thermo de hoff 06

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Chapter 6Statistical Thermodynamics

Notes on

Thermodynamics in Materials Science

by

Robert T. DeHoff

(McGraw-Hill, 1993).

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Combinatorial AnalysisConsider a system of N particles that are

allowed to occupy r states.

Microstate --- The description of the system that provides the state of each particle.

Number of possible microstates = Nr

!!...!...!!

!

321 rij nnnnn

N

Macrostate --- The description of how many particles, ni, are in each of the r states.

Number of microstates in a macrostate:

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

N=3, r=3E1=1 E2=2 E3=3 Esum Omega %

I 3 0 0 3 1 3.7%II 2 1 0 4 3 11.1%III 2 0 1 5 3 11.1%IV 1 2 0 5 3 11.1%V 0 3 0 6 1 3.7%VI 1 1 1 6 6 22.2%VII 1 0 2 7 3 11.1%VIII 0 2 1 7 3 11.1%IX 0 1 2 8 3 11.1%X 0 0 3 9 1 3.7%

27 100%

N=3, r=3E1 E2 E3

I ABC - - 11 AB C -

AC B - BC A -

III AB - CAC - BBC - A

IV A BC - B AC - C AB -

V - ABC - VI A B C

A C BB A CB C AC A BC B A

VII A - BCB - ACC - AB

VIII - BC A- AC B- AB C

IX - A BC- B AC- C AB

X - - ABC

N=3 and r=3

0

1

2

3

4

5

6

7

8

2 4 6 8 10

Sum

Om

eg

a

Combinatorial Analysis

Microstates

Macrostates

DistributionN=3, r=327Nr

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

AssumptionsConsider all particles to be identical.

The net value of a macroscopic property depends on the number of particles (ni) in each state (i). Exchanging the specific identity of the particles in a state does not change the value of the property.

On average the fraction of time each particle spends in any energy state is the same.

Probability of a macrostate is equal to the fraction of time the system of particles

spends in that macrostate

Hypothesis

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Probability of MacrostatesHypothesis

Fraction of time in a macrostate = probability of that macrostate.

smicrostateoftotal

jmacrostateinsmicrostateofPj #

#

r

1

!n

N!

r P

Nr

1ii

N

jj

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Probability of Macrostates

r

1

!n

N! P

Nr

1ii

j

Sharp distribution --- Most probable state and/or those near it are observed most of the time.

N=10, r=3

0%

2%

4%

6%

8%

10%

12%

14%

16%

9 11 13 15 17 19 21 23 25 27 29 31

SumP

N=10, r=2

0%

5%

10%

15%

20%

25%

30%

9 11 13 15 17 19 21

Sum

P

N=3, r=3

0%

5%

10%

15%

20%

25%

30%

2 4 6 8 10

Sum

P

N=6, r=3

0%

5%

10%

15%

20%

25%

5 7 9 11 13 15 17 19

Sum

P

N=4, r=3

0%

5%

10%

15%

20%

25%

3 5 7 9 11 13

Sum

P

N=10, r=4

0%

2%

4%

6%

8%

10%

12%

9 13 17 21 25 29 33 37 41

Sum

P

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

N=3, r=3

0%

5%

10%

15%

20%

25%

30%

0% 20% 40% 60% 80% 100%

Sum

P

N=10, r=3

0%

2%

4%

6%

8%

10%

12%

14%

16%

0% 20% 40% 60% 80% 100%

Sum

P

N=6, r=3

0%

5%

10%

15%

20%

25%

0% 20% 40% 60% 80% 100%

Sum

PN=4, r=3

0%

5%

10%

15%

20%

25%

0% 20% 40% 60% 80% 100%

Sum

P

Plot probability as function of fractional range.

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

N-3, r=3E1 E2 E3 Esum Omega %

I 3 0 0 3 1 3.7%II 2 1 0 4 3 11.1%III 2 0 1 5 3 11.1%IV 1 2 0 5 3 11.1%V 0 3 0 6 1 3.7%VI 1 1 1 6 6 22.2%VII 1 0 2 7 3 11.1%VIII 0 2 1 7 3 11.1%IX 0 1 2 8 3 11.1%X 0 0 3 9 1 3.7%

