Two rate constants for a first‑order reaction are measured: k_1=2.5×10^-3 s^-1 at T_1=300 K and k_2=1.2×10^-2 s^-1 at T_2=350 K. Using the Arrhenius equation, calculate the activation energy E_a in kJ·mol^-1.

E_a=2.74×10^1 kJ mol^-1 (approximately 27.4 kJ mol^-1)

Two rate constants for a first‑order reaction are measured: $k_{1} = 2.5 \times 1 0^{- 3} s^{- 1}$ at $T_{1} = 300 K$ and $k_{2} = 1.2 \times 1 0^{- 2} s^{- 1}$ at $T_{2} = 350 K$ . Using the Arrhenius equation, calculate the activation energy $E_{a}$ in kJ·mol $^{- 1}$ .

medium 3 marks Class 12 · Chemistry · Chemical Kinetics

Answer

E_{a} = 2.74 \times 1 0^{1} kJ mol^{- 1}

(approximately

27.4 kJ mol^{- 1}

)

The correct answer is $E_{a} = 2.74 \times 1 0^{1} kJ mol^{- 1}$ (approximately $27.4 kJ mol^{- 1}$ ) — Round to two significant figures as required.

Step-by-step solution

Write the two‑point Arrhenius form: $ln \frac{k _{2}}{k _{1}} = \frac{E _{a}}{R} (\frac{1}{T _{1}} - \frac{1}{T _{2}})$ .
Calculate the left side: $ln (\frac{1.2 \times 1 0 ^{- 2}}{2.5 \times 1 0 ^{- 3}}) = ln (4.8) = 1.5686$ .
Compute the temperature term: $\frac{1}{T _{1}} - \frac{1}{T _{2}} = \frac{1}{300} - \frac{1}{350} = \frac{350 - 300}{300 \times 350} = \frac{50}{105000} = 4.7619 \times 1 0^{- 4} K^{- 1}$ .
Re‑arrange for $E_{a}$ : $E_{a} = \frac{R l n ( k _{2} / k _{1} )}{( 1/ T _{1} - 1/ T _{2} )}$ .
Insert $R = 8.314 J mol^{- 1} K^{- 1}$ and the numbers: $E_{a} = \frac{8.314 \times 1.5686}{4.7619 \times 1 0 ^{- 4}} J mol^{- 1} = 2.74 \times 1 0^{4} J mol^{- 1}$ .
Convert to kJ: $E_{a} = 27.4 kJ mol^{- 1}$ .
Round to two significant figures as required.

Common mistake: Forgetting to convert the temperature difference term to the correct reciprocal form or mixing up

T_{1}

and

T_{2}

in the denominator

Concept tested: Arrhenius equation and activation energy

Published 11 September 2026 · How this solution was prepared and checked

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Two rate constants for a first‑order reaction are measured: k1​=2.5×10−3s−1 at T1​=300K and k2​=1.2×10−2s−1 at T2​=350K. Using the Arrhenius equation, calculate the activation energy Ea​ in kJ·mol−1.

Step-by-step solution

Two rate constants for a first‑order reaction are measured: $k_{1} = 2.5 \times 1 0^{- 3} s^{- 1}$ at $T_{1} = 300 K$ and $k_{2} = 1.2 \times 1 0^{- 2} s^{- 1}$ at $T_{2} = 350 K$ . Using the Arrhenius equation, calculate the activation energy $E_{a}$ in kJ·mol $^{- 1}$ .