The heat production (\(Q_r\)) of a granitic rock due to decay of the radioactive elements U, Th and K having concentration \(C_U, C_{Th}, C_K\), 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 \(\alpha, \beta, \gamma\) (in \(\mu Wkg^{-1}\))?
Show Hint
In radioactive heat generation problems: Uranium contributes most, Thorium next, and Potassium least. Always remember the order: \(U > Th > K\).
Step 1: Recall radioactive heat production.
Different radioactive isotopes release heat at different rates depending on their decay constant and energy released per decay.
The three main contributors in granitic rocks are:
- Uranium (U),
- Thorium (Th),
- Potassium (K).
Step 2: Relative contributions.
- Uranium produces the largest heat per unit mass because of its relatively short half-life and high energy release.
- Thorium produces less heat compared to Uranium, but still significant.
- Potassium (\(^{40}K\)) contributes the least per unit concentration because of its long half-life and low abundance.
Step 3: Order of coefficients.
Thus, the heat production constants follow:
\[
\alpha \, (\text{for U}) > \beta \, (\text{for Th}) > \gamma \, (\text{for K})
\]
Final Answer:
\[
\boxed{\alpha > \beta > \gamma}
\]
