Stacey F.D., Davis P.M. Physics of the Earth
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Table E.3 Thermodynamic derivatives extended to second order at constant T, P, V and S
Differential
element
Constant
TPV S
@T –11 T
@P K
T
/V – K
T
¼C
V
K
S
@V 1 V – V
@S K
T
¼C
V
mC
P
/TmC
V
/T –
@U K
T
T PmC
P
VP mC
V
PV
@H K
T
(1 T) mC
P
mC
V
(1 þ) K
S
V
@F P SVP SPV TS
@G K
T
S S þK
T
VK
S
V TS
@
T
/V¼(qK
T
/qT)
P
/K
T
V
2
(2
T
K
0
T
þC
0
T
/T)
2
(
T
K
0
T
þC
0
T
/T) [K
0
S
1þq þT(
S
þ q)]
@K
T
K
T
K
0
T
/V K
T
T
¼K
T
2
(q/qP)
T
K
T
(K
0
T
T
) K
T
[K
0
T
þT(K
0
T
T
)]
@K
S
(K
T
/V)(K
0
S
þT
S
) K
S
S
K
T
(K
0
S
S
) K
S
K
0
S
@C
V
(C
V
/V)C
0
T
¼(C
V
/V)(1 q þ
T
K
0
T
)(C
P
C
0
T
C
V
C
0
S
)/T (C
V
/T)(C
0
T
C
0
S
) TC
V
(qqT)
V
@C
P
(C
P
/V)[C
0
T
þT(q þ
T
)/(1 þT)] (C
P
/T)[C
0
T
C
0
S
þT
(
T
S
þC
0
T
)]
(C
P
/T){C
0
T
(1 þT) þ[
2
T þ(T)
2
(q1) C
0
S
]/(1 þT)}
C
P
[T(q/qT)
V
þT(
S
þq)]
@ q/V q þC
0
S
/TC
0
S
/T (q C
0
S
)
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Table E.4 Relationships between derivatives
K
S
=K
T
¼ C
P
=C
V
¼ 1 þ T (E:1)
K
0
T
¼ K
0
S
ð1 þ TÞþT½3q 2 þ ð@ ln C
V
=@ ln TÞ
V
(E: 2)
K
0
S
¼ K
0
T
ð1 þ TÞTð
S
þ
T
þ qÞ (E:3)
S
¼ð1=Þð@ ln K
S
=@TÞ
P
¼ K
0
S
1 þ q C
0
S
¼ð@ lnðT=C
P
Þ=@ ln VÞ
S
(E:4)
T
¼ð1=Þð@ ln K
T
=@TÞ
P
¼ K
0
T
1 þ q þ C
0
T
¼ð@ ln =@ ln VÞ
T
¼ð
S
þ C
0
T
Þð1 þ TÞþ þ C
0
S
þ Tð2q 1Þ
(E:5)
C
0
S
¼ C
0
T
ð@ ln C
V
=@ ln TÞ
V
¼ð@=@ ln TÞ
V
(E:6)
C
0
T
¼ ð@ lnðC
V
Þ=@ ln TÞ
V
(E:7)
q
S
¼ q C
0
S
(E:8)
ð@ lnðK
T
Þ=@ ln VÞ
T
¼
T
K
0
T
¼ð1=Þð@ ln K
T
=@TÞ
V
(E:9)
ð@ lnðK
T
Þ=@ ln TÞ
V
¼ð@ lnðC
V
Þ=@ ln TÞ
V
¼ C
0
T
= (E:10)
ð@ðK
T
Þ=@TÞ
P
¼ K
T
ð@=@TÞ
V
(E:11)
ð@ lnðK
T
Þ=@ ln VÞ
S
¼ q 1(E:12)
ð@ lnðK
S
Þ=@ ln VÞ
S
¼ q 1 þ Tð
S
þ qÞ (E:13)
ð@ lnðTÞ=@ ln VÞ
T
¼
T
þ q (E:14)
ð@ lnðTÞ=@ ln VÞ
S
¼ð1 þ TÞð
S
þ qÞ (E:15)
ð@K
0
T
=@TÞ
P
¼
T
½
T
K
0
T
þð@ ln
T
=@ ln VÞ
T
(E:16)
ð@K
0
S
=@TÞ
P
¼
S
½
S
K
0
S
þð@ ln
S
=@ ln VÞ
S
(E:17)
ð@
T
=@ ln VÞ
T
¼K
T
K
00
T
þ lq þð@C
0
T
=@ ln VÞ
T
(E:18)
ð@
S
=@ ln VÞ
S
¼K
S
K
00
S
q
S
þð@q
S
=@ ln VÞ
S
(E:19)
