The wavefunctions are difficult to represent because they are complex. This problem can be solved by using linear combinationsΦm±Φp-m, which are not complex cos mo+isin m sin m 2丌 2 (cosmo-isin mo cos me 2丌 2丌 2021/8/21 Che mistry De partne nt of Fudan University l1
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 11 ( ) m = cosm + isin m 2 1 ( ) −m = cosm − isin m 2 1 m m sin m 1 − − = m m cosm 2 2 + − = The wavefunctions are difficult to represent because they are complex. This problem can be solved by using linear combinations m −m ,which are not complex
Phusical chemiatry Chapter II Atomic Structure and Spectrum 3. Principal quantum number n.n=3 onization =3-136 This quantum number can have any 6 nE. integer value from I up to infinity. 136eV 2uT2Z# E Hydro白en 136 h n is the energy required to take the electron from the ground state to n=o? All the orbitals of a given value n are said to form a single shell of the atom In hydrogen atom, all orbitals of a given n, have the same energy. ∑(2+1)=m2m2=1+3+…+(2n-1) 2021/8/21 Che mistry De partne nt of Fudan University
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 12 3. Principal quantum number n This quantum number can have any integer value from 1 up to infinity. 2 2 2 2 4 2 n h Z e En = − ( ) 1 2 0 2 1 n l l n − = + = All the orbitals of a given value n are said to form a single shell of the atom. In hydrogen atom, all orbitals of a given n, have the same energy. is the energy required to take the electron from the ground state to . En n = 2 n n = + + + − 1 3 ... (2 1)
Phusical chemiatry Chapter II Atomic Structure and Spectrum 4. Orbital quantum number l For a given value of n, this quantum number can have any integer value from O up to n-1 1 a 一 snb—+ sin 080 80 sin a0 ()ep()=1(+1)2v 1 a de 1 a-g SIn 十 (+1) Osin a0 a0 sin ap ao The orbitals with the same value n but different values of l are said to form a subshell. For historical reasons, we associate letter symbols with the value of I =0(3),1(p),2(d),300, 2021/8/21 Che mistry De partne nt of Fudan University 13
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 13 4. Orbital quantum number l For a given value of n, this quantum number can have any integer value from 0 up to n – 1. ( ) ( ) () ( ) 2 2 2 2 2 2 1 1 1 = + + M = − R r l l n l m sin sin sin ˆ , , ( ) sin sin sin 1 1 1 2 2 2 = + + l l The orbitals with the same value n but different values of l are said to form a subshell. For historical reasons, we associate letter symbols with the value of . l l=0(s), 1(p), 2(d), 3(f), …
Magnetic quantum number m For a given value ofl, this quantum number can be any integer value starting at- and going up to +. snb—+ n. .m sin 000 80 sin 0 a9 R()(0x()=(+1)vy o0 1a2① Osin 0 a0 a0 sin* ap ao =l(+1) Myum=-ih r(roe(e)=mhR(rOOo 2021/8/21 Che mistry De partne nt of Fudan University 14
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 14 ( ) ( ) () ( ) 2 2 2 2 2 2 1 1 1 = + + M = − R r l l n l m sin sin sin ˆ , , ( ) sin sin sin 1 1 1 2 2 2 = + + l l ( ) ( ) () ( ) ( ) () Mz nlm i R r = mR r ˆ = − For a given value of l, this quantum number can be any integer value starting at –l and going up to +l. Magnetic quantum number ml
Problem Use hydrogenic orbitals to calculate the mean radius of a is orbital a Hydrogen atom is in its 4d state. The atom decays to a lower state by emitting a photon. find the possible photon energies that may be observed. Give your answers in eV 2021/8/21 Che mistry De partne nt of Fudan University 15
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 15 Problem: Use hydrogenic orbitals to calculate the mean radius of a 1s orbital. A Hydrogen atom is in its 4d state. The atom decays to a lower state by emitting a photon. Find the possible photon energies that may be observed. Give your answers in eV