Problem 73 - Chemistry: The Central Science

Question

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, \(\mathrm{pH}\), temperature, and many other factors. In a certain oil spill, hydrocarbons from the oil disappeared with a first-order rate constant of \(2 \times 10^{-6} \mathrm{~s}^{-1}\). How many days did it take for the hydrocarbons to decrease to \(10 \%\) of their initial value?

Step-by-Step Solution

Answer

It took approximately 13.33 days for the hydrocarbons to decrease to \(10\%\) of their initial value using first-order decay kinetics with a rate constant of \(2\times 10^{-6} \mathrm{~s}^{-1}\).

Step 1: Set up the equation for first-order decay kinetics

Our starting point is the general formula for first-order decay kinetics: \[C_t = C_0e^{-kt}\] Where \(C_t\) is the concentration of hydrocarbons at time t, \(C_0\) is the initial concentration of hydrocarbons, \(k\) is the first-order rate constant, and \(t\) is the time in seconds. Since we want to find the time when the remaining hydrocarbons are equal to \(10\%\) of their initial value (\(0.1C_0\)), our equation becomes: \[0.1C_0 = C_0e^{-kt}\]

Step 2: Simplify the equation by dividing by \(C_0\)

To simplify the equation, we can divide both sides of the equation by the initial concentration of hydrocarbons \(C_0\): \[\frac{0.1C_0}{C_0} = \frac{C_0e^{-kt}}{C_0}\] This gives us the following equation: \[0.1 = e^{-kt}\]

Step 3: Take the logarithm of both sides

We need to solve for time \(t\), so we'll take the natural logarithm of both sides of the equation. This will allow us to remove the exponential from the equation: \[\ln{0.1} = \ln{e^{-kt}}\] Using the logarithmic property that states \(\ln{e^x} = x\), we get: \[\ln{0.1} = -kt\]

Step 4: Solve for time \(t\)

Now we will divide by the rate constant \(-k\) to obtain the time \(t\): \[t = -\frac{\ln{0.1}}{k}\] Now we can plug in the given value of \(k = 2\times 10^{-6} \mathrm{~s}^{-1}\): \[t = -\frac{\ln{0.1}}{2\times 10^{-6}\mathrm{~s}^{-1}}\]

Step 5: Calculate the time and convert to days

Calculate the value for time in seconds: \[t \approx 1.1513\times 10^6 \mathrm{~s}\] Now we want to convert time to days by dividing by the number of seconds in a day \((24 \mathrm{h}/\mathrm{d})(60\mathrm{min}/\mathrm{h})(60\mathrm{s}/\mathrm{min}) = 86,400\,\mathrm{s/d}\): \[t \approx \frac{1.1513\times 10^6\,\mathrm{s}}{86,400\,\mathrm{s/d}} \approx 13.33\,\mathrm{days}\] Therefore, it took approximately 13.33 days for the hydrocarbons to decrease to \(10\%\) of their initial value.

Key Concepts

Understanding the First-Order Rate Constant

The first-order rate constant, denoted as 'k' in chemical kinetics, is a crucial factor in determining how quickly a reactant is consumed in a first-order reaction. This type of reaction is characterized by a rate that is directly proportional to the concentration of one of the reactants. In the context of bioremediation, the first-order rate constant represents how efficiently bacteria can degrade pollutants over time.

When we measure the speed at which bacteria 'eat' hydrocarbons from an oil spill, we find that the rate at which these hydrocarbons disappear is related to their current concentration through the rate constant. A higher 'k' value means that the bioremediation is more effective, and the pollutants decrease more rapidly. In our exercise, the rate constant is given as \(2 \times 10^{-6} \mathrm{~s}^{-1}\), indicating the rate at which the concentration of hydrocarbons is decreasing at any given moment. Understanding this constant is key for predicting the time required for a pollutant to reduce to a certain level, which is essential for environmental management and cleanup efforts.

Decay Kinetics in Environmental Chemistry

Decay kinetics describe how a substance diminishes over time. Specifically, it refers to the mathematical representation of the decay process. In bioremediation, decay kinetics let us model the reduction of pollutants, such as hydrocarbons, as they are broken down by bacteria.

For a first-order decay process, as observed in many bioremediation scenarios, the concentration of the pollutant decreases exponentially. This is crucial, as it allows us to predict the future concentration of contaminants in the environment based on its current level and the first-order rate constant. In the exercise, we assume that the decay kinetics follow a first-order model, which simplifies our calculations and enables accurate environmental monitoring. By understanding decay kinetics, environmental chemists can optimize bioremediation strategies and gauge the success of pollution cleanup efforts.

Exponential Equations and Environmental Cleanup

Exponential equations are prevalent in various scientific disciplines, including chemistry, where they model processes such as radioactive decay and concentration reduction in reactions. These equations have the form \(y = a \cdot e^{bx}\), where 'e' is the base of the natural logarithm, and 'a' and 'b' are constants. In our case, the exponential equation models the decrease of hydrocarbon concentration in an oil spill during bioremediation.

The equation \(C_t = C_0e^{-kt}\) allows us to determine the remaining concentration (\(C_t\)) after a specific time 't' using the original concentration \(C_0\), and the first-order rate constant 'k'. By rearranging and solving the equation, as shown in the original exercise, students can determine how long it will take for the hydrocarbons to decrease to any desired percentage of the initial value. This exponential model is essential for planning clean-up operations and setting realistic environmental recovery timelines.