A(g) -> B(g) + C(g) is a first order reaction. Time t infinity Psystem Pt Pinfinity The reaction was started with reactant A only. Which of the following expression is correct for rate constant k ?

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Chemistry Q57 – JEE-MAIN 2025

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Home Chemistry Chemical KineticsQuestion #284

A (g) \to B (g) + C (g)

is a first order reaction.

\begin{array}{ccc} Time & t & \infty \\ P_{system} & P_{t} & P_{\infty} \end{array}

The reaction was started with reactant A only. Which of the following expression is correct for rate constant

k

Clue 1 of 3: Stoichiometry and Pressure

Relate the partial pressures of reactants and products to the total system pressure at different times.

🥷

Ninja Strategy•Initial Condition Check

For the integrated rate law $k = \frac{1}{t} \ln \frac{[A]_{0}}{[A]_{t}}$ , at $t = 0$ , the argument of the logarithm must be 1 (since $[A]_{t} = [A]_{0}$ ), making $\ln (1) = 0$ . Only option 2 satisfies this condition when $P_{t} = P_{A}^{0}$ and $P_{\infty} = 2 P_{A}^{0}$ at $t = 0$ .

🧠 Full Solution Path

Step 1: Relate Pressures to Reaction Progress✦ Active

For the reaction $A (g) \to B (g) + C (g)$ , if $P_{A}^{0}$ is the initial partial pressure of A, then at time $t$ , the partial pressures are $P_{A} = P_{A}^{0} - x$ , $P_{B} = x$ , and $P_{C} = x$ . The total pressure at time $t$ is $P_{t} = P_{A}^{0} + x$ . At $t = \infty$ , all A is consumed, so $P_{A} = 0$ , $P_{B} = P_{A}^{0}$ , $P_{C} = P_{A}^{0}$ . The total pressure at $t = \infty$ is $P_{\infty} = 2 P_{A}^{0}$ .

P_{A}^{0} = \frac{P_{\infty}}{2}

x = P_{t} - P_{A}^{0} = P_{t} - \frac{P_{\infty}}{2}

P_{A} = P_{A}^{0} - x = \frac{P_{\infty}}{2} - (P_{t} - \frac{P_{\infty}}{2}) = P_{\infty} - P_{t}

Step 2: Apply First-Order Integrated Rate Law○ Expand

The integrated rate law for a first-order reaction in terms of partial pressures is $k = \frac{1}{t} \ln \frac{P_{A}^{0}}{P_{A}}$ .

Step 3: Substitute and Simplify○ Expand

Substitute the expressions for $P_{A}^{0}$ and $P_{A}$ derived in Step 1 into the rate law.

k = \frac{1}{t} \ln \frac{\frac{P_{\infty}}{2}}{P_{\infty} - P_{t}}

k = \frac{1}{t} \ln \frac{P_{\infty}}{2 (P_{\infty} - P_{t})}

This matches option 2.

💡 Teacher's Secret Hint

Ensure careful algebraic manipulation, especially with fractions and signs.

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Chemistry Question 57 – JEE-MAIN 2025

Clue 1 of 3: Stoichiometry and Pressure

🧠 Full Solution Path