Ch. 1: Atoms: The Quantum World

CHEM 4A: General Chemistry with Quantitative Analysis

Fall 2009

Instructor: Dr. Orlando E. Raola

Santa Rosa Junior College

Overview

1.1The nuclear atom

1.2 Characteristics of electromagnetic radiation

1.3 Atomic spectra

1.4 Radiation, quanta, photons

1.5 Wave-particle duality

1.6 Uncertainty principle

An electron will be ejected when hν > Φ because Ek,electron will benon-zero

frequency velocity

The energy of a photon is conserved.

Ephoton = Ekinetic,electron + Work Function of metal

hν = 12

mev2 + Φ

WARNING

The following material contains heavy mathematical machinery, including integrals and differential equations. The purpose is to show you how scientist arrived at very important conclusions that will allow you to understand everyday chemistry. You do not have to memorize or even attempt to write down all the numerous mathematical expressions. DO NOT RUN AWAY. THEY ARE PERFECTLY TAME AND BEYOND THIS POINT, EVERYTHING IS DOWNHILL!!!!

1

2m

ev 2 =hν −Φ

y =mx+b

Constructiveinterference(peak + peak)

Destructiveinterference

(peak + trough)

Diffraction Pattern of Electrons

Waves show diffraction…

Small angle x-ray diffraction on colloidal crystal, from http://www.chem.uu.nl/fcc/www/peopleindex/andrei/andrei.htm

Electrons show diffraction…

Electron diffraction taken from a crystalline sample, from http://www.matter.org.uk/diffraction/electron/electron_diffraction.htm

therefore electrons are waves!

λ =

hmv

=hp

ill defined location

well defined momentum

well defined location

ill defined momentum

Heisenberg Uncertainty Principle (1927)

Heinsenberg’s Uncertainty Principle

As a result from the analysis of many experiments and thoughtful theoretical derivations, Heinsenberg (1927) expressed the principle that the momentum and the position of a particle cannot be determined simultaneously with arbitrary precision. In fact the product of the uncertainties in these two variables is always at least as large as Planck constant over 4.

ΔpΔx≥

h2

Heisenberg Uncertainty Principle (1927)

In its mathematical expression:

€

Δp Δx ≥1

2h

Example 1.7

mΔv Δx=h2

Δx=h

2mΔv

=1.054571628 ×10 -34 J ⋅s

2 ⋅1.0 ×10−3 kg⋅2.0 ×10−3 m⋅s−1

=1.054571628 ×10 -34kg⋅m2 ⋅s−2 ⋅s2 ⋅1.0 ×10−3 kg⋅2.0 ×10−3 m⋅s-1

=2.6 ×10−29 m

At a node:

•Ψ2 = 0 (no electron density)

•Ψ passes through 0

electron density

The Born interpretation

Erwin Schrödinger

Features of the equation:

• Solutions exist for only certain cases.

• The left side is often written as HΨ.

• H is known as the “hamiltonian”.

The Schrödinger equation

−h2m

d2ψdx2

+V(x)ψ =Eψ

H ψ =Eψ

The Particle-in-a-box problem

For the conditions in the box V(x) = 0 everywhere, energy is only kinetic, and

−

h2m

d2ψdx2

=Eψ

has solutions ψ(x) =Asinkx+Bcoskx

which gives an expression for E

E =

k2h2

8 2m

The Particle-in-a-box problem

From the boundary conditions

the other boundary condition ψ(L) =0

makes

E =

k2h2

8mL2

ψ(0) =0

we get B = 0

k =

nL

and the expression for E becomes

The Particle-in-a-box problem

To find the constant A, we apply the normalization condition, since the particle has to be somewhere inside the box:

ψ(x)2dx=A2 sin2

0

L

∫0

L

∫nxL

⎛

⎝⎜⎞

⎠⎟dx=1

and then

ψ n =

2L

⎛

⎝⎜⎞

⎠⎟

12

sinnxL

⎛

⎝⎜⎞

⎠⎟n=1,2,3...

A =

2L

⎛

⎝⎜⎞

⎠⎟

12

and the wavefunction for the particle in a box is

,...2,1sin2

)(ø2

1

=⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛= nLx∂n

Lxn

Particle in a Box

values of n

Changing the Box

Lsmall Llarge

As L increases:

• energies of levels decrease

• separations between levels decrease

wavefunction (Ψ)

probability density (Ψ2)

lowest density

highest density

Locating Nodes

Ψ passes through 0 Ψ2 = 0

Number of nodes = n – 1

radius

colatitude

azimuth

Spherical polar coordinates

General formula of wavefunctions for the hydrogen atom

ψ(r,θ,ϕ ) =R(r)Y (θ,ϕ )

For n = 1

ψ(r,θ,ϕ ) =2e

−ra0

a0

32

×1

212

=e

−ra0

a03( )

12

a0 =4ε0h

2

mee2

General formula of wavefunctions for the hydrogen atom

ψ(r,θ,ϕ ) =R(r)Y (θ,ϕ )

For n = 2 and

ψ(r,θ,ϕ ) =1

2 6

1

a0

52

r e−

r2a0 ×

34

⎛

⎝⎜⎞

⎠⎟

12

sinθ cosφ=14

12a0

5

⎛

⎝⎜

⎞

⎠⎟r e

−r

2a0 sinθ cosφ

E

2=−

14

hℜ

Quantum numbers

n: principal quantum number

determines the energy

indicates the size of the orbital

: angular momentum quantum number, relates to the shape of the orbital

m : magnetic quantum number, possible orientations of the angular momentum around an arbitrary axis.

l

l

principal quantum number

orbital angular momentumquantum number

magneticquantum number

Electron probability in the ground-state H atom.