ð@C
P
=@PÞ
T
¼ð@ð=Þ=@ ln TÞ
P
(E:20)
468 AP P EN D I X E
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Appendix F
An Earth model: mechanical properties
Table F.1 Selected details of the Preliminary Reference Earth Model (PREM) by Dziewonski and
Anderson (1981)
Region
Radius
(km) V
P
(m s
1
) V
S
(m s
1
) (k g m
3
) K
S
( GPa) ( GPa) P (GPa) g (m s
2
)
Inner core 0 11 266.20 3667.80 13 088.48 1425.3 176.1 0.4407 363.85 0
200 11 255.93 3663.42 13 079.77 1423.1 175.5 0.4408 362.90 0.7311
400 11 237.12 3650.27 13 053.64 1416.4 173.9 0.4410 360.03 1.4604
600 11 205.76 3628.35 13 010.09 1405.3 171.3 0.4414 355.28 2.1862
800 11 161.86 3597.67 12 949.12 1389.8 167.6 0.4420 348.67 2.9068
1000 11 105.42 3558.23 12 870.73 1370.1 163.0 0.4428 340.24 3.6203
1200 11 036.43 3510.02 12 774.93 1346.2 157.4 0.4437 330.05 4.3251
1221.5 11 028.27 3504.32 12 763.60 1343.4 156.7 0.4438 328.85 4.4002
Outer core 1221.5 10 355.68 0 12 166.34 1304.7 0 0.5 328.85 4.4002
1400 10 249.59 0 12 069.24 1267.9 0 0.5 318.75 4.9413
1600 10 122.91 0 11 946.82 1224.2 0 0.5 306.15 5.5548
1800 9985.54 0 11 809.00 1177.5 0 0.5 292.22 6.1669
2000 9834.96 0 11 654.78 1127.3 0 0.5 277.04 6.7715
2200 9668.65 0 11 483.11 1073.5 0 0.5 260.68 7.3645
2400 9484.09 0 11 292.98 1015.8 0 0.5 243.25 7.9425
2600 9278.76 0 11 083.35 954.2 0 0.5 224.85 8.5023
2800 9050.15 0 10 853.21 888.9 0 0.5 205.60 9.0414
3000 8795.73 0 10 601.52 820.2 0 0.5 185.64 9.5570
3200 8512.98 0 10 327.26 748.4 0 0.5 165.12 10.0464
3400 8199.39 0 10 029.40 674.3 0 0.5 144.19 10.5065
3480 8064.82 0 9903.49 644.1 0 0.5 135.75 10.6823
D
00
3480 13 716.60 7264.66 5566.45 655.6 293.8 0.3051 135.75 10.6823
3600 13 687.53 7265.75 5506.42 644.0 290.7 0.3038 128.71 10.5204
------------ 3630 13 680.41 7265.97 5491.45 641.2 289.9 0.3035 126.97 10.4844
Lower
mantle
3630 13 680.41 7265.97 5491.45 641.2 289.9 0.3035 126.97 10.4844
3800 13 447.42 7188.92 5406.81 609.5 279.4 0.3012 117.35 10.3095
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Table F.1 (cont.)
