(a) Explain how the spatial localisation of waves leads to the quantisation ofencrgy. (b) Starting from the de Broglie relationship and the equation for kinetic energy,show that the allowed energy

levels for a wave on a circle are: E=\frac{n^{2} h^{2}}{2 m C^{2}} where C is the circumference of the circle and n is an integer. The R-clectrons of benzene (CHg) can be treated as waves localised on a “circle"with a circumference that equals six times the carbon-carbon bond length. In thissimplified model, the lowest energy level (n=0) is singly degenerate, and all theother levels are doubly degenerate. Assuming a carbon-carbon bond length of 140 pm, usc the equation derived(c)in (b) to calculate the energies of the first four allowed levels for the T-clectrons of benzenc. (d) Calculate the wavelength of the lowest energy transition in the UVabsorption spectrum of the x-clectrons of benzene. (c) Compare the results from (c) and (d) with: O the measured value of 180 nm for the lowest energy transition in theUV spectrum of benzene; (i) the measured values of 9.4 eV and 12.3 eV for the ionisation potentialsfrom the occupied benzene R-orbitals. What is the most likely explanation for any discrepancies? \left[\text { Mass of an electron }=9.1 \times 10^{-31} \mathrm{~kg} ; 1 \mathrm{pm}=1 \times 10^{-12} \mathrm{~m} ; 1 \mathrm{~J}=6.24 \times 10^{18} \mathrm{eV}\right]