lecture 6 (ch. 5): quantum numbers and electron configurations

Orbitals (Ch 5)

Lecture 6Suggested HW:

5, 7, 8, 9, 10, 17, 20, 43, 44

Bohr’s Theory Thrown Out

• In chapter 4, we used Bohr’s model of the atom to describe atomic behavior

• Unfortunately, Bohr’s mathematical interpretation fails when an atom has more than 1 electron (but, it is still a convenient model).

• Also, Bohr has no explanation of why electrons simply don’t fall into the positively charged nucleus.

• This failure is due to violation of the Uncertainty principle.

• The uncertainty principle is a cornerstone of quantum theory.

• It asserts that:

“You can NOT measure accurately both the position and momentum of an electron simultaneously, and this uncertainty is a fundamental property of the act of measurement itself”

• In other words, there is a well defined limit to what we can possibly know about particles as small as electrons.

• This limitation is a direct consequence of the wave-nature of electrons

Uncertainty Principle

• Consider an electron

• If you wish to locate the electron. To see the electron, we must use a photon

• When the electron and photon interact, there is a change in velocity of the electron due to collision with the photon

• Thus, the act of measuring the position results in a change in its momentum, and therefore its energy

λλ’

Uncertainty Principle

• Bohr’s model conflicts with the uncertainty principle because if the electron is set within a confined orbit, you know both its momentum and position at a given moment. Therefore, it violates the Uncertainty Principle and can not hold true.

Why Does the Bohr Model Fail?

Contrast Between Bohr’s Theory and Quantum Mechanics

• The primary differences between Bohr’s theory and quantum mechanics are:

– Bohr restricts the motion of electrons to exact, well-defined orbits

– In quantum mechanics, the location of the electron is not known. Instead, we describe the PROBABILTY DENSITY, or the likelihood that an electron will be found in some region of volume around the nucleus.

– Both models agree that electrons within different distances of the nucleus (shells) have different energies.

• This is directly in line with the uncertainty principle.

• We CAN NOT locate an electron accurately

• We CAN calculate a probability of an electron being in a certain region of space in the atom

• From these calculations, we get ORBITALS– An orbital is a theoretical, 3-D “map” of the places

where an electron could be.

Probability Density

• An orbital is defined by 4 quantum numbers

n (principle quantum number)L (azimuthal quantum number)mL (magnetic quantum number)ms (magnetic spin quantum number)

Quantum Numbers

• n = 1, 2, 3…..etc. These numbers correlate to the distance of an electron from the nucleus. In Bohr’s model, these corresponded to the “shell” orbiting the nucleus.

• n determines the energies of the electrons

• n also determines the orbital size. As n increases, the orbital becomes larger and the electron is more likely to be found farther from the nucleus

1. The Principle Quantum Number, n

• L dictates the orbital shape

• L is restricted to values of 0, 1….(n-1)

• Each value of L has a letter designation. This is how we label orbitals.

2. The Angular Momentum Quantum Number, L

Value of L 0 1 2 3Orbital type

s p d f

• Orbitals are labeled by first writing the principal quantum number, n, followed by the letter representation of L

Labeling Orbitals from “L”

• The 3rd quantum number, mL, relates to the spatial orientation of an orbital

• mL can assume all integer values between –L and +L

• Number of possible values of mL gives the number of orbitals of a given type in a specified “shell”

3. The Magnetic Quantum Number, mL

3. The Magnetic Quantum Number, mL

• Imagine standing in the center of an enclosed volume that contains an electron

• Now, imagine taking a picture of the electron every ten seconds for an entire day.

• If you superimposed the photos together, you would have a statistical representation of how likely the electron is to be found at each point.

Envisioning Electron Density Distribution Within An Orbital

• When n=1, the wavefunction that describes this state (Ψ1) only depends on r, the distance from the nucleus.

• Because the probability of finding an electron only depends on r and not the direction, the probability density is spherically symmetric

• For n=1, only s-orbitals are allowed

• Since L=0, mL can only be 0• This single value of mL indicates

that each shell contains a single s-orbital

S-orbitals

s-orbital

1s 2s

x

y

z

x

y

z

x

y

z3s

As n increases, the electron is more likely to be found at distances further from the nucleus, so the size of the orbital increases.

S-orbitals

• P orbitals exist in all shells where n> 2.

• For a p orbital, L =1.

• Therefore, mL = -1, 0, 1. Three values of mL means there are 3 p-orbitals in each “shell” n> 2.

• In p-orbitals, electron density is concentrated in lobes around the nucleus along either the x, y, or z axis (These are labeled as px, py, and pz respectively)

P-orbitals

• D-orbitals exist in all shells where n>3

• L= 2, so mL can be any of the following:

-2,-1,0,1,2

• Thus, there are five d-suborbitals in every shell where n>3

D-orbitals

• Can a 2d orbital exist?

