Question

Bioremediationis 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 axygen 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 $10\mathrm{\%}$ of their initial value?

Short answer

It took approximately 1.33 days for the hydrocarbons to decrease to 10% of their initial value in this bioremediation process.

Step-by-step solution

Recall the first-order rate equation

The first-order rate equation is given by: $A_t=A_0\ast e^{-kt}$, where $A_t$ is the amount left at time $t$, $A_0$ is the initial amount, $k$ is the rate constant, and $t$ is the time.

Set up the equation with the given information

We are given that the hydrocarbons decreased to 10% of their initial value, which means that $A_t=0.1\ast A_0$. Also, the rate constant is given as $k=2\times {10}^{-6}{\mathrm{s}}^{-1}$. Plugging these values into the equation, we have: $0.1=e^{-kt}$.

Solve for time (t)

To solve for $t$, first, take the natural logarithm of both sides of the equation: $\ln (0.1)=-kt$. Then, divide both sides by $-k$ to obtain the time in seconds: $t=\frac{\ln (0.1)}{-k}$.

Plug in the value of k and calculate t

The given value of $k$ is $2\times {10}^{-6}{\mathrm{s}}^{-1}$. Substituting this value into the equation for time: $t=\frac{\ln (0.1)}{-2\times {10}^{-6}}$. After calculating, we find that $t\approx 1.15129\times {10}^5$ seconds.

Convert time to days

In order to convert the time from seconds to days, we need to divide the time in seconds by the number of seconds in a day. There are $60$ seconds in a minute, $60$ minutes in an hour, and $24$ hours in a day, so there are $60\times 60\times 24$ seconds in a day. Therefore, we can convert the time to days by dividing the time in seconds by the number of seconds in a day: $\frac{1.15129\times {10}^5\mathrm{s}}{60\times 60\times 24\mathrm{s}/\mathrm{day}}\approx 1.33$ days.

Final Answer

It took approximately 1.33 days for the hydrocarbons to decrease to 10% of their initial value in this bioremediation process.

Key concepts

First-Order Rate Equation

The first-order rate equation is a mathematical formula often used to describe chemical reactions that proceed at a rate proportional to the concentration of one reactant. This equation is given by $A_t=A_0\times e^{-kt}$. Here, $A_t$ refers to the quantity of reactant remaining at time $t$, $A_0$ is the initial quantity of the reactant, $k$ represents the rate constant, and $e$ is the base of the natural logarithm. This equation assumes that the reaction rate at any given time depends directly on the concentration of one chemical species. It's a helpful way to predict how long it will take for a substance to diminish to a certain level, which is essential in processes like bioremediation. Knowing and understanding this equation enables us to predict how quickly environmental contaminants, like hydrocarbons, break down over time.

Rate Constant

The rate constant $k$ is a parameter crucial in defining the speed of a chemical reaction. In first-order processes such as hydrocarbon degradation, $k$ is expressed in units of reciprocal seconds $(s^{-1})$. For any first-order reaction, a larger value of $k$ indicates a faster reaction, meaning that the reactant’s concentration decreases rapidly. Conversely, a smaller value implies a slower process. The value of $k$ can be influenced by various environmental factors that may include temperature, the presence of catalysts or inhibitors, and the nature of the reactants themselves.In environmental chemistry, determining the rate constant helps scientists understand how effectively a contaminant can be reduced in the environment. For instance, in the provided exercise, the rate constant for hydrocarbon degradation is $2\times {10}^{-6}{\mathrm{s}}^{-1}$, indicating how the pollutant concentration decreases over time under given conditions.

Environmental Chemistry

Environmental chemistry is the scientific study of the chemical and biochemical phenomena that occur in natural places. It involves understanding how elements and compounds behave in the atmosphere, water, soil, and living organisms. By analyzing how chemical reactions affect the air, water, and soil, scientists can develop strategies to minimize harmful effects from chemical releases. Key focuses of environmental chemistry include analyzing pollution sources and impacts, understanding the fate of chemicals in the environment, and finding sustainable ways to remediate pollution. In cases like bioremediation, environmental chemistry is critical because it helps us comprehend how microorganisms, such as bacteria, utilize chemicals like hydrocarbons to "repair" and cleanse their ecosystems. Understanding these dynamics allows us to use natural processes to remediate contaminated sites effectively.

Hydrocarbon Degradation

Hydrocarbon degradation is a process where microorganisms break down hydrocarbons, such as those found in oil spills, into less harmful substances. This natural process is a vital component of bioremediation, helping to restore ecosystems affected by pollution. Factors affecting hydrocarbon degradation include oxygen levels, temperature, pH, and the presence of specific microbial communities capable of digesting hydrocarbons. Bioremediation through hydrocarbon degradation relies on the metabolic capabilities of these microorganisms. Such biochemical processes can transform toxic hydrocarbon chains into simpler molecules like carbon dioxide and water, reducing environmental risks. Successful hydrocarbon degradation requires a delicate balance of favorable conditions and can be an effective strategy for cleaning up oils and other organic pollutants in environments impacted by human activities.