Atomic Structure and Modern Quantum Theory. Atomic timeline: Balmer’s formula for the hydrogen line spectrum in 1885 Photoelectric effect discovered in.

Atomic Structure and Modern Quantum Theory

Atomic timeline:• Balmer’s formula for the hydrogen line

spectrum in 1885• Photoelectric effect discovered in 1887• X-rays discovered in 1895• Radioactivity discovered in 1896• Electron discovered in 1897• Planck’s quantum hypothesis applied to

blackbody radiation in 1900 (beginning of quantum theory)

• Einstein explains photoelectric effect using quantum hypothesis in 1905

• Plum-pudding model of atom in 1907• Rutherford’s α-scattering experiment in 1909• Rutherford’s model of atom in 1911• Bohr’s model of atom in 1913• deBroglie’s hypothesis on the wave nature of

matter in 1924• Heisenberg’s Uncertainty Principle in 1925• Schrodinger’s Wave Equation in 1926

(Quantum Mechanics is fully accepted and successfully explains atomic phenomena)

Primer for the Electromagnetic Spectrum

– As atoms are too tiny to see directly, most of our ideas and theories of the atom are based on experiments where we analyze light which is absorbed or emitted by atoms.

– So we need to go over the basics of the electromagnetic spectrum, the light spectrum, or electromagnetic radiation.

Light Waves

• Until recently, light has been considered to be a wave, so what are the characteristics of a wave?

Light Waves• Speed or velocity of light, c: 3.00x108

m/sec• Wavelength, λ: length of 1 wave in m, nm• Frequency, ν: how fast the wave oscillates

up and down in s-1 or Hz. ν = c/λ• Amplitude: height of wave (relates to

intensity of light)• Energy, E: E = hν, so relates to ν and to λ

Electromagnetic Spectrum- The spectrum ranges from the longest wavelength

(lowest frequency and lowest energy) for the radio waves to the shortest wavelength (highest frequency and highest energy) for the gamma rays.

– The visible light spectrum, the portion we see, is a tiny portion from about 380 nm (violet) to 780 nm (red). To remember the color sequence, know ROY G BIV (red, orange, yellow, green, blue, indigo, violet, where red is the lowest energy and violet is the highest energy).

The Beginning of Quantum Mechanics

• Classical mechanics and electromagnetic theory stipulated that light was a wave.

• Several experiments had puzzling results, contradicting classical theory.

• The photoelectric effect, first observed in 1887, contradicted classical mechanics.

• The number of electrons emitted was proportional to intensity, but their energies were not proportional to intensity.

• Einstein explained this in 1905.

Photoelectric Effect

Einstein proposed the following:• Light radiation is composed of quantum bits of

energy called photons, therefore light has particle characteristics as well as wave characteristics.

• So an electron in the metal surface can only escape the surface if it absorbs a photon of sufficient energy.

• There is a one-to-one ratio between an electron and a photon: one photon can transfer energy to one electron.

• If a photon has insufficient energy, the electron cannot escape the metal.

• If it has just the right energy (ω, the work or potential energy of the electron), then the electron may escape.

• If the photon has excess energy, the excess energy goes to the kinetic energy of the free electron.

• In 1911, Rutherford proposed his model of the atom where electrons orbit the nucleus.

• But by classical electromagnetic theory, the atom would be unstable, with the electrons spiraling into the nucleus.

• Also, electrons would emit a continuous light spectra as their orbit decayed.

• However, we see line spectra for elements, so electrons in an element emit only light of certain frequencies.

Atomic Theory

• Bohr said that the energy of an electron may have certain values, or it may only have certain orbits with specified radii.

• These are called “allowed energy states” and “allowed orbits”.

• He solved for the energy of an electron:

E = -RH/n2, where RH is Rydberg’s constant and n is an integer

Bohr’s Model of the Hydrogen Atom (or the Quick Fix)

• The lowest energy orbit (n=1 or E = -RH) is called the ground state and is closest to the nucleus.

• In order for an electron to change orbits, it must absorb or release energy of a certain frequency.

• If an electron changes orbits, then this is an electronic transition.

Bohr’s Model of the Hydrogen Atom

• If an electron jumps from the ground state to a higher energy state or orbit, then it is has jumped to an “excited state” and it had to have absorbed a certain amount of energy to make this transition.

