Transcript Unit 3: Atomic Theory & Quantum Mechanics Section A.6

Unit 3: Atomic Theory & Quantum
Mechanics
Section A.6 – A.7
In which you will learn about:
•The quantum mechanical model
•Heisenberg uncertainty principle
•Orbitals and their shapes
•Quantum numbers
A.6 The Quantum Mechanical Model of
the Atom
 Scientists in the mid-1920s, by then convinced that the Bohr
atomic model was incorrect, formulated new and innovative
explanations of how electrons are arranged in atoms.
 In 1924, a French graduate student in physics named Louis de
Broglie (1892-1987) proposed an idea that eventually
accounted for the fixed energy levels of Bohr’s model
Electrons as Waves
 De Broglie had been thinking that Bohr’s quantized electron orbits
had characteristics similar to those of waves.
 Imagine the path of an electron around a circle of fixed radius
 Notice that in figure (a) there is an odd number of waves, and in
(b) there is an even number of waves
 (a) works out perfectly and (b) does not
Electrons as Waves Cont’d
 Only multiples of half-wavelengths are possible on a plucked
guitar string because the string is fixed at both ends
 These finite waves for musical instruments and circles led de
Broglie to ask an interesting question
 If light can act as both a wave and particle, can a particle (like an
electron) act like a wave?
De Broglie Equation
 The de Broglie equation predicts that all moving particles have
wave characteristics
 It also explains why it is impossible to notice the wavelength of a fast-
moving car – a car moving at 25 m/s and weighing 910 kg would
have a wavelength of 2.9 x 10-38 m (too small to be seen or detected!)
 For comparison, an electron moving at the same speed has a
wavelength of 2.9 x 10-5 m (which can be easily measured).
 De Broglie knew that if an electron has wavelike motion and is
restricted to circular orbits of fixed radius, only certain
wavelengths, frequencies and energies are possible.
 This is summed up in the equation λ = h/mv, where λ is still
wavelength, h is still Planck’s constant, m is mass and v is velocity
(speed).
 No, we will NOT be doing calculations with this equation.
The Heisenberg Uncertainty Principle
 Werner Heisenberg (1901-1976) showed that it is impossible
to take any measurement of an object without disturbing the
object
 Scientists try to locate electrons by bombarding them with
photons of light, but once the electron is hit, it moves to a new
location with a new speed!
 In other words, the act of observing the electron produces a
significant, unavoidable uncertainty in the position and
motion of the electron
 The Heisenberg uncertainty principle states that it is
fundamentally impossible to know precisely both the velocity
and position of a particle at the same time.
The Real Deal with Heisenberg
 So we can’t know exactly where or how fast an electron is
moving
 Which means that it is impossible to assign fixed paths for
electrons like the circular orbits in Bohr’s model.
 The only quantity that can be known is the probability for an
electron to occupy a certain region around the nucleus.
The Schrödinger Wave Equation
 In 1926, Austrian physicist Erwin Shrödinger (1887-1961) furthered
the wave-particle theory proposed by de Broglie.
 Shrödinger derived an equation that treated the hydrogen atom’s
electron as a wave
 Shrödinger’s new model for the hydrogen atom seems to apply equally
well to atoms of other elements
 This is where Bohr’s model failed
 The atomic model in which electrons are treated as waves is called
the wave mechanical model of the atom, or the quantum
mechanical model of the atom.
 This model limits an electron’s energy to certain values (like Bohr)
 Makes no attempt to describe the electron’s path around the nucleus
Shrödinger is way too complicated
 The Shrödinger wave equation is too complex to be
considered here
 Mrs. Pford didn’t deal with it until junior year of college and
she had to use Calculus III-level math to solve it!
 Solutions to the equation are called wave functions
 Wave functions are related to the probability of finding the
electron within a certain volume of space around the nucleus
Electron’s probable location
 The wave function predicts a three-dimensional region
around the nucleus, called an atomic orbital, which describes
the electron’s probable location.
 An atomic orbital is like a fuzzy cloud in which the density at a
given point is proportional to the probability of finding the
electron at that point
Density Maps
 The density map can be thought of as a time-exposure
photograph of the electron moving around the nucleus
 The electron cloud = all the probably places ONE electron
COULD be
 The electron cloud ≠ all of the electrons in an atom
surrounding the nucleus
 To overcome the inherent uncertainty about the electron’s
location, chemistry arbitrarily draw an orbital’s surface to
contain 90% of the electron’s total probability distribution.
