Chemical Kinetics Trick — Relation Between Pt and P∞ Explained ⚡

 

❓ Question

A(g) → B(g) + C(g) is a first-order reaction.
The reaction is followed using pressure method.

Let:

• $P_t$ = total pressure at time $t$

• $P_\infty$ = total pressure at infinite time

• Reaction started with only A

Which of the following expressions gives the correct rate constant $k$?

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🖼️ Question Image

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✍️ Short Solution

To derive the expression for $k$, use pressure as a measure of extent of reaction.

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🔹 Step 1 — Initial condition

Reaction:

$$A \rightarrow B + C$$

Initial pressure of A:

$$P_A = P_0$$

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🔹 Step 2 — Pressure at time $t$

Let $x$ be the amount (in pressure units) of A decomposed at time $t$.

Then:

• Pressure of A remaining = $P_0 - x$
  

• Pressure of B formed = $x$
  

• Pressure of C formed = $x$
  

Total pressure at time $t$:

$$P_t = (P_0 - x) + x + x = P_0 + x$$

So:

$$x = P_t - P_0$$

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🔹 Step 3 — Total pressure at infinite time

When reaction goes to completion: A is fully consumed → $x = P_0$

So:

$$P_\infty = P_0 + P_0 = 2P_0$$

Thus:

$$P_0 = \frac{P_\infty}{2}$$

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🔹 Step 4 — Write first-order rate equation

For a first-order reaction:

$$k = \frac{1}{t}\ln\frac{[A]_0}{[A]_t}$$

In terms of pressures:

$$[A]_0 = P_0$$ $$[A]_t = P_0 - x = P_0 - (P_t - P_0) = 2P_0 - P_t$$

But since $2P_0 = P_\infty$:

$$[A]_t = P_\infty - P_t$$

Thus:

$$k = \frac{1}{t} \ln\left( \frac{P_0}{P_0 - x} \right) = \frac{1}{t} \ln\left( \frac{P_\infty/2}{P_\infty - P_t} \right)$$

Simplify:

$$k = \frac{1}{t} \ln\left( \frac{P_\infty}{2(P_\infty - P_t)} \right)$$

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🧮 Image Solution

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✅ Conclusion & Final Answer

✅ Correct rate constant expression:

$$\boxed{\,k=\frac{1}{t}\ln\!\left(\frac{P_\infty}{2(P_\infty-P_t)}\right)\,}$$

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