5.111 Lecture Summary #6 Monday, September 15, 2014

Transcription

1 5.111 Lectue Summay #6 Monday, Septembe 15, 014 Readings fo today: Section 1.9 Atomic Obitals. Section 1.10 Electon Spin, Section 1.11 The Electonic Stuctue of Hydogen. (Same sections in 4 th ed.) Read fo Lectue #7: Section 1.1 Obital Enegies (of many-electon atoms), Section 1.1 The Building-Up Pinciple. (Same sections in 4 th and 5 th ed.) Topics: I. Wavefunctions (Obitals) fo the Hydogen Atom II. Shape and Size of S and P Obitals III. Electon Spin and the Pauli Exclusion Pinciple I. WAVEFUNCTIONS (ORBITALS) FOR THE HYDROGEN ATOM Solving the Schödinge Equation povides values fo E and Ψ(,θ,φ). n A total of quantum numbes ae needed to descibe a wavefunction in D. 1. n pincipal quantum numbe n = detemines binding enegy (enegy level o shell). l angula momentum quantum numbe l = l is elated to n, detemines angula momentum, descibes subshell, shape of obital lagest value of l = n 1. m magnetic quantum numbe m = m is elated to l, detemines behavio in magnetic field, descibes the specific obital To descibe an obital, we need to use all thee quantum numbes: Ψ nlm (,θ,φ) The wavefunction descibing the gound state is. Using the teminology of chemists: The Ψ 100 obital is instead called the obital. n designates the shell o enegy level (1,, ) l designates the subshell (shape of obital) (s, p, d, f ) m designates obital oientation (specific obital) (p x, p y, p z ) = 0 obital = 1 obital = obital = obital fo = 1: m = 0 is p z obital, m = ±1 ae the p x and p y obitals 1

2 State label wavefunction obital H atom E n H atom E n [J] n = 1 = 0 ψ J m = 0 n = = 0 ψ J m = 0 n = = 1 ψ J m = +1 n = = 1 10 ψ 10 R H / J m = 0 n = = ψ 1-1 R H / J m = -1 What is the coesponding obital fo a 5,1,0 state? Fo a hydogen atom, obitals with the same n value have the same enegy: E = -R H /n. having the same enegy Fo any pinciple quantum numbe, n, thee ae degeneate obitals in hydogen (o any othe 1 electon atom). IN THEIR OWN WORDS MIT gaduate student Benjamin Ofoi-Okai discusses how enegy levels elate to eseach in nanoscale MRI (magnetic esonance imaging), a technique that allows -D imaging of biological molecules, such as poteins, and viuses. Image fom "Behind the Scenes at MIT. The Dennan Education Laboatoy. Licensed unde a Ceative Commons Attibution-NonCommecial-ShaeAlike License.

3 THE PHYSICAL INTERPRETATION OF A WAVEFUNCTION The pobability of finding a paticle (the electon!) in a defined egion is popotional to the squae of the wavefunction. [Ψ nlm (,θ,φ)] = PROBABLITY DENSITY = pobability of finding an electon pe unit volume at, θ, φ IIA. SHAPE OF S ORBITAL To conside the shapes of obitals, we can ewite the wavefunction Ψ nlm as the poduct of a adial wavefunction, R nl ( ), and an angula wavefunction Y lm (θ,φ) Ψ nlm (,θ,φ)] = R nl ( ) x Y lm (θ,φ) adial x angula wavefunctions (a) adial wave functions (b) angula wave functions n l R nl () l m l Y l,ml (θ, ϕ) 1 0! 0 0! 1 e!!"/!! 4π a! x a 0 e a 0 a 4π a a 0 a 0 a 0 9a 0 y a 0 e 4π e a 0 z 4π 1/ 1/ sin θ cos φ 1/ sin θ sin φ 1/ cos θ whee a 0 = (a constant) = 5.9 pm fo a gound state H-atom: Fo all s obitals (1s, s, s, etc.), the angula wavefunction, Y, is a. s obitals ae spheically symmetical independent of and. Thee ae thee common plots used to help us visualize an s obital: (1) Pobability density Ψ plot of s obitals in which density of dots epesents pobability density; () Wavefunction plotted again (distance fom nucleus); () Radial pobability distibution as a function of adius.

4 RADIAL PROBABILITY DISTRIBUTION (RPD) epots on the pobability of finding an electon in a spheical shell of thickness d at a distance fom oigin. Maximum pobability o most pobable value of is denoted. mp fo a 1s H atom = a 0 = 5.9 pm = 0.59 x m = 0.59Å a 0 Boh adius s s 1s s Radical Nodes ψ 1s ψ s ψ s a 0 1.9a 0 7.1a 0 R P D R P D R P D NODE: A value fo, θ, o φ fo which Ψ (and Ψ ) =. Image by MIT OpenCouseWae. Adapted fom Oxtoby, D., et al. Pinciples of Moden Chemisty, fifth edition. Thomson Books/Cole, 00. ISBN: RADIAL NODE: A value fo fo which Ψ (and Ψ ) = 0. In othe wods, a adial node is a distance fom the adius fo which thee is no pobability of finding an electon. 4

5 To calculate the numbe of adical nodes n 1 l 1s: = 0 adial nodes s: adial nodes s: adial nodes 4p: adial nodes IIB. THE SHAPE OF P ORBITALS Figue by MIT OpenCouseWae. Unlike s obitals, p obitals have θ, φ dependence. P obitals spheically symmetical. P obitals consist of two lobes (of opposite sign) sepaated by a plane on which Ψ = 0 (and Ψ = 0). Thee is zeo pobability of finding a p-electon in a nodal plane. Thus, thee is pobability of finding a p-electon at the nucleus. Pobability density maps of p obitals: Ψ pz Ψ Ψ p x py X Y X Y X Y Nodal planes: xy yz xz Nodal planes (planes that have no electon density) aise fom angula nodes in the wavefunction. ANGULAR NODE: A value fo at which Ψ (and Ψ ) = 0. In geneal, an obital has: n 1 total nodes angula nodes adial nodes s: total nodes, angula nodes, adial nodes p: total nodes, angula nodes, adial nodes d: total nodes, angula nodes, adial nodes 5

6 IIC. ORBITAL SIE Figue by MIT OpenCouseWae. As n inceases (fom 1 to to ), the obital mp size. As l inceases (fom s to p to d) fo a given n, the obital mp size. Only electons in s states have a substantial pobability of being vey close to nucleus. This means that although the size (also called the boundy suface) of s obitals is lage than p o d obitals, s-electons ae the shielded. III. ELECTRON SPIN: THE FOURTH QUANTUM NUMBER A fouth quantum numbe descibes the spin of an electon within an obital: the spin magnetic quantum numbe, Thee is no classical analogy to spin. An electon can have two spin states: m s = (spin up) o m s = (spin down). m s completes the desciption of an and is NOT dependent on the obital. 6

7 So we can descibe a given obital using thee quantum numbes (n, l, m l ) and a given electon using 4 quantum numbes (n, l, m l, m s ). Ψ nlml descibes an Ψ nlml m s descibes an PAULI EXCLUSION PRINCIPLE No two electons can be in the same obital and have the same spin. No two electons in the same atom can have the same quantum numbes. Within each obital, electons ae paied (one spin up and one spin down). One obital can hold no moe than two electons. 7

8 MIT OpenCouseWae Pinciples of Chemical Science Fall 014 Fo infomation about citing these mateials o ou Tems of Use, visit: