Radioactive decay and first-order kinetics

The equations developed above for stable elements can therefore be worked out for radioactive elements or chemical reactions once the reactivity factor xt has been changed into ott + / Th The residence time x of the element i in the system is now 

For a pair made of a radioactive isotope i and a stable isotope j of the same element (e.g., 14C/12C), it can be safely assumed that af = ctj. In this case, their ratio at steady-state may be written 

In a well-stirred reservoir at steady-state, we can calculate the residence time of the element from 

Broecker and Li (1970) and Broecker (1974) found that the 14C/12C ratio in the deep ocean was 84 percent of this ratio in the pre-bomb surface ocean. Assuming that surface carbon (dissolved and falling debris) is the only source of deep ocean carbon, calculate the residence time tc of this element in the deep-ocean. The 14C decay constant is 1.2 x 10 4a 1. 

From equation (7.2.19), we calculate the residence time of 14C in the deep ocean as 

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Schultz, E. (1997). Dice-shaking as an analogy for radioactive decay and first-order kinetics. Journal of Chemical Education, 74, 505-507. 

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