NAME_______________________________

MSE 508: Solid State Thermodynamics Department of Materials Science and Engineering

Boise State University Spring 2011

Practice Final Exam

May, 2011

Problem Total Points Points Obtained

Grand Total:

2

1. Compute the change in the entropy as a function of T and P when one mole of kypton (Kr) is compressed from an initial temperature and pressure of Ti and Pi, respectively, and a final condition of temperature and pressure of Tf and Pf, respectively. The heat capacity of krypton is given by Cp. Assume krypton is an ideal gas. The idea is to calculate the change in entropy as a function of the final and initial temperatures and pressures and a function of R. a. Make the calculation using the classical thermodynamics approach.

3

b. Repeat the calculation using the results of statistical thermodynamics.

c. Compare your answers. Are they the same? Why or why not?

4

2. Using the definition of S and , where: lnbS k and

0

!

!ri i

n

n

,

use Sterling’s approximation, given by: ln ! lnx x x x ,

to derive:

0

lnr

iB i

i

nS k n

n

.

5

3. In class, we have seen that the translational partition function of a monatomic gas is:

3/ 22

translational

kTV

m

. (1)

Using the partition function, you have calculated the entropy to be: 3/ 2

2 3ln

2o o

kTS N k V N k

m

(2)

Using the Classical Thermodynamics definition of entropy, S, find the coefficient relationship for pressure. Use equation (2) to derive an equation for pressure. Comment on your answer. Show all work. (10 pts)

6

4. Prove that for an ideal gas that the internal free energy for one gas atom is 3/2 kbT. That is, the thermal energy for an ideal gas atom is U = 3/2 kbT.

7

5. Describe (you do not have to derive the full model, but provide a step-by-step description) how you would determine the heat capacity using Einstein’s approach.

8

6. For Einstein’s model of heat capacity, sketch the plot of heat capacity versus T/ФE. Then add to the plot heat capacity data that is typical (i.e., normal). Describe the differences between the two plots and why

9

7. Consider a two component (A and B) system in which the binary phase diagram is a eutectic phase diagram. Sketch a Gibb's Free Energy of Mixing (Gmix) versus composition (XB) diagram at the eutectic temperature. Assume that all the Free Energy Curves are normalized to the same reference state. Label all important temperatures and compositions on your sketch. Use the tangent method to show that your diagram is correct. (5 pts)

10

8. Given ;mixG and ;mixG :(10 pts)

a. Normalize to the reference state ; . Your answer should be simplified

algebraically as much as possible without using assumptions. Assume that your initial states follow the simple regular solution model. SShhooww aallll wwoorrkk.

b. Sketch the Gibb's Free Energy versus Temperature plots for each Gibb's Free Energy of phase transitions you derived in your answer. Label all important aspects of the plot.

c. Sketch your answer of both normalized Gibb's Free Energy states on a mixG

versus composition. Label all important aspects of the plot. d. Describe the solution models we have covered. ( Where does the Redlich-Kister

model belong in your category? Describe the Redlich-Kister.

11

9. On a ∆Gmix verses 2

pX plot, sketch ;mix

G and ;L

mixG curves at some

constant temperature with a tangent line, drawn from 2 2

0 to 1 ,p pX X connecting

the two curves and showing the equilibrium conditions. On the p

mixG axis, provide

two simple equalities (i.e., of the form ? ?

? ?G G ) with one at

20 p

X and another

one at 2

1pX .

10. Using the equilibrium conditions from above, derive the two equations with two unknowns for determining the β + L two phase boundary curves.

12

11. Given the information for and in figure 10.13 of DeHoff, do the following (Note that the y-axis label should read 2mD ):

a. Plot 2mD as a function of X2 to obtain both plots in figure 10.13.

b. Plot mixGD as a function of X2 at the given temperatures.

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c. Determine the values of 12X a and 2

2X a for the spinodal region shown in figure

10.12 at temperature the given in figure 10.13. d. Determine whether or not the values of 1

2X a and 22X a are at the maximum and

minimum, respectively, for 2mD .

e. Comment on your findings using the plots to help support your comments.

14

12. Very near future technology for cutting edge metal-oxide-semiconductor field effect transistors (MOSFETs) require a higher dielectric constant (k) material for the gate oxide of the MOSFET (see cross sectional view below).

