Question

Activated carbon is used to remove a pollutant from wastewater in a mixed batch reactor, which follows first-order reaction kinetics. At a reaction rate of \(0.38 \text{/day}\), the time (in days) required to remove the pollutant by 95\% is ______ (rounded off to 1 decimal place).

Hint:

First-order reaction kinetics are commonly used to describe processes where the rate of reaction is proportional to the concentration of one reactant. Understanding these kinetics is essential for designing and operating chemical reactors.

Answer 1

Step 1: Apply the first-order kinetics equation. For a first-order reaction, the concentration of a reactant decreases exponentially over time. The equation for this reaction is: \[ C = C_0 e^{-kt} \] where \( C \) is the final concentration, \( C_0 \) is the initial concentration, \( k \) is the reaction rate, and \( t \) is time. Step 2: Set up the equation for 95\% removal. To find the time to remove 95\% of the pollutant, set \( C \) to 5\% of \( C_0 \): \[ 0.05 C_0 = C_0 e^{-kt} \] \[ e^{-kt} = 0.05 \] Step 3: Solve for \( t \). Take the natural logarithm of both sides: \[ -kt = \ln(0.05) \] \[ t = \frac{\ln(0.05)}{-k} = \frac{\ln(0.05)}{0.38} \] \[ t \approx \frac{-2.9957}{0.38} \approx 7.88 \text{ days} \]