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: = [PFree] = 100 nM, and [Site] = 10 Kd 0.5∙10–6, μM. Problem 18. Particles in a Box Problem and Conjugated Polyenes The energy levels of in molecules with conjugated bonds can be calculated with -electrons 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 as -electrons independent in a This model is useful for determining the energies of -electrons “particles box.” and the electronic spectra of such molecules as ethylene or molecules with conjugated double bonds. In this problem, use the in a model to describe the states of -electron “particle box” ethylene as well as linear and cyclic conjugated molecules. The particle in a box model yields the energy levels for by treating them as -electrons moving freely along the length of the conjugated An example of a hydrocarbon with a -bonds. non-branched chain of conjugated is shown below. -bonds trans-1,3,5-hexatriene The allowed quantum states occur for electronic wavefunctions that have wavelengths of  = where is an integer starting with = 1 and is the length of the molecule. The nL/2, n n L effective molecule lengths are = 289 pm for ethylene and = 867 pm for L L The allowed energy states for the -electrons are given by Eq. 1. (Eq. 1) In Eq. 1, is the quantum number for the energy state and is an integer between 1 and n ∞, is the constant in is the mass of the electron in kilograms and is the length of h me L Planck’s J∙s, 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 in 1,3,5-hexadiene. -electrons
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.
44th_International_Chemistry_Olympiad-_Preparatory_Problems.docx