One potential method to form a new gate oxide material on Si is to first grow the current gate oxide material, SiO2, which is nearly defect free. Then deposit a metal that will form, upon annealing (heating), a new metal oxide. Two potential high-k oxide materials candidates are yttrium oxide (Y2O3) and aluminum oxide (Al2O3). (10 pts) a. Considering an annealing temperature of 700oC, which of the two oxides would

you choose for the above process EExxppllaaiinn. b. What is the Go for the metal oxide you chose? c. What is the equilibrium constant, K, of the reaction of the metal oxide you chose? d. What is the minimum partial pressure of oxygen at the vapor-oxide interface at

the process temperature such that the new oxide will not decompose?

n-type Si Wafer

SiO2 SiO2Gate oxide

Source Contact Drain ContactGate Contact

source drainchannel

-- Vgate VDrain

drain++ ++ ++ + +++ + ++ ++ + + ++++ ++ + + +++ ++ ++ + +++ + ++ ++ + + ++++ ++ + + +

Gate Oxide MOSFET XMOSFET X--sectionsection

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13. Draw a series of free energy plots (~4-6) at key temperatures with tangent lines for 3 phases (e.g., FCC, BCC & Liquid). Using these plots, draw the complimentary T vs XB binary equilibrium phase diagram with all relevant aspects of the plot. Label all relevant aspects of the plot.

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14. Describe the key aspects and outcomes of the Quasi-Chemical Model.

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Ch. 12: Capillarity Effects in Thermodynamics 15. What are the conditions for equilibrium for a multicomponent system in which the

phases and are in equilibrium and the interface between them is convex relative to ? Equations and a brief description are adequate to answer this question.

16. Compute the pressure in a 100 micron diameter Ni droplet formed in supercooled Ni vapor at 1060oC. The vapor pressure of Ni at this temperature is half an atmosphere. The specific interfacial free energy of Ni may be taken as 2280 ergs/cm2.

18

Ch. 7: Unary Heterogenous Systems & Clausius-Clapeyron Equation 17. Compute the vapor pressure of liquid Cu over a flat surface at 1400K.

CCuu ddaattaa:: Tboil 2830 K Tmelt 1358 K Hvap 314 kJ/mole

Ch. 12: Capillarity Effects in Thermodynamics

18. Compute the equilibrium vapor pressure inside a 0.5 micron diameter bubble of Cu vapor suspended in liquid Cu with molar volume of 7.94 cc/mole (Appendix C, p.492) at 1400K. The specific interfacial free energy is 1360 ergs/cm2.

19

Ch. 11-12 Gaskell: Reactions Involving Gases and Condensed Phases 19. Lithium bromide vapor dissociates according to the reaction:

( ) ( ) 2,( )

1

2g g gLiBr Li Br

At what temperature does the partial pressure of Li reach the value of 10-5 atm when the gas is heated at a constant total pressure of 1 atm?

20

20. Consider the reaction, at pressure p, between 1 mole of SO2 gas and ½ mole of O2

gas to form an equilibrium mixture of SO2, SO3 and O2 at 1000 K given by the reaction:

SO2(g) + ½ O2(g) = SO3(g) Answer the following questions showing all steps in solving each problem.

a. Using x as your variable for moles, provide a table showing the initial amount of moles and the amount of moles upon reaction and derive an equation for the total number of moles in the system, nT, as a function of x. Fully simplify your answer. (10 points)

b. Derive equations for the partial pressure for SO2, SO3 and O2 as a function of the

total pressure, p, and of x. Fully simplify your answers. (10 points)

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c. Derive an equation for the equilibrium constant, Kp, as a function of x and total pressure, p. Fully simplify your answer. Briefly describe how you would calculate Kp and determine x. State all assumptions. (10 points)

d. Derive a polynomial equation as a function of x, P, and Kp. Fully simplify your

answer. (5 points)

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21. A CO2-CO-H2O-H2 gas moisture at a total pressure of 1 atm exerts a partial

pressure of oxygen of 10-7 atm at 1600oC. In what ratio where the CO2 and H2 mixed to produce the gas with this oxygen pressure?

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22. To what temperature must MgCO3(s) be heated in an atmosphere containing a partial pressure of CO2 of 10-2 atm to cause decomposition of the carbonate? :

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Ch. 12: Capillarity Effects in Thermodynamics 23. Using the equilibrium condition between two phases for which H = 0:

(H=curvature) d = d

Derive the Clausius-Claypeyron equation for the following cases: a. General case for two phases (Hint: TB 4 1st equation in column 1)

25

b. Condensed-vapor phase case (Hint: TB 4 3rd equation in column 1)

26

24. Using the equilibrium condition between two phases for which H 0: (H=curvature)

d = d Derive the modified Clausius-Claypeyron equation for the following cases: c. Isobaric case (assume that = solid and = vapor phase)

27

d. Isothermal case (TB 4 3rd equation in column 1) i. Transformation = solid to liquid

ii. Transformation = liquid to vapor

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e. DATA:

29

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