Radial probability distribution

Allowable Combinations of Quantum Numbers

l = 0, 1, …, (n – 1) ml = l, (l – 1), ..., -l

No two electrons in the same atom have the same four quantum numbers.

Higher probability of finding an electron

Lower probability of finding an electron

most probable radii

The most probable radius increases as n increases.

radialnodes

boundary surface

• 90% likelihood of finding electron within

radial nodes

Wavefunction (Ψ) is nonzero at the nucleus (r = 0).

For an s-orbital, there is a nonzero probability density (Ψ2) at the nucleus.

n = 1l = 0

no radial nodes

n = 2l = 0

1 radial node

n = 3l = 0

2 radial nodes

2p-orbital

n = 2l = 1, 0, or -1

no radial nodes

1 nodal plane

Plot of wavefunction is for yellow lobe along blue arrow axis.

The three p-orbitals

nodal planes

The labels “x”, “y”, and “z” do not correspond directly to ml values (-1, 0, 1).

nodal planes

The five d-orbitals

n = 3, 4, …

l = 2, 1, 0, -1, -2

dark orange (+)

light orange (–)

The seven f-orbitals

n = 4, 5, …

l = 3, 2, 1, 0, -1, -2, -3

dark purple (+)

light purple (–)

Allow

ed

su

bsh

ells

Allowed orbitals

2 electrons per orbital

Maximum of 32 electrons for n = 4 shell

Silver atoms(with one unpaired electron)

Atoms with one type of electron spin

Atoms with other type of electron spin

Stern and Gerlach Experiment: Electron Spin

Spin States of an Electron

Spin magnetic quantum number (ms) has two possible values:

Relative Energies of Orbitals in a Multi-electron Atom

After Z = 20, 4s orbitals have higher energies than 3d orbitals.

Z is the atomic number.

Probability maxmima for orbitals within a given shell are close together.

A 3s-electron has a greater probability of being found near the nucleus than 3p- and 3d-electrons due to contribution of peaks located closer to the nucleus.

Paired spins

Parallel spins

Lower energy

Higher energy

Electron Configurations: H and He

1s electron (n, l, ml, ms)• 1, 0, 0, (+½ or –½)

1s electrons (n, l, ml, ms)• 1, 0, 0, +½• 1, 0, 0, –½)

Electron Configurations: Li and Be

1s electrons (n, l, ml, ms)• 1, 0, 0, +½• 1, 0, 0, –½

2s electron*

• 2, 0, 0, +½

* one possible assignment

1s electrons (n, l, ml, ms)• 1, 0, 0, +½• 1, 0, 0, –½

2s electrons• 2, 0, 0, +½• 2, 0, 0, –½

Electron Configurations: B and C

1s electrons (n, l, ml, ms)• 1, 0, 0, +½• 1, 0, 0, –½

2s electrons• 2, 0, 0, +½• 2, 0, 0, –½

2p electron*• 2, 1, +1, +½

* one possible assignment

1s electrons (n, l, ml, ms)• 1, 0, 0, +½• 1, 0, 0, –½

2s electrons• 2, 0, 0, +½• 2, 0, 0, –½

2p electrons*• 2, 1, +1, +½• 2, 1, 0, +½

* one possible assignment

subshell being filled

Filling order for orbitals

maximum number of electrons in subshell

The Hydrogen atom: atomic orbitals

The potential in a hydrogen atom can be expressed as

Schrödinger (1927) found that the exact solutions for his equation give expression for the energy as

V(x) =−

e2

4ε0r

E =−

hℜn2

ℜ =mee

4

8h3ε02

n=1,2,3....

An atomic orbital is specified by three quantum numbers.

n the principal quantum number - a positive integer

ℓ the angular momentum quantum number - an integer from 0 to n-1

mℓ the magnetic moment quantum number - an integer from -ℓ to +ℓ

Quantum Numbers and Atomic Orbitals

1.Principal (n = 1, 2, 3, . . .) - related to size and energy of the orbital.

2.Angular Momentum (ℓ = 0 to n 1) - relates to shape of the orbital.

3.Magnetic (mℓ = ℓ to ℓ) - relates to orientation of the orbital in space relative to other

orbitals.

4.Electron Spin (ms = +1/2, 1/2) - relates to the spin states of the electrons.

Quantum Numbers

Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals

Name, Symbol(Property) Allowed Values Quantum Numbers

Principal, n(size, energy)

Angular

momentum, ℓ(shape)

Magnetic, mℓ

(orientation)

Positive integer(1, 2, 3, ...)