Region
Radius
(km) V
P
(m s
1
) V
S
(m s
1
) (k g m
3
) K
S
( GPa) ( GPa) P (GPa) g (m s
2
)
4000 13 245.32 7099.74 5307.24 574.4 267.5 0.2984 106.39 10.1580
4200 13 015.79 7010.53 5207.13 540.9 255.9 0.2957 95.76 10.0535
4400 12 783.89 6919.57 5105.90 508.5 244.5 0.2928 85.43 9.9859
4600 12 544.66 6825.12 5002.99 476.6 233.1 0.2898 75.36 9.9474
4800 12 293.16 6725.48 4897.83 444.8 221.5 0.2864 65.52 9.9314
5000 12 024.45 6618.91 4789.83 412.8 209.8 0.2826 55.9 9.9326
5200 11 733.57 6563.70 4678.44 380.3 197.9 0.2783 46.49 9.9467
5400 11 415.60 6378.13 4563.07 347.1 185.6 0.2731 37.29 9.9698
------------ 5600 11 065.57 6240.46 4443.17 313.3 173.0 0.2668 28.29 9.9985
5600 11 065.57 6240.46 4443.17 313.3 173.0 0.2668 28.29 9.9985
5701 10 751.31 5945.08 4380.71 299.9 154.8 0.2798 23.83 10.0143
Transition
zone
5701 10 266.22 5570.20 3992.14 255.6 123.9 0.2914 23.83 10.0143
------------ 5771 10 157.82 5516.01 3975.84 248.9 121.0 0.2909 21.04 10.0038
5771 10 157.82 5516.01 3975.84 248.9 121.0 0.2909 21.04 10.0038
5871 9645.88 5224.28 3849.80 218.1 105.1 0.2924 17.13 9.9883
------------ 5971 9133.97 4932.59 3723.78 189.9 90.6 0.2942 13.35 9.9686
5971 8905.22 4769.89 3543.25 173.5 80.6 0.2988 13.35 9.9686
6061 8732.09 4706.90 3489.51 163.0 77.3 0.2952 10.20 9.9361
6151 8558.96 4643.91 3435.78 152.9 74.1 0.2914 7.11 9.9048
Low velocity
zone
6151 7989.70 4418.85 3359.50 127.0 65.6 0.2797 7.11 9.9048
6221 8033.70 4443.61 3367.10 128.7 66.5 0.2796 4.78 9.8783
6291 8076.88 4469.53 3374.71 130.3 67.4 0.2793 2.45 9.8553
LID 6291 8076.88 4469.53 3374.71 130.3 67.4 0.2793 2.45 9.8553
6346.6 8110.61 4490.94 3380.76 131.5 68.2 0.2789 0.604 9.8394
Crust 6346.6 6800.00 3900.00 2900.00 75.3 44.1 0.2549 0.604 9.8394
------------ 6356 6800.00 3900.00 2900.00 75.3 44.1 0.2549 0.337 9.8332
6356 5800.00 3200.00 2600.00 52.0 26.6 0.2812 0.337 9.8332
------------ 6368 5800.00 3200.00 2600.00 52.0 26.6 0.2812 0.030 9.8222
Ocean 6368 1450.00 0 1020.00 2.1 0 0.5 0.030 9.8222
6371 1450.00 0 1020.00 2.1 0 0.5 0 9.8156
470 AP P EN D I X F
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Table F.2 Elastic properties of the core from equation of state fitting to PREM
r (km) P (GPa) K
S
(GPa) K
0
KK
00
(GPa)
0
K
00
(kg m
3
)
0 363.85 1444.03 3.3203 0.7118 174.04 0.1996 0.3061 13 082.19
200 362.90 1440.83 3.3207 0.7129 173.86 0.1997 0.3066 13 073.59
400 360.03 1431.17 3.3220 0.7165 173.23 0.2003 0.3081 13 047.66
600 355.28 1415.19 3.3243 0.7225 172.19 0.2013 0.3107 13 004.66
800 348.67 1392.98 3.3275 0.7311 170.76 0.2027 0.3144 12 944.53
1000 340.24 1364.66 3.3318 0.7427 168.94 0.2045 0.3193 12 867.21
1200 330.05 1330.43 3.3373 0.7574 166.73 0.2069 0.3257 12 772.67
