Page 132 - Physical Principles of Sedimentary Basin Analysis

114 Heat flow
Table 6.2. The abundance of the elements U, Th and K in some rocks. The first
data set is taken from Van Schmus (1995), the second is from Perry et al. (2006),
the third is from Rudnick and Fountain (1995) and the fourth is from Turcotte
and Schubert (1982).

U Th K S total
Name (ppm) (ppm) (%) (μWm −3 ) C Th /C U C K /C U
Granite (1) 3.4 50 4.45 4.8 14.7 1.3
Granite (2) 3.0 24 3.67 2.8 8.0 1.2
Granite (low Ca) 4.7 20 4.2 3.0 4.3 0.9
Marine mud 2.8 12.2 2.96 1.9 4.4 1.1
Shale 3.1 9.5 2.2 1.7 3.0 0.7
Sandstone 1.7 5.5 1.07 0.9 3.2 0.6
Peridotite 0.005 0.01 0.001 0.002 2.0 0.2
Ultramafic intrusives 0.24 0.85 0.69 0.19 3.5 2.9
Gneiss, metasediments 1.86 6.89 1.93 1.15 3.7 1.0
Gabbro 0.23 0.9 0.27 0.15 3.9 1.2
Av. continental crust 1.6 5.8 2.0 1.0 3.6 1.3
Av. oceanic crust 0.9 2.7 0.4 0.47 3.0 0.4
Undepleted mantle 0.02 0.1 0.02 0.01 5.0 1.0

per unit time for an unstable nucleus. Equation (6.42) is straightforward to integrate, and
gives the number of atoms as a function of time

N = N 0 e −λt (6.43)
where N 0 is the number of atoms at time t = 0. It is often interesting to know how long it
takes for the number of atoms to be reduced by half. We then have to solve the equation
1 −ln2
e −λt 1/2 = = e (6.44)
2
which has the solution t 1/2 for the half-life
ln2 0.693
t 1/2 = = . (6.45)
λ λ
The heat production for each radioactive isotope decays with its own time constant and the
total heat production per unit mass of rock is
−λ i t
H(t) = C i H i e (6.46)
i
where the index i is according to Table 6.1. Figure 6.3 shows the heat production for the
average continental crust (see Table 6.2) for the past 4 Ga. The total heat production was
14% higher in the crust 500 Ma years ago, and 2 times higher 2.5 Ga ago. The decay in