• In this case, a higher energy state or excited state means that the energy is less negative, so ΔE = Ef - Ei is positive.


• If an electron in an excited state falls back to a lower orbit or energy state, then it must release energy of a certain frequency.

• This is called “relaxation”.• In this case, ΔE is negative. The energy

released is photons of light of a certain frequency, and this is seen as a line in the line spectrum.


• When n = ∞, the electron has been removed from the atom altogether (ejected from atom like the photoelectric effect).

• Note that the energy of an electron that has just been removed is 0: -RH/∞2 = 0.


• The photon light energy absorbed or released when an electron changes orbits is calculated using Planck’s equation

• E = hν• ΔE = hνphoton = RH(1/ni

2 - 1/nf2)

• where h is Planck’s constant 6.626x10-34 J•s (6.626x10-27 erg•s) and n is an integer


• When a photon is absorbed, ΔE is positive meaning that the electron gained energy to jump to an excited state.

• If a photon is released, ΔE is negative, meaning the electron emitted light energy to fall back to a lower energy state.

• ΔE = hνphoton = RH(1/ni2 - 1/nf

2)


• You can also solve for the frequency and wavelength of the light absorbed or released.

• You will get a negative number for the frequency or wavelength if light is emitted.

• The negative sign just means that a photon of light with a wavelength ν has been emitted.

• ν = ΔE/h = (RH/h)(1/ni2 - 1/nf

2)• And λ = c/ ν


• The problem with this model?• Mathematically, it only works for

hydrogen as it is a 1-electron atom!• Bohr also knew something was wrong

with his fixed “orbits”.


• In 1924, de Broglie hypothesized the wave nature of electrons and other particles by saying that a particle’s wavelength is related to its mass and velocity by Planck’s constant.

• λ = h/mv

Modern Quantum Theory

• This gave electrons wave properties just as light has particle properties

• Diffraction studies of electrons showed diffraction patterns, as would be expected from waves. The theory was correct!


Diffraction Pattern of Waves

Diffraction Pattern of Electrons

• In 1925, Heisenberg stated his famous Uncertainty Principle.

• The position and momentum (mv = p) of an electron cannot both be known simultaneously with certainty.

• To find the position, we change p; and to find the momentum, we can’t know the position!


• This also means that electrons do not reside in orbits which were clearly defined.

• Electrons reside in areas called orbitals.• We speak of the probabilities of finding

an electron at any place around the nucleus.


• In 1926, Schrodinger’s Wave Equation successfully explained atomic phenomena.

• This theory and equation treated electrons both as particles and as waves.

• Even for Hydrogen, the math is complex and requires at least calculus.


• Instead of electron orbits, the wave equations yields wave functions, ϕ and its square ϕ2

• These give information about allowed electron orbitals and energies.


• Like the Bohr model, the electron’s energy and orbitals are quantized and have certain allowed states.

• Unlike the Bohr model, the orbitals in which the electrons reside are not completely known; instead the square of the wave functions give the probability of finding an electron in a location.


• ϕ2 is therefore also called the probability density.

• These electron probability densities can be drawn or represented several ways– electron density with dots showing the region of

most probability of finding the electron; – 2-D or 3-D contour drawings of electron orbitals

showing the 90% probability; – electron probability/radial graphs.


Quantum Numbers&

Atomic Orbitals

Orbitals & Electrons• Electrons surround the nucleus• Electrons are found in orbitals,

which are regions in space where electrons are most probably found

• Orbitals may be determined mathematically using the Schrodinger Wave Equation

• e- in orbitals are completely described by 4 numbers, called quantum numbers

Quantum Numbers

• n, the principal quantum number• l, the angular momentum quantum

number (also called azimuthal quantum number or the orbital shape quantum number)

• ml, the magnetic quantum number

• ms, the spin quantum number

The Principal Quantum Number

• n : values range from 1, 2, 3, ...• n determines which shell, energy

level, or period an electron is in• Ex: If n =1, then is an electron in the

first energy level (1st period)• n determines the size of the orbital:

the bigger n is, the larger the orbital• n determines the likely distance of an

electron from the nucleus

n and size of orbital• The larger n is, the

larger the orbital and the further an electron is likely to be found from the nucleus