1221.5 328.85 1326.36 3.3379 0.7691 166.46 0.2071 0.3264 12 761.17
1221.5 328.85 1301.35 3.3171 0.7034 12 163.35
1400 318.75 1267.81 3.3227 0.7182 12 068.11
1600 306.15 1225.90 3.3300 0.7378 11 946.76
1800 292.22 1179.46 3.3387 0.7612 11 809.17
2000 277.04 1128.70 3.3489 0.7889 11 654.83
2200 260.68 1073.81 3.3609 0.8182 11 482.94
2400 243.25 1015.11 3.3749 0.8609 11 292.85
2600 224.85 952.87 3.3914 0.9077 11 083.58
2800 205.60 887.40 3.4110 0.9641 10 855.02
3000 185.64 818.09 3.4344 1.0329 10 602.94
3200 165.12 748.33 3.4625 1.1180 10 328.76
3400 144.19 675.51 3.4970 1.2253 10 029.29
3480 135.75 645.93 3.5129 1.2762 9901.97
Table F.3 Elastic properties of the lower mantle from equation of state fitting to PREM
r (km) P (GPa) K
S
(GPa) K
0
KK
00
(GPa)
0
K
00
(kg m
3
)
3480 135.75 667.17 3.0790 1.5086 298.95 1.0438 0.9579 5566.89
3600 128.71 645.43 3.0955 1.5622 291.56 1.0543 0.9883 5507.52
3630 126.97 640.04 3.0997 1.5733 289.72 1.0569 0.9928 5492.63
3800 117.35 610.11 3.1246 1.6596 279.48 1.0726 1.0472 5408.72
4000 106.39 575.69 3.1563 1.7686 267.62 1.0926 1.1160 5309.63
4200 95.76 541.96 3.1911 1.8922 255.89 1.1146 1.1940 5209.55
4400 85.43 508.80 3.2297 2.0341 244.25 1.1389 1.1238 5108.10
4600 75.36 476.06 3.2730 2.1992 232.65 1.1663 1.3877 5004.69
4800 62.52 443.62 3.3221 2.3949 221.02 1.1972 1.5112 4898.69
5000 55.90 411.40 3.3786 2.6035 209.34 1.2329 1.6428 4789.64
5200 46.46 379.31 3.4446 2.9209 197.55 1.2745 1.8431 4676.95
5400 37.29 347.27 3.5231 3.2874 185.60 1.3241 2.0743 4559.93
5600 28.29 315.14 3.6187 3.7666 173.42 1.3844 2.3767 4437.64
5701 23.83 298.88 3.6756 4.0689 167.17 1.4203 2.5675 4373.62
AN EARTH MODEL: MECHANICAL PROPERTIES 471
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Appendix G
A thermal model of the Earth
Table G.1 Thermal properties of the core
r (km) T (K) T
M
(K) q (10
6
K
1
) C
P
(J K
1
kg
1
) (W m
1
K
1
)
0 5030 5330 1.387 0.1025 9.015 693 36
200 5029 5321 1.387 0.1027 9.033 693 36
400 5027 5294 1.388 0.1034 9.088 694 36
600 5023 5250 1.388 0.1045 9.181 695 36
800 5017 5188 1.389 0.1060 9.314 697 36
1000 5010 5107 1.390 0.1081 9.490 700 36
1200 5001 5012 1.391 0.1107 9.713 703 36
1221.5 5000 5000 1.391 0.1110 9.740 703 36
1221.5 5000 5000 1.390 0.1294 10.314 794 29.3
1400 4946 4890 1.391 0.1327 10.525 794 29.3
1600 4877 4772 1.393 0.1371 10.805 796 29.2
1800 4799 4629 1.395 0.1423 11.135 797 29.1
2000 4711 4545 1.398 0.1485 11.525 799 29.1
2200 4614 4452 1.401 0.1558 11.985 800 29.0
2400 4507 4261 1.405 0.1645 12.528 802 28.9
2600 4390 4057 1.409 0.1748 13.172 804 28.8
2800 4263 3840 1.415 0.1872 13.940 806 28.7
3000 4123 3609 1.421 0.2022 14.865 808 28.6
3200 3972 3367 1.429 0.2205 15.991 811 28.5
