Question

Suppose \(1.0 \mathrm{~m}^{3}\) of oil is spilled into the ocean. Find the area of the resulting slick, assuming that it is one molecule thick and that each molecule occupies a cube \(0.50 \mu \mathrm{m}\) on a side.

Short answer

The area of the slick is \(2 \times 10^6 \, m^2\).

Step-by-step solution

Calculate the volume of one molecule

Given that a molecule occupies a cube with a side length of \(0.50 \, \mu m\), we first convert the side length into meters for consistency. \(1 \, \mu m = 1 \times 10^{-6} \, m\), so \(0.50 \, \mu m = 0.50 \times 10^{-6} \, m\). The volume of one molecule is \( (0.50 \times 10^{-6} \, m)^3 \).

Calculate the total volume for one molecule

The formula for the volume of a cube is side length cubed. Thus, the volume of one molecule is: \[(0.50 \times 10^{-6} \, m)^3 = 0.125 \times 10^{-18} \, m^3 = 1.25 \times 10^{-19} \, m^3\].

Determine the number of molecules needed

The total oil volume is \(1.0 \, m^3\). To find the number of molecules, divide the total volume by the volume of one molecule: \[\frac{1.0 \, m^3}{1.25 \times 10^{-19} \, m^3} = 8 \times 10^{18}\]. Therefore, there are \(8 \times 10^{18}\) molecules in the oil spill.

Calculate the area covered by the molecules

Since the oil slick is one molecule thick, the area can be found by multiplying the area occupied by one molecule by the total number of molecules. An individual molecule occupies an area given by its cross-sectional square (side length squared) which is \[(0.50 \times 10^{-6} \, m)^2 = 0.25 \times 10^{-12} \, m^2\]. Multiply this by the number of molecules:\[8 \times 10^{18} \times 0.25 \times 10^{-12} \, m^2 = 2 \times 10^6 \, m^2\].

Key concepts

Volume Calculation

Calculating the volume of molecules is crucial for understanding how they occupy space. When dealing with molecules, it's important to note that they can take various shapes, such as spheres, cubes, or other geometrical forms. In this exercise, we're focusing on molecules that occupy cube-shaped spaces.
To begin, we need to convert relevant measurements into a consistent unit, typically meters in physics. Given a cube with a side length of \(0.50 \, \mu m\), converting micrometers to meters involves using \(1 \, \mu m = 1 \times 10^{-6} \, m\). Therefore, the side length becomes \(0.50 \times 10^{-6} \, m\).
The volume \(V\) of a cube is calculated using the formula:

• \(V = \text{side length}^3\)

The calculation follows with:

• \( (0.50 \times 10^{-6} \, m)^3 = 0.125 \times 10^{-18} \, m^3 \)
• Result: \( 1.25 \times 10^{-19} \, m^3 \)

This result represents the volume for a single molecule. By understanding and calculating the volume of individual molecules, we can more accurately assess how they contribute to larger structures in physical systems.

Molecular Dimensions

Understanding molecular dimensions is key in molecular physics, as it allows us to grasp how microscopic entities behave in macroscopic environments. In many science problems like this one, molecules are often idealized as geometrical shapes for simplification.
Here, molecules are treated as tiny cubes, each with a side length of \(0.50 \, \mu m\). This square form provides a straightforward way to estimate space requirements: when you picture a molecule as a cube, you can easily calculate its footprint by squaring the side length.
An important takeaway is:

• The area occupied by one molecule, a square in this case, is the side length squared.
• Thus, for our molecule: \((0.50 \times 10^{-6} \, m)^2 = 0.25 \times 10^{-12} \, m^2 \).

This helps relate the microscopic world of molecules to broader spatial concepts. By envisioning molecules as distinct volumes and areas within a defined framework, their interactions, and the space they need, become much clearer. This simplification aids in predicting outcomes, such as how much an oil spill may cover in a given scenario.

Area Estimation

When working with molecular layers, estimating area becomes crucial, especially in calculating coverage effects like oil spills. Once we know the number of molecules and the area each occupies, we can determine the total area affected.
Given that each molecule covers an area of \(0.25 \times 10^{-12} \, m^2\), and there are \(8 \times 10^{18}\) molecules in our spill,

• we calculate the total area by multiplying the number of molecules by the area per molecule.

Thus:

• \(8 \times 10^{18} \times 0.25 \times 10^{-12} \, m^2 = 2 \times 10^6 \, m^2 \).

This value represents the entire area of the oil slick created by these molecules at one molecule thickness. Understanding these spatial calculations helps to estimate the reach of molecularly thin oils on water surfaces. It highlights the relevance of resolution and precision in molecular studies, impacting environmental science and engineering realms.