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(A) The overall rate constant $k$ is given by $k = (\frac{k_1 k_2}{k_3})^{2/3}$.Using the Arrhenius equation $k = A e^{-E_a/RT}$,we substitute the exp

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$140$

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(A) The overall rate constant $k$ is given by $k = (\frac{k_1 k_2}{k_3})^{2/3}$.Using the Arrhenius equation $k = A e^{-E_a/RT}$,we substitute the expressions for each rate constant:$e^{-E_a/RT} = [\frac{e^{-E_{a1}/RT} 	imes e^{-E_{a2}/RT}}{e^{-E_{a3}/RT}}]^{2/3}$Taking the natural logarithm on both sides:$-\frac{E_a}{RT} = \frac{2}{3} [-\frac{E_{a1}}{RT} - \frac{E_{a2}}{RT} + \frac{E_{a3}}{RT}]$$E_a = \frac{2}{3} [E_{a1} + E_{a2} - E_{a3}]$Substituting the given values:$E_a = \frac{2}{3} [180 + 80 - 50] = \frac{2}{3} [210] = 140 \ kJ/mol$.

$Step$	$Rate \ constant$	$Activation \ energy$
$Step-1$	$k_1$	$E_{a1} = 180 \ kJ/mol$
$Step-2$	$k_2$	$E_{a2} = 80 \ kJ/mol$
$Step-3$	$k_3$	$E_{a3} = 50 \ kJ/mol$

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