A first order reaction is 50% completed after 30 minutes.This implies that the time required to complete 90% of the reaction is?
1 Answer
Explanation:
The integrated rate law for a first-order reaction looks like this
#color(blue)(ln( A/A_0) = - k * t)" "# , where#
Now, I'll assume that you're not familiar with the equation that describes the half-life of a first-order reaction.
As you know, the half-life of a chemical reaction tells you how much time is needed for the concentration of a reactant to reach half of its initial value.
Simply put, the time needed for
In your case, you know that your first-order reaction is
#A = 1/2 * A_0 -># half of the initial concentration remains after one half-life
Plug this into the above equation and solve for
#ln( (1/2 * color(red)(cancel(color(black)(A_0))))/(color(red)(cancel(color(black)(A_0)))) ) = - k * "30 min"#
This is equivalent to
#- k* "30 min" = ln(1/2)#
#-k * "30 min" = overbrace(ln(1))^(color(blue)(=0)) - ln(2)#
#k = ln(2)/"30 min" = 2.31 * 10^(-2)"min"^(-1)#
Now that you know the rate constant for this reaction, use the same equation again, only this time use
#A = 1/10 * A_0 -># equivalent to#90%# completion
and solve for
#ln( (1/10 * color(red)(cancel(color(black)(A_0))))/(color(red)(cancel(color(black)(A_0)))) ) = - 2.31 * 10^(-2)"min"^(-1) * t#
This is equivalent to
#t = (color(red)(cancel(color(black)(-))) ln(10))/(color(red)(cancel(color(black)(-))) 2.31 * 10^(-2)"min"^(-1)) = "99.68 min"#
Rounded to one sig fig, the answer will be
#t = color(green)("100 min")#