Environmental Chemistry

Reaction Kinetics & Decay

First-order decay, half-life and rate constants — the kinetics behind BOD, disinfection and pollutant persistence.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Most environmental decay processes (BOD exertion, radioactive decay, chlorine die-away, contaminant degradation) follow first-order kinetics: the rate is proportional to the amount present, giving an exponential decline. The half-life — the time to fall to one-half — is constant and independent of the starting concentration.

The Core Formula Matrix

First-order decay:

C = C_{0} e^{- k t}

Rate law:

- \frac{d C}{d t} = k C

Half-life:

t_{1/2} = \frac{ln 2}{k} = \frac{0.693}{k}

Rate constant from half-life:

k = \frac{0.693}{t _{1/2}}

The ‘IIT Traps’

⚠
**Half-life is independent of $C_{0}$ ** for first-order reactions only.
⚠
Use natural log (ln), not log₁₀, in $C = C_{0} e^{- k t}$ .
⚠
k carries units of time⁻¹ (e.g. d⁻¹); keep time units consistent.

📚 Standard references

Chemistry for Environmental Engineering and Science — Sawyer, McCarty & Parkin · Reaction Kinetics

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · NAT

A first-order reaction has a rate constant

k = 0.231 d^{- 1}

. Its half-life is _____ days.

Q2MEDIUM2 Marks · NAT

A pollutant decays first-order with

k = 0.2 d^{- 1}

from

C_{0} = 100 mg/L

. After 5 days the concentration is _____ mg/L.

Q3HARD2 Marks · MCQ

For a first-order reaction, the time required for the concentration to fall to one-quarter of its initial value equals: