THE STRUCTURE OF THE ATOMS/QUANTUM NUMBERS · 2014. 9. 23. · 7 QUANTUM NUMBERS The principal quantum number (n) It is known that the principal quantum number defines the energy,

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THE STRUCTURE OF THE ATOMS/QUANTUM NUMBERS

UP, Medical School, Dept. Biophysics

September 2014

ATOMOS = INDIVISIBLE

‘All that exists are atoms and empty space;

everything else is merely thought to exist.’

Democritus, 415 B.C.

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THOMSON MODEL (1902)

„Plum pudding”

Joseph John Thomson

1897 - electron

THE ATOM MODELSAtoms are stable

Their chemical properties show periodicity (Mendeleev 1869)

After excitation they emit light, and their emission spectra is linear

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1

4

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nR

n: 3,4,5……

R: Rydberg constant

(R = 10 973 731.6 m-1)

Johann Jacob Balmer’s

empirical formula (1885):

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RUTHERFORD MODEL (1911)Ernest Rutherford

RUTHERFORD’S CONCLUSIONS1. The majority of matter is „empty space”!

2. The positive charge is concentrated into a tiny space (nucleus ~10-15 m).

3. Electrons are revolving around the nucleus, like planets around the Sun.

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BOHR’S MODEL

Bohr’s postulates:

1. Electrons in an atom can only have defined orbits. The formula defining

the radius of the allowed orbits is:

2. When the electron jumps from one allowed orbit to another, the energy

difference of the two states is emitted as a photon with the energy of hν:

Niels Bohr

12 EEfh

mv

hnnr

hnmrvL

2

2

Stationary wave!

Jsh 34106.6 Planck constant

THE CONCLUSIONS OF THE BOHR’S MODEL

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13

EE

1. Radius of the 1st orbit: r1 = 5.3 ·10-11 m (Bohr-radius)

r2 = 4r1, r3 = 9r1….. rn = n2r1

2. Energy of the first orbit: E1 = -13.6 eV

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12

EE

2

1

n

EEn

(because it is bound)

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THE FRANK-HERTZ EXPERIMENTThe proof of the Bohr’s model

Atoms can absorb only precisely given amounts of energy. The Hg

atoms e.g. 4,9 eV. The 4,9 eV is equals to the energy difference

between the ground state and the first excited state of a Hg atom.

QUANTUM MECHANICAL ATOM MODEL

Matter wave – wave

function ()

Described by the

Schrödinger’s equation

Probability of

occurrence of an

electron: 2

The position of the ground state electron of a hydrogen

atom, around the nucleus.

The density of the spots is proportional to the finding

probability of the electron.

The graph shows Ψ2 in the function of the distance

measured from the nucleus.

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Heisenberg’s uncertainty principle (1927)

It is impossible to precisely determine the position and the momentum of the

particle at the same time. The multiplication of the uncertainty (error) of two

measurements at the same time is always higher than h / 4 :

4

hpx x

The relation gives a limit of principle: the multiplication of

the measured uncertainty of the two quantities can not be

smaller than h / 4.

QUANTUM NUMBERS

It is already known from the Bohr’s atom model that the energy of the

electrons is quantized so they can have only one value. The energy values

are determined by the n principal quantum number.

The quantum mechanics is proved that there are sublevels of the given

energy levels that is why the n principal quantum number is not enough and

more other quantum numbers are needed.

Quantum numbers describe values of conserved quantities in the

dynamics of the quantum system. They often describe specifically the

energies of electrons in atoms, but other possibilities include angular

momentum, spin etc.

QUANTUM MECHANICAL ATOM MODEL

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QUANTUM NUMBERSThe principal quantum number (n)

It is known that the principal quantum number defines the energy, and anenergy value belongs to every n value ( n → En ). The electrons with given n valuesare forming shells which are named with K, L, M, etc. letters. There can be more otherstates inside a shell which states are determined by the orbital quantum number.

Bohr had predicted the positions of orbits with amazing accuracy but did not takecount that this is not the only position of electron, this is the place where theelectron can be found with the highest probability.

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1

n

EEn

The orbital quantum number (l)It defines the magnitude of the angular momentum of an electron.

