Bacterial Growth and Decay Graphs in Chemical Kinetics

 

❓ Question

A person's wound was exposed to bacteria and bacterial growth started.

Later, antibacterial medicine was applied and bacterial decay rate became proportional to:

$$r \propto N^2$$

Given:

• Before medicine → bacterial growth follows first order kinetics
• $N$ = number of bacteria

Identify the correct set of graphs for “before” and “after” medicine application.

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🖼 Question Image

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✍️ Short Concept

This question is purely about:

👉 Shape of kinetic graphs
👉 Order of reaction

You only need to identify:

• First order growth graph
• Second order decay graph

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🔷 Step 1 — Before Medicine: First Order Growth 💯

Given:

$$\frac{dN}{dt} \propto N$$

This is first order growth.

So bacteria increase:

$$N = N_0 e^{kt}$$

Graph nature:

✅ Exponential increase
✅ Curve rises rapidly with time

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🔷 Step 2 — After Medicine: Second Order Decay

Given:

$$r \propto N^2$$

So:

$$-\frac{dN}{dt} = kN^2$$

Integrated form:

$$\frac{1}{N} = \frac{1}{N_0} + kt$$

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🔷 Step 3 — Shape of Decay Graph

For second order decay:

• $N$ decreases non-linearly
• Rapid decrease initially
• Then slows down gradually

Important graph:

$$\frac{1}{N} \text{ vs } t$$

gives straight line.

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🔷 Step 4 — Graph Identification Trick

Before medicine:

✅ Exponential growth curve

After medicine:

✅ Second order decay curve
OR
✅ Straight line in $1/N$ vs $t$

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🔷 Step 5 — JEE Trap Alert 🚨

❌ Confusing first order decay with second order decay

❌ Assuming all decay graphs are exponential

Remember:

First order:

$$\ln N \text{ vs } t = \text{straight line}$$

Second order:

$$1/N \text{ vs } t = \text{straight line}$$

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✅ Final Takeaway

Before medicine:

$$\boxed{\text{Exponential growth}}$$

After medicine:

$$\boxed{\text{Second order decay}}$$

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⭐ Golden JEE Insight

Graph questions in kinetics are mostly about:

👉 Recognising integrated rate law forms

Not heavy calculations.