33) The half-life of a first-order reaction is 13 min. If the initial concentration of reactant is 0.13 M. it takes __ min for it to decrease to 0.085 M. A) 12 B) 10. C) 8.0 D) 11 E) 7.0 Answer: C

The half-life of a first-order reaction is the time it takes for the concentration of the reactant to decrease to half of its initial concentration. In this case, the half-life is given as 13 minutes. The question asks how long it takes for the concentration to decrease from 0.13 M to 0.085 M.<br /><br />To solve this, we can use the formula for first-order kinetics:<br /><br />\[<br />C = C_0 \times e^{-kt}<br />\]<br /><br />Where:<br />- \(C\) is the final concentration<br />- \(C_0\) is the initial concentration<br />- \(k\) is the rate constant<br />- \(t\) is the time<br /><br />First, we need to find the rate constant \(k\). We can use the half-life formula for a first-order reaction:<br /><br />\[<br />t_{1/2} = \frac{\ln(2)}{k}<br />\]<br /><br />Given that \(t_{1/2} = 13\) minutes, we can solve for \(k\):<br /><br />\[<br />k = \frac{\ln(2)}{13} \approx 0.0533 \text{ min}^{-1}<br />\]<br /><br />Now, we can use the first-order kinetics formula to find the time \(t\) it takes for the concentration to decrease to 0.085 M:<br /><br />\[<br />0.085 = 0.13 \times e^{-0.0533 \times t}<br />\]<br /><br />Solving for \(t\), we get:<br /><br />\[<br />t \approx 8.0 \text{ minutes}<br />\]<br /><br />Therefore, the correct answer is C) 8.0 minutes.