Calculate the activation energy for a reaction of which rate constants becomes 4 times when temperature changes from $30^{0} C$ to $50^{0} C$ [R= 8.314 $J m o l^{- 1} K^{- 1}$ ]

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Solution

$l n \frac{k^{'}}{k} = \frac{E_{a}}{R} \times [\frac{T^{'} - T}{T T^{'}}]$
$T = 30^{o} C = 30 + 273 K = 303 K$
$T^{'} = 50^{o} C = 50 + 273 K = 323 K$
$l n \frac{k^{'}}{k} = \frac{E_{a}}{R} \times [\frac{T^{'} - T}{T T^{'}}]$
$l n \frac{4 k}{k} = \frac{E_{a}}{8.314 J / m o l / K} \times [\frac{323 - 303}{303 \times 323}]$
$1.386 = 2.46 \times 10^{- 5} \times E_{a}$
$E_{a} = 56400 J / m o l$
$E_{a} = 56.4 k J / m o l$
because $1 k J = 1000 J$
Note: Since rate constant becomes 4 times, $k^{'} = 4 k$

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