Electron Arrangement in Atom

1. Historic Basis2. Modern Theory3. Three Quantum Numbers as Locators

LECTURE FifteenCHM 151 ©slg

Topics:

WHERE THE ELECTRONS ARE.....

We are going to examine in historical succession the ideas and experiments that led to the modern atomic theory and sophisticated placement of theelectrons about the nucleus.

The current theory, based on quantum mechanics, places the electrons around the nucleus of the atom in “ORBITALS,” regions corresponding to allowed energy states in which an electron has about 90% probability of being found.

Historical Events, Nature of

Electromagnetic Radiation

1. 1864 James Maxwell: Wave motion of electromagnetic radiation 2. 1885 Rydberg, Balmer: Wavelength of atomic spectra

3. 1900 Max Planck: Quantum theory of radiation, packets of specific energy

4. ~1905 Einstein: Particle- like properties of radiation, “photons”

James Maxwell described all forms of radiation in terms of oscillating (wave like) electric and magneticfields in space. The fields are propagated at right angles to each other.

All “forms of radiation” include visible light but also, x-rays, radioactivity, microwaves, radio waves: allare described today as electromagnetic radiation.

The waves have characteristic frequency and wavelength, and travel at a constant velocity in a vacuum, 3.0 X 10 8 m/s.

wavelength

node

crest

trough

crest

trough

node

cycle

Wave description:

Frequency: # cycles / sec

c =

speed light, vacuum = wavelength X frequency

3.00 X 108 ms-1 = , m X , s-1 (hertz)

hertz, Hz, s-1 # cycles per second Same as m/s

= c/ = c/

Important Relationship, all electromagnetic radiation:

Rearranging:

A red light source exhibits a wavelength of 700 nm, and a blue light source has a wavelength of 400 nm.What is the characteristic frequency of each of these light sources?

Red light: 700 nm = ? m = ? s-1

700 nm 1m = 700 X 10-9 m = 7.00 X 10-7 m 109 nm

= c / = 3.00 X 108 ms-1 = 4.29 X 1014 s-1

7.00 X 10-7 m

= 4.29 X 1014 cycles per sec = 4.29 X 1014 Hz or s-1

Blue Light: 400 nm = ? m = ? s-1

400 nm 1m = 400 X 10-9 m = 4.00 X 10-7 m 109 nm

= c / = 3.00 X 108 ms-1 = 7.50 X 10 14 s-1

4.00 X 10-7 m

= 7.50 X 1014 cycles per sec = 7.50 X 1014 Hz

RED light: 700 nm, 4.29 X 1014 Hz

BLUE light: 400 nm, 7.50 X 1014 Hz

Point to remember: the shorter the wavelength, thehigher the frequency: the longer the wavelength, the lower the frequency.

longerlower

shorterhigher

GROUP WORK

Microwave ovens sold in US give off microwave radiation with a frequency of 2.45 GHz. What is the wavelength of this radiation, in m and in nm?

= c/ 2.45 GHz = ? m = ? nm 109 Hz (s-1) = 1 GHz c = 3.00 X 108 ms-1

1. Convert GHz to Hz; call Hz “s-1”

2. Calculate wavelength, , in m3. Convert m to nm (109 nm = 1 m)

Max Planck made a major step forward with his theory that energy is not continuous but rather is generated in small, measurable packets he called quantum (which refers back to the Latin, meaning bundle).

He related the energy of the quantum to its frequency or wavelength as below:

Energy quantum = h x radiation = h x c

radiation

h is Planck’s constant, 6.63 X 10-34 joule sec

c, speed of light = Wavelength, X frequency, = c = c

Wavelength, frequency, energy relationships:

energy of photon = h, Planck’s constant x

E = h = h c

higher shorter

Sample Calculations: E = h = h c

Blue light, = 4.00 X 10-7 m

E = 6.63 X 10-34 joule s x 3.00 X 108 ms-1

4.00 X 10-7 mE = 4.97 X 10-19 joule

Microwave oven, = 2.45 X 109 Hz or s-1

E = 6.63 X 10-34 joule s x 2.45 X 109 s-1

E = 1.62 X 10-24 joule

The relationships expressed by this equationinclude the following:

Energy of a quantum is directly proportional to thefrequency of radiation: high frequency radiation isthe highest energy radiation (x rays, gamma rays)

Energy of radiation is inversely proportional to its wavelength: long waves are lowest in energy, short waves are highest. Radio waves, microwavesrepresent low energy forms of radiation.

