Understanding the Rate of BOD Exertion

Biochemical Oxygen Demand (BOD) is a measure of the amount of oxygen consumed by microorganisms in decomposing organic matter in water. When we talk about the rate of BOD exerted, we are referring to how quickly this oxygen is being used up at any given moment.

The process of BOD exertion is typically modeled as a first-order reaction. In a first-order reaction, the rate of the reaction is directly proportional to the concentration of the reactant remaining. In the context of BOD, the 'reactant' is the remaining oxygen-demanding organic matter.

Let's define some terms:

• \(L_a\): Ultimate BOD (total oxygen demand if the reaction went to completion).
• \(Y_t\): BOD satisfied at time \(t\) (the amount of oxygen consumed up to time \(t\)).
• \(L_t\): BOD remaining at time \(t\) (the oxygen demand that still needs to be satisfied).

The relationship between these is \(L_a = Y_t + L_t\).

Relationship between Rate of BOD Exerted and BOD Remaining

The rate at which BOD is exerted at time \(t\) is proportional to the amount of organic matter available for decomposition at that time. This amount of available organic matter is directly related to the BOD remaining ($L_t$).

Mathematically, the rate of decrease of the remaining BOD ($L_t$) is proportional to $L_t$ itself:

where \(K\) is the reaction rate constant (typically base e, \(K_1\), or base 10, \(K\)). The negative sign indicates that \(L_t\) is decreasing over time.

The rate of BOD exerted is the rate at which oxygen is consumed, which is equivalent to the rate at which the remaining BOD is being satisfied. This rate can be expressed as the rate of change of \(Y_t\), or the negative rate of change of \(L_t\):

From the first-order reaction equation, we have \(-\frac{dL_t}{dt} = K \cdot L_t\).

Therefore, the rate of BOD exerted at any time \(t\) is given by:

This equation clearly shows that the rate of BOD exerted is directly proportional to \(L_t\), which is the BOD remaining at that time.

Analyzing the Options

• Option 1: Directly proportional to BOD satisfied (\(Y_t\)) - This is incorrect. As time increases, \(Y_t\) increases, but the rate of BOD exertion generally decreases (because \(L_t\) decreases).
• Option 2: Directly proportional to BOD remaining (\(L_t\)) - This aligns with the first-order kinetics model as shown above. The rate is \(K \cdot L_t\).
• Option 3: Inversely proportional to BOD satisfied (\(Y_t\)) - This is incorrect. There isn't a simple inverse relationship.
• Option 4: Inversely proportional to BOD remaining (\(L_t\)) - This is incorrect. The rate is directly, not inversely, proportional to \(L_t\).

Based on the first-order kinetics model commonly used for BOD exertion, the rate at which BOD is exerted at any time is directly proportional to the amount of BOD remaining to be satisfied.