N=4, r=3E1 E2 E3 Esum Omega %

I 4 0 0 4 1 1.2%II 3 1 0 5 4 4.9%III 3 0 1 6 4 4.9%IV 2 2 0 6 6 7.4%V 1 3 0 7 4 4.9%VI 2 1 1 7 12 14.8%VII 0 4 0 8 1 1.2%VIII 2 0 2 8 6 7.4%IX 1 2 1 8 12 14.8%X 0 3 1 9 4 4.9%XI 1 1 2 9 12 14.8%XII 1 0 3 10 4 4.9%XIII 0 2 2 10 6 7.4%XIV 0 1 3 11 4 4.9%XV 0 0 4 12 1 1.2%

N=6, r=3E1 E2 E3 Esum Omega %

I 6 0 0 6 1 0.1%II 5 1 0 7 6 0.8%III 5 0 1 8 6 0.8%IV 4 2 0 8 15 2.1%V 3 3 0 9 20 2.7%VI 4 1 1 9 30 4.1%VII 4 0 2 10 15 2.1%VIII 2 4 0 10 15 2.1%IX 3 2 1 10 60 8.2%X 1 5 0 11 6 0.8%XI 3 1 2 11 60 8.2%XII 2 3 1 11 60 8.2%XIII 0 6 0 12 1 0.1%XIV 3 0 3 12 20 2.7%XV 1 4 1 12 30 4.1%XVI 2 2 2 12 90 12.3%XVII 0 5 1 13 6 0.8%XVIII 1 3 2 13 60 8.2%XIX 2 1 3 13 60 8.2%XX 0 4 2 14 15 2.1%XXI 2 0 4 14 15 2.1%XXII 1 2 3 14 60 8.2%XXIII 0 3 3 15 20 2.7%XXIV 1 1 4 15 30 4.1%XXV 1 0 5 16 6 0.8%XXVI 0 2 4 16 15 2.1%XXVII 0 1 5 17 6 0.8%XXVIII 0 0 6 18 1 0.1%

N=10, r=3E1 E2 E3 Esum Omega %

I 10 0 0 10 1 0.00%II 9 1 0 11 10 0.02%III 9 0 1 12 10 0.02%IV 8 2 0 12 45 0.08%V 8 1 1 13 90 0.15%VI 7 3 0 13 120 0.20%VII 8 0 2 14 45 0.08%VIII 7 2 1 14 360 0.61%IX 6 4 0 14 210 0.36%X 7 1 2 15 360 0.61%XI 6 3 1 15 840 1.42%XII 5 5 0 15 252 0.43%XIII 7 0 3 16 120 0.20%XIV 4 6 0 16 210 0.36%XV 6 2 2 16 1260 2.13%XVI 5 4 1 16 1260 2.13%XVII 3 7 0 17 120 0.20%XVIII 6 1 3 17 840 1.42%XIX 5 3 2 17 2520 4.27%XX 4 5 1 17 1260 2.13%XXI 2 8 0 18 45 0.08%XXII 6 0 4 18 210 0.36%XXIII 3 6 1 18 840 1.42%XXIV 5 2 3 18 2520 4.27%XXV 4 4 2 18 3150 5.33%XXVI 1 9 0 19 10 0.02%XXVII 2 7 1 19 360 0.61%XXVIII 3 5 2 19 2520 4.27%XXIX 5 1 4 19 1260 2.13%XXX 4 3 3 19 4200 7.11%XXXI 0 10 0 20 1 0.00%XXXII 1 8 1 20 90 0.15%XXXIII 2 6 2 20 1260 2.13%XXXIV 5 0 5 20 252 0.43%XXXV 3 4 3 20 4200 7.11%XXXVI 4 2 4 20 3150 5.33%XXXVII 0 9 1 21 10 0.02%XXXVIII 1 7 2 21 360 0.61%XXXIX 2 5 3 21 2520 4.27%

XL 4 1 5 21 1260 2.13%XLI 3 3 4 21 4200 7.11%XLII 0 8 2 22 45 0.08%XLIII 4 0 6 22 210 0.36%XLIV 1 6 3 22 840 1.42%XLV 3 2 5 22 2520 4.27%XLVI 2 4 4 22 3150 5.33%XLVII 0 7 3 23 120 0.20%XLVIII 3 1 6 23 840 1.42%XLIX 2 3 5 23 2520 4.27%

