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Wave Nature of Matter

Light has wave-like and particle-like properties

Can matter have wave and particle properties?

de Broglie’s hypothesis: matter has wave-like properties in addition to the expected particle like propertiesaddition to the expected particle-like properties

Confirmed by electron diffraction experiments

de Broglie proposed that electrons moving around the nucleus have wave-like properties

Wave Nature of MatterSince electrons have wave-like properties, each electron has an associated wavelength

where λ = wavelength

h = Planck’s constantmvhλ =

m = mass

v = velocity

(mv = momentum)

What is the wavelength associated with an electron of mass m = 9.109 x 10-28 g that travels at 40.0 % of the speed of light?

Wave Nature of MatterIf all matter has wave-like properties (and an associated wavelength), why don’t we notice it?

If you run 15 km/hr, what’s your wavelength?

The wavelength is inversely proportional to mass.

The wavelength of everyday objects is extremely small because its mass is large.

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Find out de Broglie wavelength (in m) of an electron of m=9 ×10-28 g; v= 1 ms-1

Α. λ = 7 x 10-4

B. λ = 7 x 1014

C. λ = 7 x 10-29

D λ 7 105

h= 6.63×10-34 J = 6.63×10-34 kg.m2s-1

D. λ = 7 x 105

Find out de Broglie wavelength (in m) of an electron of m=9 ×10-28 g; v= 5.9 ×106 ms-1

Α. λ = 1 x 10-4

B. λ = 1 x 1014

C. λ = 1 x 1010

D. λ = 1 x 10-10


Find out de Broglie wavelength (in m) of a baseball of m= 142 g; v= 25.0 ms-1

Α. λ = 2 x 10-4

B. λ = 2 x 1014

C. λ = 2 x 10-34

D λ 2 10 59

h= 6.63×10-34 J = 6.63×10-34 kg.m2s-1

D. λ = 2 x 10-59

Find out de Broglie wavelength (in m) of the Earth of m = 6 ×1027 g; v = 3.0 ×104 ms-1

Α. λ = 4 x 10-4

B. λ = 4 x 1014

C. λ = 4 x 1010

D. λ = 4 x 10-63

The Quantum Mechanical Model

Waves don’t have a discrete position!Spread out through spaceCannot pinpoint one specific location

Since electrons have wave-like properties, we cannot know their exact position, and velocity at any given time.

Bohr’s model ignores the wave properties of electrons

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Macroscopic world: position and velocity (momentum) of a particle can be determined to infinite precision.

Quantum mechanical world : an uncertainty associated with each Q ymeasurement.

Bohr’s Model

Bohr’s model suggests that the electron has the lowest energy at

Α. Furthest point from the nucleusB. Can not predictC. Close to the nucleus D T b t d b th f l E hD. To be computed by the formula E=hν

Bohr’s model suggests emission of light occurs becauseΑ. An electron jumps to a higher energy level B. An electron jumps to a lower energy levelC. An electron jumps from the ground stateD. Electrons collide with each other

An incorrect statement about ‘n’ in Bohr’s model is

Α. It is called principle quantum numberB. It does not relate to the energy of an electronC. Energy increases as n increases D St bilit i d

Bohr’s Model

D. Stability increases as n decreases

Bohr’s model suggests absorption of light occurs because

Α. An electron jumps to a higher energy level B. An electron jumps to a lower energy levelC. An electron jumps from the ground stateD. Electrons collide with each other

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The Quantum Mechanical ModelHeisenberg Uncertainty Principle: The exact position (location) and exact momentum (mass × velocity) in space of an object cannot be known simultaneously.

If you know the momentum of an electron, you can’t know its exact l tilocation.

Electrons don’t move in well-defined circular orbits around the nucleus.

In 1926 Schrödinger developed an equation that incorporates both the particle-like and wave-like behavior of electrons.

Heisenberg’s Uncertainty Principle

πhumx4

≥Δ⋅Δ

Uncertainty in position

Uncertainty in momentum

Heisenberg’s Uncertainty Principle

πhumx4

≥Δ⋅Δ

An electron moving near an atomic nucleus has a speed 6 × 106 ms-1 ± 1%. What is the uncertainty in position ( )? xΔ

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Schrödinger wave equationdescribes the total energy of an electron in an atom based on its location and the electrostatic attraction/repulsion

Solving the Schrödinger wave equation leads to a series of mathematical functions called wave functions (ψ)

Although ψ does not have any physical meaning, the square of the wave


g ψ y p y g, qfunction (ψ 2) describes the probability of finding the electron at a given location

These wave functions are called orbitals and have a characteristic energy and shape.

