Question

A first order reaction take 100 minutes for completion of 60 Decomposition of 60% of reaction find the time when 90% of reaction complete?

Answer 1

Approximately `251.3` minutes.

Explanation:

The exponential decay function models the number of moles of reactants remaining at a given time in first-order reactions. The following explanation calculates the decay constant of the reaction from the given conditions, hence find the time it takes for the reaction to reach `90%` completion.

Let the number of moles of reactants remaining be `n(t)`, a function with respect to time.

`n(t)=n_0*e^(-lambda*t)`

where `n_0` the initial quantity of reactant particles and `lambda` the decay constant. The value `lambda` can be calculated from the number of moles of reactants left at a given time. The question states that there are `(1-60%)=40%=0.40` of reactant particles left at time `t=100 color(white)(l) "min"`. Letting `n_0=1 color(white)(l)"mol"`,

`1.00 color(white)(l)"mol"*e^(-lambda*100 color(white)(l) "min")=0.40 color(white)(l)"mol"`

`-lambda*100 color(white)(l) "min"=ln((0.40 color(white)(l)color(red)(cancel(color(black)("mol"))))/(1.00 color(white)(l)color(red)(cancel(color(black)("mol")))))`

Therefore `lambda=-(ln(0.40))/(100 color(white)(l) "min")~~9.162*10^(-3) color(white)(l)"min"^(-1)`

Let `n(t)=(1-90%)*1.00 color(white)(l)"mol"=0.10 color(white)(l)"mol"` and solve for `color(darkblue)(t)`:

`1.00 color(white)(l)"mol"*e^(-lambda*color(darkblue)(t))=0.10 color(white)(l)"mol"`

`-lambda*color(darkblue)(t)=ln((0.10 color(white)(l)color(red)(cancel(color(black)("mol"))))/(1.00 color(white)(l)color(red)(cancel(color(black)("mol")))))`

`t=-(ln(0.10))/(lambda)=-(ln(0.10))/(9.162*10^(-3) color(white)(l)"min"^(-1))=251.3 color(white)(l)"min"`

That is: it takes approximately `251.3` minutes for the reaction to complete by `90%`.

See Also
There's a neat explanation for the expression of the number of moles of reactant particles that remains at time `t` on Chemistry LibreText. See https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Kinetics/Rate_Laws/The_Rate_Law