Rate equation for first order reaction is as follows:
![t=\frac{2.303}{k}log\frac{A_{0}}{A_{t}}](https://tex.z-dn.net/?f=t%3D%5Cfrac%7B2.303%7D%7Bk%7Dlog%5Cfrac%7BA_%7B0%7D%7D%7BA_%7Bt%7D%7D)
Here, k is rate constant of the reaction, t is time of the reaction,
is initial concentration and
is concentration at time t.
The rate constant of the reaction is
.
(a) Let the initial concentration be 100, If 90% of the chemical is destroyed, the chemical present at time t will be 100-90=10, on putting the values,
![t=\frac{2.303}{k}log\frac{A_{0}}{A_{t}}=\frac{2.303}{0.1 day^{-1}}log\frac{100}{10}=23.03 days](https://tex.z-dn.net/?f=t%3D%5Cfrac%7B2.303%7D%7Bk%7Dlog%5Cfrac%7BA_%7B0%7D%7D%7BA_%7Bt%7D%7D%3D%5Cfrac%7B2.303%7D%7B0.1%20day%5E%7B-1%7D%7Dlog%5Cfrac%7B100%7D%7B10%7D%3D23.03%20days)
Thus, time required to destroy 90% of the chemical is 23.03 days.
(b) Let the initial concentration be 100, If 99% of the chemical is destroyed, the chemical present at time t will be 100-99=1, on putting the values,
![t=\frac{2.303}{k}log\frac{A_{0}}{A_{t}}=\frac{2.303}{0.1 day^{-1}}log\frac{100}{1}=46.06 days](https://tex.z-dn.net/?f=t%3D%5Cfrac%7B2.303%7D%7Bk%7Dlog%5Cfrac%7BA_%7B0%7D%7D%7BA_%7Bt%7D%7D%3D%5Cfrac%7B2.303%7D%7B0.1%20day%5E%7B-1%7D%7Dlog%5Cfrac%7B100%7D%7B1%7D%3D46.06%20days)
Thus, time required to destroy 99% of the chemical is 46.06 days.
(c) Let the initial concentration be 100, If 99.9% of the chemical is destroyed, the chemical present at time t will be 100-99.9=0.1, on putting the values,
![t=\frac{2.303}{k}log\frac{A_{0}}{A_{t}}=\frac{2.303}{0.1 day^{-1}}log\frac{100}{0.1}=69.09 days](https://tex.z-dn.net/?f=t%3D%5Cfrac%7B2.303%7D%7Bk%7Dlog%5Cfrac%7BA_%7B0%7D%7D%7BA_%7Bt%7D%7D%3D%5Cfrac%7B2.303%7D%7B0.1%20day%5E%7B-1%7D%7Dlog%5Cfrac%7B100%7D%7B0.1%7D%3D69.09%20days)
Thus, time required to destroy 99.9% of the chemical is 69.09 days.