Deep Earth Exam Question
a) Briefly outline the current evidence that the inner core of the Earth is composed of
iron alloyed with light elements.
[5]
b) What is an “equation of state”.
[5]
c) The Birch-Murnaghan 3rd order equation-of-state is given by:
7
5
2




   3

3     3    3  3 '
P  K 0       1  ( K 0  4)    1 
2    0    0   4
 0 





The equation-of-state parameters for iron carbide, Fe3C are K0 = 173 GPa, ρ0 =
7.7819 Mgm-3, K’=5.79.
(i) Use this EOS to work out the pressure for densities of 12.6, 12.65, 12.7, 12.75
and 12.8 Mgm-3; insert your answers into the table provided. [25]
(ii) The incompressibility as a function of pressure may be given by:
K P  K0  K ' P
Use this relation to work out the incompressibility the pressures calculated in c(i)
above; insert your answers into the table provided.
[10]
(iii) The incompressibility as a function of temperature may be given by:
K P ,T  K P 
dK
T
dT
If dK/dT = -0.03 GPaK-1, calculate KP,T at your pressure values, assuming a core
temperature of 5500K.
[10]
(iv) The adiabatic incompressibility is related to the isothermal incompressibility
via:
K S  K P ,T (1  T )
For an assumed thermal expansion coefficient of 4.34 x 10-5 K-1 and a Grüneisen
parameter of 1.5, calculate KS at each pressure.
[10]
(v) On the graph provided, plot these KS values as a function of density; also
include on your plot the seismological bounds for the inner core which are KS =
1343.3 GPa at a density of 12.76 Mgm-3 and KS = 1425.3 GPa at a density of
13.08 Mgm-3).
[20]
(vi) (a) On the basis of your plot comment on the likelihood of there being carbon
in the inner core; b) what are the major assumptions made in your calculations; c)
how this result is affected by these assumptions.
[15]
Table for part (c):
ρ (Mgm-3)
12.6
12.65
12.7
12.75
12.8
P (GPa)
KP (GPa)
KP,T (GPa)
KS (GPa)
Graph for part (c):
3000
2800
2600
KS GPa
2400
2200
2000
1800
1600
1400
1200
1000
12.5
12.6
12.7
12.8
12.9
-3
Density Mgm
13
13.1
13.2
Answers:
a) from seismology: density < pure Fe therefore light element; cosmochemistry says
iron cores have light elements; mineral physics matches light element density with
S,Si, C, O as most likely; mineral physics measures solubilities of light elements
into Fe. [2,1,1,1]
b) relates volume or density to P/T; includes gas law, BM 1-3, logarithmic, universal,
Murnaghan, etc.[2, 3@1 for any/each of rest].
c) (i) see below [5@ 5 each]
(ii) see below [5@ 2 each]
(iii) see below [5@ 2 each]
(iv) see below [5@ 2 each]
(v) see below [5@ 3 each plus 2@2.5 each]
(vi) a) not likely as lines far apart [4] b) assumption in choice of EOS, no use of
K’’, neglect of dK/dT as f(P), neglect of dα/dP, neglect of dγ/d(P,T),
assumption of core T=5500K [6@ 1 each] c) although the numbers will be
different, the effect on the story is probably very little as the effects of P and T
will inversely offset each other; the differences in the numbers will be an
illustration of the uncertainties [5].
P
331.1487542
337.6408634
344.2106818
350.8586725
357.5852987
rho
12.6
12.65
12.7
12.75
12.8
K (P)
2090.351
2127.941
2165.98
2204.472
2243.419
K(P,T)
1925.351287
1962.940599
2000.979847
2039.471714
2078.41888
Ks
2614.72
2665.77
2717.43
2769.70
2822.60
3000
2800
2600
KS GPa
2400
2200
2000
1800
1600
1400
1200
1000
12.5
12.6
12.7
12.8
12.9
-3
Density Mgm
13
13.1
13.2