• Can a 1p orbital exist?

• Can a 4s orbital exist?

Examples

Stern- Gerlach experiment

• A beam of Ag atoms was passed through an uneven magnetic field. Some of the atoms were pulled toward the curved pole, others were repelled.

• All of the atoms are the same, and have the same charge. Why does this happen?

4. The Magnetic Spin Number, ms

• Spinning electrons have magnetic fields. The direction of spin changes the direction of the field. If the field of the electron does not align with the magnetic field, it is repelled.

• Thus, because the beam splits two ways, electrons must spin in TWO opposite directions with equal probability. We label these “spin-up” and “spin-down”

Two possible orientations: ms =

4. The Magnetic Spin Number, ms

NO TWO ELECTRONS IN THE SAME ATOM CAN HAVE THE SAME 4 QUANTUM NUMBERS!!!

Quantum numbers: 1, 0 , 0, + ½ 1, 0, 0, – ½ * Allowed

Quantum numbers: 1, 0, 0, + ½ 1, 0, 0 , + ½ * Forbidden !!

1s 1s

Pauli Exclusion Principle

n l

ml ms

2 1

1

0

-1

21

21

21

21

21

21

n l ml ms2 1 1 + ½

2 1 1 - ½

2 1 0 + ½

2 1 0 - ½

2 1 -1 + ½

2 1 -1 - ½

-1 0 1

Spin up: + ½ Spin down: - ½

Representation of the three 2p-orbitals

Example: What Are The Allowed Sets of Quantum Numbers For An Electron In A 2p Orbital? (n, L, mL, ms)

• List all possible sets of quantum numbers in the n=2 shell?

• n = 2

• L = 0, 1

• mL = 0 (L=0) = -1, 0, 1 (L=1)

• ms = +/- ½

n L mL mS

2 0 0 + ½

2 0 0 - ½

2 1 -1 + ½

2 1 -1 - ½

2 1 0 + ½

2 1 0 - ½

2 1 1 + ½

2 1 1 - ½

S

P

Example: List ALL Possible Sets of Quantum Numbers In the n=2

• As previously stated, the energy of an electron depends on n.

• Orbitals having the same n, but different L (like 3s, 3p, 3d) have different energies.

• When we write the electron configuration of an atom, we list the orbitals in order of energy according to the diagram shown on the left (the periodic table is aligned to depict this).REMEMBER: S-orbitals can hold no more than TWO

electrons. P- orbitals can hold no more than SIX, and D-orbitals can hold no more than TEN electrons.

Electron Configurations

• Write the electron configurations of N, Cl, and Ca

𝑁 (7𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠) :1𝑠22𝑠22𝑝3

𝐶𝑙 (17𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠) :1𝑠22𝑠22𝑝63𝑠23𝑝5

𝐶𝑎(20𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠):1 𝑠22𝑠22𝑝63𝑠23𝑝64 𝑠2

Energy

Example

• If we drew the orbital representations of N based on the configuration in the previous slide, we would obtain:

1s

2s

2p

Energy

For any set of orbitals of the same energy, fill the orbitals one electron at a time with parallel spins. (Hund’s Rule)

Example, contd.

• In chapter 4, we learned how to write Lewis dot configurations. Now that we can assign orbitals to electrons, we can write proper valence electron configurations.

Noble Gas Configurations

• Give the noble gas configurations of:• K• K+

• Cl-• Zn• Sr

Group Examples

• When we fill orbitals in order, we obtain the ground state (lowest energy) configuration of an atom.

• What happens to the electron configuration when we excite an electron?

• Absorbing light with enough energy will bump a valence electron into an excited state.

• The electron will move up to the next available orbital. This is the 1st excited state.

Excited States

• Ground state Li: 1s2 2s1

• 1st excited state Li: 1s2 2s0 2p1

• 2nd excited state Li: 1s2 2s0 2p0 3s1

1s

2s

2p

3s

Ground state

1st excited state

2nd excited state Energy

Example

ns1

1

2

3

4

5

6

7

ns2ns2

np1ns2

np2ns2

np3ns2

np4ns2

np5

ns2

np6

ns2 (n-1)dx

• As you know, the d-orbitals hold a max of 10 electrons

• These d-orbitals, when possible, will assume a half-filled, or fully-filled configuration by taking an electron from the ns orbital

• This occurs when a transition metal has 4 or 9 valence d electrons

4s

3d

Unfavorable4s

3d

Favorable

[Ar] 4s1 3d5Example: Cr [Ar] 4s2 3d4

Transition Metals