To determine the fraction of the maximum oxygen consumption (Biochemical Oxygen Demand, or BOD) that has occurred in four days, we can use the following first-order kinetic equation for BOD:

$\text{BOD}_t = L_0 \left(1 - e^{-k t}\right)$

where:

• $\text{BOD}_t$ is the amount of oxygen consumed after time $t$.
• $L_0$ is the ultimate BOD, or the total amount of oxygen that will be consumed when the waste is completely degraded.
• $k$ is the BOD rate constant.
• $t$ is the time in days.

The fraction of the maximum oxygen consumption after time $t$ is given by:

$\frac{\text{BOD}_t}{L_0} = 1 - e^{-k t}$

Given:

• $k = 0.115 \, \text{d}^{-1}$
• $t = 4 \, \text{days}$

Substituting the values:

$\frac{\text{BOD}_t}{L_0} = 1 - e^{-0.115 \times 4}$

Let's calculate this.The fraction of the maximum oxygen consumption that has occurred in four days is approximately 0.369 or 36.9%.

Would you like more details or have any questions? Here are five related questions to consider:

1. How does temperature affect the BOD rate constant?
2. What is the significance of the BOD rate constant in environmental engineering?
3. How would the fraction of oxygen consumption change if the time period was longer?
4. What are some methods to determine the BOD rate constant experimentally?
5. How is the ultimate BOD ($L_0$) typically determined?

Tip: The BOD rate constant $k$ can vary depending on environmental conditions such as temperature and the nature of the organic matter in the waste.