Angular momentum:

The angular momentum of a body which is revolving around an r radius orbital

with v speed is a vectored quantity.

Its value is L = mvr.

Its direction is perpendicular to the

plane of the velocity.

The angular momentum resulting from

the movements of the electrons on

their orbital can only be:

where h is the Planck constant and l is the orbital quantum number, whichcan be an integer between 0 and n-1.

Example: n = 2; l = 0 (2s state): L = 0l = 1 (2p state):

QUANTUM NUMBERS

2)1(

hllL

22

hL

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The magnetic quantum number (m)

where m is the magnetic quantum number which values are whole

numbers between -l and +l. This determines the direction of the angular

momentum definitely.

It defines the direction of the angular momentum of an electron. That is why

the angular momentum can be set only in given directions. The projection of

the angular momentum on the direction of the outer magnetic field can only be:

QUANTUM NUMBERS

How can it define the angular momentum:

Example: if n = 2; l = 0, 1; m = -1, 0, +1

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hmLz

ZEEMAN EFFECT I

When an atom turns from aninitial higher energy level toa stationary level with lowerenergy then the energydifference can be emitted asa photon. This may give aline in the visible spectrum.

In the presence of an external magnetic field, these different states willhave different energies due to having different orientations of the magneticdipoles in the external field, so the atomic energy levels are split into alarger number of levels and the spectral lines are also split. The rate of splitis proportional to the applied magnetic field. The new lines appearsymmetrically on the right and on the left side of the original line. This isthe so-called Zeeman effect (normal Zeeman effect).

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The spin quantum number (s)

It defines the value of the spin angular momentum of the electron. It is

imagined as the electron (like the Earth) not just revolving around its orbit

but it is spinning around its own axis. The electron’s own angular

momentums can only be:

where s is the spin quantum number. The spin quantum number can only be ½.

It does not defines other sublevels.

2)1(

hssS

QUANTUM NUMBERS

The magnetic spin quantum number (ms)

It defines the direction of spin angular momentum of an electron.

The projection of the angular momentum on the direction of the outer

magnetic field (z) can only be:

where ms is the magnetic spin quantum number which is ½ or -½, so

the spin (owned angular momentum) can be set only in two directions.

2

hmS sz

QUANTUM NUMBERS

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THE STERN-GERLACH EXPERIMENT

The Stern-Gerlach experiment involves sending a beam of particles through an

inhomogeneous magnetic field and observing their deflection. The particles

passing through the Stern-Gerlach apparatus are deflected either up or down

by a specific amount. This result indicates that spin angular momentum is

quantized (it can only take on discrete values), so that there is not a continuous

distribution of possible angular momenta.

http://www.youtube.com/watch?v=rg4Fnag4V-E

THE STERN-GERLACH EXPERIMENT

Conclusions:

1st The experiment proves that the angular momentum is quantized.2nd Why is the beam deflected into two beams?

if l=0 => m=0 => no deflectionif l=1 => m=0, 1 => deflects into three beams

(that is why a two-beam deflection can not caused by direction of the angular momentum)

Phipps and Taylor reproduced the effect using hydrogen atoms in their ground state in 1927.

1925 – Uhlenbeck and Goudsmit formulated their hypothesis of the existence of the electron spin.

1922

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THE EINSTEIN-DE HAAS EFFECTA freely suspended body consisting of a ferromagnetic material

acquires a rotation when its magnetization changes.

Because of the change of the

external magnetic field

mechanical rotation of the

ferromagnetic material is

happened associated with the

mechanical angular momentum,

which, by the law of conservation

of angular momentum, must be

compensated by an equally large

and oppositely directed angular

momentum inside the

ferromagnetic material.

spin angular momentum is indeed of the same nature as the angular

momentum of rotating bodies

QUANTUM NUMBERS

Quantum

number

Symbol Quantized value Values

Principle n Energy 1,2,3…

Orbital l Value of angular

momentum

0,1……n-1

Magnetic m Direction of angular

momentum

-l, -l+1…0…l-1,

l

Spin s Value of own angular

momentum

½

Magnetic spin ms Direction of own angular

momentum

–½, +½

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QUANTUM NUMBERS

http://dilc.upd.edu.ph/images/lo/chem/Quantum/quantum.swf