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Atomic Theory and Spectral Lines -- Chemical Physics

matter is composed of atoms – size ~ 10-8 cm - density ~ 1023/cm3

109 different atoms identified - 92 stable (occur naturally)- 17 transuranic (created artificially)

pattern found by Mendeleev in 1869 (periodic table) led to our currentlyaccepted model of atoms:

- an atoms is a nucleus (10-14 m) surrounded by a cloud of electrons (10-10 m)- a nucleus comprises a number of protons with an almost equal number of neutrons

- atomic or chemical properties depend on the electrons (ie on Z, the charge)(Z = charge on the nucleus = number of protons = number of atomic electrons since atoms are neutral)

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reminder:

office hours next week only tuesday 2-4

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periodic table of the elements:

in astrophysics everything exceptH and He is considered a metal

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adding neutrons to a nucleus (or taking them away) does not affect the nuclear charge (or number of electrons) so chemically the atom is not different

it does affect the nuclear properties (stability etc)

Isotopes

- specify an element by Z- specify an isotope by Aa complete description requires both

Z is implied by the historical name eg 14C A = 14 Z = 6 - carbon with 2 extra neutronss

hydrogen p+e 11H

deuterium p+n+e 12H (in heavy water)

tritium p+n+n+e 13H

these are all isotopes of hydrogen but are not separate elements they have the same chemical properties but different nuclear properties(therefore things like nuclear burning in stars are different).

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the number of neutrons is not arbitrary- too different from the proton number results in instability

A – Z = neutrons

Z = protons

too many protons

too many neutrons

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Bohr-Rutherford Model of the Atom

Rutherford (McGill!) discovered the nucleus in experiments where he scattered alpha particles (helium nuclei – small,dense, + charge)off thin foils of material and saw that most went right through,some showed large deflections, and some bounced right back.

he hypothesized that each atom comprised a dense nucleusorbited by electrons like planets around the sun - mostly space

- the electrostatic (Coulomb) force between the electrons and the nucleus was the ‘gravity’

major problem with this idea:- electrons which go in circles are accelerating- accelerated charges radiate energy- therefore electrons will lose energy and spiralinto the nucleus – all matter collapses in an instant

- also doesn’t explain discrete lines in spectra

p

eeg hydrogen

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Bohr’s hypothesis: classical physics does not apply: quantum theory

electrons only orbit in particular orbits with L (angular momentum)equal to where n is an integer and h is Planck’s constant

in this case, with centripetal force equaling the Coulomb force

we have:

but so

solving for r we get:

is a constant for a given atom

is the principal quantum number

the radius of the orbit increases as the square of the principal quantum number

h n∂ nh/ =2

22 /)(/ reZermv =22 / mvZer =

mvrnL == h 22 )/()( rnmv h=

22 )//( rnmZer h=)/(/)( 22222 mZenmZenr hh ==

mZe22 /h

n

π

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another route to Bohr’s quantum condition: wave/particle duality

wavelength of an orbiting electron (non-relativistic) p=momentum

circumference must be an integer number of wavelengths (think standing wave)

the only orbits which can exist have

using the equality of centripetal and Coulomb forces

we get

mvpph /2/2/ hh ππλ ===

λπ nr =2

mvnr /22 hππ =mvnr /h=

222 // rZermv =222 )/()/()( rnrZemmv h==

mZenr 222 /h=

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energy of an electron at radius r

rZemvE /2/ 22 −=kinetic potential

use )/(/ 22222 mZenmvZer h==

rZerZerZeE 2//2/ 222 −=−= (negative – electrons are bound)2222 2/)( hnmZeE −=

1 /1 2 =−∝ nnE has the tightest binding (E is large and negative)

all orbits are bound0 →∞→ En

0>E continuum of unbound states

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Bohr part II:

(a) radiation in the form of a single discreet quantum (photon) is emittedor absorbed as the electron jumps from one orbit to another

(b) the energy of the radiated photon equals the energy difference between orbits

photons are emitted when the electron goes from a higher energy orbit (na)to a lower energy orbit (nb) (na > nb)

E(na) = E(nb) + h

photons are absorbed to cause electrons to go from a lower energy orbit toa higher energy orbit (nb > na)

