Question 1 of 5: Diffusion, Hydrolysis, Capillary Rise, Sorption-Retardation, and Contaminant Attenuation Processes

Question 1: Diffusion, Hydrolysis, Capillary Rise, Sorption-Retardation, and Contaminant Attenuation Processes (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. (a) Free-water diffusion coefficient $D_0=2\times10^{-5}\ \text{cm}^2/\text{s}$ for Cl⁻, porosity $n=0.3$, tortuosity $\tau=1.22$. (b) Hydrolysis rate coefficient $k=0.001\ \text{day}^{-1}$. (c) Tube radius $r=0.25\ \text{mm}$, interfacial tension $\sigma=72\ \text{dyne/cm}$, completely wetting ($\theta=0$). (d) Porewater velocity $v=16\ \text{cm/day}$, $K_d=6.6\ \text{mL/g}$, porosity $n=0.37$, solids density $\rho_s=2.64\ \text{g/cm}^3$.

Find. (a) Effective diffusion coefficient $D_e$. (b) Time for 99% hydrolysis. (c) Capillary-rise height. (d) Retarded contaminant transport velocity. (e) The physical/chemical distinction between retarding and mass-reducing attenuation processes.

Approach. Each part applies a single closed-form relation directly to the given data — the tortuosity correction (a), first-order decay inverted for time (b), the Young-Laplace/Jurin capillary-rise law (c), and the linear sorption retardation factor (d).

1. Part (a) — effective diffusion coefficient. The tortuosity correction scales the free-water value down by porosity and up by the reciprocal of tortuosity (the actual diffusive path through the pore network is longer and partly obstructed by solids): $$D_e=\frac{D_0\,n}{\tau}=\frac{(2\times10^{-5})(0.3)}{1.22}=\boxed{4.92\times10^{-6}\ \text{cm}^2/\text{s}}.$$
2. Part (b) — time for 99% hydrolysis. Hydrolysis follows first-order decay, $C/C_0=e^{-kt}$; 99% hydrolyzed means $C/C_0=0.01$: $$t_{99}=\frac{\ln(1/0.01)}{k}=\frac{\ln(100)}{0.001}=\boxed{4605\ \text{days}}\ (12.6\ \text{years}).$$
3. Part (c) — capillary rise. Convert $\sigma=72\ \text{dyne/cm}=0.072\ \text{N/m}$. With a fully wetting fluid ($\cos\theta=1$), Jurin's law gives $$h=\frac{2\sigma\cos\theta}{\rho g r}=\frac{2(0.072)}{(998)(9.81)(0.25\times10^{-3})}=\boxed{0.0588\ \text{m}\ (58.8\ \text{mm})}.$$
4. Part (d) — retarded transport velocity. Dry bulk density from solids density and porosity: $\rho_b=\rho_s(1-n)=(2.64)(0.63)=1.663\ \text{g/cm}^3$. Retardation factor and transport velocity: $$R=1+\frac{\rho_b}{n}K_d=1+\frac{1.663}{0.37}(6.6)=\boxed{30.7},\qquad v_c=\frac{v}{R}=\frac{16}{30.7}=\boxed{0.522\ \text{cm/day}}.$$ Even a modest $K_d$ produces heavy retardation here, because the medium's low porosity concentrates a large solid mass per unit pore volume.

Part (e) — retarding vs. mass-reducing processes. A useful line runs through every attenuation mechanism a contaminant plume experiences: does it merely slow the contaminant down relative to the groundwater (spreading its mass over more pore volume, or temporarily storing it out of the flowing phase) without destroying or removing any of it, or does it actually remove mass from the system? Purely retarding processes are physical and reversible: linear sorption/desorption onto aquifer solids (the $K_d$-based retardation of part (d) itself), mechanical dispersion (velocity heterogeneity spreads the plume without changing total mass), and diffusion into low-permeability matrix blocks (matrix diffusion) all slow the apparent centre-of-mass velocity or dilute the peak concentration, but the same total mass of contaminant is still present in the system and can in principle re-mobilize (desorb, diffuse back out) later — this is why a "pump-and-treat" remedy on a heavily sorbed contaminant shows classic tailing/rebound behaviour. Mass-reducing processes are chemical, biological, or physical-removal mechanisms that permanently destroy the parent compound or take it out of the aquifer system: biodegradation (aerobic or anaerobic biotransformation to CO₂/CH₄/daughter products), abiotic hydrolysis (part (b) of this question) and other abiotic transformations (reductive dechlorination, hydrolysis, oxidation), volatilization to the atmosphere through the unsaturated zone, and engineered removal (pump-and-treat extraction, excavation). A practical corollary for remedial design: a retarding process alone (sorption, dispersion) is never a permanent solution — it buys time and lowers peak concentration, but only a genuine mass-reducing process (natural attenuation via biodegradation, or an engineered removal) actually shrinks the total contaminant inventory over the long term.