The Bohr model used a single quantum number (n) to describe an orbit, the Schrödinger model uses three quantum numbers: n, l, and ml to describe an orbital.

Orbitals and Quantum Numbers

An orbital:describes a specific distribution of electron density in spacehas a characteristic energyhas a characteristic shape

is described by three quantum numbers: n, l, ml

can hold a maximum of 2 electronsA fourth quantum number (ms) is needed to describe each electron in an orbital


Principal quantum number (n):integral values

n = 1, 2, 3, 4, ……

describes the energy of the electrongyas n increases, the energy of the electron increasesas n increases, the average distance from the nucleus increasesas n increases, the electron is more loosely bound to the nucleus

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Angular momentum quantum number (l):also known as the Azimuthal quantum numberintegral values

l = 0, 1, 2, 3,….,(n-1)

lExample: If n = 3, then l = 0, 1, or 2.

If n = 4, then l = 0, 1, 2, or 3.

– defines the shape of the orbital

Orbitals and Quantum NumbersAngular momentum quantum number (l)

The value for l from a particular orbital is usually designated by the letters s, p, d, f, and g:

Value of l 0 1 2 3 4Letter used s p d f g

An orbital with quantum numbers of n = 3 and l = 2 would be a 3d orbital

An orbital with quantum numbers of n = 4 and l = 1 would be a 4p orbital


Magnetic quantum number (ml ):

describes the orientation in space of the orbital

integral values between l and -lIf l = 1 then ml = 1 0 -1If l 1, then ml 1, 0, -1

If l = 2, then ml = 2, 1, 0, -1,-2

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When n = 3, the values of l can be

Α. +3, +2, +1, 0 B. +3, +2, +1, 0, -1, -2, -3C. +2, +1, 0D +2 +1 0 -1 -2


D. +2, +1, 0, 1, 2

When n = 4, the values of l can be

Α. +3, +2, +1, 0 B. +3, +2, +1, 0, -1, -2, -3C. +2, +1, 0D. +2, +1, 0, -1, -2

When l = 2, the values of ml can beΑ. +3, +2, +1, 0 B. +3, +2, +1, 0, -1, -2, -3C. +2, +1, 0D. +2, +1, 0, -1, -2


When l = 3, the values of ml can beΑ. +3, +2, +1, 0 B. +3, +2, +1, 0, -1, -2, -3C. +2, +1, 0D. +2, +1, 0, -1, -2

What type of orbital is designated by n = 3, l = 2 ?Α. 3s B. 2pC. 3dD. 4s


What type of orbital is designated by n = 4, l = 0? Α. 3s B. 2pC. 3dD. 4s

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What type of orbital is designated by n = 2, l = 1 ?Α. 3s B. 2pC. 3dD. 4s


What type of orbital is designated by n = 3, l = 0? Α. 3s B. 2pC. 3dD. 4s

Summary

Quantum Values Property Description

number

n 1,2,3…. size shell

l 0..…. n-1 shape subshell

ml -l…..0…..+l orientation orbital

The number of subshells in a shell = n

The number of orbitals in a subshell = 2l+1

The number of orbitals in a shell = n2

The shapes of Orbitalss-orbital- what’s value of l ?

Spherical in shapethe size of the s-orbital increases with increasing n

1s

2s

3s

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Orbital Shapes

Any nodal planes?Nodal planes are defined as those planar areas where the electron density is low

p-orbital- what’s value of l ? What are the values of ml ?

Three p orbitals dumbbell shaped

same size and energy within same shell (same n value)different spatial orientation

Orbital Shapes

ml = -1 ml = 0 ml = 1

Orbital ShapesWhere are the nodal planes?

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Orbital Shapes

d-orbitals-what’s value of l ? What are the values of ml ?five d orbitals are present in each shell where n > 3same energy within same shelldifferent shapes different orientation in space

Orbital Shapes

Where are the nodal planes?

Third Shell Orbitals

3dEn

3s

3pShell n = 3

ergy

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What are the quantum numbers associated with the following subshells:

1s

2p

Subshells n values l values ml values

2p

3p

3d

5f

Summary

Compare and contrast the Bohr and quantum mechanical models

Summary

An electron has a 100% probability of being somewhere

ORBITAL: The region in space where an electron is likely to be found

The usual pictures of orbitals show the region where the electron will be found 90% of the time

http://www.falstad.com/qmatom/

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Summarynodes: regions in space where the electron can’t be foundradial probabilities: the probability of finding an electron a certain distance from the nucleus

Different orbitals have different shapes and sizes different energies

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