Chapter 7 Quantum Theory and the Electronic Structure of Atoms Lecture Solutions PDF

Chapter 7: Quantum Theory and the Electronic Structure of Atoms 1  Quantum Theory and The Electronic Structure of Atoms Before 1900: Classical physics (bouncing ball explanation) could describe the pressure of a gas, but not what holds mo...

Chapter 7: Quantum Theory and the Electronic Structure of Atoms 1  Quantum Theory and The Electronic Structure of Atoms Before 1900: Classical physics (bouncing ball explanation) could describe the pressure of a gas, but not what holds molecules together Properties of atoms and molecules are NOT governed by the same physical laws of large objects (such as gravity and movement) Max Planck (1900) discovered that atoms and molecules emit energy in discrete quantities (quanta) Energy is not continuous Nobel Prize in Physics for his quantum theory (1918) 2  Waves A wave is a vibrating disturbance by which energy is transmitted Wavelength (λ): the distance between identical points on successive waves (units: m) Frequency (ν): the number of waves that pass through a point in 1 second (units: cycles/sec, sec-1 s-1, Hertz, Hz) Amplitude (A): the vertical distance from a wave’s midline to a peak or trough (units: none) Speed (u, in m/s): u = λν 3 Waves 4  Electromagnetic Radiation Electromagnetic waves (Maxwell, 1873) have an electric field component and a magnetic field component Same wavelength and frequency, but travel perpendicular to each other Electromagnetic radiation is the emission and transmission of energy in the form of waves Electromagnetic waves (light) travel at 3.00 x 108 m/s = 186,000 miles/sec = c λν = c = speed of light = 3.00 x 108 m/s c ν= λ is usually given in nm: convert to m to match with c!!! λ 5 Example: What is the wavelength (in m and nm) of an x-ray with a frequency of 6.28 x 1017 Hz? λν = c c 3.00 x 108 m/s λ= = = 4.78 x 10 −10 m ν 6.28 x 1017 s−1 −10 1 nm 4.78 x 10 mx = 4.78 x 10−1 nm 10 −9 m 6  Electromagnetic Radiation 1 ν ∝ so if ν↑then λ↓ (becomes more energetic) λ 7 8  Planck’s Quantum Theory 1900 (Planck) The smallest quantity of energy that can be emitted (or absorbed) in the form of electromagnetic radiation is a “quantum” These are discrete packets of energy The energy of a single quantum: E = hν = hc/λ where h = Planck’s Constant = 6.63 x 10-34 J s Energy is always emitted in multiples of hν (2 hν, 3 hν, but not 1.3 hν) 9  The Photoelectric Effect Einstein, 1905, published while working in a patent office in Berne, Switzerland Nobel Prize in Physics for this in 1921. Confirmed Planck’s theory: In metals, electrons are held by attractive forces To remove them, light of high energy (high ν) is needed There is a minimum ν (threshold frequency) to eject the electrons E = hν If hν > E (due to an increase in v), then electrons will also have kinetic energy hν = KE + W where W is the work function, how strongly the electrons are held by the metal ↑νlight, ↑KE emitted (e-) The number of electrons ejected is proportional to the light intensity, but the energies of the ejected electrons are not proportional (same v) Beam of light is a stream of particles called photons Light possesses both wavelike and particle like behavior (particle-wave duality) 10 Example: The energy of a photon is 5.87 x 10–20 J. What is its wavelength in nm? hc hc E = hv = → λ= λ E (6.63 x 10−34 J ∙ s)(3.0 x 10 8 m/s) λ= (5.87 x 10−20 J) −6 1 nm 3.39 x 10 m x =3.39 x 103 nm 10−9 m 11  Bohr’s Theory of the Hydrogen Atom Niels Bohr (1913): Received Nobel Prize in Physics in 1922 Emission Spectra: A continuous or line spectra of radiation emitted by substances In the sun and heated solids, the emission is continuous In a gas line spectra are the light emission only at specific wavelengths every element has a unique