Tags
Bohr correspondence principle, Free particle, Particle in 1D box, Particle in 3D box, Particle in a box, Quantum mechanicl tunneling
(A)PARTICLE IN A BOX
(I) PARTICLE IN A ONE DIMENSIONAL (1D) BOX
En =n2h2/8ml2
Ψn = (2/l)½Sin (nπ/l)x
Discussions:
- The value of “n” can not be zero as it will mean the total energy, E is zero; which is not possible.
- Hence, ZPE, E1=h2/8ml2 ; for n = 1(maximum degeneracy is one-non degenerate)
- The integral values of n (1, 2, 3, …) characterizes the energy levels , En (E1 , E2 , E3 , ……) and the wave functions, yn (y1, y2, y3,…………).
- The energy levels are not equally spaced and the wave functions will have nodes and anti nodes depending on n values.
- Double bonds n. The л⇒ л* transition is n ⇒ n+1.The number of MO’s = 2n, of which 50% levels viz., n are occupied.
Application 1D box model to conjugated olefins:
- Mass of the electron, me = 9.11×10-31 kg ; h = 6.625 x 10-34Js; kb = 1.38 x 10-23JK-1
- Ethylene: l = 1.33 Ǻ = 1.33×10-10 m (lmax=75nm); NB:1nm = 1x 10 -9 m.
- 1, 3-Butadiene: l = 2(1.33) + 1(1.54) = 4.2Ǻ = 4.2×10-10 m (lmax =117nm).
- 1, 3, 5-Hexatriene (Three double bonds): l = 3(1.33) + 2(1.54) = 7.07Ǻ = 7.07×10-10 m (lmax = 6 nm). Transition 3 → 4 (6 MOs)
- A conjugated olefin with 12 double bonds (Box length, l = 12 double bond distances + 11 single bond distances = 12(1.33) + 11(1.54) Ǻ) = 32.9 Ǻ = 32.9 x 10-10m; (λmax = 1427nm). Transition n : 12 →13 (12 MOs)
- The absorption increases to visible region making the compound colored.
- There could be an outer distribution of electron density little away from the extreme carbons-Tunneling effect.
- The bond angles are 120º . Hence, the zig-zag orientation and further twisting due to C-C free rotations (conformations) will make the length of the molecule shorter.
The file, in the following Link (LINK-1): QC-3.1: Part in a Box (1D), gives an account of the concepts of a particle in a 1D Box mentioned above.
LINK-1:QC-3.1. Part in a Box (1D)
(II) PARTICLE IN A 2D-BOX ( xy plane)
E= Ex + Ey
Ψ = ΨxΨy = (2/lx)½ (2/ly)½ sin(nxπx/lx)sin(nyπx/ly)
= {2/A½}sin(nxπx/lx)sin(nyπx/ly)
(III.a) PARTICLE IN A 3D- BOX (Rectangular box); lx ≠ ly ≠ lz
Ex= nx2h2/8mlx2
E= Ex + Ey+ Ez = [(nx2/ lx2)+ (ny2/ ly2) + (nz2/ lz2)]( h2/8m)
ZPE =(3h2 /8ml2) (maximum degeneracy is six)
Ψx = (2/lx)½ sin(nxπx/lx)
Ψ = ΨxΨyΨz = (2/lx)½ (2/ly)½ (2/lz)½ sin(nxπx/lx) (nyπy/ly) (nzπz/lz)
=(8/V)½ sin(nxπx/lx) (nyπy/ly) (nzπz/lz)
II.b) PARTICLE IN A 3D BOX ((Cubical) ; lx= ly = lz = l
E = (nx2 + ny2+ nz2) (h2/ 8ml2)
Ψ = (8/V)½ sin(nxπx/lx) (nyπy/ly) (nzπz/lz)
The file, in the following Link (LINK-2): QC-3.2: Particle in a Box (2D & 3D), gives an account of the concepts of of a particle in a 2D & 3D Boxes mentioned above.
LINK-2:QC-3.2. Part in a Box (2D & 3D)
(B) QUANTUM EFFECTS
(IV)QUANTUM MECHANICAL TUNNELING
- The probability of finding the particle outside the box is not zero.
- There is penetration of the probability of the particle outside box and go into the
classically forbidden region.
- Such quantum mechanical tunneling behavior is more predominant as the classical behavior of the system is reduced.
- Radiation (α , β or γ emission ) is due to quantum mechanical tunneling
(V) FREE PARTICLES
- Un like an electron in an atom or molecule, the free particle is not bound to any external force not even to gravitational force and can move within the container without any restriction. Hence, its potential energy is constant and may be taken as zero.
- There are no restrictions to k. Hence, the free particle has any energy in a continuous form as translational energy.
(VI) BOHR’S CORRESPONDENCE PRINCIPLE
In the limit of classical sized system, the quantum mechanical results had to go over in to the classical results.
That is the quantum mechanical results become identical to the classical results when the quantum number describing the system becomes large.
Ground state energy: n = 1
In the case of a particle in a box, as the particle becomes heavier and the dimension of the box larger, the energy levels become closer. Thus energy becomes continuous as expected by classical mechanics.
The file, in the following Link (LINK-3): QC-3.3: Quantum effects , gives an account of the quantum concepts mentioned above.
LINK-3:QC-3.3. Quantum Effects
Application of the model of particle in a box
(i) Study of the behavior of electron in a molecule containing π-electrons(conjugated systems) as a particle confined to the π-electron axis
(ii) Study of the transnational property of molecules(in the field of thermodynamics) in gas phase
Illustrations:
- Calculate the value of n necessary to give energy of ½kT for O2 molecule at 298K in a box of length (a) 1nm (b) 1cm.
- Compare the quantum mechanical behavior & calculate the ground state energy of an electron in a box of length 2Ả (QM behavior)
- Compare the result for a particle of mass 0.1 mg in a box of length 1 cm. (Classical behavior)
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https://dradchem.wordpress.com/2015/06/22/properties-of-chemistry-poc-contents-2/