Question

An egg takes 4.0 minutes to boil at sea level where boiling point of water is $T_1$ K, whereas it takes 8.0 minutes to boil on a mountain top where boiling point of water is $T_2$ K. The activation energy for the reaction that takes place during boiling of egg is

• A. $0.693 \frac{T_2 - T_1}{T_1 T_2}$
• B. $0.693 \frac{T_1 - T_2}{T_1 T_2}$
• C. $0.693 R \frac{T_1 T_2}{T_2 - T_1}$
• D. $0.693 R \frac{T_1 T_2}{T_1 - T_2}$

Hint:

Arrhenius tricks: \begin{itemize} \item Time $\propto 1/k$ \item Use $\ln 2 = 0.693$ \item Cross-multiply temperatures carefully \end{itemize}

Answer 1

The Correct Option is D

Concept: Use Arrhenius equation: \[ k = A e^{-E_a/RT} \] Time $\propto \frac{1}{k}$ Step 1: Rate ratio \[ \frac{k_1}{k_2} = \frac{t_2}{t_1} = \frac{8}{4} = 2 \] Step 2: Arrhenius relation \[ \ln\left(\frac{k_1}{k_2}\right) = \frac{E_a}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \] \[ \ln 2 = \frac{E_a}{R}\left(\frac{T_1 - T_2}{T_1 T_2}\right) \] Since $\ln 2 = 0.693$: \[ E_a = 0.693 R \frac{T_1 T_2}{T_1 - T_2} \]