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Chapter 18: Problem 83

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 \(2 \times 10^{-6} \mathrm{s}^{-1} .\) At that rate, how many days would it take for the hydrocarbons to decrease to 10\(\%\) of their initial value?

Short Answer

Expert verified
It would take approximately 13.34 days for the hydrocarbons to decrease to 10% of their initial value at the given first-order rate constant of \(2 \times 10^{-6} \mathrm{s}^{-1}\).

Step by step solution

01

Set up the first-order reaction equation

Apply the first-order reaction equation using the given information: \[ln \frac{[A]_0}{[A]} = kt\] Since we want the hydrocarbons to decrease to 10% of their initial value, \([A] = 0.1[A]_0\). Substitute this into the equation: \[ln \frac{[A]_0}{0.1[A]_0} = kt\]
02

Solve for time t

Now, we can simplify the equation and solve for \(t\): \[ln(1/0.1) = kt\] \[ln(10) = kt\] Now, substitute the given first-order rate constant \(k = 2 \times 10^{-6} \mathrm{s}^{-1}\): \[t = \frac{ln(10)}{2 \times 10^{-6} \mathrm{s}^{-1}}\]
03

Calculate the time t

Use a calculator to find the value of \(t\): \[t \approx \frac{2.303}{2 \times 10^{-6} \mathrm{s}^{-1}} \approx 1.1515 \times 10^6 \mathrm{s}\]
04

Convert the time to days

To convert the time from seconds to days, divide by the number of seconds in a day (86,400 seconds): \[t \approx \frac{1.1515 \times 10^6 \mathrm{s}}{86400 \frac{\mathrm{s}}{\mathrm{day}}} \approx 13.34 \mathrm{days}\] Therefore, it would take approximately 13.34 days for the hydrocarbons to decrease to 10% of their initial value.

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Most popular questions from this chapter

The average concentration of carbon monoxide in air in an Ohio city in 2006 was 3.5 ppm. Calculate the number of CO molecules in 1.0 L of this air at a pressure of 759 torr and a temperature of \(22^{\circ} \mathrm{C}\).

Write balanced chemical equations for each of the following reactions: (a) The nitric oxide molecule undergoes photodissociation in the upper atmosphere. (b) The nitric oxide molecule undergoes photoionization in the upper atmosphere. (c) Nitric oxide undergoes oxidation by ozone in the stratosphere. (d) Nitrogen dioxide dissolves in water to form nitric acid and nitric oxide.

Why is rainwater naturally acidic, even in the absence of polluting gases such as \(\mathrm{SO}_{2} ?\)

In 1986 an electrical power plant in Taylorsville, Georgia, burned \(8,376,726\) tons of coal, a national record at that time. (a) Assuming that the coal was 83\(\%\) carbon and 2.5\(\%\) sulfur and that combustion was complete, calculate the number of tons of carbon dioxide and sulfur dioxide produced by the plant during the year. (b) If 55\(\%\) of the SO \(_{2}\) could be removed by reaction with powdered CaO to form \(\mathrm{CaSO}_{3},\) how many tons of \(\mathrm{CaSO}_{3}\) would be produced?

Why is the photodissociation of \(\mathrm{N}_{2}\) in the atmosphere relatively unimportant compared with the photodissociation of \(\mathrm{O}_{2} ?\)
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