Question

A solution is 0.100 M in substances A and B, which react by a first-order reaction. If the reaction is 15.0%

complete in 6.75 min, what is its half-life under these conditions? What would be the half-life if A and B

were each at 0.200 M, and how long would it take for 15.0% completion of the reaction?

Answer 1

v=kC - rate of a first-order reaction. k-rate constant of reaction.

v=-dC/dt

-dC/dt=kC

dC/C=-kdt

$\int$dC/C=-$\int$kdt

lnC-lnC₀=-kt оr ln(C/C₀) =-kt

C/C₀=0.85 (C=C₀-0.15C₀=0.85C₀); t=6.75min (the reaction is 15.0% complete in 6.75 min)

ln(0.85)=-k*6.75min

k=-ln(0.85) / 6.75min=0.024min^-1

t=-ln(C/C₀) / k

T_1/2=-ln((C₀/2)/C₀)/k or T_1/2=ln2/k - the half-life does not depend on initial concentration (as well as the 15% completion time of the reaction, as well as the time of any degree of completion of first-order reactions)

T_1/2=0.693/k=0.693/0.024min^-1=28,88min.

For the above reasons, the half-life of A and B at an initial concentration of 0.200M, and the reaction completion time of 15.0% will be the same as for a concentration of 0.100M.