課程名稱︰物理化學一-熱力學 課程性質︰必修 課程教師︰鄭原忠 開課學院：理學院 開課系所︰化學系 考試日期（年月日）︰103/10/31 考試時限（分鐘）：110分鐘 試題 : Physical Chemistry I: Thermodynamics Mid-term Exam #1 Date: 10/31/2014 1. (35%) The van der Waals equation of state is given by the equation _2 _ (P + a/V)(V - b) = RT (a) Show that the behavior of a van der Waals gas approaches ideal gas behavior in the limit at a constant temperature. _ (b) Find V of a van der Waals gas at a constant T in the limit P → ∞ (high P limit). What does the result mean? (c) Calculate the second virial coefficient in terms of volume (B) for a van der Waals gas in terms of the van der Waals constants. (hint: 1/(1 - x) = 1 + x + x^2 + ...). (d) Give the Boyle temperature of a van der Waals gas. (e) Find the critical temperature of a van der Waals gas in terms of the van der Waals constants. (f) Draw an isotherm of the van der Waals equation at a temperature lower than the critical temperature. Note that some of the points on this curve are unphysical. Point out these points on the curve that you draw and explain why. (g) Consider P = P(n,V,T) for the van der Waals gas. Write down the total differential of the pressure P. 2. (20%) In a reversible adiabatic expansion of an ideal gas with γ = C_P / C_V independent of temperature, the pressure and volume are related by γ PV = const. (a) Show that the work of adiabatic expansion from P_1, V_1 to P_2, V_2 is w = (P_2V_2 - P_1V_1)/(γ - 1) (b) Calculate the temperature increase and final pressure of helium if a mole of helium is compressed adiabatically and reversibly from 44.8 L at 0 ℃ to 22.4 L. (Hint: assume helium an ideal gas). 3. (25%) Answer the following questions regarding Joule-Thomson expansion: (a) Describe Joule-Thompson expansion and sketch an experimental apparatus that demonstrates this phenomenon. (b) Show that the process is isoenthalpic. (c) Define the Joule-Thompson coefficient μ_JT. Show that μ_JT = 0 for an ideal gas. (d) Describe the temperature dependence of the Joule-Thompson coefficient of a real gas. Explain the trend? (e) Given that the inversion temperature for helium is 43 K, describe what happens if a cylinder of pressurized helium is opened to blow helium gas on a thermometer at the room temperature. 4. (10%) Calculate q, w, ΔS, ΔU, and ΔH for mixing 2 mol of H2 with 1 mol of O2 at 300K under conditions where the gases are ideal and no chemical reaction occurs. 5. (20%) The P-V diagram on the right depicts a cycle of a heat engine that uses 1/2 mole of an ideal gas as its working fluid. The curved part a-b of the cycle is an adiabatic process. Answer the following questions (R = 0.082 L atm K^-1 mol^-1): http://ppt.cc/VCAw (a) How much heat enters this gas per cycle, and where does it happen? How much heat leaves this gas in a cycle, and where does it occur? (b) How much work does this engine do in a cycle? What is the thermal efficiency of the engine? 6. (10%) Consider a composite system with two compartments linking by a thermal conductor. Answer the following questions: http://ppt.cc/KE3a (a) Use the second law to show that T^(1) = T^(2) at equilibrium. (hint: at equilibrium, entropy is maximum therefore any small displacement in energy should not change the entropy in the first order). (b) Show that Clausius' statement of the second law: "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time. Can be derived from the maximum entropy principle. --

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