L 1 5 4 23 1260 2.13%LI 3 0 7 24 120 0.20%LII 0 6 4 24 210 0.36%LIII 2 2 6 24 1260 2.13%LIV 1 4 5 24 1260 2.13%LV 2 1 7 25 360 0.61%LVI 1 3 6 25 840 1.42%LVII 0 5 5 25 252 0.43%LVIII 2 0 8 26 45 0.08%LIX 1 2 7 26 360 0.61%LX 0 4 6 26 210 0.36%LXI 1 1 8 27 90 0.15%LXII 0 3 7 27 120 0.20%LXIII 1 0 9 28 10 0.02%LXIV 0 2 8 28 45 0.08%LXV 0 1 9 29 10 0.02%LXVI 0 0 10 30 1 0.00%

27Nr

81Nr

729Nr

049,59Nr

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

N=10, r=4

0%

2%

4%

6%

8%

10%

12%

0% 20% 40% 60% 80% 100%

Sum

P

N=10, r=3

0%

2%

4%

6%

8%

10%

12%

14%

16%

0% 20% 40% 60% 80% 100%

Sum

P

N=10, r=2

0%

5%

10%

15%

20%

25%

30%

0% 20% 40% 60% 80% 100%

Sum

PPlot probability as

function of fractional range.

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

N=10, r=2I E1 E2 Esum Omega %II 10 0 10 1 0.10%III 9 1 11 10 0.98%IV 8 2 12 45 4.39%V 7 3 13 120 11.72%VI 6 4 14 210 20.51%VII 5 5 15 252 24.61%VIII 4 6 16 210 20.51%IX 3 7 17 120 11.72%X 2 8 18 45 4.39%XI 1 9 19 10 0.98%XII 0 10 20 1 0.10%

024,1Nr

N=10, r=4E1 E2 E3 E4 Esum Omega %

I 10 0 0 0 10 1 0.000%II 9 1 0 0 11 10 0.001%III 9 0 1 0 12 10 0.001%IV 8 2 0 0 12 45 0.004%V 9 0 0 1 13 10 0.001%VI 8 1 1 0 13 90 0.009%VII 7 3 0 0 13 120 0.011%VIII 8 0 2 0 14 45 0.004%IX 8 1 0 1 14 90 0.009%X 7 2 1 0 14 360 0.034%XI 6 4 0 0 14 210 0.020%XII 8 0 1 1 15 90 0.009%XIII 7 2 0 1 15 360 0.034%XIV 7 1 2 0 15 360 0.034%XV 6 3 1 0 15 840 0.080%XVI 5 5 0 0 15 252 0.024%XVII 8 0 0 2 16 45 0.004%XVIII 7 0 3 0 16 120 0.011%XIX 7 1 1 1 16 720 0.069%XX 4 6 0 0 16 210 0.020%XXI 6 3 0 1 16 840 0.080%XXII 6 2 2 0 16 1,260 0.120%XXIII 5 4 1 0 16 1,260 0.120%XXIV 3 7 0 0 17 120 0.011%XXV 7 0 2 1 17 360 0.034%XXVI 7 1 0 2 17 360 0.034%XXVII 6 1 3 0 17 840 0.080%XXVIII 6 2 1 1 17 2,520 0.240%XXIX 5 4 0 1 17 1,260 0.120%XXX 4 5 1 0 17 1,260 0.120%XXXI 5 3 2 0 17 2,520 0.240%XXXII 2 8 0 0 18 45 0.004%XXXIII 7 0 1 2 18 360 0.034%XXXIV 6 0 4 0 18 210 0.020%XXXV 3 6 1 0 18 840 0.080%XXXVI 6 2 0 2 18 1,260 0.120%XXXVII 6 1 2 1 18 2,520 0.240%XXXVIII 4 5 0 1 18 1,260 0.120%XXXIX 5 2 3 0 18 2,520 0.240%XL 5 3 1 1 18 5,040 0.481%

