An experiment shows that a concentration of 0.1 g/m³ of free available chlorine yield a 99% kill of bacteria in 8 minutes. Calculate disinfection rate constant (1 /min)? Assume that Chick's Law and Watson's Law hold with n = 1.

Question

Prepp Team · Accepted Answer

Disinfection Kinetics Explained

Disinfection is a critical step in treating water and wastewater, designed to inactivate or destroy harmful microorganisms like bacteria. The efficiency of a disinfectant is determined by factors such as its concentration, the duration of contact time, and the susceptibility of the target microorganisms.

Chick's Law Application

Chick's Law provides a mathematical model to understand how disinfectants affect microbial populations. It posits that the rate at which microorganisms are killed is directly proportional to the number of viable microorganisms remaining and the concentration of the disinfectant.

The law can be expressed mathematically as:

$$ \frac{dN}{dt} = -k C^n N $$

Where:

$ \frac{dN}{dt} $ represents the rate of change in the number of microorganisms over time.
$ N $ is the population of viable microorganisms at time $ t $.
$ k $ is the intrinsic disinfection rate constant.
$ C $ is the concentration of the disinfectant.
$ n $ is the coefficient of dilution, reflecting how sensitive the microorganisms are to changes in disinfectant concentration.

Watson's Law complements Chick's Law by relating contact time and concentration for a specific level of kill, often stated as $ t C^n = K $, where $ K $ is a constant. The problem specifies $ n = 1 $, simplifying the relationship.

Calculate Disinfection Rate Constant

The question requires us to calculate the disinfection rate constant given specific experimental conditions. We are provided with:

Disinfectant concentration: $ C = 0.1 $ g/m³.
Contact time: $ t = 8 $ minutes.
Required bacterial kill: 99%.
Coefficient of dilution: $ n = 1 $.

When $ n = 1 $, the integrated form of Chick's Law is commonly written as:

$$ \ln\left(\frac{N_0}{N_t}\right) = k \cdot C \cdot t $$

Here, $ N_0 $ is the initial number of bacteria and $ N_t $ is the number of bacteria remaining after time $ t $. The question asks for the disinfection rate constant in units of $1/\text{min}$. This implies we should calculate the value of $k$ using the numerical value of the concentration.

Step 1: Determine the Bacteria Ratio

A 99% kill means that only 1% of the initial bacteria remain viable. This gives us the ratio:

$$ \frac{N_0}{N_t} = \frac{1}{0.01} = 100 $$

Step 2: Calculate the Natural Logarithm

We need the natural logarithm of this ratio:

$$ \ln(100) \approx 4.60517 $$

Step 3: Substitute Values into Chick's Law Equation

Plug the values into the integrated form of Chick's Law:

$$ 4.60517 = k \cdot (0.1) \cdot (8) $$

Step 4: Solve for the Rate Constant (k)

Rearrange the equation to find $k$:

$$ k = \frac{\ln(100)}{C \times t} $$

Using the numerical value of concentration $ C = 0.1 $:

$$ k = \frac{4.60517}{0.1 \times 8} $$ $$ k = \frac{4.60517}{0.8} $$ $$ k \approx 5.75646 $$

Step 5: Final Result

The calculated disinfection rate constant is approximately 5.756 /min. This numerical value is derived using the principles of Chick's Law and the provided experimental data, matching the required units.

Prepp Team · Answer

4.706 /min

Prepp Team · Answer

2.056 /min

Prepp Team · Answer

7.056 /min