GATE-2018 (24) :Wyllie – Time average equation

17. The heat production $(Q_r)$ of a granitic rock due to the decay of the radioactive elements U, Th, and K having concentrations C_u, C_Th, and C_K, respectively, is given by the expression

$Q_r=\alpha C_u+\beta C_{Th} +\gamma C_K $

Which one of the following correctly represents the relation between the magnitude of coefficients a, b, g (in mWkg^−1)?

A)$\alpha >\beta >\gamma$

B)$\alpha < \beta> \gamma$

C)$\alpha > \beta < \gamma$

D)$\alpha < \beta < \gamma$

Solution:

The amount of heat generated per second by these elements (in mWkg^−1 ) are: natural uranium, 95.2; thorium, 25.6, and natural potassium 0.00348 (Rybach, 1976; 1988).

The heat $Q_r$ produced by radioactivity in a rock that has concentrations $C_u, C_{Th}, C_K$, respectively, of these elements is

$Q_r=95.2 C_u+ 25.6 C_{Th} +0.00348 C_K $---(1)

Based on the above equation $\alpha >\beta >\gamma$

Concentration (p.p.m by weight) of granite $C_U$=4.6, $C_{Th}$=18, and $C_K$=33,000

Substitute these values in the above equation (1)  

$Q_r=437.92+ 460.8 +114.84 $

$Q_r=1013.56 $ ---(2)

The answer is option (A)

_References: 

_{1. Fundamentals of Geophysics by William Lowrie.}