First Order Growth and Second Order Decay Trick

 

❓ Concept Question

How do graphs change when:

• Bacterial growth follows first order kinetics
• Bacterial decay follows second order kinetics after medicine?

---

🖼 Concept Image

---

✍️ Short Concept

This concept is based on:

👉 Growth kinetics
👉 Decay kinetics
👉 Identifying graph behaviour from integrated rate laws.

---

🔷 Step 1 — Before Medicine: Bacterial Growth 💯

Given:

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

This represents:

$$\text{First order growth}$$

Solution form:

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

Graph behaviour:

$$N \text{ vs } t$$

👉 Exponential increase
👉 Upward rising curve

---

🔷 Step 2 — After Medicine: Bacterial Decay

Given:

$$r \propto N^2$$

So:

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

This represents:

$$\text{Second order decay}$$

---

🔷 Step 3 — Integrated Form of Second Order

Integrated equation:

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

This is equation of straight line:

$$y = mx + c$$

So graph:

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

👉 Straight line
👉 Slope = $k$

---

🔷 Step 4 — Concept Lock

Before medicine:

$$\text{First order growth}$$
$$N \text{ vs } t \rightarrow \text{Exponential curve}$$

After medicine:

$$\text{Second order decay}$$
$$\frac{1}{N} \text{ vs } t \rightarrow \text{Straight line}$$

---

✅ Final Takeaway

👉 First order growth:

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

👉 Second order decay:

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

---

⭐ Golden JEE Insight

Kinetics graph questions are mostly solved by:

👉 Recognising integrated equation forms

No lengthy calculations needed.