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

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