Understanding Biochemical Oxygen Demand (BOD)

Biochemical Oxygen Demand (BOD) is a measure of the amount of dissolved oxygen required by aerobic biological microorganisms to break down organic material present in a water sample at certain temperature over a specific time period.

The question asks to find the ultimate BOD ($L_0$) of a sewage sample. The ultimate BOD represents the total amount of oxygen required for the complete decomposition of the organic matter.

We are given the 5-day BOD ($BOD_5$) at 20°C, which is the amount of oxygen consumed over 5 days at 20°C. We are also given the de-oxygenation constant ($k$). The relationship between BOD at any time $t$ ($BOD_t$) and the ultimate BOD ($L_0$) is given by the formula, assuming a first-order reaction rate.

The formula relating $BOD_t$, $L_0$, and the de-oxygenation constant is typically given as:

$\text{BOD}_t = L_0 (1 - e^{-kt})$ (using base-e constant $k$)

or

$\text{BOD}_t = L_0 (1 - 10^{-k't})$ (using base-10 constant $k'$)

Often, in practical problems, the given constant is the base-10 constant ($k'$) unless specified otherwise, especially when its value is relatively high like 0.8 per day. Let's assume the given de-oxygenation constant of 0.8 per day is the base-10 constant $k'$.

Calculating Ultimate BOD ($L_0$)

We can calculate the ultimate BOD ($L_0$) using the given 5-day BOD ($BOD_5$), time ($t$), and the de-oxygenation constant ($k'$).

1. Identify the given values:
   • 5-day BOD ($\text{BOD}_5$) = 180 ppm
   • Time ($t$) = 5 days
   • De-oxygenation constant ($k'$) = 0.8 per day (assuming base-10)
2. Write down the formula for BOD using the base-10 constant:

   $\text{BOD}_t = L_0 (1 - 10^{-k't})$

3. Substitute the given values into the formula:

   We are given $\text{BOD}_5 = 180$ ppm and $t=5$ days, $k'=0.8$ per day.

   $180 = L_0 (1 - 10^{-(0.8 \times 5)})$

4. Solve the equation for $L_0$:

   First, calculate the term in the exponent:

   $0.8 \times 5 = 4$

   So, the equation becomes:

   $180 = L_0 (1 - 10^{-4})$

   Calculate $10^{-4}$:

   $10^{-4} = 0.0001$

   Substitute this back into the equation:

   $180 = L_0 (1 - 0.0001)$

   $180 = L_0 (0.9999)$

   Now, solve for $L_0$:

   $L_0 = \frac{180}{0.9999}$

   Calculating the value:

   $L_0 \approx 180.018$ ppm

The calculated ultimate BOD is approximately 180.018 ppm. Looking at the options, 180 ppm is the closest value, which suggests either a slight rounding or that the 5-day BOD value itself was rounded in the problem statement leading to this result.

Key Terms in BOD Calculation

Based on the calculation using the de-oxygenation constant as a base-10 value (0.8 per day), the ultimate BOD is approximately 180 ppm, which matches one of the provided options.