quantum numbers (2)

7/27/2019 Quantum Numbers (2)

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Atomic structure


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Thomson model of the

atom


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Rutherford experiment (withGeiger and Marsden)


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Rutherfords model of the

atom (planetary)


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Bohr model of the atom


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Werner Heisenberg


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Particle in a Box


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Particle in a box


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ndescribes the size of the orbital

1, 2, 3, and so on

nucleus

electron

The principal qunatum number n describes the energy of an orbital


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Orbital Quantum Number

l describes the shape of the orbital

any integer between 0 and n - 1


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Magnetic Quantum Numbers

ms is the spin quantum number. It describes theorientation of the electron spin

ms = +1/2 or ms=-1/2

ml the magnetic quantum number. It describes the

orientation in space

-l to +l


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Quantum NumbersDefinitions n principal quantum # 1,2,3 determines energy

l angular momentum q. # 0,1,..,n-1 angular dependence ml magnetic quantum # 0,1,.. ,l orientation in space

s spin q. # 1/2 spin magnetic moment ms magnetic spin q. # 1/2 orientation of spin

magnetic moment in spaceOrbital nodesAngular nodes: lRadial nodes: n-l-1Examples1. The 14 wave functions for 14 electrons that can be accommodated in 5f

orbitals have the following quantum numbers:

n = 5l = 3ml = -3, -2, -1, 0, 1, 2, 3s = 1/2mS = +1/2, -1/2


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Angular and Radial Part of the Wave

Function


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All nodes

n-1

Angular nodes

l

Radial nodes

n-l-1


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Orbital symbol

n,,m = n (-symbol)directionExamples:

1,0,0 = 1s2,1,0 = 2pzLimits on quantum #s

for energy level n0, 1, 2, (n-1);

s, p, d,

m 0, 1, , Example:n = 2, = ?, m = ?

= 0, m = 0 = 1, m = 1, 0, -1

What we now have:


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What we now have:

1) Energy levels for electron in an atom

depends only on n!!!

2) Each level has n2

orbitals with that energyDegeneracy

all ,m with same n have same energy

(e.g., n = 2, degeneracy = 4: 2s, 2px, 2py,

2pz)

3) Orbital: mathematical function that

gives wave-like properties:

phase, direction

4) Square Orbital:

Probability distribution ofelectron position in that orbital

We will use plots of orbitals to show these

properties

and to determine properties of the atom


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Radial wave/probability functions


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Size of Hydrogen1s orbital

e- is found 90% of

the time from

r = 0 2.6 ao (.14 nm)

90% of probability

contour line

No angular dependence!!

ao

= 5.29 x 10-11 m

e(-r/ao)

2.6 ao

Radial function

Figure 16 19: 1 2 3 orbitals


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Figure 12.18:

1s, 2s, 3s orbitals

Figure 16.19: 1s, 2s, 3s orbitals

(no angular dependence!!)

(idea of relative size)

+-+

90% Probability contours showing relative

size of orbitals

phases of

wave function ()

radial probability distribution (r22) = probability of findingelectron at a distance r from the center of the nucleus

+-+

++ -

++-

Understanding radial distribution
http://localhost/var/www/apps/conversion/AppData/Local/Temp/Atom3secondpartraddistfunc2009Lec7.ppt


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Angular representation of the porbitals

= 1 ; now there IS angular dependence

m = 1, 0, -1 : 3 orbitals in 3 different directions

Phases given as + and - signs

r

e-

+


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Exercises:

1. Consider an atom which has the

following electron configuration on the

4th shell: 4d10. Find the quantum numbersthat describe this orbital.

2. How many radial nodes, angular nodes

are there in a 3pz, 4s, 5d orbitals?


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quantum numbers (2)

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