E(nb) + h= E(na)

frequency of emitted (or absorbed) photon:

ab = (E(na) - E(nb) )/h

na is not necessarily nb +/- 1

n = 1 is called the ground state – lower energy states are not possible

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Hydrogen spectral lines

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eg Hydrogen Z=1

222

42 1'

12)(

nR

nh

menE −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

πenergy of the nth level

R’ = 2.18 x 10-18 Joules = 13.6 eV (Rydberg energy)

this is often expressed in terms of wavelengths =c /λk = 1/λ= wave number

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−===

2222

1111'1

abab

ab

abab nn

Rnnhc

R

ck

υ

λR = 10.97 m-1 (Rydberg constant)

na and nb are two levels in the atom na > nb

for every nb there is an infinite series of nas

na = nb+1, nb+2, nb+3. . .

the series are named after the people who discovered them

Lyman nb = 1 (ultraviolet)

Balmer nb = 2 (found first since it is visible)

Paschen nb = 3

Brackett nb = 4

Pfund nb = 1

remember: E=h2222 2/)( hnmZeE −=

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Balmer series:nb = 2 na = 3 λ = 656.3 nm H

na = 4 H

na = 5 H

Lyman series:nb = 1 na = 2 λ = 121.6 nm L

na = 3 L

na = 4 L

historical names

can also have theselines in absorption

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Energy Levels

ground state

excited states

E = R’ = 13.6 eV

unbound states(continuum)

Each atom has a characteristic energy level diagram(good for identifying which atom it is)

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Excitation raise from na to nb with na < nb

radiative excitation - absorption of a photon of the correct energy- produces absorption lines

source

absorber

spectrograph

flux

wavelength

flux

wavelength

without absorber one getsa continuous spectrum

with absorber one getsa spectrum with absorption lines

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excited states are unstable (10-8 seconds lifetime, typically)so why don’t the atoms in the absorber de-excite with no loss of photons and hence no absorption lines?

two reasons:

geometry – decays photons go in all directions so loss of intensity

combinatorics – several de-excitation paths usually available

initial fluxre-emitted flux

λ3

λ2

λ1

absorb λ1

emit λ2 + λ3

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collisional excitation - no photons are absorbed; inelastic collisions of the atomwith other atoms or electrons (Coulomb interaction)- atom gains some of the projectile’s kinetic energy and has its energy level raised

e

e

vi

vf

h = 1/2m (vi

2 - vf2)

the atom eventually de-excites through photon emissionwe see ‘emission lines’

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de-excitation

radiative – emission of photon(s) 10-8 seconds typically

collisional – super-elastic collision: excited atom is hit by a particleor atom which then gains energy from the the collision

forbidden transitions – special form of radiative de-excitation – longlifetimes since they violate quantum mechanics rules to firstorder – have to proceed in a more complicated way which takesmore time. Observation of these implies low temperature andlow density of the region. Otherwise collisions would de-excitethe atoms much sooner.

generally only seen in astrophysics!

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⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−===

2222

1111'1

abab

ab

abab nn

Rnnhc

R

ck

υ

λ

Ionization

bound electrons can be liberated from the atom if enough energyis supplied (by a photon or collision)

E > E (binding energy)

X is an atom

X + energy X+ + e-

ion electron

Nomenclature used: hydrogen neutral H or HI

ionized H+ or H II

oxygen neutral O

ionized O+ or O II

twice ionized O++ or O III++ etc is cumbersome after 3 or 4 electrons have been removed (atoms can be fully stripped) – Roman Numerals are preferred (eg Fe IX)

E

na = ∞

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Energy needed to ionize is greater than or equal to the energy state of the atom

potential ionization thecalled isenergy minimum The

energy kinetic aselectron by the off carried is excess the

required, minimum an thegreater th is E If

)()( n to state from go energy to Need

−∞>∞= nEEEn

flux

wavelength

flux

wavelength

spectrum at source spectrum after absorber

λ threshold

for λλthreshold E > IP (ionization potential)

so absorption occurs at all wavelengths – one gets a broad depletioninstead of an absorption line

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Emission continua exist by inverse analogy; if there is a plasma(ions plus electrons) some recombination can occur if the electronemits a photon of E = KE + IP

KE = electron’s kinetic energyIP = ionization potential of the level into which the electron will fall (not necessarily the ground state)