XLI 4 4 2 0 18 3,150 0.300%XLII 1 9 0 0 19 10 0.001%XLIII 7 0 0 3 19 120 0.011%XLIV 2 7 1 0 19 360 0.034%XLV 6 0 3 1 19 840 0.080%XLVI 3 6 0 1 19 840 0.080%XLVII 6 1 1 2 19 2,520 0.240%XLVIII 5 1 4 0 19 1,260 0.120%XLIX 5 3 0 2 19 2,520 0.240%L 3 5 2 0 19 2,520 0.240%LI 5 2 2 1 19 7,560 0.721%LII 4 4 1 1 19 6,300 0.601%LIII 4 3 3 0 19 4,200 0.401%LIV 0 10 0 0 20 1 0.000%LV 1 8 1 0 20 90 0.009%LVI 2 7 0 1 20 360 0.034%LVII 6 1 0 3 20 840 0.080%LVIII 6 0 2 2 20 1,260 0.120%LIX 2 6 2 0 20 1,260 0.120%LX 5 0 5 0 20 252 0.024%LXI 5 1 3 1 20 5,040 0.481%LXII 3 5 1 1 20 5,040 0.481%LXIII 5 2 1 2 20 7,560 0.721%LXIV 4 4 0 2 20 3,150 0.300%LXV 4 2 4 0 20 3,150 0.300%LXVI 3 4 3 0 20 4,200 0.401%LXVII 4 3 2 1 20 12,600 1.202%LXVIII 0 9 1 0 21 10 0.001%LXIX 1 8 0 1 21 90 0.009%LXX 1 7 2 0 21 360 0.034%LXXI 6 0 1 3 21 840 0.080%LXXII 2 6 1 1 21 2,520 0.240%LXXIII 5 0 4 1 21 1,260 0.120%LXXIV 4 1 5 0 21 1,260 0.120%LXXV 5 2 0 3 21 2,520 0.240%LXXVI 3 5 0 2 21 2,520 0.240%LXXVII 2 5 3 0 21 2,520 0.240%LXXVIII 5 1 2 2 21 7,560 0.721%LXXIX 3 3 4 0 21 4,200 0.401%LXXX 4 3 1 2 21 12,600 1.202%LXXXI 4 2 3 1 21 12,600 1.202%LXXXII 3 4 2 1 21 12,600 1.202%LXXXIII 0 9 0 1 22 10 0.001%LXXXIV 0 8 2 0 22 45 0.004%LXXXV 1 7 1 1 22 720 0.069%LXXXVI 6 0 0 4 22 210 0.020%LXXXVII 4 0 6 0 22 210 0.020%LXXXVIII 1 6 3 0 22 840 0.080%LXXXIX 2 6 0 2 22 1,260 0.120%XC 5 0 3 2 22 2,520 0.240%XCI 3 2 5 0 22 2,520 0.240%XCII 5 1 1 3 22 5,040 0.481%XCIII 2 5 2 1 22 7,560 0.721%XCIV 2 4 4 0 22 3,150 0.300%XCV 4 1 4 1 22 6,300 0.601%XCVI 4 3 0 3 22 4,200 0.401%XCVII 3 4 1 2 22 12,600 1.202%XCVIII 4 2 2 2 22 18,900 1.802%XCIX 3 3 3 1 22 16,800 1.602%C 0 8 1 1 23 90 0.009%

C I 0 7 3 0 23 120 0.011%C II 1 7 0 2 23 360 0.034%C III 3 1 6 0 23 840 0.080%C IV 1 6 2 1 23 2,520 0.240%C V 5 1 0 4 23 1,260 0.120%C VI 1 5 4 0 23 1,260 0.120%C VII 4 0 5 1 23 1,260 0.120%C VIII 5 0 2 3 23 2,520 0.240%C IX 2 3 5 0 23 2,520 0.240%C X 2 5 1 2 23 7,560 0.721%C XI 3 4 0 3 23 4,200 0.401%C XII 4 2 1 3 23 12,600 1.202%C XIII 4 1 3 2 23 12,600 1.202%C XIV 2 4 3 1 23 12,600 1.202%C XV 3 2 4 1 23 12,600 1.202%C XVI 3 3 2 2 23 25,200 2.403%C XVII 0 8 0 2 24 45 0.004%C XVIII 3 0 7 0 24 120 0.011%C XIX 0 7 2 1 24 360 0.034%C XX 0 6 4 0 24 210 0.020%C XXI 2 2 6 0 24 1,260 0.120%C XXII 1 6 1 2 24 2,520 0.240%C XXIII 5 0 1 4 24 1,260 0.120%C XXIV 1 4 5 0 24 1,260 0.120%C XXV 2 5 0 3 24 2,520 0.240%C XXVI 1 5 3 1 24 5,040 0.481%C XXVII 3 1 5 1 24 5,040 0.481%C XXVIII 4 0 4 2 24 3,150 0.300%C XXIX 4 2 0 4 24 3,150 0.300%C XXX 4 1 2 3 24 12,600 1.202%C XXXI 2 3 4 1 24 12,600 1.202%C XXXII 2 4 2 2 24 18,900 1.802%C XXXIII 3 3 1 3 24 16,800 1.602%C XXXIV 3 2 3 2 24 25,200 2.403%C XXXV 0 7 1 2 25 360 0.034%C XXXVI 2 1 7 0 25 360 0.034%C XXXVII 0 6 3 1 25 840 0.080%C XXXVIII 1 6 0 3 25 840 0.080%C XXXIX 3 0 6 1 25 840 0.080%C XL 1 3 6 0 25 840 0.080%C XLI 5 0 0 5 25 252 0.024%C XLII 0 5 5 0 25 252 0.024%C XLIII 1 5 2 2 25 7,560 0.721%C XLIV 2 2 5 1 25 7,560 0.721%C XLV 4 1 1 4 25 6,300 0.601%C XLVI 1 4 4 1 25 6,300 0.601%C XLVII 4 0 3 3 25 4,200 0.401%C XLVIII 3 3 0 4 25 4,200 0.401%C XLIX 2 4 1 3 25 12,600 1.202%C L 3 1 4 2 25 12,600 1.202%