spectrum “fingerprint” 12  Emission Spectra Examples Noncontinuous spectrum (only certain discrete wavelengths included) Continuous spectrum (light of all wavelengths included) 13 14  Emission Spectra Examples Na K Li Ba We can identify elements by a characteristic color of light they produce when heated. The colors we see are derived from especially bright lines in their emission spectrum 15  Emission Spectra Energetically excited atoms only emit radiation in discrete energies corresponding to the atom’s electronic energy levels (The energies of the electrons are quantized) When an electron moves from a higher to a lower energy orbit it emits a photon, or light n is a “quantum number” with values of 1, 2, 3… The energy difference between levels with different n values is calculated by using the Rydberg equation 1 1 nf = final energy level ΔE = hν = RH( 2 − ) ni n2f ni = initial energy level RH = 2.18 x 10–18 J As n → 0, ΔE increases; as n → ∞, ΔE decreases 16  Emission Spectra n = 1: ground level (lowest energy state) n = 2, 3…excited state or level Higher energy than the ground state (The farther away from the nucleus, the less tightly the electrons are held) Each spectral line in an emission spectrum corresponds to a specific transition Radiant energy absorbed by an atom causes the electron to move to a higher energy state Radiant energy is emitted (as a photon) when the electron moves back to a lower energy state 17 Example: What is the wavelength (in nm) of a photon emitted during a transition from ni = 6 to nf = 3 in the H atom? 1 1 −18 1 1 ΔE = RH ( − ) = 2.18 x 10 J( − ) = −1.82 x 10 −19 J (emitted) n2 i n 2 f 6 2 32 ch (3.0 x 108 m/s)(6.63 x 10−34 Js) λ= = = 1.09 x 10 −6 m = 1.09 10 3 nm ΔE 1.82 x 10−19 J 18  The Dual Nature of the Electron Why is an electron restricted to specific fixed distances? de Broglie (1924): electrons behave as both waves (energy) and matter (position) particles with wavelike properties h λ= mu Where m is the mass in kg and u is the velocity of a particle in m/s 19  Problems with Bohr’s model Only explained emission spectra for hydrogen (1 e⎻) Only explained particle-like properties of electrons (it also behaves wavelike) Locating an electron in an atom is not easy Electrons do not orbit in well defined “Bohr” paths 20  Schrodinger Equation and Heisenberg Uncertainty Principle Schrodinger Equation Nobel Prize in Physics in 1926 Complicated mathematical techniques used to describe the behavior and energies of the particle-like and wavelike behavior of subatomic particles Heisenberg Uncertainty Principle Nobel Prize in Physics in 1932 It is impossible to know simultaneously both the momentum (mν = p) and the position (radius, position) of a particle with certainty 21  Quantum Mechanics Electron density: the probability that an electron will be found in a particular region of an atom Atomic orbital: the wave function of an electron or the distribution of electron density (shape) Electrons in atoms with many electrons are assumed to behave as the single electron (H). The Schrödinger equation can only be solved for hydrogen 22  Quantum Numbers (Form: n, l, ml, ms) Each orbital can be described mathematically by a “wave function” that is characterized by a set of quantum numbers The first three numbers used to describe the distribution of electrons in atoms (n, l, ml) The fourth number is used to describe the spin of electrons (ms) Electrons in multi-electron atoms can be classified into a series of: Shells → subshells → orbitals n l ml 23  n: Principle Quantum Number Shell Related to energy of the shell and to the distance of electrons from the nucleus (size) Possible values of n = 1, 2, 3, 4, … (quantized) The larger the n, the greater average distance of electrons from the nucleus ⸫ the larger the