For a decomposition reaction the values of rate constant k at two… - Vidyadip

Question

For a decomposition reaction the values of rate constant $k$ at two different temperatures are given below:
$k _1=2.15 \times 10 L mol ^{-1} s ^{-1} \text { at } 650 K$
$k _2=2.39 \times 10 L mol ^{-1} s ^{-1} \text { at } 700 K$
Calculate the value of activation energy for this reaction. $\left( R =8.314 J K ^{-1} mol ^{-1}\right)$

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Answer

Here
$k _1=2.15 \times 10 L mol ^{-1} s ^{-1} \text { at } 650 K$
$T _1=650 K$
$T _2=700 K \text { and }$
$k _2=2.39 \times 10 L mol ^{-1} S ^{-1} \text { at } 700 K$
$R =8.314 J K ^{-1} mol ^{-1}$
Using the formula
$\log \frac{k_2}{k_1}=\frac{ E _{ a }}{2.303 R }\left[\frac{ T _2- T _1}{ T _1 T _2}\right]$
$\log \frac{2.39 \times 10^{-7}}{2.15 \times 10^{-8}}=\frac{E_a}{2.303 \times 8.314}\left[\frac{700-650}{650 \times 700}\right]$
$\log 1.111 \times 10 =\frac{E_a}{19.147} \times \frac{50}{455000}$
$1.0457 =\frac{E_a}{19.147} \times \frac{1}{9100}$
$E _{ a } =182202.812 J \text { or } 182.203 kJ$

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