3.012 Quiz 4 3.012 12.15.03 Fall 2003 100 points total 1. Thermodynamics. (50 points) a. Shown below is a portion of the nickel-titanium binary phase diagram. Using the diagram, answer the questions below. X B1 X B2 3.012 Quiz 4a 1 of 6 12/20/03
i. (7 points) For the two compositions marked X B1 and X B2 at T = 1200 C, fill in the table below with the phases present, phase compositions, and phase fractions. For the phase fractions, mark composition points on the T = 1200 C isotherm with letters and write the phase fractions as whole numbers or as ratios of line AB segments, i.e.. Note that the composition in mole fraction (i.e. atomic %) is AC given along the top x-axis of the phase diagram. The chart is filled in as follows (letter points are marked on the copy of the phase diagram given below): Overall composition Phases present Composition of phase Phase fraction X B = X B1 0.48 f = 1 X B1 X B2 L X B 0.51 L X B 0.60 f = BC AC f L = AB AC ii. (5 points) Mark the invariant point(s) on the diagram with a filled circle and the congruent transition(s) with a triangle (exclude the congruent transitions expected at the pure A and pure B compositions). Write below the equilibria that exist at the invariant point(s) you marked: The phase diagram shows one congruent point (the melting of phase to liquid) and two invariant points- a eutectic and a peritectic: L β + L + δ 3.012 Quiz 4a 2 of 6 12/20/03
A B C X B1 X B2 3.012 Quiz 4a 3 of 6 12/20/03
iii. (8 points) In the space below, draw a reasonable plot of molar free energy vs. composition (X B ) which shows the molar free energy of the phases liquid, β,, and δ at T = 1200 C. Draw common tangents where appropriate to indicate regions of two-phase equilibrium. If you need a couple of tries to get it right, be sure to mark the right graph clearly! β δ L µ A,o µ B,o β + L L + + L L L X B iv. (5 points) On the free energy diagram you drew above, mark the chemical potentials µ A and µ B for a sample with composition X B1 at T = 1200 C. Via the ibbs-duhem equation, the chemical potentials are the endpoints of the tangent to at X B = X B1. 3.012 Quiz 4a 4 of 6 12/20/03
b. The simplest model for paramagnetic materials was proposed by Madame Curie. In this model, we consider that atoms in a magnetic field have two magnetic energy levels, which correspond to the degree of freedom that the magnetic dipoles may be aligned either parallel or anti-parallel with the applied field. For this simple system the energies of the two states may be taken as: ε 1 = 0 ε 2 = 2µ B o Where µ o here is not chemical potential, but rather the magnitude of the magnetic dipole moment. B is the applied magnetic field strength. The magnetic moment of the atom in each of the two states is: µ = µ µ 2 = µ 1 o o i. (5 points) Calculate the partition function of one atom of the Curie paramagnetic material. all _ states ε j ε 1 ε 2 2µ o B q = e = e + e = 1+ e j ii. (6 points) What are the probabilities of finding the atom in each of the two energy levels as a function of temperature? ε 1 e 1 p 1 = = q 1+ e p 2 = ε 2 2µ ob 2µ o B e e = q 2µ ob 1+ e 3.012 Quiz 4a 5 of 6 12/20/03
iii. (7 points) What value does the average magnetic moment of an atom in the Curie model approach as temperature approaches infinity? (Show your work). all _ states e e e µ = p µ j j = µ 1 p 1 + µ 2 p 2 = µ 1 + µ 2 = µ o j 1+ e 1+ e 1+ e 1+ e ε 1 ε 2 2µ ob 2µ o B µ µ o 1 e = o 2µ ob 2µ ob 2µ ob 2µ ob From the above equation, as T, the average magnetic moment approaches zero. c. (7 points) Lattice models of materials can be used to predict the properties of many thermodynamic systems. The regular solution lattice model which we derived in class is found to break down (it is unable to make correct predictions) for materials where the two components of the solution are strongly interacting either with one another or with themselves- e.g. through strong hydrogen bonding or electrostatic interactions. Explain why the regular solution model should fail in such situations. Both the entropy of mixing and enthalpy of mixing calculated for the regular solution model rely on an assumption of perfectly random mixing- i.e. mixing is unaffected by the interactions between molecules in the system. In strongly interacting systems, this assumption will break down, and cause the mean field expressions for contacts in the system to be invalid. For example, if A and B interact more strongly than A with A or B with B, then the system will favor clustering of A molecules next to B molecules. In this situation the entropy of the system cannot be calculated by a simple counting of ways the lattice could be filled, and the number of A-B contacts will be greater than predicted by a mean field approximation. 3.012 Quiz 4a 6 of 6 12/20/03