Lecture 7: Quantum Numbers and Electron Configurations

Dr Harris9/10/12HW: Ch 5: 2, 5, 7, 17, 19, 24, 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 (nonetheless, it still serves as a useful visual representation of an atom)

• 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.

Bohr’s model2πr = nλ n= 1, 2, 3…𝐸𝑛=−2.1799 𝑥10− 18 𝐽

𝑛2

Heisenburg’s 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”

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

Heisenburg’s Uncertainty Principle

• Consider an electron

• If you wish to locate the electron within a distance Δx, we must use a photon with a wavelength which is equal to or less than Δx

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

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

λλ’

Heisenburg’s Uncertainty Principle

• If the minimum uncertainty (precision) in x is defined as Δx, and that of p is Δp:

(Δx)(Δp)

• This indicates the limit of the accuracy of trying to simultaneously determine both an electron’s position and momentum

• This uncertainty is very significant when applied to atomic and subatomic particles

Example• An electron is moving at a speed of 5.0 x 106 m/s. You want to

measure the position of an electron within 5 x 10-11 m. Estimate the uncertainty in the momentum, then the velocity.

(∆𝑥 )(∆𝑝 )≥ h4 𝜋

(∆ 𝑝)≥(6.626 𝑥10− 34 𝐽𝑠)4𝜋 (5𝑥 10−11𝑚)

(∆ 𝑝 )≥1𝑥10−24𝑘𝑔𝑚𝑠−1

𝑚 (∆𝑉 )≥1𝑥10−24𝑘𝑔𝑚𝑠−1

(∆𝑝)≥ h4𝜋 (∆𝑥 )

(∆𝑉 )≥1𝑥 106𝑚𝑠−1

We assume the mass of the electron to be exact (m= 9.109 x 10-31 kg), so the uncertainty is only in the velocity

(∆ 𝑝)=𝑚(∆𝑉 )

Very large uncertainty (20% of actual speed)

Why does Bohr’s Model Fail?

• 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 can not hold true.

Schrodinger Equation and Wave Mechanics• From the uncertainty principle came the Schrodinger equation,

which is now the central equation of a branch of physics known as quantum mechanics

• This equation is consistent with both the wave and particle nature of electrons• Schrondinger treats the electron as a circular wave around the

nucleus

• You don’t need to know how to use this equation, but you should understand the importance of Ψ (wavefunction), which we will discuss in a later slide

�̂�Ψ=𝐸Ψ

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, wavefunctions Ψ(x,y,z) are used to describe an electron’s wave characteristics

• The square of the wavefunction, Ψ2, gives the PROBABILTY DENSITY, or the probability that an electron will be found in some region of volume around the nucleus.

Probability Density???

• 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 Ψ2, we get ORBITALS• An orbital is a theoretical, 3-D “map” of the places where an

electron could be.

Orbitals and the Quantum Numbers

• 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)

1st quantum number ‘n’: Principal Quantum Number

• 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

2nd Quantum Number ‘L’: Aziumthal Quantum Number

• 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.

Orbital Labeling

Value of L 0 1 2 3

Orbital type s p d f

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

3rd quantum number ‘mL’: Magnetic quantum number

• The 3rd quantum number, mL, determines 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 each orbital type in a given shell

Magnetic Quantum Number

S-orbitals

• 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 (same in all 3 dimensions). The wavefunction is denoted Ψ1s .

• n=1 can only be an s -orbital

Since L= 0 for s orbital, mL can only be 0. Hence, each “shell” only has one s-orbital

1s 2s

x

y

z

x

y

z

x

y

z3s

Spheres shown above represent the volume of space around the nucleus in which an electron may be found. Images below are cross sections with probabilities plotted versus distance from the nucleus. Regions where an electron has 0% probability of existing is called a NODE.

node 2 nodes

P-orbitals

px

pz

py

• Unlike s-orbitals, p-orbitals are not spherically symmetric. Instead, electron density is concentrated in lobes around the nucleus along either the x, y, or z axis.

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

• For a p orbital, L =1.

• Therefore, mL = -1, 0, 1. Hence, there are 3 arrangements of p-orbitals in each shell n> 2

D-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-orbitals in every shell where n>3

• 4 of them have a ‘four-leaf clover’ shape. The fifth is represented as two lobes along the z axis with a doughnut shaped region of density around the nucleus

Questions:

• Can a 2d orbital exist?

• Can a 1p orbital exist?

• Can a 4s orbital exist?

The 4th and final quantum number ‘ms’: Magnetic Spin Number

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?

The 4th and final quantum number ‘ms’: Magnetic Spin Number

• 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, 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 =

Example: What are the allowed quantum numbers of a 2p electron?

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 3 p-orbitals

Pauli Exclusion Principle

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

Example

• 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

Electron Configurations• 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.

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.

Example

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

𝑁 (7𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠) :1𝑠22𝑠22𝑝3

𝐶𝑙 (17𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠) :1𝑠22𝑠22𝑝63𝑠23𝑝5

𝐶𝑎(20𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠):1 𝑠22𝑠22𝑝63𝑠23𝑝64 𝑠2

Energy

Example, contd.

• 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)

Excited States

• 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 atom?

• 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.

Example: Multiple Excited states of Li

• 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

ns1

1

2

3

4

5

6

7

ns2ns2

np1ns2

np2ns2

np3ns2

np4ns2

np5

ns2

np6

ns2 (n-1)dx

Valence Electron Configurations

Noble Gas Configurations

• 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.

Group Examples

• Write noble gas electron configurations for As+

• Write the noble gas electron configuration for the 1st and 2nd excited states of Mg+

Transition Metals

• 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