Question

Oil spreads in a thin layer on water called an "oil slick." How much area in \(\mathrm{m}^{2}\) will \(200 \mathrm{~cm}^{3}\) of oil cover if it forms a layer \(0.5\) nm thick?

Short answer

The oil will cover an area of \(4 \times 10^5 \text{ m}^2\).

Step-by-step solution

Convert Units

First convert the volume of oil from \(\text{cm}^3\) to \(\text{m}^3\). We know that \(1 \text{ cm}^3 = 1 \times 10^{-6} \text{ m}^3\). Thus, \(200 \text{ cm}^3\) is equal to \(200 \times 10^{-6} \text{ m}^3\).

Understand the Relation Between Thickness and Area

The volume of oil is the product of the area it covers and the thickness of the layer. This relationship is given by \(\text{Volume} = \text{Area} \times \text{Thickness}\).

Convert Thickness to Meters

Next, convert the thickness of the oil layer from nanometers (nm) to meters (m). We know that \(1 \text{ nm} = 1 \times 10^{-9} \text{ m}\). Therefore, \(0.5 \text{ nm} = 0.5 \times 10^{-9} \text{ m} = 5 \times 10^{-10} \text{ m}\).

Solve for Area

Using the formula from Step 2 and the converted values, solve for the area: \[ \text{Area} = \frac{\text{Volume}}{\text{Thickness}} \]. Substituting the values, we get \[ \text{Area} = \frac{200 \times 10^{-6} \text{ m}^3}{5 \times 10^{-10} \text{ m}} = \frac{200 \times 10^{-6}}{5 \times 10^{-10}} \text{ m}^2 = 4 \times 10^{5} \text{ m}^2 \].

Key concepts

Unit Conversion

Before starting any calculation, it's essential to make sure all units are consistent. If you have measurements in different units, convert them to the same unit. In this problem, we initially have the volume in cubic centimeters (\text{cm}^3) and need to convert it to cubic meters (\text{m}^3).
One cubic centimeter is equal to one millionth of a cubic meter: \(1 \text{ cm}^3 = 1 \times 10^{-6} \text{ m}^3\).
So, converting 200 cubic centimeters of oil: \(200 \text{ cm}^3 = 200 \times 10^{-6} \text{ m}^3 \).
This step ensures the volume will be in the correct unit when we solve the final problem. A small mistake in unit conversion can lead to incorrect results.

Area Calculation

To find the area covered by the oil slick, use the relationship between volume, area, and thickness. This relationship is given by the formula: \( \text{Volume} = \text{Area} \times \text{Thickness} \).
If you know the volume and thickness, rearrange the formula to solve for the area: \(\text{Area} = \frac{\text{Volume}}{\text{Thickness}} \).
This formula tells us how to proceed once the volume and thickness have the same unit (both in meters). This allows us to calculate the area the oil slick will cover on the water surface.

Volume and Thickness Relationship

The connection between volume, area, and thickness is crucial in understanding how physical quantities relate. Here is what each part represents:

• Volume: The total amount of oil, which in our example is 200 cubic centimeters or \(200 \times 10^{-6} \text{ m}^3\).
• Area: The surface area the oil will cover, which we need to find.
• Thickness: The depth of the oil layer, given as 0.5 nanometers (nm).

The relationship is straightforward: if you know the total volume of oil and how deep it spreads, you can find the area by dividing the volume by the thickness. It’s a simple way to assess how a given volume of a substance spreads out over a surface.

Nanometers to Meters Conversion

Often in science, different units are used to describe the same quantity. Understanding how to convert between these units is crucial. Nanometers are very small units often used to measure tiny distances, like the thickness of an oil layer.
One nanometer is one billionth of a meter: \(1 \text{ nm} = 1 \times 10^{-9} \text{ m}\).
Given the thickness 0.5 nm, we convert it to meters: \(0.5 \text{ nm} = 0.5 \times 10^{-9} \text{ m}\), which simplifies to \(5 \times 10^{-10} \text{ m}\).
Converting units from nanometers to meters standardizes our units, ensuring that the calculations are correct and the resulting area is accurate.