0 to n-1

-ℓ,…,0,…,+ℓ

1

0

0

2

0 1

0

3

0 1 2

0

0-1 +1 -1 0 +1

0 +1 +2-1-2

Sample Problem 7.5

SOLUTION:

PLAN:

Determining Quantum Numbers for an Energy Level

PROBLEM: What values of the angular momentum (ℓ) and magnetic (mℓ)

quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals are allowed for n = 3?

Follow the rules for allowable quantum numbers found in the text.

l values can be integers from 0 to n-1; mℓ can be integers from -ℓ through 0 to + ℓ.

For n = 3, ℓ = 0, 1, 2

For ℓ = 0 mℓ = 0

For ℓ = 1 mℓ = -1, 0, or +1

For ℓ= 2 mℓ = -2, -1, 0, +1, or +2

There are 9 mℓ values and therefore 9 orbitals with n = 3.

Sample Problem 7.6

SOLUTION:

PLAN:

Determining Sublevel Names and Orbital Quantum Numbers

PROBLEM: Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers:

(a) n = 3, ℓ = 2 (b) n = 2 ℓ= 0 (c) n = 5, ℓ = 1 (d) n = 4, ℓ = 3

Combine the n value and ℓ designation to name the sublevel.

Knowing ℓ, we can find mℓ and the number of orbitals.

n ℓ sublevel name possible mℓ values # of orbitals

(a)

(b)

(c)

(d)

3

2

5

4

2

0

1

3

3d

2s

5p

4f

-2, -1, 0, 1, 2

0

-1, 0, 1

-3, -2, -1, 0, 1, 2, 3

5

1

3

7

1s 2s 3s

The 2p orbitals.

Representation of the 1s, 2s and 3s orbitals in the hydrogen atom

Representation of the 2p orbitals of the hydrogen atom

Representation of the 3d orbitals

Representation of the 4f orbitals

Types of Atomic Orbitals

Levels and sublevels

When n = 1, then ℓ = 0 and mℓ = 0Therefore, in n = 1, there is 1 type of subleveland that sublevel has a single orbital

(mℓℓ has a single value 1 orbital)

This sublevel is labeled s (“ess”)

Each level has 1 orbital labeled s, and it is SPHERICAL in shape.

s orbital are spherical

Dot picture of electron cloud in 1s orbital.

Surface density4πr2ψ versus distance

Surface of 90% probability sphere

1s orbital

2s orbitals

3s orbital

p orbitals

When n = 2, then ℓ = 0 and 1Therefore, in n = 2 levell there are

2 types of orbitals — 2 sublevels

For ℓ = 0 mℓ = 0 this is a s sublevel

For ℓ = 1 mℓ = -1, 0, +1

this is a p sublevel with 3 orbitals

planar node

Typical p orbital

When l = 1, there is When l = 1, there is a a PLANAR PLANAR NODENODE through the through the nucleusnucleus

p Orbitals

The three p orbitals lie 90o apart in space

2px Orbital 3px Orbital

d Orbitals

When n = 3, what are the values of ℓ?

ℓ = 0, 1, 2 and so there are 3 sublevels in level n=3.

For ℓ = 0, mℓ = 0 s sublevel with single orbital

For ℓ = 1, mℓ = -1, 0, +1 p sublevel with 3 orbitals

For ℓ = 2, mℓ = -2, -1, 0, +1, +2

d sublevel with 5 orbitals

d Orbitals

s orbitals have no planar node (ℓ = 0) and so are spherical.

p orbitals have ℓ = 1, and have 1 planar node,

and so are “dumbbell” shaped.

This means d orbitals (with ℓ = 2) have 2 planar nodes

typical d orbital

planar node

planar node

3d3dxyxy Orbital Orbital

3d3dxzxz Orbital Orbital

3d3dyzyz Orbital Orbital

3d3dxx22

- y- y22 Orbital Orbital

3d3dzz22 Orbital Orbital

f — Orbitalsf — Orbitals

One of 7 possible f orbitals.

All have 3 planar surfaces.

Can you find the 3 surfaces here?

f — Orbitalsf — Orbitals

Spherical NodesSpherical Nodes

•Orbitals also have spherical Orbitals also have spherical nodesnodes•Number of spherical nodes Number of spherical nodes = n - l - 1 = n - l - 1•For a 2s orbital:For a 2s orbital: No. of nodes = 2 - 0 - 1 = 1 No. of nodes = 2 - 0 - 1 = 1

2 s orbital

Summary of Quantum Numbers of Electrons in Atoms

Name Symbol Permitted Values Property

principal n positive integers(1,2,3,…) orbital energy (size)

angular momentum

ℓ integers from 0 to n-1 orbital shape (The ℓ values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.)

magnetic mℓ integers from -ℓ to 0 to +ℓ orbital orientation

spin ms+1/2 or -1/2 direction of e- spin

The 3d orbitals

One of the seven possible 4f orbitals.

Schematic representation of the energy levels of the hydrogen atom