Chick–Watson Inactivation

In 1908 Harriet Chick published in the Journal of Hygiene the observation that disinfection behaves like a chemical reaction: plot the surviving fraction on a logarithmic axis against time and you get a straight line. Organisms do not die all at once at some threshold; they die at a constant proportional rate, so each equal interval of time removes the same fraction of whatever is left. In the same volume H. E. Watson added the piece Chick's law was missing — that concentration does not simply multiply the rate, but enters raised to an exponent — and the pair of them has carried disinfection theory ever since.

Written for the log reduction, it is $\mathrm{LR} = k\,C^{n}\,t$. Three of those symbols are ordinary. $C$ is the disinfectant concentration, $t$ is the contact time, and $\mathrm{LR}$ is the base-10 log reduction achieved. The interesting one is $n$, Watson's dilution coefficient, and what it measures is which of your two knobs is worth turning. When $n = 1$, concentration and time trade one for one — halve the residual, double the time, get the same kill — and the model collapses to exactly the CT product the regulations use. Free chlorine sits close enough to 1 that the regulatory shortcut works, which is not a coincidence: the framework was built on the disinfectant it works for. When $n$ is well above 1 the disinfectant is concentration-driven, and a stronger residual buys more than a longer basin. When it is below 1, the reverse, and the concrete is the better purchase.

$k$ is unit-bound, and this is the honesty problem of the page. It carries the units $(\text{mg/L})^{-n}$ per minute, which is not a rate you can convert, because the exponent on the concentration changes the dimensions. A $k$ fitted at $n = 1$ cannot be used at $n = 0.8$; a $k$ taken from a paper that worked in different concentration units is a different number entirely, and papers very often do not say. This page takes $k$ on the trade basis — milligrams per litre and minutes — and no calculator can guess what basis a bare number came from.

Which is a good reason to fit your own. A $k$ measured on your water beats any published figure, because it silently absorbs everything the model leaves out: the organism, the pH, the temperature, the particle load, the mixing in your jar. But fit it honestly. One pair of numbers cannot recover both $k$ and $n$ — that is two unknowns and one equation, and putting one point through a two-parameter model is a coincidence, not a fit. Run several concentrations at several times, plot log survival against time to get the slopes, then plot those slopes against log concentration to get $n$ from the second slope.

And then be sceptical of the line itself. Chick's law assumes every organism is equally susceptible and equally exposed, and real survival curves say otherwise in two visible ways. At the start there is often a shoulder, a lag while the disinfectant works through clumps, through cell walls, through whatever is shielding the target. At the end there is a tail: the line flattens and further contact buys almost nothing.

The tail is the half that matters, and it is worth being blunt about why. The tail is the resistant fraction — organisms shielded inside particles, aggregated into clumps, or simply tougher than their siblings — and it is precisely the fraction that survives to reach a customer's tap. Extrapolating a fitted straight line past the data you actually collected is optimistic in exactly the direction you cannot afford to be optimistic in. If your bench work stops at 4 logs, you know about 4 logs. The model will happily print 8 and it does not know anything about them.

Various refinements exist to bend the line — Hom's model puts an exponent on time as well, and delayed-Chick–Watson forms build in a shoulder — and all of them cost you another fitted parameter. For plant work the honest position is usually the simple model plus a healthy distrust of its extremes, rather than a more elaborate model fitted to the same thin data.

One organism is outside all of this. Cryptosporidium resists free chlorine at every CT a drinking-water plant can physically deliver, and no value of $k$ worth writing down will change that. Its control comes from filtration, from UV, or from ozone. That is the reason the modern treatment train looks the way it does, and it is why the log-credit framework separates the pathogens rather than treating disinfection as one number.