課程名稱︰普通化學甲下 課程性質︰系定必修 課程教師︰蘇志明 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰2008/03/21 考試時限（分鐘）：180 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. The Schrodinger equation for a particle with mass m moving in one-dimensi- onal space under the action of a potential V(x) is h^2 d^2Ψ(x) _ _________ ________ + V(x)Ψ(x) = E*Ψ(x) 8π^2 * m 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 function Ψ over the entire 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 functions should be normalized. (c) Consider a ping-pong ball moving in a one-dimensional box with a length of 1 meter. 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 the Schrodinger's world? With this total energy, what would be the "energy gap" between the neighboring quantum states? (d) Assuming that the mass of the ping-pong ball is exact to be 1g, and the accuracy of the speed is determined to be within ±1. 0 *10^(-6)m/sec .From Heisenberg's uncertainty principle, what would be the measurement uncertainty for the position of the ping-pong ball set by the fundamental physical law? 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 serching for the explanation of the black body radiation, a phrase so-called "ultraviolet catastrophe" emerged. What does it mean? 3. For hydrogen-like atoms, the wave functions for the 1s, 2s and 3s states 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 Now consider the case of a simple hydrogen atom and answer the following questions: (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 radius position of the nodal surfaces (they are spherical surfaces) 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 position). (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) In the electrolysis of an aqueous solution of Na2SO4, What reactions occur at the anode and the cathode (assuming standard conditions)? S2O8 2- + 2e- ---------> 2SO4 2- E0 = 2.01 V O2 + 4H+ + 4e- ---------> 2H2O E0 = 1.23 V 2H2O + 2e- ---------> H2 + 2OH- E0 = -0.83 V Na+ + e- ---------> Na E0 = -2.71 V (b) When water containing a small amount (~0.01M) of sodium sulfate is electrolyzed, measurement of the volume of gases generated consistently gives a result that the volume ration of hydrogen to oxygen is not quite 2:1. To what do you attribute this discrepancy? Predict whether the measured ratio is greater than or less than 2:1. 5. A zinc-copper battery is constructed as follows: Zn | Zn2+ (0.10 M) || Cu2+ (2.5M) | Cu The mass of each electrode is 200g. The standard half-reactions are: Zn2+ + 2e- ---------> Zn(s) E0 = -0.76 V Cu2+ + 2e- ---------> Cu(s) E0 = 0.34 V (a) Calculate the cell potential when this battery is first connected. (b) Calculate the cell potential after 10.0 A of current has flowed for 10.0 hour. (Assume each half-cell contains 1.00L of solution and also assume the battery behaves ideally.) (c) Calculate the mass of each electrode after 10.0 h. (d) How long can this battery deliver a current of 10.0 A before it goes dead? 6. A galvanic cell is based on the following half-reactions: Fe2+ + 2e- ---------> Fe(s) E0 = -0.440V 2H+ + 2e- ---------> H2(g) E0 = 0.000V In this cell, the iron compartment contains an iron electrode and [Fe2+] = 1.00 * 10^(-3) M, and the hydrogen compartment contains a platinum electrode, P(H2(g)) = 1.00atm and weak acid, HA, at an initial concentration of 1.00 M. If the observed cell potential is 0.333 V at 25℃, calculate the Ka value for the weak acid HA at 25℃. 7. The following two diagrams are a standard hydrogen electrode, and the mercury cell for production of chlorine and sodium hydroxide. Fill out the bland blocks labeled by A, B, C, D, etc.. ----------------------------------------------------------------------------- Some constants and equations: Mass of an electron = 9.11* 10^(-31) kg Charge of an electron = 1.60 * 10^(-19) C Speed of light = 3.00 * 10^8 m Planck's constant h = 6.63 * 10^(-34) Js --

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