MAT E 640
Advanced Thermodynamics in Materials
Department of Chemical and Materials Engineering
Final Exam
December 19, 2011 180 minutes
Answer all questions. State any assumptions you make and explain your answers. The total number of marks is 70.
1. Choose true or false to the following statements. (14 marks)
T F Based on a statistical approach to entropy, the equilibrium state represents the
most probable distribution for that system.
T F The entropy for all real materials goes to zero as the temperature approaches absolute zero.
T F Increasing under-cooling can always promote the nucleation rate, but suppress the growth rate.
T F The Gibbs free energy change associated with the mixing of different ideal
gases at constant temperature is only due to a change in the configurational entropy of the gases.
T F For the van der Waals equation of state, the constant b is a correction factor for the finite volume of the particles in a real gas, while the constant a is a correction factor for temperature.
T F The process of liquid-liquid phase separation always occurs by nucleation and growth.
T F The critical nuclei size for homogeneous nucleation is larger than that for heterogeneous nucleation at the same temperature and pressure.
2. A binary solution has activity coefficients, γ1 and γ1, given by
where x1 and x2 are the mole fractions, where Tm is the melting
point and β = 4 ln5 . (Note, this is not a regular solution since α is not a constant)
a) What is the free energy of mixing, ΔGM, upon forming the binary solution (in terms of R, T, α, and x1)? (4 Marks)
b) Find the change in partial molar enthalpy of component 1 upon mixing of this binary solution (in terms of R, α, β, Tm, and x1), H1 − H10 . Is this quantity positive or negative? What does this tell you about the interaction between components 1 and 2? (6 Marks)
c) What are the two spinodal compositions for this solution at T = 0.8Tm ? (4 Marks)
3. For a subregular solution, solution model is given by:
a) Determine ΔGM(X)S (3 Marks)
b) Determine ΔHM(X)S and ΔSM(X)S (4 Marks)
c) Find ΔG 1(X)S and ΔG 2(X)S (4 Marks)
d) Determine activities a1 and a2 . (4 Marks)
4. A pressure-volume isotherm is shown below for the CO2 van der Waals gas.
a) What is the approximate value of the equilibrium vapor pressure? How do you determine this value? [3 marks]
b) What is the pressure of the system when the molar volume for the liquid is 0.125L? [2 marks]
c) What are the molar volumes of the liquid and vapor when the two phases are in equilibrium? [3 marks]
5. For the Au-Sn phase diagram given below draw schematics of plausible molar free energy curves showing the common tangent construction as a function of composition, x Sn, at T1=350 。C (30 at.%< x Sn<100 at.%) and T2=700 。C (0 at.%< x Sn<100 at.%). (10 Marks)
6. Use above Au-Sn phase diagram, predict the Ωl and estimate the heat (enthalpy) of melting for pure Sn (assuming the liquid solution is regular). (9 Marks)