ChemE 240, Homework 4

Assigned: February 8, 2007

Topics: Equilibrium and stability criteria, the VDW equation of state, the Maxwell
construction method.

1) Derive the stability criterion for:

2) Consider two phases of a 1-component system. Let’s call them α and β. They have
the following Helmholtz free energies, where n, ε, and νo are positive constants.

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nε ⎛ν ⎞ nε ⎛ν ⎞
Aα = b(T )⎜⎜ o ⎟⎟ Aβ = c(T )⎜ o ⎟
⎜ν ⎟
2 ⎝ν α ⎠ 3 ⎝ β ⎠

At coexistence, the following is true for the molar volumes of each phase:

9 ⎛c⎞ 3⎛c⎞
να = ⎜ ⎟ν o ν β = ⎜ ⎟ν o
16 ⎝ b ⎠ 4⎝b⎠

Now, let’s make a specific choice for b(T) and c(T), where θ is also positive.

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 2 3
⎛T ⎞ ⎛T ⎞
b(T ) = 2 − ⎜ ⎟ c(T ) = 2 − ⎜ ⎟
⎝θ ⎠ ⎝θ ⎠

a) Calculate the latent heat, ΔHαβ= T(Sβ - Sα) for the transition from phase α to phase β.
Express your result in terms of the ratio b/c and other system properties.

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b) Sketch the curve along which phases α and β coexist in the plane of pressure and
temperature. Focus on the behavior near T = θ. Also, note that b/c ≈ 1 for T near θ.
Describe the phase transition(s) that would occur as temperature is increased at constant
pressure.

3) Consider the van der Waals equation of state:

a) Explain how you would find the values of the pressure, temperature and molar volume
at the critical point (Pc, Tc and Vc respectively) in terms of the constants a and b. You
don’t need to do the math, just show me your method.

∂p ⎛ ∂2 p ⎞
At the critical point we have ⎛⎜ ⎞⎟ = 0 and ⎜⎜ 2 ⎟⎟ = 0 . Solve the VDC equation for p
⎝ ∂v ⎠ T ⎝ ∂v ⎠ T
and take these derivatives. Solve for the constants a and b in terms of Tc and vc and then
plug back into the VDW equation to solve for Pc.

b) Using the Maxwell Construction method discussed in class determine the molar
volumes of the equilibrium phases of oxygen at T = Tc/1.05 and T = Tc/1.15.
If possible construct the binodal curve for oxygen on a P-v phase diagram. Note that for
oxygen a = 1.378 atm-L2 mol-2 and b = 0.03183 L mol-1. Also, for oxygen Pc=50.37 atm,
Tc=156.3 K, and vc=0.0955 L/mol.

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b)

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4) Problem 2.18 in Chandler, including application of the Gibbs adsorption isotherm.

5) The first programming assignment involves exploring random number generators and
arrays using Matlab. By the end of this assignment you should be able to plot variables
and write short M-files. Here’s the assignment:

a) Generate a sequence of N random numbers uniformly distributed on the interval (0,1).

b) Calculate their mean and standard deviation. How do these change as N is varied?
Written comments are fine here (I don’t need to see your code).

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c) Compute the following function, starting at m=0, where the brackets indicate average
quantities: g (m) = xn x n+ m . What do you expect this result to be? Is it possible to
calculate the result by hand? How do the simulated results compare? Make a plot of
g(m) and comment on it. Print out your code and a plot of g(m) to give to me.

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