6 Westrich and Berner 1984 suggest that the oxidation of org

6. Westrich and Berner (1984) suggest that the oxidation of organic matter can be considered a first-order re- action, with the differential rte equation written as: dG/dt =-k,G. They assume three forms of organic matter, behaving with three different and decreasing rates of oxidation. (See Section 2.7). The third form is assumed recalcitrant and unreactive. Middieburg (1989) questions this approach and suggests that all or- ganic matter will oxidize, given sufficient time. He proposes a single rate law for the oxidation of organic matter, whether aerobic or anaerobic, with a rate constant that is itself a function of time as given by the equation: logok (y i)::-0.95 log,u\' (y)-0.81 for which r= 0.987. Assuming Middleburg\'s approach applies to the oxidative breakdown of sewage sludge, calculate how long it should take the fresh sewage sludge at a dis its original weight. posal site to decompose to l % of

Solution

log K = -0.95 log t – 0.81

log K = -0.95 log t + log 0.155

log K = log (0.155 t ^-0.95)

K = 0.155 t ^-0.95

dG/dt = -K G

dG/dt = -0.155 t ^-0.95 G

Integrating between t = 0 and t = t we get,

ln(G/G₀) = -0.155 t^0.05 / 0.05|^t₀

ln(G/G₀) = -3.098 t^0.05    … (1)

Final concentration is 1% of initial concentration

G = 0.01 G₀

Substituting in equation (1) we get,

-4.6 = -3.098 t^0.05

t^0.05 = 1.49

Time taken to decompose to 1% of original weight, t = 2781.7 y