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MSE 308 Problem Set 10: Thermodynamics of Materials - Einstein Model and Gibbs Free Energy, Assignments of Thermodynamics

A problem set from the materials science and engineering (mse) 308 course focused on thermodynamics of materials. Students are required to repeat the derivation of einstein's crystal model, calculate and plot partition function and specific heat for various temperatures using a non-excel mathematical program, comment on the similarities and differences between statistical and classical thermodynamics, derive the tangent method equation, and prove the gibbs free energy equation is zero using given volumes.

Typology: Assignments

Pre 2010

Uploaded on 08/18/2009

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MSE 308 Dept. of Materials Science & Engineering
Thermodynamics of Materials Spring 2005/Bill Knowlton
Problem Set 10
1. We derived Einstein’s famous crystal model in class. Repeat the exercise explaining all
assumptions and steps.
2. For each Einstein temperature,
θ
E, of 100K, 200K, 300K, and 500K, use a computer
program other than Excel to calculate and plot the:
a. partition function
b. Cv
for the Einstein model as a function of given E
hv
Tk
θ
=. Consider the system as a simple
cubic Einstein crystal. Of interest is the range of temperatures from 0-500K. Your plot
should be similar to figure 6.1 in Gaskell. Use a mathematical program that is NOT
Excel. Your plots should be labeled thoroughly.
3. Comment on the similarities and differences between statistical thermodynamics and
classical thermodynamics.
4. Comment on the partition function.
5. Derive the equation that we obtained for the Tangent method. Explain any assumptions
and your steps.
6. The change in volume of each component for a particular solution two component system
are given by
2
12
2
212
(1 2 )
2
o
o
VaX X
VaXX
∆=
∆=
1
Strictly using the following form of the Gibb’s-Duhem equation:
2
1
0
kk
i
Xd B
=
=
prove the Gibb’s-Duhem equation is zero using the volumes given.
1 of 1

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MSE 308 Dept. of Materials Science & Engineering Thermodynamics of Materials Spring 2005/Bill Knowlton

Problem Set 10

  1. We derived Einstein’s famous crystal model in class. Repeat the exercise explaining all assumptions and steps.

2. For each Einstein temperature, θ E , of 100K, 200K, 300K, and 500K, use a computer

program other than Excel to calculate and plot the: a. partition function b. Cv

for the Einstein model as a function of given (^) E

hv T k

θ =. Consider the system as a simple

cubic Einstein crystal. Of interest is the range of temperatures from 0-500K. Your plot should be similar to figure 6.1 in Gaskell. Use a mathematical program that is NOT Excel. Your plots should be labeled thoroughly.

  1. Comment on the similarities and differences between statistical thermodynamics and classical thermodynamics.
  2. Comment on the partition function.
  3. Derive the equation that we obtained for the Tangent method. Explain any assumptions and your steps.
  4. The change in volume of each component for a particular solution two component system are given by 2 1 2 2 2 1 2

o o

V a X X V a X X

1

Strictly using the following form of the Gibb’s-Duhem equation: 2

1

k k^0 i

X d B

∑^ ∆^ =

prove the Gibb’s-Duhem equation is zero using the volumes given.

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