First Order Reaction Using Pressure Data

 

❓ Concept Question

For a gas-phase first order reaction:

$$A(g) \rightarrow 2B(g) + C(g)$$

How do we use pressure data to calculate the rate constant k?

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🖼 Concept Image

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✍️ Short Concept

In gas reactions:

$$P \propto n$$

👉 Pressure acts like concentration

So we can directly apply first order equations using pressure.

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🔷 Step 1 — Reaction Type 💯

$$A(g) \rightarrow 2B(g) + C(g)$$

Total moles:

$$1 \rightarrow 3$$

👉 Total pressure increases with time.

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🔷 Step 2 — Infinity Pressure Concept

At:

$$t = \infty$$

All A converted.

So:

$$P_\infty = 3P_0$$

🔥 This is the MOST IMPORTANT relation.

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🔷 Step 3 — Pressure at Time t

At time $t$:

Total pressure:

$$P_t = \text{Pressure of A + products}$$

Unreacted A decreases exponentially:

$$P_A \propto e^{-kt}$$

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🔷 Step 4 — First Order Shortcut Formula

Direct formula using pressure:

$$\boxed{ k = \frac{2.303}{t} \log \frac{P_\infty - P_0}{P_\infty - P_t} }$$

👉 No need to separately calculate concentration.

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🔷 Step 5 — Common JEE Traps

❌ $P_\infty$ ko initial pressure samajh lena

❌ Stoichiometry ignore kar dena

❌ Total pressure = pressure of A maan lena

👉 Always separate total vs partial pressures.

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✅ Final Takeaway

For gas-phase first order reactions:

$$\boxed{P_\infty = n_{final} \times P_0}$$
$$\boxed{k = \frac{2.303}{t} \log \frac{P_\infty - P_0}{P_\infty - P_t}}$$

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⭐ Golden JEE Insight

First order reaction:

👉 Always follows exponential decay

👉 Reaction theoretically never completes fully