Bioremediation is the process by which bacteria repair their environment in response, for example, to an oil spill. The efficiency of bacteria for 'eating' hydrocarbons depends on the amount of oxygen in the system, pH, temperature, and many other factors. In a certain oil spill, hydrocarbons from the oil disappeared with a first-order rate constant of 2 * 10 s. At that rate, how many days would it take for the hydrocarbons to decrease to 10% of their initial value?

Name: Chemistry: The Central Science
Author: Brown
ISBN: 9780134414232

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Identify that the problem involves first-order kinetics, which can be described by the equation: \( [A] = [A]_0 e^{-kt} \), where \([A]\) is the concentration at time \(t\), \([A]_0\) is the initial concentration, \(k\) is the rate constant, and \(t\) is time.

Recognize that the problem asks for the time it takes for the concentration to decrease to 10% of its initial value, meaning \([A] = 0.1[A]_0\).

Substitute \([A] = 0.1[A]_0\) into the first-order kinetics equation: \( 0.1[A]_0 = [A]_0 e^{-kt} \).

Simplify the equation by dividing both sides by \([A]_0\), resulting in \( 0.1 = e^{-kt} \).

Take the natural logarithm of both sides to solve for \(t\): \( \ln(0.1) = -kt \). Substitute the given rate constant \(k = 2 \times 10^{-1} \text{s}^{-1}\) and solve for \(t\) in seconds, then convert \(t\) to days.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Kinetics

First-order kinetics refers to a reaction rate that is directly proportional to the concentration of one reactant. In this context, the rate at which hydrocarbons are consumed by bacteria is dependent on their concentration. The mathematical representation involves a rate constant and can be described using the equation ln([A]0/[A]) = kt, where [A]0 is the initial concentration, [A] is the concentration at time t, k is the rate constant, and t is time.

Half-Life

The half-life of a substance is the time required for its quantity to reduce to half its initial value. For first-order reactions, the half-life is constant and independent of the initial concentration. Understanding half-life is crucial for determining how long it will take for the hydrocarbons to decrease to a specific percentage of their initial amount, such as 10%.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of bioremediation, as bacteria consume hydrocarbons, the concentration decreases exponentially over time. This concept is essential for calculating the time required for the hydrocarbons to reach a certain percentage of their original concentration, as it allows for the application of logarithmic functions to solve for time.

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