ChemE 240, Homework 3 Solutions Assigned: February 1, 2007 Due: February 8, 2007 Topics: Equilibrium and stability criteria. 1. Determine the sign of when the magnetization density is given by

.

( ) nVSHT ,,/∂∂

)(/ 010 TTbbVM −+=

2. At low magnetizations the free energy of the magnet is given by

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21/ kmnA =

At high values of the magnetization the free energy of the magnet is given by

cmmhnA +−= 20)(

21/

a) Determine the equations of state H=H(T,m) for the two phases.

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b) Determine the chemical potentials ( ) VMTnA ,,/∂∂=μ for the two phases.

c) Show that HmnA −= /μ in both cases.

d) For a fixed temperature, find the magnetic field strength at which there is a discontinuity in m. Express your answer in terms of k, h, c and m0. 2

e) For that same temperature, determine the size of the discontinuous change in m. Express your answer in terms of k, h, c and m0.

3) For a rubber band with the entropy, S, related to the energy E>0, and the length according to

γ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=

2

0

12ll

nEcnS

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where 1ol l ≤ , c>0. The requirements of positive tension, , and stability set upper and lower bounds on the possible value of the exponent γ (i.e.,

0f ≥

L Uγ γ γ< < ). Determine Lγ and Uγ .

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4) Examine the following phase diagrams. For each, not if it is plausible or not. If not, explain why.

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