3400 3808 3112 1.438 0.2432 17.386 814 28.4
3480 3739 3007 1.443 0.2539 18.040 815 28.3
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Table G.2 Thermal properties of the lower mantle
r (km) T (K) q (10
6
K
1
)
S
" C
P
(J K
1
kg
1
)
3480 3739 1.1412 0.2770 11.290 1.2156 5.3816 1203
3600 2838 1.1447 0.2894 11.590 1.2405 5.4189 1191
3630 2740 1.1454 0.2926 11.667 1.2469 5.4289 1190
3800 2668 1.1515 0.3117 12.123 1.2848 5.4771 1191
4000 2596 1.1591 0.3365 12.708 1.3337 5.5170 1192
4200 2525 1.1676 0.3644 13.357 1.3879 5.5547 1193
4400 2452 1.1757 0.3919 14.149 1.4415 5.5633 1195
4600 2379 1.1881 0.4324 14.904 1.5172 5.6315 1196
4800 2302 1.2008 0.4748 15.848 1.5961 5.6730 1198
5000 2227 1.2154 0.5245 16.959 1.6872 5.7057 1201
5200 2144 1.2335 0.5860 18.255 1.7975 5.7556 1203
5400 2060 1.2548 0.6602 19.833 1.9285 5.8001 1206
5600 1974 1.2815 0.7549 21.801 2.0922 5.8465 1209
5701 1931 1.2972 0.8136 23.007 2.1921 5.8190 1214
Table G.3 Thermal properties of the continental upper mantle
r (km) T (K) (10
6
K
1
) C
P
(J K
1
kg
1
)
5701 2010 1.10 20.6 1200
5771 1985 1.11 21.3 1202
5871 1948 1.13 24.1 1209
5971 1907 1.15 27.4 1217
5971 1853 1.02 24.9 1202
6061 1817 1.04 26.9 1206
6151 1780 1.06 28.8 1210
6151 1719 0.96 30.2 1205
6256 1282 1.04 31.9 1197
6332 880 1.13 33.5 1186
6332 880 1.07 53.0 1208
6371 300 1.15 40.0 850
A THERMAL MODEL OF THE EARTH 473
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Appendix H
Radioactive isotopes
Table H.1 Naturally occurring, long-lived isotopes
Isotope
Percentage
of element
Decay
mechanism
Decay constant
(year
1
)
Half life
(years)
Final daughter
product
40
K 0.01167 85.5%
10.5%K
0.001%
þ
5.544 10
10
1.250 10
940
Ca
40
Ar
40
Ar
50
V 0.25
K
1.6 10
16
6 10
15 50
Cr
50
Ti
87
Rb 27.8346 1.42 10
11
4.88 10
10 87
Sr
115
In 95.77 1.4 10
15
5 10
14 115
Sn
123
Te 0.87 K 5.8 10
14
1.2 10
13 123
Sb
138
La 0.089
K
6.3 10
12
1.1 10
11 138
Ce
138
Ba
142
Ce 11.05 1.4 10
16
5 10
15 138
Ba
144
Nd 23.87 2.9 10
16
2.4 10
15 140
Ce
146
Sm 0.03 6.93 10
9
1.0 10
8 142
Nd
147
Sm 15.07 6.54 10
12
1.06 10
11 143
Nd
148
Sm 11.27 5.8 10
14
1.2 10
13 144
Nd ¼>
140
Ce
149
Sm 13.84 1.7 10
15
4 10
14 145
Nd
152
Gd 0.20 6.3 10
15
1.1 10
14 148
Sm ¼>
144
Nd
156
Dy 0.0524 3.5 10
15
2 10
14 152
Gd
176
Lu 2.60 1.87 10
11
3.71 10
10 176
Hf
174
Hf 0.163 3.5 10
16
2 10
15 170
Yb
187
Re 63.93 1.5 10
11
4.6 10
10 187
Os
190
Pt 0.0127 1.16 10
12
6 10
11 186
Os
204
Pb 1.364 4.95 10
18
1.4 10
17 200
Hg
232
Th 100 6 þ4 4.9475 10
11
1.4010 10
10 208
Pb
235
U 0.7201 7 þ4 9.8485 10
10
7.0381 10
8 207
Pb
238
U 99.2743 6 þ4
5.4 10
5
%
fission
1.551 25 10
10
4.4683 10
9 206
Pb
Note:
235
Uand
238
U abundances do not add to 100% because
234
U occurs in the decay series of
238
U.