View CD ROM sliding spectra here

Einstein took the next step in line by using Planck’s quantum theory to explain the photoelectric effectin which high frequency radiation can cause electrons to be removed from atoms.

Einstein decided that light has not only wave- like properties typical of radiation but also particle- likeproperties. He renamed Planck’s energy quantum asa “photon”, massless particles with the quantizedenergy/frequency relationships described by Planck.

“Quantized” refers to properties which have specific allowed values only.

It was discovered in this time frame that each element which was subjected to high voltage energy source in the gas state would emit light.

When this light is passed through a prism, instead of obtaining a continuous spectrum as one obtains for white light, one observes only a few distinct lines of very specific wavelength.

Each element emits when “excited” its own distinct “lineemission spectrum” with identifying wavelengths.

The discovery of emission lines led to calculations relating their wavelengths by both Johann Balmer and Johannes Rydberg. Class view spectra here....

Historical Events, the Nature of Electron

1. 1804 Dalton: Indivisible atom 2. 1897 Thomson: Discovery of electrons 3. 1904 Thomson: Plum Pudding atom

4. 1909 Rutherford: The Nuclear atom

5. 1913 Bohr: Planetary atom model, e’s in orbits

The Plum Pudding Atom: Positive matter with electrons embedded likeraisins in a pudding

JJ Thompson’s Picture of the atom:

The Rutherford Nuclear Atom:Mass and positive charge in tiny nucleusin center of atoms; electrons dispersedoutside nucleus

Rutherford’s Picture of the Atom:

Bohr combined the ideas we have met to presenthis “planetary” model of the atom, with the electrons circling the nucleus like planets around the sun:

Bohr used all the ideas to date:

• electron in the atom outside the tiny positive nucleus

• excited elements emit specific wavelengths of energy only

• radiation comes in packets of specific energy and wavelength

Bohr’s atom placed the electrons in energy quantized orbitsabout the nucleus and calculated exactly the energy of theelectron for hydrogen in each orbit.

1

2

3

4

5

6

n, integer values for shells around nucleus, = 1-6-->infinity

n=1, lowest energy orbit n=6, highest energy orbit pictured

each orbit is quantized: has an energy of a specific frequency only

allowed transitions shown by arrows: e may be "excited" to higher orbitonly if energized by photon of energy of matching frequency; it falls back to lower shell emitting the energy it has gained in form of light.

The wavelength of the light emitted represents the energy differencebetween the orbits

2

3

4

left: e excited, photon ofcorrect, matching energy

right: e returning to origin, emitting light: line spectra

e

ee

e

12

3

4

1

e

e

e

e

Bohr also predicted that each shell or orbit about thenucleus would have its occupancy limited to 2n2 electrons, where n = the orbit number.

Many of Bohr’s ideas, in modified form, remain in thepresent day quantum mechanics description of atomic structure.

Bohr was able to calculate exactly the energy values for the hydrogen spectrum using his model; however the calculations only worked for one electron systems and did not explain the electronic behavior of larger atoms.

GROUP WORK

Predict maximum occupancy (2n2 )of each shell:

n=1 = _____ e’s n=5 = _____ e’s

n=2 = _____ e’s n=6 = _____ e’s

n=3 = _____ e’s n=7= _____ e’s n=4 = _____ e’s

PlanckEnergy is “quantized”, comes in packets with energy hv

EinsteinEnergy can interact with matter, photoelectric effect, “photon”

BohrPhotons of energy can interact with electrons in orbits of lowest possible energy around the nucleus and “excite” e’s to higher energy orbits. The e’s give off this energy as light, spectral lines as they return to “ground” state.

Summation:

Electron as matter/energy particle

1. 1925 DeBroglie: Matter Waves 2. 1926 Heisenberg’s Uncertainty Principle

3. 1926 Schroedinger’s Wave Equation and Wave Mechanics 4. Modern Theory: Use of wave equation to describe electron energy/probable location in terms of three quantum numbers.

DeBroglie next suggested that all matter moved in wavelike fashion, just like radiation. Large macroscopic matter (moving golf balls, raindrops, etc) have characteristic wavelengths associated with their motion but the wavelengths are too tiny to be detectable or significant.

Electrons, on the other hand have very significant wavelengths in comparison to their size.

Einstein gave radiation matter- like, particle properties; DeBroglie gave matter wave- like properties.

Heisenberg’s Uncertainty Principle:

If an electron has some properties that are wave likeand others that are like particles, we cannot simultaneously describe the exact location of theelectron and its exact energy. The accurate determination of one changes the value of the other.

