Quantum Quantum MechanicsMechanics

Chapter 7 §4-5Chapter 7 §4-5

The de Broglie The de Broglie RelationRelation

19241924

All matter has a wave-like All matter has a wave-like nature…nature…Wave-particle duality… Wave-particle duality…

All matter and energy exhibit wave-like and particle-like properties.

The de Broglie The de Broglie RelationRelation

The de Broglie Equation The de Broglie Equation relates the wavelength of a relates the wavelength of a particle to its momentum.particle to its momentum.

λ =h

ρ=h

mυ

λ Wavelength

h Planck’s constant 6.626x10-34 J•s

υ Velocity, m/s

m Mass, kg

The de Broglie The de Broglie RelationRelation

Compare the wavelengths of (a) an Compare the wavelengths of (a) an electron traveling at a speed of one-electron traveling at a speed of one-hundredth the speed of light with (b) hundredth the speed of light with (b) that of a baseball of mass 0.145 kg that of a baseball of mass 0.145 kg having a speed of 26.8 m/s (60.0 mi/hr). having a speed of 26.8 m/s (60.0 mi/hr).

λ =6.63x10−34 J • s

9.11x10−31kg( ) 3.0x106 ms( )= 2.43x10−10m

(a) the electron What is the mass of an electron?

What is the electron speed if it is one-hundredth the speedof light?

9.11x10−31kg

3.0x106 ms

The de Broglie The de Broglie RelationRelation

Compare the wavelengths of (a) an Compare the wavelengths of (a) an electron traveling at a speed of one-electron traveling at a speed of one-hundredth the speed of light with (b) hundredth the speed of light with (b) that of a baseball of mass 0.145 kg that of a baseball of mass 0.145 kg having a speed of 26.8 m/s (60.0 mi/hr). having a speed of 26.8 m/s (60.0 mi/hr).

(b) the baseball

λ =6.63x10−34 J • s

0.145kg( ) 26.8 ms( )= 1.71x10−34m

The de Broglie The de Broglie RelationRelation

Compare the wavelengths of (a) an Compare the wavelengths of (a) an electron with (b) that of a baseball. electron with (b) that of a baseball. What does that mean?(a)The electron (2.43x10-

10 m)

(a)The baseball (1.71x10-

34 m)

The Schroedinger The Schroedinger EquationEquation

Schroedinger combined Planck’s photons, Schroedinger combined Planck’s photons, Einstein’s wave-particle duality, and de Einstein’s wave-particle duality, and de Broglie’s idea that all energy and matter Broglie’s idea that all energy and matter follow the wave particle duality into one follow the wave particle duality into one equation (the wave function) for the equation (the wave function) for the electron:electron:

ih

δδt

ψ r, t( ) =−h2

2m∇2ψ +V r( )ψ r, t( )

No, you don’t have to memorize it.

This created the basis for Quantum mechanics.

Quantum MechanicsQuantum Mechanics

Quantum Mechanics of an atom are Quantum Mechanics of an atom are divided into four quantum numbers:divided into four quantum numbers:

nn

mm

mmss

l

l

Principle Quantum Number – the number that represents the energy level

Angular Momentum Quantum Number – Azimuthal – the number that represents the subshell

Magnetic Quantum Number – the number that representsthe orbital within the subshell

Spin Quantum Number – the number that representsthe electron’s spin

Quantum MechanicsQuantum Mechanics

Quantum Mechanics of an atom are Quantum Mechanics of an atom are divided into four quantum numbers:divided into four quantum numbers:

nn

mm

mmss

l

l

Electron Spin:ms = –1/2 OR +1/2

Energy level: n = 1 -∞

Subshell: Based on which energy level the electron is in; = 0 - (n-1) lOrbital: Based on which subshell the electron is in;m = – - + l l l

Quantum MechanicsQuantum Mechanics

Energy level

n

Subshell Orbitalm

Electron Spin

ms

1 0 (s) 0 –1/2 OR +1/2

2 0 (s) 0 –1/2 OR +1/2

2 1 (p) –1, 0, +1 –1/2 OR +1/2

3 0 (s) 0 –1/2 OR +1/2

3 1 (p) –1, 0, +1 –1/2 OR +1/2

3 2 (d) –2, –1, 0, +1, +2 –1/2 OR +1/2

4 0 (s) 0 –1/2 OR +1/2

4 1 (p) –1, 0, +1 –1/2 OR +1/2

4 2 (d) –2, –1, 0, +1, +2 –1/2 OR +1/2

4 3 (f) –3, –2, –1, 0, +1, +2, +3

–1/2 OR +1/2

l l

Let’s Practice Let’s Practice

Determine the quantum numbers for…Determine the quantum numbers for…NaNa

First, write out the electron configuration.1s2 2s2 2p6

Next, write out the four quantum numbers for the lastelectron in the electron configuration:n =

=

m =

ms =

l

l

3s1

3 Since the energy level is 3

0 Since the subshell is s, which is indicated by the number 0

0 Since the orbital is in the s subshell, so the only possible value is 0.

±1/2 Spin must follow Pauli’s exclusion principle

Let’s Practice Let’s Practice

Determine the quantum numbers for…Determine the quantum numbers for…WW

First, write out the electron configuration.1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14

Next, write out the four quantum numbers for the lastelectron in the electron configuration:n =

=

m =

ms =

l

l

5d4

5 Since the energy level is 5

2 Since the subshell is d, which is indicated by the number 2

+1 Since the orbital is in the 4th orbital in the d subshell: –2. –1, 0 +1, +2.

±1/2 Spin must follow Pauli’s exclusion principle

Let’s Practice Let’s Practice

Determine the quantum numbers for…Determine the quantum numbers for…BrBr

First, write out the electron configuration.1s2 2s2 2p6 3s2 3p6 4s2 3d10 Next, write out the four quantum numbers

for the lastelectron in the electron configuration:n =

=

m =

ms =

l

l

4p5

4 Since the energy level is 4

1 Since the subshell is p, which is indicated by the number 1

0 Since the orbital is in the 2nd orbital in the p subshell: –1, 0 +1.

±1/2 Spin must follow Pauli’s exclusion principle