Kinesin molecules move independently of each other. Assume that kinesin molecules bound on the microtubule sites and free kinesin molecules in solution are in a dynamic equilibrium. Use the following parameters: Kd = 0.5∙10–6, [PFree] = 100 nM, and [Site] = 10 μM. Problem 18. Particles in a Box Problem and Conjugated Polyenes The energy levels of -electrons in molecules with conjugated bonds can be calculated with varying degrees of accuracy, depending on the complexity of the model. The most sophisticated and accurate approaches involve complex theoretical methods for solving the multi-particle Schrödinger equation. A simplified yet still powerful approach is to treat the -electrons as independent “particles in a box.” This model is useful for determining the energies of -electrons and the electronic spectra of such molecules as ethylene or molecules with conjugated double bonds. In this problem, use the “particle in a box” model to describe the -electron states of ethylene as well as linear and cyclic conjugated molecules. The particle in a box model yields the energy levels for -electrons by treating them as moving freely along the length of the conjugated -bonds. An example of a hydrocarbon with a non-branched chain of conjugated -bonds is trans-1,3,5-hexatriene shown below. The allowed quantum states occur for electronic wavefunctions that have wavelengths of  = nL/2, where n is an integer starting with n = 1 and L is the length of the molecule. The effective molecule lengths are L = 289 pm for ethylene and L = 867 pm for The allowed energy states for the -electrons are given by Eq. 1. (Eq. 1) In Eq. 1, n is the quantum number for the energy state and is an integer between 1 and ∞, h is the Planck’s constant in J∙s, me is the mass of the electron in kilograms and L is the length of the box in meters. Use two significant figures for your calculations. a) Use the particle in a box model to determine the following: i. the first two energy levels for the -electrons in ethylene; ii. the first four energy levels for the -electrons in 1,3,5-hexadiene.
b) For each species, fill the energy levels with the -electrons, keeping in mind the Pauli principle for electron pairing. Identify the quantum number n of the highest occupied energy level of each species. c) Use the highest occupied and lowest unoccupied energy levels to predict the wavelength of light that can be used to excite a -electron from the highest energy occupied state to the lowest energy unoccupied state for each species. d) The molecule in carrots that makes them appear orange is -carotene. Use the particle in a box model to predict the energy gap between the highest occupied state and the lowest unoccupied state. Use this energy to determine the maximum wavelength for absorption of light by -carotene. Use a length for -carotene of L = 1850 pm. trans--carotene Some molecules have cyclic conjugated -systems. Benzene and coronene are examples of such molecules. For molecules with “circular” -electron distributions, the quantized energy levels are given by Eq. 2. (Eq. 2) In this case, the quantum number n has integer values between 0 and and R is the radius of the ring in meters. Unlike the linear particle in a box problem, the circular problem allows for both positive and negative integer values for n for clockwise and counterclockwise motion. Also, for the circular problem, n = 0 is an eligible quantum state. For this problem, assume that the ring radii are 139 pm for benzene and 368 pm for coronene. e) Describe the benzene’s -electron system using the particle-in-the-ring equation for energy levels. Draw a diagram depicting all occupied energy levels as well as the lowest-unoccupied energy level. When building the energy levels, keep in mind the Pauli principle for electron pairing and that there may be several states with the same energy referred to as degenerate states. Make sure that you use the right number of  electrons. Use two significant figures in your answers. f) Now, draw a similar energy level diagram for coronene and calculate the quantized energy values for the occupied energy levels and the lowest unoccupied energy level. Use two significant figures in your answers.
7 Actinides 90 Th 232.0 91 Pa 231.0 92 U 238.0 93 Np (237.1) 94 Pu (244.1) 95 Am (243.1) 96 Cm (247.1) 97 Bk (247.1) 98 Cf (251.1) 99 Es (252.1) 100 Fm (257.1) 101 Md (258.1) 102 No (259.1) 103 Lr (260.1)