ch. 1: atoms: the quantum worldsrjcstaff.santarosa.edu/~oraola/chem1alect/lect...problem: give the...
TRANSCRIPT
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Ch. 1: Atoms: The Quantum World
CHEM 4A: General Chemistry with Quantitative Analysis
Fall 2009 Instructor: Dr. Orlando E. Raola
Santa Rosa Junior College
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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
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An electron will be ejected when hν > Φ because Ek,electron will be non-zero
frequency velocity
The energy of a photon is conserved.
Ephoton = Ekinetic, electron + Work Function of metal
hν = 12mev
2 + Φ
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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!!!!
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12
mev2 = hν − Φ
y = mx + b
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Constructive interference (peak + peak)
Destructive interference
(peak + trough)
Diffraction Pattern of Electrons
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Waves show diffraction…
Small angle x-ray diffraction on colloidal crystal, from http://www.chem.uu.nl/fcc/www/peopleindex/andrei/andrei.htm
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Electrons show diffraction…
Electron diffraction taken from a crystalline sample, from http://www.matter.org.uk/diffraction/electron/electron_diffraction.htm
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therefore electrons are waves!
λ =
hmv
=hp
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ill defined location
well defined momentum
well defined location
ill defined momentum
Heisenberg Uncertainty Principle (1927)
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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.
Δp Δx ≥
2π
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Heisenberg Uncertainty Principle (1927)
In its mathematical expression:
�
Δp Δx ≥ 12
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Example 1.7
mΔv Δx =
2
Δx =
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
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At a node:
• Ψ2 = 0 (no electron density)
• Ψ passes through 0
electron density
The Born interpretation
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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”.
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The Schrödinger equation
−
2md 2ψdx2 +V(x)ψ = Eψ
Hψ = Eψ
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The Particle-in-a-box problem
For the conditions in the box V(x) = 0 everywhere, energy is only kinetic, and
−
2md 2ψdx2 = Eψ
has solutions ψ (x) = Asinkx + B coskx
which gives an expression for E
E =
k 2h2
8π 2m
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The Particle-in-a-box problem
From the boundary conditions
the other boundary condition ψ (L) = 0
makes
E =
k 2h2
8mL2
ψ (0) = 0
we get B = 0
k =
nπL
and the expression for E becomes
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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
∫nπ x
L⎛⎝⎜
⎞⎠⎟
dx = 1
and then
ψ n =
2L
⎛⎝⎜
⎞⎠⎟
12
sin nπ xL
⎛⎝⎜
⎞⎠⎟
n = 1,2,3...
A =
2L
⎛⎝⎜
⎞⎠⎟
12
and the wavefunction for the particle in a box is
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,...2,1sin2)(ø21
=⎟⎠⎞⎜
⎝⎛⎟
⎠⎞⎜
⎝⎛= n
Lx∂n
Lxn
Particle in a Box
values of n
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Changing the Box
Lsmall Llarge
As L increases:
• energies of levels decrease
• separations between levels decrease
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wavefunction (Ψ)
probability density (Ψ2)
lowest density
highest density
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Locating Nodes
Ψ passes through 0 Ψ2 = 0
Number of nodes = n – 1
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radius
colatitude
azimuth
Spherical polar coordinates
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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
2π12
=e
−r
a0
πa03( )
12
a0 =4πε0
2
mee2
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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) = 12 6
1
a0
52
r e−
r2a0 ×
34π
⎛⎝⎜
⎞⎠⎟
12
sinθ cosφ =14
12πa0
5
⎛
⎝⎜
⎞
⎠⎟ r e
−r
2a0 sinθ cosφ
E2 = −
14
hℜ
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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.
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principal quantum number
orbital angular momentum quantum number
magnetic quantum number
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Electron probability in the ground-state H atom.
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Radial probability distribution
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Allowable Combinations of Quantum Numbers
l = 0, 1, …, (n – 1) ml = l, (l – 1), ..., -l
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No two electrons in the same atom have the same four quantum numbers.
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Higher probability of finding an electron
Lower probability of finding an electron
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most probable radii
The most probable radius increases as n increases.
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radial nodes
boundary surface
• 90% likelihood of finding electron within
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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.
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n = 1 l = 0
no radial nodes
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n = 2 l = 0
1 radial node
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n = 3 l = 0
2 radial nodes
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2p-orbital
n = 2 l = 1, 0, or -1
no radial nodes
1 nodal plane
Plot of wavefunction is for yellow lobe along blue arrow axis.
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The three p-orbitals
nodal planes
The labels “x”, “y”, and “z” do not correspond directly to ml values (-1, 0, 1).
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nodal planes
The five d-orbitals
n = 3, 4, …
l = 2, 1, 0, -1, -2
dark orange (+)
light orange (–)
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The seven f-orbitals
n = 4, 5, …
l = 3, 2, 1, 0, -1, -2, -3
dark purple (+)
light purple (–)
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Allo
wed
su
bsh
ells
Allowed orbitals
2 electrons per orbital
Maximum of 32 electrons for n = 4 shell
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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
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Spin States of an Electron
Spin magnetic quantum number (ms) has two possible values:
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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.
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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.
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Paired spins
Parallel spins
Lower energy
Higher energy
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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)
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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
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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
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subshell being filled
Filling order for orbitals
maximum number of electrons in subshell
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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
4πε0r
E = −
hℜn2 ℜ =
mee4
8h3ε02 n = 1,2,3....
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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
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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
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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
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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.
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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
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1s 2s 3s
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The 2p orbitals.
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Representation of the 1s, 2s and 3s orbitals in the hydrogen atom
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Representation of the 2p orbitals of the hydrogen atom
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Representation of the 3d orbitals
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Representation of the 4f orbitals
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Pauli Exclusion Principle
In a given atom, no two electrons can have the same set of four quantum numbers (n, ℓ, mℓ, ms). Therefore, an orbital can hold only two electrons, and they must have opposite spins.
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Types of Atomic Orbitals
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Levels and sublevels
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s orbital are spherical
Dot picture of electron cloud in 1s orbital.
Surface density 4πr2y versus distance
Surface of 90% probability sphere
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1s orbital
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2s orbitals
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3s orbital
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p orbitals
When n = 2, then ℓ = 0 and 1 Therefore, 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
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p Orbitals
The three p orbitals lie 90o apart in space
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2px Orbital 3px Orbital
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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
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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
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One of 7 possible f orbitals.
All have 3 planar surfaces.
Can you find the 3 surfaces here?
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2 s orbital
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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
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Experimental observation of the spin of the electron (Stern and Gerlach, 1920)
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A comparison of the radial probability distributions of the 2s and 2p orbitals
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The radial probability distribution for an electron in a 3s orbital. The radial probability distribution for the 3s, 3p, and 3d orbitals.
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The 3d orbitals
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One of the seven possible 4f orbitals.
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Schematic representation of the energy levels of the hydrogen atom