Orbitals (Ch 5) Lecture 6 Suggested 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 (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.

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 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 momentum 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 λ λ’λ’ 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 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) m L (magnetic quantum number) m s (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 L0123 Orbital typespdf Orbitals are labeled by first writing the principal quantum number, n, followed by the letter representation of L Labeling Orbitals from “L”

The 3 rd quantum number, m L, relates to the spatial orientation of an orbital m L can assume all integer values between –L and +L Number of possible values of m L gives the number of sub- orbitals of each orbital type in a given shell 3. The Magnetic Quantum Number, m L

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 ( Ψ 1 ) 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, m L can only be 0 This single value of m L indicates that each shell contains a single s-orbital S-orbitals s-orbital

1s 2s x y z x y z x y z 3s The blue spheres represent a volume of space around the nucleus where an electron may be found. A region within an orbital with 0% probability of housing an electron is called a NODE. node 2 nodes S-orbitals

P orbitals exist in all shells where n> 2. For a p orbital, L =1. Therefore, m L = -1, 0, 1. Three values of m L means there are 3 p- suborbitals 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 p x, p y, and p z respectively) P-orbitals

D-orbitals exist in all shells where n>3 L= 2, so m L 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, m s

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” 4. The Magnetic Spin Number, m s

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 Pauli Exclusion Principle

nl mlml msms nlmlml msms 211+ ½ 211- ½ 210+ ½ 210- ½ 21+ ½ 21- ½ 01 Spin up: + ½ Spin down: - ½ Representation of the 3 p-orbitals Example: What Are The Allowed Sets of Quantum Numbers For An Electron In A 2p Orbital? (n, L, m L, m s )

List all possible sets of quantum numbers in the n=2 shell? n = 2 L = 0, 1 m L = 0(L=0) = -1, 0, 1 (L=1) m s = +/- 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. 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 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 1 st excited state. Excited States

Ground state Li:1s 2 2s 1 1 st excited state Li: 1s 2 2s 0 2p 1 2 nd excited state Li:1s 2 2s 0 2p 0 3s 1 1s 2s 2p 3s Ground state 1 st excited state 2 nd excited state Energy Example

ns ns 2 np 1 ns 2 np 2 ns 2 np 3 ns 2 np 4 ns 2 np 5 ns 2 np 6 ns 2 (n-1)d x

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 Unfavorable 4s 3d Favorable [Ar] 4s 1 3d 5 Example: Cr  [Ar] 4s 2 3d 4 Transition Metals