Answer :
Final answer:
The rate constant k for the first-order reaction is calculated as -0.022228 min⁻¹, using the integrated rate law. To determine when the reaction would be 90% complete, the time calculated is approximately 103.43 minutes.
Explanation:
The student's question involves calculating a rate constant (k) for a first-order reaction and determining the time for the reaction to be 90% complete, using given logarithmic values. From the integrated rate law for a first-order reaction, which is ln([A]0/[A]) = kt, where [A]0 is the initial concentration and [A] is the concentration at time t, we can determine the rate constant k. Since the reaction is 40% complete in 80 minutes, 60% of the initial concentration remains. Using the logarithmic values: ln(0.60) = (ln(10^-1)+ln(6)) = -(ln(10)+ln(6)) = -1 - 0.7782 = -1.7782.
Now, we can calculate the rate constant k: (-1.7782) = k(80 minutes), so k = -1.7782 / 80 = -0.022228 min⁻¹. To find the time for the reaction to be 90% complete (where only 10% remains), we set up the equation: ln(0.10) = -1 - 0.0000 (since ln(1) = 0), giving us -2.3026 = k(t) which yields t = -2.3026 / k. Substituting for k, we find t to be approximately 103.43 minutes.