---
title: "Rate constant K at 25°C; ultimate BOD"
subject: "Sanitary Engineering"
chapter: 5
key: "semester_6/sanitary_engineering:bod-k-ultimate-293-56"
source: https://www.bctengineer.com/q/semester_6/sanitary_engineering/bod-k-ultimate-293-56
site: BCT Engineering
---

# Rate constant K at 25°C; ultimate BOD

Sanitary Engineering · Chapter 5: Characteristics and Examination of Wastewater

- Asked in: 2068 Magh (old course), 10 marks
- Video: [17 7 Variation of Reaction rate constant and Ultimate BOD with Temperature](https://www.youtube.com/watch?v=EwUYf3HbyYo) — Muzaffar Zainal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/sanitary_engineering/bod-k-ultimate-293-56
- Whole chapter: https://www.bctengineer.com/md/semester_6/sanitary_engineering/chapter-5

## Question

The following observations were made on a sewage sample at 20°C: BOD5 at 20°C = 293.55 mg/l; BOD1 at 20°C = 56.15 mg/l. Calculate rate reaction constant K at 25°C and ultimate first stage BOD at 30°C.

## Short answer

Given BOD5=293.55 and BOD1=56.15 mg/L at 20°C, ratio >5 indicates inconsistent data. Using standard K20=0.1/day, K25=0.1258/day. Ultimate BOD=429.3 mg/L (same at 30°C).

## Answer

#### Check of the data
For a first-order reaction, $y_t=L(1-10^{-kt})$, the ratio $y_5/y_1$ cannot exceed 5 (it is 5 only when $k\to0$). Here

$$
\frac{y_5}{y_1}=\frac{293.55}{56.15}=5.228>5
$$

so the two readings cannot both belong to a single first-order curve; one of the printed values is doubtful. Therefore $K_{20}$ cannot be found from the pair. Instead, the more reliable 5-day value is used together with the standard rate constant $K_{20}=0.1$ day⁻¹ (base 10), $\theta=1.047$.

#### Step 1: Rate constant at 25°C
$$
K_{25}=K_{20}(1.047)^{25-20}=0.1\times1.047^{5}=0.1258\ \text{day}^{-1}
$$

#### Step 2: Ultimate first-stage BOD
$$
L=\frac{y_5}{1-10^{-K_{20}\times5}}=\frac{293.55}{1-10^{-0.5}}=\frac{293.55}{0.6838}=429.3\ \text{mg/L}
$$

The ultimate BOD does not depend on temperature, so the ultimate BOD at 30°C is the same, 429.3 mg/L (only the rate constant changes with temperature; $K_{30}=0.1\times1.047^{10}=0.1583$ day⁻¹, so the 5-day BOD at 30°C would be 359.9 mg/L).

**Answer: $K_{25}$ = 0.1258 day⁻¹ (base 10); ultimate BOD ≈ 429.3 mg/L.**

## Check yourself

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

1. Why can K20 not be found from y5 = 293.55 and y1 = 56.15?
   A. y5/y1 = 5.23 exceeds 5, which is impossible for first-order BOD
   B. The ratio is below 1
   C. Temperatures differ
   D. The values are too small
   Answer: A. For first-order BOD, y5/y1 cannot exceed 5. (Key point: Data check)

2. What constant is assumed instead?
   A. K20 = 0.23 per day
   B. K20 = 0.1 per day (base 10)
   C. K20 = 1
   D. K20 = 0.5
   Answer: B. The standard value is used with the reliable 5-day figure. (Key point: Assumed constant)

3. What is K25?
   A. 0.1 x 1.047^25
   B. 0.1 / 1.258
   C. 0.1 x 1.047^5 = 0.1258 per day
   D. 0.1 x 25
   Answer: C. Exponent equals 25 - 20 = 5. (Key point: K25)

4. How does ultimate BOD at 30°C compare with that at 20°C?
   A. It is higher
   B. It is lower
   C. It is 359.9 mg/L
   D. It is the same, about 429 mg/L
   Answer: D. L = 293.55/0.6838 = 429.3 mg/L, independent of temperature. (Key point: Ultimate BOD at 30°C)

## Related in Sanitary Engineering

- [Ultimate BOD and rate constant](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Abod-ultimate-rate-constant-100-210) — Sanitary Engineering, chapter 5
- [BOD5 and ultimate BOD, 5% dilution](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Abod-dilution-5pct-blank) — Sanitary Engineering, chapter 5
- [BOD1 at 22°C to BOD5 at 30°C](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Abod-1day-22c-310-to-5day-30c) — Sanitary Engineering, chapter 5
- [First and second stage BOD](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Afirst-second-stage-bod-400) — Sanitary Engineering, chapter 5
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- [BOD1 at 23°C to BOD5 at 30°C](https://www.bctengineer.com/question.md?key=semester_6%2Fsanitary_engineering%3Abod-1day-23c-105-to-5day-30c) — Sanitary Engineering, chapter 5

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