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.]
[Given: $ N $ = No. of bacteria, $ t $ = time, bacterial growth follows 1st order kinetics.]
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The Correct Option is B
Approach Solution - 1
The given problem deals with understanding bacterial growth and decay, followed by the application of an antibacterial treatment. Let's analyze both scenarios using first-order kinetics and decay proportionality.
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Bacterial Growth (Before Treatment):
- Bacterial growth follows first-order kinetics, which means the rate of increase in the number of bacteria is directly proportional to the number of bacteria present.
- Mathematically, this can be expressed as: \(\frac{dN}{dt} = kN\), where k is the rate constant for growth.
- The solution of this differential equation for the number of bacteria N over time t is: N = N_0 e^{kt}, where N_0 is the initial number of bacteria.
- This results in an exponential growth curve.
-
Bacterial Decay (After Treatment):
- Once the antibacterial medicine is applied, the bacteria begin to decay, and the rate of decay is given as proportional to the square of the existing number of bacteria.
- This can be represented by: \(\frac{dN}{dt} = -rN^2\), where r is the decay constant.
- The solution for this second-order differential equation is: \(\frac{1}{N} = r\cdot t + \frac{1}{N_0}\).
- This implies a hyperbolic decay curve.
The given correct graph:
shows an exponential increase followed by a rapid decrease, indicating the bacterial growth and subsequent decay after treatment. Hence, it correctly represents the 'before' and 'after' situations given in the problem.
Other options are incorrect because they either do not match the exponential growth or hyperbolic decay patterns correctly.
Approach Solution -2
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.
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