg’s Uncertainty Principle Cannot simultaneously measure position (x) and momentum (p) of a small particle    x.  p > h/4     x = uncertainty in position   p = uncertainty in momentum p = mu, so p  E

Heisenberg’s Uncertainty Principle As  p  0,  x becomes large In other words, If E (or p) of e- is specified, there is large uncertainty in its position Unlike Bohr Model  x.  p > h/4 

Wave Mechanics(Schrodinger) Wave mechanics = deBroglie + Heisenberg + wave eqns from physics Leads to series of solutions (wavefunctions,  ) describing allowed E n of the e-  n corresponds to specific E n Defines shape/volume (orbital) where e- with E n is likely to be  n   gives probability of finding e- in a particular space

probability density falls off rapidly as distance from nucleus increases Where 90% of the e - density is found for the 1s orbital Ways to Represent Orbitals (1s)  1s  

Quantum Numbers Q# = conditions under which  n  can be solved Bohr Model uses a single Q# (n) to describe an orbit Wave mechanics uses three Q# (n, l, m l ) to describe an orbital

Three Q#s Act As Orbital ‘Zip Code’ n = e- shell (principal E level) l = e- subshell or orbital type (shape) m l = particular orbital within the subshell (orientation)

l = 0 (s orbitals) l = 1 (p orbitals) these have different m l values Orbital Shapes

these have different m l values l = 2 (d orbitals) Orbital Shapes

Energy of orbitals in a 1 e- atom Three quantum numbers (n, l, m l ) fully describe each orbital. The m l distinguishes orbitals of the same type. n=1 n=2 n=3 E 1s 2s2p 3p3d3s orbital l = 0l = 1l = 2

Spin Quantum Number, m s In any sample of atoms, some e- interact one way with magnetic field and others interact another way. Behavior explained by assuming e- is a spinning charge

m s = -1/2m s = +1/2 Spin Quantum Number, m s Each orbital (described by n, l, m l ) can contain a maximum of two e-, each with a different spin. Each e- is described by four quantum numbers (n, l, m l, m s ).

Energy of orbitals in a 1 e- atom E 1s 2s2p 3p3d3s orbital

Filling Order of Orbitals in Multielectron Atoms

The Quantum Periodic Table l = 0 l = 2 l = 1 l = 3 n s block d block p block f block

More About Orbitals and Quantum Numbers

n = principal Q# n = 1,2,3,… Two or more e- may have same n value e- are in the same shell n =1: e- in 1 st shell; n = 2: e- in 2 nd shell;... Defines orbital E and diameter n=1 n=2 n=3

l = angular momentum or azimuthal Q# l = 0, 1, 2, 3, … (n-1) Defines orbital shape # possible values determines how many orbital types (subshells) are present Values of l are usually coded l = 0: s orbital l = 1: p orbital l = 2: d orbital l = 3: f orbital A subshell l = 1 is a ‘p subshell’ An orbital in that subshell is a ‘p orbital.’

m l = magnetic Q# m l = + l to - l Describes orbital orientation # possible m l values for a particular l tells how many orbitals of type l are in that subshell If l = 2 then m l = +2, +1, 0, -1, -2 So there are five orbitals in the d (l=2) subshell

Problem: What orbitals are present in n=1 level? In the n=2 level? n(l) 1s one of these 2s one 2p three If n = 1 l = 0 (one orbital type, s orbital) m l = 0 (one orbital of this type) Orbital labeled 1s If n = 2 l = 0 or 1 (two orbital types, s and p) for l = 0, m l = 0 (one s orbital) for l = 1, m l = -1, 0, +1 (three p orbitals) Orbitals labeled 2s and 2p

Problem: What orbitals are present in n=3 level? If n = 3 l = 0, 1, or 2 (three types of orbitals, s, p,and d) l = 0, s orbital l = 1, p orbital l = 2, d orbital m l for l = 0, m l = 0 (one s orbital) for l = 1, m l = -1, 0, +1 (three p orbitals) for l = 2, m l = -2, -1, 0, +1, +2 (five d orbitals) Orbitals labeled 3s, 3p, and 3d n(l) 3s one of these 3p three 3d five

Problem: What orbitals are in the n=4 level? Solution One s orbital Three p orbitals Five d orbitals Seven f orbitals