To solve this problem we need to know and use two different equations. The first is the equation for the half life of a second order reaction. It is:
where t is the half life, k is the second order rate constant, and [A'] is the initial concentration of A. We can use this equation to calculate the rate constant k:
k = 3.827 L/mol*sec
We can now use the integrated rate law for a second order reaction to calculate the time required for [A] to drop to 1/7 of the initial value.
Solving for t:
`t=(([A'])/([A]) - 1)(1/(k[A']))`
Now we can input the values and solve for t. [A'] is the initial concentration (13.198 M). [A] is 1/7 of the initial value (1.885 M). And k is the rate constant that we calculated above. Inputting all of these values into the above equation gives t=1,188.31 seconds. So the answer is that it takes 1,188.31 seconds for the concentration to drop to 1/7 of its initial value.