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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