The Bohr atom tried to describe exact energy andposition for the e’s around the nucleus and workedonly for H.

Born’s interpretation of Heisenberg’s Principle:

If we want to make an accurate statement about theenergy of an electron in the atom, we must acceptsome uncertainty in its exact position. We can onlycalculate probable locations where an electrons is tobe found.

Schroedinger’s wave equation describes the electron asas a moving matter wave, and results in a picture in which we place electrons in probable locations aboutthe nucleus based on their energy.

Schroedinger’s Wave Equation

The mathematics employed by Schroedinger to describethe energy and probable location of the electron aboutthe nucleus is complex and only recently been solvedfor larger atoms than hydrogen.

However, it yields a description of the atom whichaccounts for the differences between the elements. IT WORKS!

Schroedinger’s wave equation describes theelectrons in a given atom in terms of probableregions of differing energies in which an electron is most likely to be found.

We call the regions “orbitals” rather than “orbits”, and each is centered about the nucleus.

The description of each orbital is given in the form of three “quantum numbers”, which give an address like assignment to each orbital. Thequantum numbers are in the form of a series ofsolutions to the wave equation.

Heisenberg:

Uncertainty Principle: cannot determine simultaneously the exact location and energy of an electron in atom

Schroedinger:

Wave equation to calculate probable location of e’s around nucleus using dual matter/wave properties of e’s.

Three quantum numbers from equation locate e’s of various energies in probable main shells, subshells, orbitals.

Summation:

The Quantum Numbers

“Locators, which describe each e- about the nucleus in terms of relative energy and probable location.”

The first quantum number, n, locates each electron in a specific main shell about the nucleus. The second quantum number, l , locates the electron in a subshell within the main shell.

The third quantum number, ml , locates the electron ina specific orbital within the subshell.

“n”, the Principal quantum number:

• Has all integer values 1 to infinity: 1,2,3,4,... • Locates the electron in an orbital in a main shell about the nucleus, like Bohr’s orbits • describes maximum occupancy of shell, 2n2.

The higher the n number:• the larger the shell • the farther from the nucleus• the higher the energy of the orbital in the shell.

Locator #1, “n”, the first quantum number

1

23

4

5

6

7

"n" MAIN SHELLS ABOUT THE NUCLEUS

Locator #2, “l”, the second quantum number

• locates electrons in a subshell region within the main shell

• limits number of subshells to a value equal to n (1 in 1st shell, 2 in 2nd shell, 3 in 3rd shell etc)

• only four types of subshells are found to be occupied in unexcited, “ground state” of atom

• These subshell types are known by letter: “s” “p” “d ” “f”

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

1s

2s 2p

3s 3p 3d

4s 4p 4d 4f

5s 5p 5d 5f (5g)

6s 6p 6d (6f 6g 6h)

7s (7p 7d 7f 7g 7h 7i)

Lowest energy, smallest shell

Highest,biggest

And More About “l”:

AS WELL AS LOCATION, this Q# is quite important in describing a relative energy value for each of these regions , and the shape of each orbital in the subshell .

We will see shortly that each subshell type, s, p, d, f contain orbitals of unique shape.

The “l” Q# distinguishes between the subshells in terms of energy, s <p <d <f

Locator #3, “ml”, the third quantum number

“ml”, the third quantum number, specifiesin which orbital within a subshell an electronmay be found.

It turns out that each subshell type contains a uniquenumber of orbitals, all of the same shape and energy.

This third number completes the description of wherea electron is likely to be found around the nucleus:

All electrons can be located in an orbital within a subshell within a main shell. To find that electron one need a locating value for each: the “n” number describes a shell (1,2,3...) the “l” number describes a subshell region (s,p,d,f...) the “ml” number describes an orbital within the region

The Third Q#, ml continued

“ml” values will describe the number of orbitals within a

subshell, and give each orbital its own unique “address”:

s subshell p subshell d subshell f subshell

1 orbital 3 orbitals 5 orbitals 7 orbitals

Now that we have found places to put our electrons,in orbitals within subshells within shells, let’s take a look at the shapes of the various types of orbitals.

The “orbital shapes” are simply enclosed areas ofprobability for an electron after a three dimensionalplot is made of all solutions for that electron from thewave equation.

Each orbital within a subshell is centered about thenucleus and extends out to the boundaries of itsmain shell. Its exact orientation within the subshelldepends on the value of its ml number.

all s orbitals

all p orbitals

d orbitals

“To be continued...”