Electron Arrangement - Part 1

Chapter 8Some images Copyright © The McGraw-Hill Companies, Inc.

Brad Collins

Properties of Waves

Wavelength (λ) is the distance between identical points on successive waves.

Amplitude is the vertical distance from the midline of a wave to the peak or trough.

8.1

Properties of Waves

Frequency (ν) is the number of waves that pass through a particular point in 1 second (Hz = 1 cycle/s).

The speed (u) of the wave = λ x ν

8.1

(λ)

Maxwell (1873), proposed that visible light consists of electromagnetic waves.

Electromagnetic radiation is the emission and transmission of energy in the form of electromagnetic waves.

Speed of light (c) in vacuum = 3.00 x 108 m/s

8.1

ALL electromagnetic radiation: c = λ∙ν

8.1

Types of Electromagnetic (EM) Radiation

A photon has a frequency of 6.0 x 104 Hz. Convert this frequency into wavelength (nm). Does this frequency fall in the visible region?

λ

νc = λ∙ν λ = c ÷ ν λ = 3.00 x 108 m/s ÷ 6.0 x 104 s–1 λ 5.0 x 103 m 1m = 1 x 109 nm λ = 5.0 x 103 m × 109 nm/m λ = 5.0 × 1012 nm

8.1

Particle Behavior of Light• Max Planck (1900) found that energy is related to wavelength. • Shorter wavelengths have higher

frequencies and higher energy • Expressed as E = constant × frequency (ν) • Planck’s constant (h) is the smallest unit of

energy • Energy exists in discrete units with

energies that are integer multiples of hν. • Energy is gained or lost in integer (n)

multiples of h∙ν • ∆E = n × h∙ν

Energy (light) is emitted or absorbed in discrete units (quanta). 8.1

• The photoelectric effect is the ability of light to dislodge electrons from a metal surface.

• If light shines on the surface of a metal, there is a point at which electrons are ejected from the metal.

• The electrons will only be ejected once the threshold frequency is reached.

• Below the threshold frequency, no electrons are ejected.

• Above the threshold frequency, the number of electrons ejected depends on the intensity of the light.

• The photoelectric effect provides evidence for the particle nature of light -- “quantization”.

Photoelectric Effect

8.1

Particle Nature of Light “Photoelectric Effect” Solved by Einstein, 1905

Light has dual nature • Wave nature, c = λ∙ν • Particle nature, E = h∙ν

h

KE

A photon is a “particle” or “quanta” of lightA photon has energy, E = h∙ν

The kinetic energy (KE) of the ejected electron: KE(electron) = hν(photon) – BE(electron)

Electrons are held in the metal with binding energy, BE

To eject an electron, a photon must have more energy than the BE of the electron.

8.1

•Radiation composed of only one wavelength is called monochromatic.

•Radiation that spans a whole array of different wavelengths is called continuous.

•White light can be separated into a continuous spectrum of colors.

•Elements do not emit light in a continuous spectrum

The Bohr ModelEmission Spectra of Elements

8.2

8.2

Continuous Spectrum of Light

White Light

Note that there are no dark regions in the continuous

spectrum that would correspond to different lines.

Line Emission Spectrum of Hydrogen Atoms

Line Spectrum of Elements

8.2

8.2

The Rydberg Equation• 1888 - Johannes Rydberg found hydrogen

emission lines could be described using an equation: !

!

• where RH is a constant = 1.097 x 107 m–1 • and nA and nB are two whole number

integers • Based on measurements - empirical

• Why does this equation work?

1λ ( –=

1nA2

1nA2 )

8.2

Bohr Model of the Atom• Neils Bohr - 1913

• Believed electron energy is quantized • Electrons can only have specific,

quantized energy values, En • *En = – B/n2

• B is constant = 2.18 x 10–18 J • n is one of the quantized energy

levels, n = 1, 2, 3, etc. • Light is emitted when an electron

moves from a higher energy value to a lower one.

*Sign is negative to indicate that electrons in the atom have lower energy than free electrons (infinitely far from the nucleus)

Ground State

Excited State

8.2

E = hν

E = hν

8.2

nf = 1

ni = 2

nf = 1

ni = 3

nf = 2

ni = 3

8.2

Bohr Model of the AtomE(photon) = ∆E = Ef – Ei

Ef = –B/n2f

Ei = –B/n2i

∆E = –B/n2f – (–B/n2i)

∆E = B × (1/n2i – 1/n2f)

E(photon) = h∙ν, ν = c/λ, so: E(photon) = h∙c/λ λ = h∙c/E(photon) λ = (6.63 × 10–34 J∙s × 3.00 × 108 m/s)/1.55× 10–10 J λ = 1.280 × 10–6 m * 109 nm/m λ = 1280 nm

Calculate the wavelength (in nm) of a photon emitted by a hydrogen atom when its electron drops from the n = 5 state to the n = 3 state.

E(photon) = ∆E = B × (1/n2i – 1/n2f) E(photon) = 2.18 × 10–18 J × (1/25 – 1/9) E(photon) = 2.18 × 10–18 J × (–0.0711) E(photon) = –1.55 × 10–19 J

8.2

Wavelength of Photon

Energy of Photon

Why is electron energy quantized?

