课程名称︰普通化学甲下 课程性质︰ 课程教师︰苏志明 开课学院：工学院 开课系所︰化工系 考试日期（年月日）︰2007/4/11 考试时限（分钟）： 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1. The general Schrodinger equation for a particle with mass m in one=dimensional space under the action of a potential V(x) is h^2 d^2Ψ _ ＿＿ ＿＿＿ + V(x)Ψ = EΨ 2m dx^2 (a) Consider the particle in a one-dimensional box problem. The potential is V(x) = ∞ x≦0 = 0 0 < x < L = ∞ x≧L Write the Schrodinger equation for this special case, and also describe the expected general form of the eigen-wave funtion Ψ over the intire coordinate space. (b) Starting from the trial function Ψ = Asin(kx) + Bcos(kx) and the boundary values imposed by the above potential, obtain the eigen-wavefunction Ψ and eigen-energy E. Note that the final wave function should be normalized. (c) Sketch the probability distributions of the first and second quantum states over the entire box space. (d) Consider a ping-pong ball moving in a 1 meter one-dimensional box. Assuming that the ball is a point mass of 1g, and moving with a speed of 1m/sec, what would be the corresponding quantum number of this dynamical system in Schrodinger's world? 2. (a) What is black body radiation? Why the radiation is emphasized by the adjective "black"? How do you construct a black body radiation source (or device)? (b) In the course of searching for the explanation of the black body radiation, a phrase so-called "ultraviolet catastrophe" emerged. What does it mean? (c) What is the major scientific contribution of Louis de Broglie to physics? 3. For hydrogen-like atoms, the wave functions for the 1s and 3s are Ψ(1s) = π^(-1/2) * (Z/a)^(3/2) * e^(-σ) Ψ(2s) = 1/4 * (2π)^(-1/2) * (Z/a)^(3/2) * (2-σ) * e^(-σ/2) Ψ(3s) = 1/81 * (3π)^(-1/2) * (Z/a)^(3/2) * (27-18σ+2σ^2) * e^(-σ/3) in which σ = Zr / a, a = 5.29*10^(-11) m (a) Determine the radius of the most probable radial probability of the 1s state, and the radii of the maximum radial probability of the 2s state(there are more than 1 local maximum for the 2s state). (b) We could define the size of the hydrogen 1s orbital as being the sphere with a radius that enclosed 90% of the total electron probability. From the above 1s wave function, calculate this radius. (c) Calculate the position of the nodes for the 2s and 3s states. (d) Calculate the relative probability ratio of finding the electron between the 2s and 3s states centered at the nucleus(i.e. the origin posotion). (e) For the 2s and 3s states, calculate their relative probability ratio of finding the electron in a shell located at a radius distance of 300 pm. 4. (a) What does the abbreviation VSEPR stand for? Write down its full English expression. Also, explain the basic working rules of the VSEPR model. (b) Naming and also drawing the five basic geometric structures adapted in the VSEPR model. 5. The following two diagrams are the Downs cell for the electrolysis of molten sodium chloride, and the mercury cell for production of chlorine and sodium hydroxide. Fill out the blank blocks labeled with A, B, C, D, etc. (有图) ------------------------------------------------------------------------------ Some constants: Charge of an electron = 1.60 * 10^(-19) C Mass of an electron = 9.11 * 10^(-31) kg Planck's constant h = 6.63 * 10^(-34) Js Speed of light = 3.00 * 10^(8) m ∫(sin^2(ax)dx = x/2 - sin(2ax)/4a ∫(cos^2(ax)dx = x/2 + sin(2ax)/4a ∫e^(ax) dx = e^(ax) / a ∫xe^(ax) dx = e^(ax) * (ax-1) / a^2 ∫x^m * e^(ax) dx = (x^m * e^(ax))/a - (m/a) * ∫x^(m-1) * e^(ax) dx --

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