Question

Oil spreads on water to form a film about \(100 \mathrm{nm}\) thick (two significant figures). How many square kilometers of ocean will be covered by the slick formed when one barrel of oil is spilled \((1\) barrel \(=31.5\) U.S. gal)?

Short answer

Answer: 1.19 square kilometers.

Step-by-step solution

Convert oil film thickness to meters

The thickness of the oil film is given in nanometers. We need to convert this to meters to later calculate the area of oil slick. 1 nanometer is equal to \(1\times10^{-9}\) meters, so the thickness of the oil film in meters is: \((100 \mathrm{nm}) \times (1\times10^{-9} \frac{\mathrm{m}}{\mathrm{nm}}) = 1\times10^{-7} \mathrm{m}\)

Convert gallons to cubic meters

First, we need to convert the volume of a barrel of oil in gallons to cubic meters. We know that 1 US gallon is equal to 0.00378541 cubic meters. Therefore: \((31.5 \mathrm{gal}) \times (0.00378541 \frac{\mathrm{m^3}}{\mathrm{gal}}) = 0.11919 \mathrm{m^3}\)

Calculate the area of the oil slick in square meters

Now we will divide the volume of a barrel of oil in cubic meters (0.11919 \(\mathrm{m^3}\)) by the thickness of the oil film in meters (\(1\times10^{-7}\) \(\mathrm{m}\)) to find the area of the oil slick formed in square meters. Area in square meters: \(\frac{0.11919 \mathrm{m^3}}{1\times10^{-7} \mathrm{m}} = 1.19\times10^6 \mathrm{m^2}\)

Convert the area to square kilometers

Finally, we will convert the area of the oil slick from square meters to square kilometers. We know that 1 square kilometer is equal to \(1\times10^6\) square meters. Therefore: Area in square kilometers: \(\frac{1.19\times10^6 \mathrm{m^2}}{1\times10^6 \frac{\mathrm{m^2}}{\mathrm{km^2}}} = 1.19 \mathrm{km^2}\) So, the slick formed when one barrel of oil is spilled will cover 1.19 square kilometers of ocean.

Key concepts

Oil Slick Area Measurement

When considering environmental impacts like an oil spill, accurately measuring the affected area is crucial. The challenge lies in determining how far a certain volume of oil will spread once it encounters water. One approach is to calculate the potential area of an oil slick by knowing the typical thickness of an oil film that spreads on the water's surface. In our example, this thickness is about 100 nanometers (nm). Knowing this, we can estimate the area that one barrel of oil could potentially cover by dividing the total volume of the oil by the film thickness. This calculation provides us with a clearer picture of the possible extent of environmental influence, which is essential for organizing cleanup efforts and assessing ecological impact.

For educational purposes, exercises like these also help students understand how to apply mathematical concepts to real-world environmental issues. Estimating the size of an oil slick gives practical context to abstract mathematical operations, enhancing learning through practical application.

Unit Conversion

Unit conversion is a fundamental skill in many scientific calculations, especially when dealing with measurements that span various systems, like the Imperial and Metric systems. Students need to grasp the concept of unit conversion to perform accurate calculations in fields such as chemistry, physics, and environmental sciences.

In the context of our oil spill scenario, the exercise requires converting the thickness of the oil film from nanometers to meters and the volume of a barrel of oil from US gallons to cubic meters. To do this, we apply known conversion factors, such as the fact that 1 nanometer is equal to \(1\times10^{-9}\) meters, and 1 US gallon equals 0.00378541 cubic meters.

Understanding and applying these conversions is critical because it ensures that all measurements are in compatible units, which is necessary for further calculations. A misstep in this conversion process can lead to incorrect results and an inaccurate understanding of the situation at hand.

Volume to Area Conversion

The conversion from volume to area is particularly intriguing because it combines dimensional analysis with practical application. In our oil spill example, the volume of oil (a three-dimensional measure) must be connected to an area (a two-dimensional measure), which involves considering the thickness of the oil layer.

To calculate the area that a certain volume of oil will cover, we divide that volume by the layer's thickness. The resulting figure reveals how spread out the oil will become once it is on the water's surface. This step is crucial for understanding the potential spread of the oil slick and planning adequate response measures. By mastering volume to area conversion, students can better interpret how substances behave in different conditions. This knowledge is essential, as it has direct implications for environmental conservation efforts and response strategies to such ecological incidents.