orbital Equal to the “row” in the Periodic Table for main group elements 24  l: Secondary Quantum Number Subshell Angular momentum quantum number Related to the shape of various subshells (orbitals) within a shell (4 shapes) Possible values of l are 0 to (n ⎼ 1) possible values of l: l = 0 1 2 3 4 (n-1) name of orbital: s p d f g alpha sharp principal diffuse fundamental spherical bowtie 2 bowties flower + doughnut (1 orbital) (3 orbitals) (5 orbitals) (7 orbitals) 25 26  l: Secondary Quantum Number The s subshell: spherical; size↑ as n↑; 1 possible arrangement The p subshell: bowtie shape; identical in shape and size; size↑ as n↑; 3 possible arrangements (px, py, pz, the orientations) 27  l: Secondary Quantum Number The d subshell: double bowtie shape and bowtie/ring; size↑ as n↑; 5 possible arrangements 28  values of n “shell” values of l “subshell” orbitals ENERGY SHAPE 1 0 1s 2 0, 1 2s, 2p 3 0, 1, 2 3s, 3p, 3d Example: If n = 2, l = 0 or 1 It is either the 2s (only one arrangement) or the 2p subshell (3 arrangements: px, py, pz) All orientations are equally possible if there is more than one orientation 29  ml : Magnetic Quantum Number (orbital) Related to spatial orientation of orbitals (arrangement) within a given subshell Wave function ѱ, probability of where to find an electron around a nucleus Possible values of ml = -l, …….0,……., +l There are (2l + 1) values of ml The number of ml values = the number of orbitals (different arrangements) within a subshell Example: In a subshell having l = 2, there what are possible values of ml? -2, -1, 0, 1, 2 30  ms: Electron Spin Quantum Number Electrons “spin on axes like the earth”, so magnetic properties can be accounted for (electrons act like little magnets when spinning) Electrons have intrinsic angular momentum – “spin” : ms Possible values: ms = + ½ and – ½ 31  Summary of Quantum Numbers Atomic Orbitals: an address of an electron in an atom. Total # orbitals in energy level = n2 and total number electrons = 2n2 n l ml Subshell # orbitals = 0 to (n-1) = (2l + 1) = n2 1 0 0 1s 1 2 0 0 2s 1 1 -1, 0, 1 2p 3 3 0 0 3s 1 1 -1, 0, 1 3p 3 2 -2, -1, 0, 1, 2 3d 5 4 0 0 4s 1 1 -1, 0, 1 4p 3 2 -2, -1, 0, 1, 2 4d 5 3 -3, -2, -1, 0, 1, 2, 3 4f 7 32 Example: Give the values of the quantum numbers associated with the orbitals in the 3p subshell n = 3, the subshell is p, so l = 1 ml = -1, 0, 1 Example: What is the total number of orbitals associated with the principal quantum number n = 4? ⸫ 1 4s orbital 3 4p orbitals 5 4d orbitals 7 4f orbitals Total: 1 + 3 + 5 + 7 = 16 33 Example: Write the quantum numbers associated with an electron in the 4d shell (n, l, ml, ms) n=4 l = 2 (d orbital) ml = -2, -1, 0, 1, 2 ms = + or – ½ (4, 2, ⎼2, ± ½) (4, 2, ⎼1, ±½) (4, 2, 0, ±½) (4, 2, 1, ±½) (4, 2, 2, ±½) 34 Example: Write the 6 quantum number sets for an electron in a 5p orbital (n, l, ml, ms) n=5 l = 1 (p orbital) ml = -1, 0, 1 ms = + or – ½ (5, 1, -1, ½) (5, 1, -1, -½) (5, 1, 0, ½) (5, 1, 0, -½) (5, 1, 1, ½) (5, 1, 1, -½) Example: Write the quantum numbers associated with an electron in the 3s shell n=3 l = 0 (s orbital) ml = 0 ms = + or – ½ (3, 0, 0, ½) (3, 0, 0, -½) 35 Example: How many electrons can the 3rd shell (n = 3) hold? n=3 l = 0 1 2 subshell 3s 3p 3d (2l + 1) # orbitals 1 3 5 # electrons 2 6 10 = 18 total Example: How many electrons can the 4th shell (n = 4) hold? n=4 l = 0 1 2 3 subshell 4s 4p 4d 4f (2l + 1) # orbitals 1 3 5 7 # electrons 2 6 10 14 = 32 total 36  Relationship to Periodic Table 1s 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 5d 6p 7s 6d 4f 5f 37  Energy of Orbitals In general: as n↑, quantum #↑, energy ↑ However, it now depends on how subshells are filled “Read” the Periodic Table to determine the energy of a subshell The energy is determined by both the principal