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

Problem Set 11 Solutions

1. In class, you were given:

1 2mixJF cX X

mole⎛ ⎞∆ = ⎜ ⎟⎝ ⎠

.

a. Find 1 2 and F F∆ ∆ . b. Check your answers by substituting them into the relation:

1 1 2 2mixF X F X F∆ = ∆ + ∆ .

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

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

2. Similar to problem 1, the Helmholtz free energy of mixing is given by: 2 2

1 2mix oJF a X X

mole⎛ ⎞∆ = ⎜ ⎟⎝ ⎠

.

a. Find 1 2 and F F∆ ∆ . b. Check your answers by substituting them into the relation:

1 1 2 2mixF X F X F∆ = ∆ + ∆ .

c. Graph 1 2 and and mixF F F∆ ∆ ∆ all on the same plot as a function of X2 over a range of ao from -5000 to 5000 J/mol plotted at 500 J/mol increments.

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

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

Part c: Blue: ∆Fmix; Red: ∆F1; Green: ∆F2 Note that the largest positive values correspond to ao = 5000 J/mol and the most negative values correspond to ao = -5000 J/mol.

0 0.2 0.4 0.6 0.8 1X2

-1500

-1000

-500

0

500

1000

1500

2000

∆F

xim,∆F 1

,∆F 2

HJ lom

L

ao: −5000 to 5000 plotted at 500Jêmol increments

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

3. In the following binary system, the partial molar heat of mixing of component 1 can be fitted by the expression: .

21 2 1o

JH a X Xmole

⎛ ⎞∆ = ⎜ ⎟⎝ ⎠

.

a. Calculate the heat of mixing and the partial molar heat of mixing of component 2.

b. Graph the partial molar heat of mixing for both components and the heat of mixing all on the same plot as a function of X2 over a range of ao from -10 to 10 J/mol plotted at 1 J/mol increments.

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

Part c: Blue: ∆Hmix; Red: ∆H1; Green: ∆H2 Note that the largest positive values correspond to ao = 10 J/mol and the most negative values correspond to ao = -10 J/mol. The fact that ∆H2 does not go to 0 when X2 is not present suggests an unrealistic solution.

0 0.2 0.4 0.6 0.8 1X2

-2

-1

0

1

2

∆H

xim,∆H 1

,∆H 2

HJ lom

L

ao: −10 to 10 plotted at 1 Jêmol increments

4. In problem 6 of the last problems set, you proved the Gibb’s-Duhem equation strictly

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

10k k

iX d B

=

∆ =∑

when given the change in volume of each component for a particular solution with two components. Make up your own problem analogous to problem 6 and provide the solution to the problem. Open ended problem.

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