Chlorination Pathogen Reduction Explained

This question asks us to determine the percentage of pathogens that remain viable after a specific contact time during a chlorination process. We are given the contact time and the rate constant for inactivation, which allows us to use a standard disinfection model.

Understanding Disinfection Kinetics

The inactivation of microorganisms (pathogens) by disinfectants like chlorine typically follows first-order kinetics. This means the rate at which pathogens are inactivated is directly proportional to the concentration of viable pathogens present at any given time. The relationship is often described by the following formula, derived from Chick's Law:

$$ N_t = N_0 \cdot e^{-Kt} $$

In this equation:

• $N_t$ is the number of pathogens remaining after time $t$.
• $N_0$ is the initial number of pathogens at the start (time $t=0$).
• $K$ is the disinfection rate constant, which indicates how quickly the disinfectant inactivates the pathogens. Its units are typically inverse time (e.g., per minute).
• $t$ is the contact time, the duration for which the pathogens are exposed to the disinfectant.

Calculating Remaining Pathogen Percentage

To find the percentage of pathogens remaining, we need to calculate the ratio of the number of pathogens remaining ($N_t$) to the initial number of pathogens ($N_0$), and then multiply by 100. This can be expressed as:

$$ \text{Percentage Remaining} = \frac{N_t}{N_0} \times 100\% $$

By substituting the kinetic formula ($N_t = N_0 \cdot e^{-Kt}$) into this expression, we get:

$$ \text{Percentage Remaining} = \frac{N_0 \cdot e^{-Kt}}{N_0} \times 100\% $$

The $N_0$ terms cancel out, simplifying the formula to:

$$ \text{Percentage Remaining} = e^{-Kt} \times 100\% $$

Applying Given Values

The problem provides us with the following information:

• The rate constant, $K = 0.046 \, \text{/min}$.
• The contact time, $t = 1 \, \text{min}$.

We can now plug these values into our derived formula:

$$ \text{Percentage Remaining} = e^{-(0.046 \, \text{/min} \times 1 \, \text{min})} \times 100\% $$

First, calculate the product $Kt$:

$$ Kt = 0.046 \times 1 = 0.046 $$

Now, substitute this back into the formula:

$$ \text{Percentage Remaining} = e^{-0.046} \times 100\% $$

Final Calculation and Result Interpretation

To find the final percentage, we calculate the value of $e^{-0.046}$:

$$ e^{-0.046} \approx 0.95503 $$

Now, convert this fraction to a percentage:

$$ \text{Percentage Remaining} \approx 0.95503 \times 100\% $$

$$ \text{Percentage Remaining} \approx 95.503\% $$

Rounding the result to two decimal places gives us 95.50%. This means that after 1 minute of contact time with the chlorine, approximately 95.50% of the initial pathogens remain viable, while the rest have been inactivated.

Identifying the Correct Option

Let's compare our calculated result with the options provided:

Our calculated value of 95.50% matches Option 2 exactly.