576,048,1Nr

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

C LI 3 2 2 3 25 25,200 2.403%C LII 2 3 3 2 25 25,200 2.403%C LIII 2 0 8 0 26 45 0.004%C LIV 0 7 0 3 26 120 0.011%C LV 1 2 7 0 26 360 0.034%C LVI 0 4 6 0 26 210 0.020%C LVII 0 6 2 2 26 1,260 0.120%C LVIII 2 1 6 1 26 2,520 0.240%C LIX 0 5 4 1 26 1,260 0.120%C LX 4 1 0 5 26 1,260 0.120%C LXI 3 0 5 2 26 2,520 0.240%C LXII 1 5 1 3 26 5,040 0.481%C LXIII 1 3 5 1 26 5,040 0.481%C LXIV 4 0 2 4 26 3,150 0.300%C LXV 2 4 0 4 26 3,150 0.300%C LXVI 1 4 3 2 26 12,600 1.202%C LXVII 3 2 1 4 26 12,600 1.202%C LXVIII 2 2 4 2 26 18,900 1.802%C LXIX 3 1 3 3 26 16,800 1.602%C LXX 2 3 2 3 26 25,200 2.403%C LXXI 1 1 8 0 27 90 0.009%C LXXII 0 3 7 0 27 120 0.011%C LXXIII 2 0 7 1 27 360 0.034%C LXXIV 0 6 1 3 27 840 0.080%C LXXV 1 2 6 1 27 2,520 0.240%C LXXVI 1 5 0 4 27 1,260 0.120%C LXXVII 0 4 5 1 27 1,260 0.120%C LXXVIII 4 0 1 5 27 1,260 0.120%C LXXIX 0 5 3 2 27 2,520 0.240%C LXXX 3 2 0 5 27 2,520 0.240%C LXXXI 2 1 5 2 27 7,560 0.721%C LXXXII 3 0 4 3 27 4,200 0.401%C LXXXIII 1 4 2 3 27 12,600 1.202%C LXXXIV 1 3 4 2 27 12,600 1.202%C LXXXV 3 1 2 4 27 12,600 1.202%C LXXXVI 2 3 1 4 27 12,600 1.202%C LXXXVII 2 2 3 3 27 25,200 2.403%C LXXXVIII 1 0 9 0 28 10 0.001%C LXXXIX 0 2 8 0 28 45 0.004%C XC 1 1 7 1 28 720 0.069%C XCI 0 6 0 4 28 210 0.020%C XCII 4 0 0 6 28 210 0.020%C XCIII 0 3 6 1 28 840 0.080%C XCIV 2 0 6 2 28 1,260 0.120%C XCV 0 5 2 3 28 2,520 0.240%C XCVI 2 3 0 5 28 2,520 0.240%C XCVII 3 1 1 5 28 5,040 0.481%C XCVIII 1 2 5 2 28 7,560 0.721%C XCIX 0 4 4 2 28 3,150 0.300%C C 1 4 1 4 28 6,300 0.601%

