Read the passage carefully and answer the questions.
The chemical reactions proceed at different rates which depend on various factors like concentration, temperature, etc. Rate law is an expression which relates the rate of reaction with concentration of various reacting species in the reaction. For a first order reaction, the concentration of the reactant after time \( t \) is related with its initial concentration by the relation, \( C = C_0e^{-kt} \). The rate constants for different orders have different expressions. For a first order reaction, \( k =\frac{2.303}{t}\log\frac{a}{a-x} \), where \( a \) is the initial concentration and \( x \) is the extent of reaction.
A first order reaction is one-fifth completed in 40 minutes. Calculate the time required for its 100% completion
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → infinite
For a first order reaction the time needed to bring the concentration down from \( [\mathrm{R}]_0 \) to \( [\mathrm{R}] \) is
\( \displaystyle t = \frac{2.303}{k}\log\frac{[\mathrm{R}]_0}{[\mathrm{R}]} \)
The concentration falls exponentially, \( [\mathrm{R}] = [\mathrm{R}]_0\,e^{-kt} \): in every equal interval the same fraction of what is left reacts, so the concentration comes closer and closer to zero but never reaches it.
From the data, one-fifth is completed in 40 min, so four-fifths of the reactant is left:
\( \displaystyle k = \frac{2.303}{40\ \mathrm{min}}\log\frac{[\mathrm{R}]_0}{\frac{4}{5}[\mathrm{R}]_0} = \frac{2.303}{40}\log 1.25 \)
\( \displaystyle k = \frac{2.303 \times 0.0969}{40} = 5.58 \times 10^{-3}\ \mathrm{min^{-1}} \)
For 100% completion \( [\mathrm{R}] = 0 \):
\( \displaystyle t = \frac{2.303}{5.58 \times 10^{-3}}\log\frac{[\mathrm{R}]_0}{0} \rightarrow \infty \)
So the time required for 100% completion of a first order reaction is infinite.
Shortcut: The value of k is not needed at all. Whatever the rate constant, \( \log([\mathrm{R}]_0/[\mathrm{R}]) \) becomes infinite as \( [\mathrm{R}] \to 0 \), so no first order reaction is ever 100% complete.