<u>Answer:</u> The time taken by the reaction is 84.5 seconds
<u>Explanation:</u>
The equation used to calculate half life for first order kinetics:
![k=\frac{0.693}{t_{1/2}}](https://tex.z-dn.net/?f=k%3D%5Cfrac%7B0.693%7D%7Bt_%7B1%2F2%7D%7D)
where,
= half-life of the reaction = 9.0 s
k = rate constant = ?
Putting values in above equation, we get:
![k=\frac{0.693}{9}=0.077s^{-1}](https://tex.z-dn.net/?f=k%3D%5Cfrac%7B0.693%7D%7B9%7D%3D0.077s%5E%7B-1%7D)
Rate law expression for first order kinetics is given by the equation:
......(1)
where,
k = rate constant = ![0.077s^{-1}](https://tex.z-dn.net/?f=0.077s%5E%7B-1%7D)
t = time taken for decay process = 50.7 sec
= initial amount of the reactant = ?
[A] = amount left after decay process = 0.0741 M
Putting values in equation 1, we get:
![0.077=\frac{2.303}{50.7}\log\frac{[A_o]}{0.0741}](https://tex.z-dn.net/?f=0.077%3D%5Cfrac%7B2.303%7D%7B50.7%7D%5Clog%5Cfrac%7B%5BA_o%5D%7D%7B0.0741%7D)
![[A_o]=3.67M](https://tex.z-dn.net/?f=%5BA_o%5D%3D3.67M)
Now, calculating the time taken by using equation 1:
![[A]=0.0055M](https://tex.z-dn.net/?f=%5BA%5D%3D0.0055M)
![k=0.077s^{-1}](https://tex.z-dn.net/?f=k%3D0.077s%5E%7B-1%7D)
![[A_o]=3.67M](https://tex.z-dn.net/?f=%5BA_o%5D%3D3.67M)
Putting values in equation 1, we get:
![0.077=\frac{2.303}{t}\log\frac{3.67}{0.0055}\\\\t=84.5s](https://tex.z-dn.net/?f=0.077%3D%5Cfrac%7B2.303%7D%7Bt%7D%5Clog%5Cfrac%7B3.67%7D%7B0.0055%7D%5C%5C%5C%5Ct%3D84.5s)
Hence, the time taken by the reaction is 84.5 seconds