CC I 3 0 3 4 28 4,200 0.401%CC II 2 1 4 3 28 12,600 1.202%CC III 2 2 2 4 28 18,900 1.802%CC IV 1 3 3 3 28 16,800 1.602%CC V 0 1 9 0 29 10 0.001%CC VI 1 0 8 1 29 90 0.009%CC VII 0 2 7 1 29 360 0.034%CC VIII 3 1 0 6 29 840 0.080%CC IX 1 1 6 2 29 2,520 0.240%CC X 0 5 1 4 29 1,260 0.120%CC XI 1 4 0 5 29 1,260 0.120%CC XII 0 3 5 2 29 2,520 0.240%CC XIII 2 0 5 3 29 2,520 0.240%CC XIV 3 0 2 5 29 2,520 0.240%CC XV 2 2 1 5 29 7,560 0.721%CC XVI 0 4 3 3 29 4,200 0.401%CC XVII 1 2 4 3 29 12,600 1.202%CC XVIII 2 1 3 4 29 12,600 1.202%CC XIX 1 3 2 4 29 12,600 1.202%CC XX 0 0 10 0 30 1 0.000%CC XXI 0 1 8 1 30 90 0.009%CC XXII 1 0 7 2 30 360 0.034%CC XXIII 3 0 1 6 30 840 0.080%CC XXIV 0 2 6 2 30 1,260 0.120%CC XXV 2 2 0 6 30 1,260 0.120%CC XXVI 0 5 0 5 30 252 0.024%CC XXVII 1 1 5 3 30 5,040 0.481%CC XXVIII 1 3 1 5 30 5,040 0.481%CC XXIX 2 1 2 5 30 7,560 0.721%CC XXX 2 0 4 4 30 3,150 0.300%CC XXXI 0 4 2 4 30 3,150 0.300%CC XXXII 0 3 4 3 30 4,200 0.401%CC XXXIII 1 2 3 4 30 12,600 1.202%CC XXXIV 0 0 9 1 31 10 0.001%CC XXXV 3 0 0 7 31 120 0.011%CC XXXVI 0 1 7 2 31 360 0.034%CC XXXVII 1 0 6 3 31 840 0.080%CC XXXVIII 1 3 0 6 31 840 0.080%CC XXXIX 2 1 1 6 31 2,520 0.240%CC XL 0 4 1 5 31 1,260 0.120%CC XLI 0 2 5 3 31 2,520 0.240%CC XLII 2 0 3 5 31 2,520 0.240%CC XLIII 1 2 2 5 31 7,560 0.721%CC XLIV 1 1 4 4 31 6,300 0.601%CC XLV 0 3 3 4 31 4,200 0.401%CC XLVI 0 0 8 2 32 45 0.004%CC XLVII 2 1 0 7 32 360 0.034%CC XLVIII 0 4 0 6 32 210 0.020%CC XLIX 0 1 6 3 32 840 0.080%CC L 2 0 2 6 32 1,260 0.120%

CC LI 1 2 1 6 32 2,520 0.240%CC LII 1 0 5 4 32 1,260 0.120%CC LIII 0 3 2 5 32 2,520 0.240%CC LIV 1 1 3 5 32 5,040 0.481%CC LV 0 2 4 4 32 3,150 0.300%CC LVI 0 0 7 3 33 120 0.011%CC LVII 2 0 1 7 33 360 0.034%CC LVIII 1 2 0 7 33 360 0.034%CC LIX 0 3 1 6 33 840 0.080%CC LX 1 1 2 6 33 2,520 0.240%CC LXI 0 1 5 4 33 1,260 0.120%CC LXII 1 0 4 5 33 1,260 0.120%CC LXIII 0 2 3 5 33 2,520 0.240%CC LXIV 2 0 0 8 34 45 0.004%CC LXV 0 3 0 7 34 120 0.011%CC LXVI 1 1 1 7 34 720 0.069%CC LXVII 0 0 6 4 34 210 0.020%CC LXVIII 1 0 3 6 34 840 0.080%CC LXIX 0 2 2 6 34 1,260 0.120%CC LXX 0 1 4 5 34 1,260 0.120%CC LXXI 1 1 0 8 35 90 0.009%CC LXXII 0 2 1 7 35 360 0.034%CC LXXIII 1 0 2 7 35 360 0.034%CC LXXIV 0 1 3 6 35 840 0.080%CC LXXV 0 0 5 5 35 252 0.024%CC LXXVI 0 2 0 8 36 45 0.004%CC LXXVII 1 0 1 8 36 90 0.009%CC LXXVIII 0 1 2 7 36 360 0.034%CC LXXIX 0 0 4 6 36 210 0.020%CC LXXX 1 0 0 9 37 10 0.001%CC LXXXI 0 1 1 8 37 90 0.009%CC LXXXII 0 0 3 7 37 120 0.011%CC LXXXIII 0 1 0 9 38 10 0.001%CC LXXXIV 0 0 2 8 38 45 0.004%CC LXXXV 0 0 1 9 39 10 0.001%CC LXXXVI 0 0 0 10 40 1 0.000%

