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Arrhenius Equation

A reaction ta...

Question

A reaction takes place in three steps with an individual rate constant and activation energy, as given below.
Rate constant Activation energy
Step 1 $k_{1}$ $E_{a_{1}} = 180 k J / m o l$
Step 2 $k_{2}$ $E_{a_{2}} = 80 k J / m o l$
Step 3 $k_{3}$ $E_{a_{3}} = 50 k J / m o l$
And overall rate constant, $k = {(\frac{k_{1} k_{2}}{k_{3}})}^{2 / 3}$ .
The overall activation energy of the reaction will be:

A

140 k J / m o l

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B

150 k J / m o l

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C

130 k J / m o l

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D

120 k J / m o l

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Solution

The correct option is A $140 k J / m o l$
$A e^{- E_{a} / R T} = {[\frac{A e^{- E_{a_{1}} / R T} \times A e^{- E_{a_{2}} / R T}}{A e^{- E_{a_{3}} / R T}}]}^{2 / 3}$
$= {[A e^{(- E_{a_{1}} - E_{a_{2}} + E_{a_{3}}) / R T}]}^{2 / 3}$
$E_{a} = \frac{2}{3} [E_{a_{1}} + E_{a_{2}} - E_{a_{3}}]$
$= \frac{2}{3} [180 + 80 - 50] = 140 k J / m o l$

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