che359_sp10_exam2_soln - ChE 359 Exam#2 Solutions prepared...

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Page 1 of 6 ChE 359 Exam #2 Solutions prepared by Professor Lo 1. (50 points) The Haber process is commonly used to make ammonia (NH 3 ) from nitrogen gas (N 2 ) and hydrogen gas (H 2 ), by passing the reactants over an iron oxide catalyst at high temperature and pressure. We can model all of these as ideal gases, where the molecules are coarse grained into spherical particles. We will use the lattice model to study the thermodynamics of mixing when the reactants are combined; we will not study the reaction itself; that will come later in the semester. a. (10 points) The reaction chamber can be modeled by a square lattice consisting of 100 lattice sites. There are 25 nitrogen particles and 75 hydrogen particles. Assume that the nitrogen particles are indistinguishable, and the hydrogen particles are indistinguishable. First, write out the factorial expression for the multiplicity W A of the heterogeneous system (i.e., the chamber is segregated into a nitrogen-only region with 25 lattice sites, and a hydrogen-only region with 75 lattice sites), and second, write out the factorial expression for the multiplicity W B of the homogeneous system (i.e., all particles are allowed to mix freely within the reaction chamber). We first calculate W A , which we know is 1 because the particles are indistinguishable. Each of the 25 nitrogen lattice sites can only be occupied by a nitrogen particle, and each of the 75 hydrogen lattice sites can only be occupied by a hydrogen particle. W A = N N n N × N H n H = 25! 25! 25 25 ( ) ! × 75! 75! 75 75 ( ) ! = 1 We then calculate W B . In this case, each of the 100 chamber sites can be occupied by either a nitrogen particle or a hydrogen particle. There are many more possibilities! W A = N N + N H n N = 100! 25! 100 25 ( ) ! = 5.35 × 10 29 b. (10 points) Use the ratio W B /W A to calculate the change in entropy [J/K] when the particles are allowed to mix freely. We use Boltzmann’s law: Δ S = S B S A = k ln W B k ln W A = k ln W B W A = 1.38 × 10 23 J K ( ) ln 5.35 × 10 29 ( ) = 9.45 × 10 22 J K
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ChE 359 SP2010 – Exam #2 Page 2 of 6 c. (10 points) First, state the probability p N that a given lattice site is occupied by a nitrogen particle, and the probability p H that a given lattice site is occupied by a hydrogen particle in the fully mixed scenario. Second, given that the energy of any one of the indistinguishable nitrogen particles is ε N and the energy of any one of the indistinguishable hydrogen particles is H ,
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che359_sp10_exam2_soln - ChE 359 Exam#2 Solutions prepared...

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