Question

A person's wound was exposed to some bacteria and then bacteria growth started to happen at the same place. The wound was later treated with some antibacterial medicine and the rate of bacterial decay ($r$) was found to be proportional with the square of the existing number of bacteria at any instance. Which of the following set of graphs correctly represents the 'before' and 'after' situation of the application of the medicine?
[Given: $ N $ = No. of bacteria, $ t $ = time, bacterial growth follows 1st order kinetics.]

• A.
• B.
• C.
• D.

Hint:

In first-order kinetics, the growth curve exhibits exponential increase. When an antibacterial treatment is applied, the growth rate slows down, resulting in a flattening of the growth curve.

Answer 1

The Correct Option is B

Step 1: Understanding Bacterial Growth Before Treatment
The initial growth of the bacteria follows 1st order kinetics. This means that the rate of bacterial growth is proportional to the current number of bacteria present.
In first-order kinetics, the graph of the number of bacteria \( N \) over time \( t \) (shown as \( \frac{N}{N_0} \)) would exhibit an exponential increase. This is characterized by a rapidly rising curve as bacteria reproduce over time. Since the rate of decay \( r \) is proportional to the square of the number of bacteria at any instant, \( r \propto N^2 \). Hence, the rate \( r \) increases as \( N \) increases.
Step 2: Introducing the Medicine
Once the medicine is introduced, its effect is to slow down the bacterial growth by increasing the decay rate. The rate of bacterial decay is now proportional to the square of the number of bacteria, i.e., \( r \propto N^2 \). As the medicine works, the number of bacteria no longer follows the exponential increase. Instead, it starts to flatten, and the growth rate slows down because of the increased bacterial decay rate.

Thus, the growth of bacteria is hindered, and the graph showing bacterial growth should start to flatten out after a certain time.
Step 3: Interpreting the Graphs
Before treatment: The graph of \( \frac{N}{N_0} \) versus \( t \) should show an upward exponential curve (exponential growth), as described earlier for first-order kinetics. After treatment: Once the medicine is applied, the bacterial growth curve flattens. The graph of rate of decay \( r \) versus the number of bacteria \( N \) should show a sharp decline, as the bacteria are no longer able to grow rapidly.
Step 4: Evaluating the Graphs
Option (1): The graphs do not correctly show the flattening of bacterial growth after the application of the medicine.
Option (2): This option correctly represents the situation, where:
The "before" graph shows an exponential increase of \( \frac{N}{N_0} \) over time. The "after" graph shows a declining rate of decay \( r \), reflecting the effect of the antibacterial treatment.
Option (3): The "after" graph incorrectly shows an increasing rate of decay, which is not correct after the medicine is applied.
Option (4): This graph shows a linear relationship for the "after" scenario, which is not consistent with bacterial decay behavior under first-order kinetics.
Therefore, Option (2) is the correct set of graphs that represents the situation before and after the application of the medicine.