576,048,1Nr

4,10 rN

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

N r rN

3 4 6.4x101

15 4 1.073741824x109

4 15 5.0625x104

50 30 7.17897987691853x1073

1,000 100

6.0x1023 1.0x1010 33106100.1 xx

000,2100.1 x

Problem 6.4

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Boltzman Hypothesis

where S is the entropy. is the of microstates in a macrostate.

The Boltzman constant, k = R/NO.

NO is Avogardo’s number.R is the ideal gas constant.

Provides a sharp extremum.Range is compressed by assuming logarithmic relation.

Average energy of particles is fixed.

ln k S

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Problem 6.5n 0 1 2 3 4 5 6 7 8 9

n+1/2 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 N U 0 0 1 2 4 2 1 0 0 0 10 45 37,800 0 1 1 2 2 2 1 1 0 0 10 45 453,600 n 0 1 0 0 -2 0 0 1 0 0 0 U 0 1.5 0 0 -9 0 0 7.5 0 0 0

0U 12lnln1

2 kkS

0 N

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Find Conditions for Equilibrium

• Find an expression for change in entropy of the system.

• Determine the constraints.

• Apply the constraints and the extremum criterion: .0)( indS

• Solve the remaining equations for the conditions for equilibrium.

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Find an expression for dS(ni)Substitute for :

r

i 1i!n

N!ln k S

Expand:

r

1ii!nln-N!ln k S

Note the Stirling approximation: x-x lnx x! ln

r

1ii

r

1iii nnlnnN-NlnN k S

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Find an expression for dS(ni)

Rearranging:

Note:

r

1iin N

r

1iii NlnNnlnn k S

and x ln - x

1 ln

r

1i

ii N

nlnn k S

Taking the derivative:

r

1i

i

N

nln k dS idn

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Isolation ConstraintsConsider an isolated system.

Closed system ---

r

1iisys n N

r

1iiisys ne U

Insulated system ---

r

1ii? V

Rigid system ---

0 d? dVr

1ii

Closed system ---

Insulated system ---

Rigid system ---

0 dnened dUr

1iii

r

1iiisys

0 dn dNr

1iisys

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Constrained Maximum EntropyApply Lagrange multipliers to constraints & add to condition for

entropy maximum.

Rearrange, raise to power of e to yield r equations:

0 dU dN dS syssys

r),1,2, (i]k

eexp[ ]

kexp[

N

n i

sys

i

Substitute for entropy and constraints:

0dnednN

nln k

r

1i

r

1iiii

r

1i

i

idn

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Constrained Maximum Entropy

r

1iisys n N

r

1i sys

i

N

n 1Apply: and

Solve for :

P1

k

eexp

kexp

1r

1i

i

k

eexp Function Partition

r

1i

i

PDefine:

Yielding:

P

ke

exp

N

ni

sys

i

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

idnPdVTdSdU By analogy.

Compare phenomenological & statistical expressions for dS to evaluate & :

Constrained Maximum Entropy

r

1iidnP

r

iii kdnedS

1

ln

sysNkdUdS Pdln

sysNT

dVT

pdU

TdS d

1

T

1 Plnk

T

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

All equilibrium thermodynamic functions can be derived if the partition function is known.

Complete expression equilibrium distribution of particles over energy levels:

Constrained Maximum Entropy

kT

e-exp

1

N

n i

sys

i

P

kT

eexp Function Partition

r

1i

i

P

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Problem 6.6

N 10 20 30 60 80 100 150 170

Exact N! 3.63E+06 2.43E+18 2.65E+32 8.32E+81 7.16E+118 9.33E+157 5.71E+262 7.26E+306

Exact lnN! 1.51E+01 4.23E+01 7.47E+01 1.89E+02 2.74E+02 3.64E+02 6.05E+02 7.07E+02

Stirling NlnN-N 1.30E+01 3.99E+01 7.20E+01 1.86E+02 2.71E+02 3.61E+02 6.02E+02 7.03E+02

