Reaction kinetics and natural decay

A mass balance tells you what goes in and out; kinetics tells you how fast things change inside. Bacteria die in a river, organic matter is consumed by microbes, chlorine reacts with pipe walls, radon decays in a basement and pesticides break down in soil, each at its own rate. Engineers need those rates to size treatment tanks, to predict how far downstream an outfall affects a river, and to estimate how long a contaminated site will take to recover. This lesson covers rate laws and reaction order, the first-order model that describes most environmental decay, half-life, the effect of temperature on rates, and how to estimate a rate constant from laboratory or field data.

Reaction rates and order

The rate of a reaction is the change in concentration per unit time. For many reactions, the rate depends on concentration raised to some power, called the order of the reaction:

Here k is the rate constant and n is the order. The minus sign shows that the concentration is decreasing. The order is usually found by experiment, not from the balanced equation, because most environmental processes involve many steps.

In a zero-order reaction the concentration falls by the same amount each day, regardless of how much is left, so it reaches zero in a finite time. In a first-order reaction it falls by the same fraction each day, so it approaches zero gradually and never quite reaches it. First-order behaviour is so common in environmental systems that it is the default assumption unless data show otherwise.

First-order decay and half-life

Integrating the first-order rate law from an initial concentration C₀ gives exponential decay:

The half-life is the time for the concentration to fall to half its initial value. Setting C/C₀ = 1/2 gives a half-life that does not depend on the starting concentration:

After one half-life, 50 per cent remains; after two, 25 per cent; after three, 12.5 per cent; after ten, about 0.1 per cent. Engineers use the same arithmetic to describe how long a spill takes to fade, how quickly a disinfectant dissipates in a water main and how fast a radioactive isotope loses its activity.

Bacterial die-off

Bacteria from sewage and animal manure gradually die in rivers and lakes because of sunlight, temperature, predation and lack of nutrients. Die-off is usually modelled as first order, with rate constants that vary widely, from about 0.5 per day in cold, turbid water to several per day in warm, clear water exposed to sunlight. Engineers use die-off rates to estimate how far downstream of a sewer overflow a beach or drinking water intake is at risk.

Temperature and reaction rates

Most chemical and biological reactions speed up as temperature rises. The Arrhenius equation describes this in terms of an activation energy, but environmental engineers usually use a simpler empirical form, often called the van 't Hoff–Arrhenius relationship:

Here k₂₀ is the rate constant at 20 °C, T is the temperature in degrees Celsius, and θ is a temperature coefficient, typically between 1.02 and 1.09. A common value for biochemical oxygen demand is 1.047, which means the rate rises about 4.7 per cent for each degree of warming, roughly doubling every 15 °C. Temperature effects matter enormously in the prairies and northern states, where river temperatures swing from near freezing in winter to more than 20 °C in summer.

Estimating a rate constant from data

Rate constants come from experiments. For a first-order reaction, taking the natural logarithm of C = C₀e^(−kt) gives ln C = ln C₀ − kt, a straight line in time. Plotting ln C against t, or C on a logarithmic axis, and fitting a line by least squares gives −k as the slope. If the points curve instead of falling on a line, the reaction is not first order, or more than one process is involved.

Joy ran a degradation test on the herbicide from lesson 3. She spiked lake water with the herbicide, kept it in the dark at the lake's summer temperature, and measured the concentration over two weeks.

# Fit a first-order rate constant by least squares on ln(C) versus t
import math
t = [0, 1, 2, 4, 7, 10, 14]                    # days
C = [50.0, 42.1, 36.4, 26.8, 17.0, 11.2, 6.4]   # ug/L
y = [math.log(c) for c in C]
n = len(t)
tm, ym = sum(t) / n, sum(y) / n
slope = sum((a - tm) * (b - ym) for a, b in zip(t, y)) / sum((a - tm) ** 2 for a in t)
icept = ym - slope * tm
k = -slope
ss_res = sum((b - (icept + slope * a)) ** 2 for a, b in zip(t, y))
ss_tot = sum((b - ym) ** 2 for b in y)
print(f"k = {k:.3f} per day, fitted C0 = {math.exp(icept):.1f} ug/L")
print(f"half-life = {math.log(2) / k:.1f} days, R^2 = {1 - ss_res / ss_tot:.4f}")

k = 0.147 per day, fitted C0 = 48.8 ug/L
half-life = 4.7 days, R^2 = 0.9994

The fit gives k ≈ 0.147 per day and a half-life of about 4.7 days, with an R² above 0.999, so the first-order model describes the data very well. Joy's earlier estimate of 0.10 per day, used in lesson 3, was conservative; with the measured rate, the lake concentration would be lower. Field rates, however, often differ from laboratory rates because sunlight, sediments and changing temperature all play a part, so Nadia keeps both values and reports the range.

Kinetics in treatment and distribution

Kinetics also drives treatment design. In 1908 Harriette Chick showed that bacteria exposed to a disinfectant die off approximately as a first-order reaction in time, and Herbert Watson added the effect of disinfectant concentration. The resulting Chick-Watson model is usually written with the product of concentration and contact time:

Here N is the number of surviving organisms, C the disinfectant concentration and t the contact time. Because only the product C × t appears, a short contact at a high dose achieves the same kill as a long contact at a low dose. This is the basis of the CT concept used to design chlorine contact tanks in lesson 7. Disinfection is usually described in log reductions: a 2-log reduction removes 99 per cent of the organisms, a 3-log reduction 99.9 per cent and a 4-log reduction 99.99 per cent.

Chlorine itself decays in a water main as it reacts with organic matter in the water and with biofilm and corrosion products on the pipe walls. Utilities model this as first-order decay, with a bulk rate measured in bottles and a wall rate that depends on pipe material and age. If the chlorine residual at the far ends of the system falls too low, utilities add booster chlorination stations or flush the mains. Iron pipes that are old and heavily tuberculated can consume chlorine many times faster than new plastic pipe, which is one reason residual monitoring is part of every distribution system's routine.

Kinetics in the real world

Environmental systems rarely behave as neatly as a single first-order reaction. Microbial growth follows saturation kinetics: at low substrate concentrations the rate is roughly first order, but at high concentrations the microbes are working as fast as they can and the rate levels off, approaching zero order. The Monod equation, used in wastewater design, captures this behaviour. Some pollutants degrade in steps, forming intermediates that may be more toxic than the parent compound; trichloroethylene, a common groundwater solvent, can break down to vinyl chloride, a known human carcinogen, before degrading further. Sorption to soil, volatilization to air and photolysis by sunlight may all act at once. Engineers handle this complexity by measuring rates under realistic conditions, using conservative values in design and checking predictions with monitoring.

Persistence is the flip side of decay. Substances with half-lives of years or decades in the environment, such as PCBs, mercury and many per- and polyfluoroalkyl substances (PFAS), accumulate in sediments and food chains. Some become more concentrated at each step up the food chain, a process called biomagnification, so that fish-eating birds and people who eat a lot of fish receive the highest doses. For these substances, preventing release is far more effective than any downstream treatment.