8.3

The Dual Nature of Electrons

Louis de Broglie, 1924 • Electron is both a particle and a

wave • Electrons behave as standing

waves • Circumference (2∙π∙r) of the

electron’s standing wave is given by: • 2∙π∙r = n∙λ = n∙h/m∙u • where

• u = velocity of electron • m = mass of electron

λ = h/muλ = 6.63 x 10-34 / (2.5 x 10-3 x 15.6)λ = 1.7 x 10-32 m = 1.7 x 10-23 nm

What is the de Broglie wavelength (in nm) associated with a 2.5 g Ping-Pong ball traveling at 15.6 m/s?

m in kgh in J•s u in (m/s)

Practice Problem

8.3

The Schrödinger Wave Equation• Heisenberg’s Uncertainty Principle: on the mass scale of atomic

particles we cannot determine the exact location AND momentum of a particle simultaneously

• If Δx is the uncertainty in position and Δmv is the uncertainty in momentum, then:

∆x × ∆mv = h/4∙π • If the position, x is known precisely (∆x is small), then the

momentum (mv) value will be known less precisely (∆mv will be large).

• Uncertainty + wave nature means we cannot describe an electron’s ‘orbit’ the same way we would a satellite or planet. • Instead we describe the ‘probability of finding an electron’ in a

region around the nucleus. (Schrödinger Wave Equation)8.3

The Quantum Model• Schrödinger’s wave equation (Ψ) describes wave and particle

nature of electrons • Ψ is a function of 4 properties:

• Ψ = fn(n, l, ml, ms)

• where n, l, ml, ms are called ‘quantum numbers’

• The 4 quantum numbers are the ‘address’ of the electron • Schrödinger’s wave equation describes the energy of an

electron, and the probability of finding an electron in a region of space outside the nucleus.

• Schrödinger’s wave equation can only be solved exactly for the hydrogen atom. • For multi-electron atoms the solution must be approximated.

8.4

Quantum Numbers• n is the Principal Quantum Number

• Same as Rydberg’s ’n’ and Bohr’s ’n’ • Defines energy level (shell) of the electron • n = 1, 2, 3, etc.

• n corresponds to the distance from the nucleus

Shells are sometimes given letter-names

8.4

Quantum Numbers• l is the Angular Momentum Quantum Number

• l describes the number and type of sub-levels (subshells) in a given shell and the shape of the volume of space that the electron occupies.

• l = 0, 1, … n-1 !!!!!

• Types of subshells l = 0, 1, are given letter names !!!

• Subshell nomenclature: • Call l = 0 orbital type in the first (n=1) energy shell, 1s

n Allowed l’s Translation1 0 one subshell in first shell2 0, 1 two subshells in second shell3 0, 1, 2 three subshells in third shell4 0, 1, 2, 3 four subshells in fourth shell

l 0 1 2 3 4 5 etcsymbol! s p d f g h …

8.4

Subshell Shapes• When l = 0, shape is spherical

l = 0 (s orbitals)

8.4

Subshell Shapes• When l = 1, shape is ‘dumbbell’

l = 1 (p orbitals)

8.4

Subshell Shapes• When l = 2, shape is (mostly) ‘cloverleaf’

l = 2 (d orbitals)

8.4

Quantum Numbers• ml is the Magnetic Quantum Number

• Describes the number of orbitals in a subshell and their orientation in space

• ml can have values of –l…0…+l

subshell Allowed ml’s Translations (l = 0) 0 one s-orbitalp (l = 1) –1, 0, 1 three p-orbitalsd (l = 2) –2, –1, 0, 1, 2 five d-orbitalsf (l = 3) –3, –2, –1, 0, 1, 2, 3 7 f-orbitals

8.4

ml = -1 ml = 0 ml = 1

ml = -2 ml = -1 ml = 0 ml = 1 ml = 28.4

8.4

Quantum Numbers and Orbitals

Quantum Numbers• ms is the Electron Spin Quantum Number

• Describes the 2 possible ‘spins’ of an electron • Spin gives rise to a magnetic field

• 2 possible fields: • Values of ms = +½, –½

ms = +½ ms = -½ 8.4

Schrödinger Consequences

• Each set of 4 quantum numbers describes the existence and energy of each electron in an atom.

• Pauli Exclusion Principal: No 2 electrons in an atom can have the same 4 quantum numbers. • In a stadium each seat is identified by section, row,

and seat numbers.

S1, R42, S15

Only 1 person per seat Only 1 electron per set of 4 quantum numbers

8.4

Schrödinger Summary• Electrons with the same n, occupy the same shell • Electrons with the same n and l, occupy the same subshell

• Electrons with the same n, l, and ml, occupy the same orbital

Practice ProblemHow many electrons can one orbital hold?!If n, l, and ml are fixed, ms can be +½ or –½ So, electrons in a single orbital can have the quantum numbers: n, l, and ml, +½ or n, l, and ml, –½ Answer: 2 electrons

8.4

How many 2p orbitals are there in an atom?

2p

n=2

l = 1

Practice Problem

‘p’-type orbitals mean l = 1, so ml = –1, 0 or +1 Answer: 3 orbitals

8.4

How many electrons can be placed in the 3d subshell?

3d

n=3

l = 2

Practice Problem

If l = 2, then ml = –2, –1, 0, +1, +2 • 5 orbitals, each with 2 electrons • Total = 10 electrons

8.4

Bohr’s Model Revisited

1s

2s 2p 2p 2p

3s 3p 3p 3p 3d

4s 4p 4p 4p

3d 3d 3d 3d

4d 4d 4d 4d 4d

n = 1

n = 2

n = 3

n = 4

Hydrogen (1 electron)

8.4

Orbitals on the same level are degenerate (all types have the same energy)

• Transitions from 3s or 3p have different ∆Evalues.

• Split energy levels lead to more emission lines.

Bohr’s Model Revisited

1s

2s

2p 2p 2p

3s

3p 3p 3p

3d4s

4p 4p 4p3d 3d 3d 3d

4d 4d 4d 4d 4d

n = 1

n = 2

n = 3

n = 4

Multi-electron

8.4

Orbitals on the same level are split (types have different energy)