Stirling N! 4.54E+05 2.16E+17 1.93E+31 4.28E+80 3.19E+117 3.72E+156 1.86E+261 2.22E+305%err -13.76% -5.72% -3.51% -1.57% -1.14% -0.89% -0.57% -0.49%

Problem 6.7i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 N ln

ni 14 18 27 38 51 78 67 54 32 27 23 20 19 17 15 500 -1.36E+03 ni 0 0 -1 -1 -2 0 1 1 2 2 1 0 -1 -1 -1 0 -1.86E+00

ni/ 14 18 26 37 49 78 68 55 34 29 24 20 18 16 14 500 -1.36E+03

ln(ni/N) ni 0.00 0.00 2.92 2.58 4.57 0.00 -2.01 -2.23 -5.50 -5.84 -3.08 0.00 3.27 3.38 3.51 1.57

KmoleJxkkS

/1057.2lnlnln 2312

1

2

KmoleJxnN

nkS i

r

i O

i

/1017.2ln 23

1

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Thermodynamic Functions in Terms of Partition Function

r

1i sys

ii N

nlnn k S

kT

e-exp

1

N

n i

sys

i

P

r

1i

P 11

lnT

1 S nkne

r

iii

sysNkU PlnT

1 S

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Thermodynamic Functions in Terms of Partition Function

Deduce F from S:

ln kT N- F sys P

Apply: T

F- S

V

VTkT

Pln

N ln k N S syssys P

TS- F U

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Thermodynamic Functions in Terms of Partition Function

Apply: T

U C

VV

VT

kT

2

22

sysVsysV

lnN

T

lnT k 2N C

PP

Apply: TS U F

VTkT

Pln

N U 2sys

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Monatomic Gas ModelAssumptions:

All particles are identical.Volume = lx x ly x lz

Energy of the system is not quantized & is equal to kinetic energies of the particles.

zzyyxx dvkTmvdvkTmvdvkTmv

lxdxdydz

lylz)2/

2(exp[)2/

2(exp[)2/

2(exp[

000P

3/2

m

kT2V

P

2vm2

1 KE

1/2

x-

2

m

kT2 dv

2exp

xvkT

m

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Thermodynamic Propertiesof Ideal Monatomic Gases

Apply

3/2

m

kT2V ln ln

P

T

1

2

3

T

ln

V

P

ln kT N- F sys P

3/2

sys m

kT2V ln kT N- F

ApplyVT

kT

Pln

N ln k N S syssys P

kN 2

3

m

kT2Vln kN S sys

3/2

sys

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

kTN2

1 U O

R2

3 kN

2

3 C OV

Thermodynamic Propertiesof Ideal Monatomic Gases

VTkT

Pln

N U 2sysApply:

Apply: T

U C

VV

Equipartition of energy:

freedomofdegreeperkT2

1 U

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Einstein’s Model of a CrystalConsider a simple cubic crystal --- 6 nearest

neighbors, 1 atom & 3 bonds per unit cell. Hypothesis --- Energy of crystal is the sum of the

energies of its bonds. The atoms vibrate around equilibrium positions as if bound by vibrating springs. Only certain vibrational frequencies are allowed in coupled springs. The energies (i) of the bonds are proportional to their vibrational frequencies ().

i = (i + 1/2) hwhere h = Planck’s constant.

The adjustable parameter is set by assuming an Einstein temperature: E = h/k

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

Einstein’s Model of a Crystal

r

0i kT

h )21

(i-exp

P

Evaluate the

partition function:

ir

i

0 kT

h-exp

kT

h

2

1-exp

PFactor:

Approximate as infinite

series:

i

i

kT

h-exp

kT

h

2

1-exp

P

kTh

exp1

1

kT

h

2

1-exp P

Substitute for infinite

series:

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

2kT

h-exp-1ln -

kT

h

2

1- ln

P

Einstein’s Model of a CrystalTake ln of both sides:For simple cubic: Osys NN 3

Apply ln kT N- F sys P

kT

hkTNO

exp1ln3hN2

3 F O

ApplyVT

kT

Pln

N ln k N S syssys P

exp-1ln k3N - exp-1

exp k3N S OO

kT

h

kThkTh

kT

h

09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

2

2

O

exp1

exp

kT

h3N C

kTh

kTh

kV

VTkT

Pln

N U 2sys

Apply

Einstein’s Model of a Crystal

kThkTh

exp1

exp1hN

2

3 U O

Apply: T

U C

VV