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）︰103/12/5 考試時限（分鐘）：110分鐘 試題 : Physical Chemistry I: Thermodynamics Mid-term Exam #2 Date: 12/5/2014 1. (25%) Consider a one-component system dU = TdS - PdV + μdn and U = TS - PV + μn. Answer the following questions. (a) Use Legendre transform to derive a thermodynamic potential that has (1/T, V, μ/T) as its natural variables. Let's call this function Y = Y (1/T, V, μ/T). Give the definition of Y and its total differential. (hint: start from the entropy). ┌ ┐2 ╭ ∂p ╮ │╭ ∂V ╮ │ (b) Show that C - C = -T(d)│───│ ││───│ │ P V ╰ ∂V ╯T,n│╰ ∂T ╯p,n│ └ ┘ 2╭ ∂(ΔG/T) ╮ (c) Show that ΔH = -T │──────│ ╰ ∂T ╯P_i,P_f;n_i,n_f _ ╭ ∂^2μ╮ (d) Show that C = -T│────│ P ╰ ∂T^2 ╯P (e) Show that within a small temperature range the natural logarithm of an equilibrium constant (ln K) is a linear function of the inverse temperature (1/T). Sketch a plot of ln K vs. 1/T for an endothermic reaction to explain how this can be used to measure the reaction enthalpy and entropy. Clearly state the approximations involved. _2 _ 2. (20%) Consider a van der Waals gas (P + a/V )(V - b) = RT. ╭ ∂S ╮ (a) Calculate │───│ for a mole of van der Waals gas. ╰ ∂P ╯T ╭ ∂U ╮ (b) Calculate │───│ for a mole of van der Waals gas. ╰ ∂V ╯T 3. (25%) At 2000 ℃ water is 2% dissociated into oxygen and hydrogen at a total pressure of 1 bar. In the following, you must use Gibbs free energy to explain your answers when applicable. (R = 8.314 x 10^-2 L bar K^-1 mol^-1) (a) Give the total balanced chemical equation. Calculate K for the equilibrium. (b) Calculate the standard Gibbs energy of formation for water at 2000 ℃. (c) Assuming that all gases are ideal, calculate the standard reaction enthalpy (Δ_r, H^0) and entropy (Δ_r, S^0) for the water dissociation reaction. (d)(10%) Predict the percentage of dissociated water at a total pressure of 1 bar at 2500 ℃. 4. (20%) In this problem, we consider vaporization of pure liquids. http://ppt.cc/KXDB (a) Drive the Clapeyron equation and show that the Clapeyron equation reduces to dP P(Δ_vap)H ── = ─────── dT RT^2 for vaporization and sublimation. (b) On the right we show experimental vapor pressures of several pure liquids as a function of temperature. Explain why the lines are linear. What does the slopes tell you? (c) Estimate the molar enthalpy of vaporization for water from this plot. (d) How do you expect the curve to change at a wider temperature range? Sketch a qualitative curve of ln P as a function of inverse temperature. 5. (20%) Let's consider liquid-vapor phase transition for water . At 100 ℃ and 1 atm, the specific heat capacity of liquid water and steam is 75.3 Jmol^-1K^-1 and 37.5 Jmol^-1K^-1, respectively. The heat of vaporization of water is 40.7 kJ/mol. The critical point of water is (217.7 atm, 374 ℃). (a) Sketch a plot of the molar enthalpy of water as a function of temperature from T = 90 ℃ to T = 110 ℃ at 1 atm. Sketch the same plot for the heat capacity. What is the order of the phase transition here? Explain. (b) Give the equation that would allow you to calculate the chemical potential from T = 90 ℃ to T = 110 ℃ at 1 atm. Sketch a plot of μ vs. T for water around 100 ℃. (c) Now qualitatively sketch a plot of the molar enthalpy of water as a function of temperature from T = 364 ℃ to T = 384 ℃ at 217.7 atm. Sketch the same plot for the heat capacity. What is the order of the phase transition here? Explain. (d) Sketch a plot of μ vs. T for water at 217.7 atm and around 374 ℃. 6. (10%) For a pure system, its chemical potential at the gas phase (μ_G), liquid phase (μ_L), and solid phase (μ_S) are functions of the temperature. Consider water at 1 atm in this question. (a) Sketch the three functions, μ_G(T), μ_L(T), and μ_S(T), for water on the same plot from T = -10 ℃ to T = 110 ℃ to explain the ice → liquid water → steam transition. You only need to qualitatively sketch the curves, but a clear explanation of the features of the curves (e.g. the signs of the slopes) is required. (b) Now consider that salt is added to the water and we further assume that the ionic compound does not go into either the gas phase or the solid phase. What happens to the chemical potentials at 1 atm? What physical phenomena are predicted by the chemical potential plot (μ_G(T), μ_L(T), and μ_S(T)) of water with salt added? --

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