A short-lived thorium isotope,
230
Th, also occurs in the decay series of
238
U. Intermediate
daughters in the U and Th decay series are not included in this table. A list of the elements
appears in Table A.3.
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Table H.2 Short-lived isotopes, produced in the upper atmosphere by cosmic rays, arriving on
the Earth with interplanetary dust or, in the case of
234
U, produced by radioactive decay, that
are useful as tracers of atmospheric, oceanic or sedimentary processes
Isotope Half-life Decay mechani sm Decay product
neutron 10.6 minutes
1
H
3
H 12.26 years
3
He
7
Be 53.3 days K
7
Li
10
Be 1.5 10
6
10
B
14
C 5730 years
14
N
22
Na 2.60 years
þ 22
Ne
32
Si 160 years
32
S via
32
P
32
P 14.3 days
32
S
33
P 25 days
33
S
35
S 87 days
35
Cl
36
Cl 3.01 10
5
years
36
Ar
37
Ar 35 days K
37
Cl
39
Ar 270 years
39
K
53
Mn 3.8 10
6
years K
53
Cr
234
U 2.47 10
5
years
230
Th ¼>
206
Pb
(half-life 75 000 years)
Table H.3 Extinct isotopes with decay products that are identifiable in meteorites or provide
isotopic clues to early Solar System processes
Isotope
Decay
mechanism Half-life (years)
Decay constant
(years
1
) Decay product
22
Na
þ
2.60 0.267
22
Ne
26
Al 85%
þ
15% K
7.2 10
5
9.7 10
726
Mg
60
Fe 3 10
5
2 10
660
Ni via
60
Co
107
Pd 6.5 10
6
1.07 10
7 107
Ag
129
I 1.6 10
7
4.2 10
8 129
Xe
146
Sm 1.0 10
8
6.9 10
9 142
Nd
182
Hf 2 9 10
6
7.7 10
8 182
W via
182
Ta
236
U 8.3 10
7
2.9 10
8 208
Pb via
232
Th
244
Pu 99.7%
0.3% fission
8.3 10
7
8.5 10
9
99.7%
208
Pb (via
232
Th)
0.3% fission products
RADIOACTIVE ISOTOPES 475
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476
– [476–476] 13.3.2008 10:51AM
Appendix I
A geologic time scale
Table I.1 A geological time scale, 2004 (see Gradstein et al., 2005). Numbers are dates of
commencements of geological periods (millions of years ago)
Phanerozoic Cenozoic Quaternary Holocene 0.01
Pleistocene 1.81
Tertiary Pliocene 5.33
Miocene 23.03
Oligocene 33.9
Eocene 55.5
Paleocene 65.5
Mesozoic Cretaceous 145.5
Jurassic 199.6
Triassic 251
Paleozoic Permian 299
Carboniferous 359
Devonian 416
Silurian 444
Ordovician 488
Cambrian 542
Proterozoic 2500
Archean 4000
Priscoan (Hadean)