quantum number and the angular momentum quantum number Total energy is dependent upon the sum of the orbital energies and the repulsion between the electrons in the orbitals 38  Energy of Orbitals “Read” the Periodic Table to determine the energy of a subshell s p 1 2 2 3 d 3 4 3 4 5 4 5 6 5 6 7 6 7 4 f 5 39  Orbital Diagrams An orbital diagram shows the atomic orbitals and the spin of the electrons in those orbitals There are at most 2 electrons in each atomic orbital Spin is shown as an up arrow or a down arrow for each electron For hydrogen there is only one electron, and it is in the 1s subshell ↑ 1s 40  Orbital Diagrams There are three principles to help draw orbital diagrams: Pauli Exclusion Principle Hund’s Rule Aufbau Principle Pauli Exclusion Principle No two electrons in an atom can have identical values of all 4 quantum numbers There is a maximum number of 2 electrons per orbital A single orbital can hold a pair of electrons with opposite “spins” ↑↑ ↑↑ ↑↓ ↑↓ 1s 2s 1s 2s X NO YES 41  Orbital Diagrams Hund’s Rule The most stable arrangement of electrons in subshells is the one with the greatest number of parallel spins Don’t pair electrons until all orbitals of that type in the same energy level, i.e., 2p, each have one electron C ↑↓ ↑↓ ↑↓ X NO 1s 2s 2p C ↑↓ ↑↓ ↑ ↑ YES 1s 2s 2p Analogy: seats on a bus People will occupy seats singly until all seats have at least one person, only then will people ‘pair up’ Electrons won’t ‘pair up’ until necessary 42  Orbital Diagrams Paramagnetism and Diamagnetism “Paired” electrons are two electrons in one orbital with opposite spins (NOT parallel spins) A single electron in an orbital is called “unpaired” Atoms with 1 or more unpaired electrons are paramagnetic (attracted by a magnet) Atoms with all spins paired are diamagnetic (repelled by a magnet) Odd number of e–: paramagnetic only; Even number of e–: para or diamagnetic C ↑↓ ↑↓ ↑ ↑ Paramagnetic 1s 2s 2p Be ↑↓ ↑↓ Diamagnetic Diamagnetic: Doubled up 1s 2s 43  Orbital Diagrams Examples of orbital diagrams: Look at the Periodic Table! C ↑↓ ↑↓ ↑ ↑ 1s 2s 2p N ↑↓ ↑↓ ↑ ↑ ↑ 1s 2s 2p O ↑↓ ↑↓ ↑↓ ↑ ↑ 1s 2s 2p K ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ 1s 2s 2p 3s 3p 4s Sc ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ 1s 2s 2p 3s 3p 4s 3d 44  Energy of Orbitals “Read” the Periodic Table to determine the energy of a subshell 45  The Shielding Effect (many electrons) Electrons in the smaller orbitals (lower energy) are closer to the nucleus The 1s electrons are closer and have lower energy than the electrons in the larger orbitals, such as 2p or 3s 2p, 3s… are “shielded” from the attractive forces of the nucleus by the 1s electrons, reducing the electrostatic attraction between the protons in the nucleus and e- in the 2s or 2p orbitals The shielding effect causes a slight increase in energy of the more distant electrons: In ease of removal of electrons: 2s (harder) < 2p (easier) ↑ energy = easier to remove e– The stability of an electron is determined by the strength of its attraction to the nucleus (2s lower in energy than 2p) 46  The Aufbau Principle (Building Up Principle) and Electron Configurations As protons are added to the nucleus to build up elements, electrons are similarly added to atomic orbitals In a neutral atom, atomic # = # protons = # electrons Electrons are added to atomic orbitals, two per orbital, in the general order of increasing principal quantum number n, with some crossovers that result from shielding Electron configurations are how the electrons are distributed among the various atomic orbitals 47  Alkali metals and alkaline earth metals fill the s orbitals last Main group elements fill the p orbitals last Transition metals fill the d orbitals last Lanthanides (4f) and actinides (5f) fill the f