---
title: "Define BOD, COD; derive BOD equation"
subject: "Sanitary Engineering"
chapter: 5
key: "semester_6/sanitary_engineering:bod-cod-derive-equation"
source: https://www.bctengineer.com/q/semester_6/sanitary_engineering/bod-cod-derive-equation
site: BCT Engineering
---

# Define BOD, COD; derive BOD equation

Sanitary Engineering · Chapter 5: Characteristics and Examination of Wastewater

- Asked in: 2071 Bhadra, 8 marks
- Video: [BOD Calculations](https://www.youtube.com/watch?v=mGaNsGvutSE) — Belinda Sturm
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/sanitary_engineering/bod-cod-derive-equation
- Whole chapter: https://www.bctengineer.com/md/semester_6/sanitary_engineering/chapter-5

## Question

Define BOD and COD and explain their significance in wastewater examination. Derive BOD equation showing relation between ultimate BOD and BOD remaining at any time, t.

## Short answer

BOD is oxygen used by bacteria to degrade biodegradable organics; COD chemically oxidizes all organics. BOD is crucial for pollution load and treatment design; COD enables quick monitoring. Derivation: first-order decay dL_t/dt = -K L_t gives L_t = L e^(-Kt) and BOD exerted y_t = L(1 - e^(-Kt)).

## Answer

**BOD (Biochemical Oxygen Demand)** is the amount of oxygen required by bacteria to oxidise (stabilise) the biodegradable organic matter in a water or wastewater sample under aerobic conditions at a specified temperature and time (standard: 5 days at 20°C). It is given in mg/L.

**COD (Chemical Oxygen Demand)** is the amount of oxygen needed to oxidise all organic matter (biodegradable and non-biodegradable) by a strong chemical oxidant, potassium dichromate ($K_2Cr_2O_7$) in boiling acid medium, expressed in mg/L. The test takes about 2-3 hours.

#### Significance of BOD
1. It measures the **strength (organic pollution load)** of sewage; raw domestic sewage has BOD₅ of 200-400 mg/L.
2. It is used to **design treatment plants** (trickling filter, activated sludge, ponds) and to find their efficiency.
3. It is used to find the **oxygen demand on a receiving stream**, and so the degree of treatment needed before discharge (self-purification calculations).
4. It is the main parameter in effluent standards (for example, effluent standards in Nepal and India limit BOD₅ to about 30-50 mg/L for discharge to inland surface water).
5. It measures only the biodegradable organic matter and gives an idea of how much will be decomposed in nature.

#### Significance of COD
1. It is a **rapid test** (hours against 5 days) and can be used for plant control.
2. It also includes non-biodegradable matter, so it is used for industrial wastes that are toxic to bacteria, where BOD cannot be measured.
3. COD is always greater than BOD. The ratio **BOD/COD** shows biodegradability: above about 0.5 the waste is easily treated biologically; below 0.3 it is difficult and contains toxic or refractory matter.
4. It is used with BOD to estimate the non-biodegradable part: $COD-BOD_u$.

#### Derivation of BOD equation
The oxidation of organic matter by bacteria follows a first-order reaction: the rate of reaction at any time is proportional to the oxygen equivalent of organic matter remaining.

Let $L$ = ultimate first-stage BOD (total organic matter at $t=0$), $L_t$ = BOD remaining at time $t$, $y_t$ = BOD exerted (satisfied) in time $t$, and $K$ = rate constant (day⁻¹).

$$
\frac{dL_t}{dt}=-K L_t
$$

$$
\int_{L}^{L_t}\frac{dL_t}{L_t}=-K\int_{0}^{t}dt\ \Rightarrow\ \ln\frac{L_t}{L}=-Kt\ \Rightarrow\ L_t=L e^{-Kt}
$$

In base 10 form, $L_t=L\,10^{-kt}$, where $k=K/2.303$.

BOD remaining after time $t$ is therefore related to the ultimate BOD by

$$
\boxed{L_t = L\,e^{-Kt}=L\,10^{-kt}}
$$

and the BOD exerted is

$$
y_t=L-L_t=L\left(1-e^{-Kt}\right)=L\left(1-10^{-kt}\right)
$$

For $k=0.1$ per day at 20°C, $y_5=0.68L$ (about two-thirds of the ultimate BOD is satisfied in 5 days). The curve of $y_t$ against $t$ rises quickly at first, and approaches $L$ slowly.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which statement correctly defines BOD?
   A. Oxygen used by bacteria to stabilise biodegradable organic matter under aerobic conditions, standard 5 days at 20°C
   B. Oxygen needed by potassium dichromate to oxidise all organic matter in boiling acid
   C. Oxygen present in the water sample after 5 days at 20°C
   D. Oxygen used by bacteria in 20 days at 5°C
   Answer: A. BOD is the oxygen consumed by bacteria in oxidising biodegradable organic matter, measured at 20°C for 5 days. (Key point: Definition of BOD)

2. What is the basis of the BOD equation derivation?
   A. The rate of oxidation is constant with time
   B. The rate of oxidation at any time is proportional to the oxygen equivalent of organic matter remaining
   C. The rate is proportional to the BOD already exerted
   D. The rate is proportional to the square of the dissolved oxygen
   Answer: B. First-order kinetics: dLt/dt = -K Lt, so the rate depends on the matter remaining. (Key point: First-order assumption)

3. What does integrating dLt/dt = -K Lt from t = 0 to t give?
   A. Lt = L (1 - e^(-Kt))
   B. Lt = L + Kt
   C. Lt = L e^(-Kt)
   D. Lt = L e^(Kt)
   Answer: C. ln(Lt/L) = -Kt, so the BOD remaining is Lt = L e^(-Kt). (Key point: Result of integration)

4. Which expression gives the BOD exerted in time t?
   A. yt = L e^(-Kt)
   B. yt = L(1 + e^(-Kt))
   C. yt = L - K t e^(-Kt)
   D. yt = L(1 - e^(-Kt))
   Answer: D. BOD exerted is the ultimate minus the remaining: yt = L - Lt = L(1 - e^(-Kt)). (Key point: BOD exerted)

## Related in Sanitary Engineering

- [BOD, COD; ultimate BOD of 480 mg/l](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Abod-cod-ultimate-bod-480) — Sanitary Engineering, chapter 5
- [BOD/COD significance; BOD at 30°C](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Abod-cod-significance-temp-conversion) — Sanitary Engineering, chapter 5
- [BOD and COD with significance](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Abod-cod-significance) — Sanitary Engineering, chapter 5
- [Rate constant and remaining BOD](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Abod-rate-250-325) — Sanitary Engineering, chapter 5
- [Wastewater examination and sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Awastewater-sampling) — Sanitary Engineering, chapter 5, asked 2×
- [Ultimate BOD from 5-day BOD at 37°C](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Abod-5day-37c-200) — Sanitary Engineering, chapter 5

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