1s

2s

3s

The Principal Quantum Number

• n also determines how many energy sublevels there are for an energy level

• The number of sublevels equals n• If n=1, there is 1 sublevel• If n=2, there are 2 sublevels• If n=3, there are 3 sublevels• If n=4, there are 4 sublevels• If n=5, there are 5 sublevels• If n=6, there are 6 sublevels

Azimuthal Quantum Number• l : values range from 0 to n-1• So if n = 1, then l = 0• So if n = 2, then l = 0 or 1• So if n = 3, then l = 0, 1 or 2• l determines the shape or type of the

orbitalValue of l 0 1 2 3 4 5

Type of

orbital

s p d f g h

Azimuthal Quantum Number• l determines the orbital type• Each orbital type corresponds to an

energy sublevel• Sublevels are also called subshells• NOTE: So n also tells you how many

orbital types are ALLOWED on a given energy level!

Azimuthal Quantum Number

• If l=0, then the orbital is a s orbital

• s orbitals have a spherical shape

• Note that for every n value or energy level, there is a s orbital

• The notation is 1s, 2s, 3s, etc.

1s Orbital (s orbitalon 1st energy level)


• If l=1, then the orbital is a p orbital

• p orbitals have a dumbell shape

• Note that for n=1, there are no p orbitals. Why?

• The notation is 2p, 3p, etc.

2p Orbital


• If l=2, then the orbital is a d orbital

• Most d orbitals have a cloverleaf shape

• Note that for n=1 or 2, there are no d orbitals. Why?

• The notation is 3d, 4d, etc.

3d Orbital


• If l=3, then the orbital is a f orbital

• f orbitals have complex shapes

• Note that for n=1, 2 or 3, there are no f orbitals. Why?

• The notation is 4f, 5f, etc.

4f Orbital

Magnetic Quantum Number• ml : values range from - l to l, increments

of 1• So if l = 0, then ml = 0• So if l = 1, then ml = -1, 0, 1• So if l = 2, then ml = -2, -1, 0, 1, 2• So if l = 3, then ml = -3, -2, -1, 0, 1, 2, 3• ml determines how many of that type of

orbital there are for each energy level• ml determines the spatial orientation of

an orbital

ml and number of orbitals• For l = 0, then ml = 0; so is only 1 s orbital

for each n value• For l = 1, then ml = -1, 0, 1; so are 3 p

orbitals for each n value (except n =1)• For l = 2, then ml = -2, -1, 0, 1, 2; so are 5

d orbitals for each n value (except n =1, 2)

• For l = 3, then ml = -3, -2, -1, 0, 1, 2, 3; so are 7 f orbitals for each n value (except n =1, 2, 3)

ml and spatial orientation

• ml tells how the orbital is aligned on the x, y, and z axis

• There are 3 p orbitals, one on the x axis, one on the y axis, and one on the z axis

ml and spatial orientation• There are 5 d orbitals, which are aligned

differently along the x, y, and z axis

Spin Quantum Number• ms : values are 1/2 or -1/2

• ms gives the spin of an electron in an orbital

• if the spin is 1/2 then this is called an up spin• if the spin is -1/2 then this is called a down spin• it is like 2 tops: one spinning clockwise and the

other spinning counterclockwise• 2 electrons in the same orbital MUST have

opposite spins and opposite ms values!

Quantum Number Relationships

Placing Electrons in Orbitals• Every orbital may hold up to 2 electrons• These 2 electrons have opposite spins:

an up spin and a down spin• You can determine how many electrons

are possible for an energy level by 2n2

• If n=1, only 2 electrons are possible• If n=2, 8 electrons are possible• If n=3, 18 electrons are possible

Placing Electrons in Orbitals• There are rules for placing electrons in

orbitals.– Hund’s Rule– Aufbau Principle– Pauli’s Exclusion Principle

• There is also an energy order which we follow in placing electrons in an orbital energy diagram.

Atomic Structure and Modern Quantum Theory. Atomic timeline: Balmer’s formula for the hydrogen line spectrum in 1885 Photoelectric effect discovered in.

Documents

intensity of light energy

highest energy

lowest energy

excess energy

photoelectric effect

potential energy

kinetic energy

right energy