 Simply, there is a 90% chance you will find an electron
somewhere within the electron cloud
A.7 Quantum Numbers
 There are four quantum numbers that are used to describe
the probable position of an electron.
 Each quantum number is usually only referenced by name or
variable, but there are also actual numbers, too
 No two electrons can have the same exact set of four
quantum numbers (more on this next time)
Principal Quantum Number (1st)
 The principal quantum number (n) indicates the relative size
and energy of atomic orbitals
 In other words, n = energy level
 n can have whole-number values ranging from 1-7.
 If quantum numbers were an address, this is like telling you
what state the electron lives in (not very specific if I want to
find it)
Angular Momentum Quantum Number
(2nd)
 The angular momentum quantum number (l) specifies the shape
of the orbital that the electron is in.
 This is sometimes referred to as the sublevel, but I find this term to
be confusing. I try to explain below (somewhat unsuccessfully?)
 Sublevel = shape of orbital or orbital type (can be s, p, d, or f)
 Orbital = specific orientation of sublevel (can be px, py, or pz depending on
the axis the density map is on)
 l can have whole number values ranging from 0 to n-1.
 If l = 0 it’s an s orbital, if l = 1  p, if l = 2  d, if l = 3  f
 See next slide for pictures of orbitals
 In the address analogy, using this number helps specify which city
the electron is in.
Before We Move On…
 Shapes of orbitals include:
Number of Orbitals
 There is only ONE type of s orbital
 There are THREE types of p orbital
 There are FIVE types of d orbital
 There are SEVEN types of f orbital (not shown in previous
slide)
Hydrogen’s First Four Principal Energy
Levels
Principal
Quantum
Number (n)
Sublevels (Types
of Orbitals)
Present
Number of
Orbitals Related
to Sublevel
Total Number of
Orbitals in
Energy Level (n2)
1
s
1
1
2
s
p
1
3
4
s
p
d
1
3
5
s
p
d
f
1
3
5
7
3
4
9
16
Magnetic Quantum Number (3rd)
 The magnetic quantum number (m)specifies which
orientation of the orbitals an electron is in
 For example, if we know the electron is in energy level 2 and it
is in the p-type orbital, we need to know exactly which porbital it is in (there are three possibilities)
 m can have integer values going from –l to +l.
 In the address analogy, this is like giving the street the
electron is on.
Spin Magnetic Quantum Number (4th)
 The spin magnetic quantum number (ms) is an inherent
property of electrons that separate them into individual
positions.
 Up until this point, two electrons can share the first 3 quantum
numbers, but since no two electrons can share ALL four, we use
spin to indentify which is which
 The electrons aren’t actually spinning! We refer to them as spin
up or spin down but these are just arbitrary terms.
 ms can only be +1/2 or -1/2
 In the address analogy, this is like finally giving the house
number where the electron is at.
Using Quantum Numbers
 If you’ve been following along with the rules…
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n = 1, 2, 3, 4, 5, 6, or 7
l = 0 up to n-1 in integers (0 =s, 1 = p, 2 = d, 3 = f)
m = - l up to +l in integers
ms = ±1/2
 Example Problem: Write the quantum numbers associated with
the first electron added to the 4f sublevel.
 ANSWER: n = 4 (given), l = 3 (known because it’s an f orbital),
m = 3 (-l up to +l in integers and in this case l = 3 – I’m choosing
3, it could be -3, -2, -1, 0, 1, 2 or 3), and ms = +1/2 (again, I’m
choosing + because it can only be for ONE electron).
And now that you’re completely
confused…Homework!
 1) Differentiate between the wavelength of visible light and the
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wavelength of a moving soccer ball.
2) List the number and types of orbitals contained in the hydrogen
atom’s first four energy levels.
3) Explain why the location of an electron in an atom is uncertain
using the Heisenberg uncertainty principle and de Broglie’s waveparticle duality. How is the location of electrons in the atom
defined?
4) Compare and contrast Bohr’s model and the quantum
mechanical model of the atom.
5) Write the numbers associated with each of the following:
 A) the fifth energy level
 B) the 6s sublevel
 C) an orbital on the 3d level