orbitals last 48  Electron Configurations How to assign electrons to atomic orbitals: 1. Each shell, n, contains subshells l, 0 to (n-1) 2. Each subshell l contains (2l + 1) orbitals, ml 3. No more than 2 e– can be in an orbital (max # e– = 2x # orbitals) 4. Max # e– in any shell = 2n2 # e– = Z (atomic number) for hydrogen: 1s1 1 = n, s = l, and 1 = # e– in subshell H: Electron configuration: 1s1 (read 1-s-1) He: Electron configuration: 1s2 (read 1-s-2) Li: Electron configuration: 1s22s1 (read 1-s-2, 2-s-1) 49  Transition Metal Electron Configurations Usually have incompletely filled subshells There is a slightly greater stability (lower energy) associated with ½ filled (d5) and completely filled (d10) subshells Therefore, put only one electron in the s shell for the 4th and 9th elements to make 5d and 10d Fill the s subshell for the 5th and 10th elements in the d subshell Fill or ½ fill the d before filling s For Cr & Mo: NO: ns2nd4 YES: ns1nd5 For Cu & Ag: NO: ns2nd9 YES: ns1nd10 50 Transition Metal Electron Configurations Mo (Zformula: General = 42) Ag (Z = 47) 1. Calculate how many electrons (use Z) 4 2 4d the Periodic Table 2. Use 5s and fill shells 4d9 5s2 3. Carefully fill partially filled d orbitals 4d5 4s1 4d10 5s1 51 Total of the superscripts = total # of e⎻ Z = atomic # = total # of e⎻ Z Atom Electron Configuration 1 H 1s1 2 He 1s2 3 Li 1s22s1 General formula: 4 Be 1s22s2 1. Calculate how many electrons (use Z) 5 B 1s22s22p1 2. Use the Periodic Table and fill shells 6 C 1s22s22p2 7 N 1s22s22p3 3. Carefully fill partially filled d orbitals 8 O 1s22s22p4 9 F 1s22s22p5 10 Ne 1s22s22p6 11 Na 1s22s22p63s1 24 Cr 1s22s22p63s23p64s13d5 25 Mn 1s22s22p63s23p64s23d5 26 Fe 1s22s22p63s23p64s23d6 52 Example: Example: Write the electronic configuration of Ge (Z = 32, group 4) Ge = 1s22s22p63s23p64s23d104p2 53 Example: Write an electron configuration for Pb (Z = 82, group 6) 1s22s22p63s23p64s23d104p65s24d105p66s24f145d106p2 54  Noble Gas Core Electron Configuration Short hand notation: Use the preceding noble gas in brackets, then add the additional electrons Ge = 1s22s22p63s23p64s23d104p2 → Ge = [Ar]4s23d104p2 [Ar] Pb = 1s22s22p63s23p64s23d104p65s24d105p66s24f145d106p2 → Pb = [Xe]6s24f145d106p2 [Xe] Valence Shell Electron Configurations Valence shell – outermost energy level – only show largest value of n (e.g., for Ge, n = 4) Ge = 4s24p2 (valence shell electron configuration, although it has 3d10 electrons) Pb = 6s26p2 (valence shell electron configuration, although it has 4d14 & 5d 10 electrons) 55  Valence shell orbital diagram Shows only the electrons in the valence shell of Ge and Pb (the shell with the highest principal quantum number) Example: Ge = 4s24p2 Ge ↑↓ ↑ ↑ 4s 4p Pb = 6s26p2 Pb ↑↓ ↑ ↑ 6s 6p Elements in the same group have the same valence shell electron configurations Example – Group 4: (all ns2np2) C 2s22p2 Si 3s23p2 Ge 4s24p2 Sn 5s25p2 Pb 6s26p2 56  Energy of Orbitals “Read” the Periodic Table to determine the energy of a subshell 57 Example: Element 88, Ra (Group 2/2A) 1. Write the expanded electron configuration 1s22s22p63s23p64s23d10 4p65s24d105p66s24f145d106p67s2 2. Write the electron configuration (EC) with the noble gas core [Rn] 7s2 3. Write the valence shell EC 7s2 4. Draw the valence shell orbital diagram: (only level 7!) ↑↓ 7s 5. Is this element diamagnetic or paramagnetic? Diamagnetic 58  Energy of Orbitals “Read” the Periodic Table to determine the energy of a subshell 59 Example: Element 18, Br (Group 7A/17) 1. Write the expanded electron configuration 1s22s22p63s23p64s23d104p5 2. Write the EC with the noble gas core [Ar] 4s23d104p5 3. Write the valence shell EC: 4s24p5 4. Draw the valence shell orbital diagram: (only level 4!) ↑↓ ↑↓ ↑↓ ↑ 4s 4p 5. Is this element diamagnetic or paramagnetic? Paramagnetic 60