In quantum mechanics, the movement of π electrons along a neutral chain of conjugated carbon atoms m — Physical Chemistry — Electrochemistry Chemistry Question
Particles in a box: polyenes
In quantum mechanics, the movement of π electrons along a neutral chain of conjugated carbon atoms may be modeled using the ‘particle in a box’ method. The energy of the π electrons is given by the following equation:
where n is the quantum number (n = 1, 2, 3, …), h is Planck’s constant, m is the mass of
Using this semi-empirical formula with B = 65.01 nm calculate the value of the wavelength (nm) for octatetraene (CH2=CH–CH=CH–CH=CH–CH=CH2).
Model Answer
From the given semi-empirical formula, the wavelength (nm) is calculated as
Derive equation 1 (an expression for the wavelength (nm) corresponding to the transfer of an electron from the HOMO to the LUMO) in terms of k and the fundamental constants, and hence calculate theoretical value of the constant Bcalc..
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Model Answer
Solve the equation to obtain: k1 = 14.92, k2 = - 0.355 (Eliminated).
Thus, k = 15.
So, the formula of polyene is:
CH2=CH–(CH=CH)13–CH=CH2
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Model Answer
For polyene with k = 15 ; ΔE = 199 kJ·mol–1.
Taking the value of k = 5; ΔE = 415 kJ·mol–1
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Levels with the same energy are said to be degenerate. Draw a sketch showing all the energy levels, including any degenerate levels, that correspond to quantum numbers having values of 1 or 2 for a cubic box.
Model Answer
E111: only a single state.
E112: triple degenerate, either nx, ny or nz can equal to 2.
E122: triple degenerate, either nx, ny or nz can equal to 1.
E222: single state.
Energy diagram:
Cubic box
E222: _____
E122: _____ _____ _____
E112: _____ _____ _____
E111: _____
Energy