Given the half life of the first order decomposition reaction is 355 s
Rate constant of the first order reaction is related to the half life by the equation,
[tex] k = \frac{0.693}{t_{\frac{1}{2}}} [/tex]
[tex] k = \frac{0.693}{355 s} [/tex]
k = 0.00195 [tex] s^{-1} [/tex]
The concentration of the substance is decreased to 1/4 th.
If we start with 1 M solution, after time t the concentration becomes 1/4th = 0.25 M
First order rate law:
[tex] [A] = [A]_{0} e^{-kt} [/tex]
[tex] [A] = 0.25 M [/tex]
[tex] [A]_{0} = 1 M [/tex]
[tex] k = 0.00195 s^{-1} [/tex]
Plugging in the values to solve for t,
[tex] 0.25 M = 1 M (e^{-(0.00195s^{-1})t}) [/tex]
[tex] ln(\frac{0.25}{1}) = - (0.00195 s^{-1})(t) [/tex]
[tex] t = \frac{ln(0.25)}{0.00195} s [/tex]
t = 710 s
