Question

A typical rate of deposit of dust ("dustfall") from unpolluted air was reported as 10 tons per square mile per month. (a) Express this dustfall in milligrams per square meter per hour. (b) If the dust has an average density of \(2 \mathrm{g} / \mathrm{cm}^{3}\), how long would it take to accumulate a layer of dust \(1 \mathrm{mm}\) thick?

Short answer

It would take approximately 4227.4 years for a 1mm layer of dust to accumulate from unpolluted air.

Step-by-step solution

Convert tons per square mile per month to mg per square meter per hour

Initially, the conversion of units is done. 1 tonne is equivalent to \(1 \times 10^{12}\) milligrams (mg), 1 square mile converts to \(2.59 \times 10^6\) square meters (m^2) and 1 month has approximately \(720\) hours. So the dustfall rate is calculated as: \((10 \times 10^{12}\) mg) / (\(2.59 \times 10^6\) m^2 x 720 hours) = approximately \(53.9\) mg/m^2 hr.

Convert a 1mm thick layer of dust into volume per area unit

Since we need to find out how long it takes for a 1mm dust layer to accumulate, let's firstly convert this layer into a volume per area unit, which matches the units used in Step 1. 1mm can be written as 0.1cm, so every square meter would have a volume of \(1m^2 \times 0.1cm = 1000cm^3\). However, since the density is given in g/cm^3 and we are dealing with milligrams, we will convert this volume to \(1 \times 10^{6}\) cm^3.

Calculate the mass of a 1mm thick layer of dust

Mass can be calculated using the volume of the dust layer and the given dust density. Substituting the found volume into the formula for mass (mass = density x volume) gives us \(2 g/cm^3 x 10^6 cm^3 = 2 \times 10^6\) g, or \(2 \times 10^{9}\) mg.

Determine the time

Time is calculated by dividing the mass of the dust layer by the dustfall rate. Therefore, \(2 \times 10^{9}\)mg / \(53.9\) mg/m^2 hr gives us \(3.7 \times 10^{7}\) hours, which converts to approximately 4227.4 years.

Key concepts

Dust Deposition

Dust deposition, often called dustfall, refers to the process of dust settling from the atmosphere onto surfaces. This natural phenomenon can vary greatly depending on factors such as location, weather, and air quality. An unpolluted area was reported to have a typical dust deposition rate of 10 tons per square mile per month. This metric is used to gauge air cleanliness and the presence of pollutants. To better grasp this rate, it's important to convert units into a more universally understandable form, hence the conversion to milligrams per square meter per hour is necessary. This provides a more precise measurement, converting tons and miles into smaller, easily relatable units (milligrams and square meters). Understanding this concept helps in tracking and predicting air quality and its impacts.

Density Calculation

Density is a physical property that defines how much mass is contained within a certain volume. For calculating density, you generally use the formula:

• Density = Mass / Volume

In this exercise, the dust's density was given as 2 grams per cubic centimeter (\(2 \, \text{g/cm}^3\)). This means that each cubic centimeter of dust weighs 2 grams. Understanding density allows us to connect the weight of an object to its size, laying the foundation for further calculations involving mass and volume.
Using this density value, we can calculate the mass of a 1-millimeter thick layer of dust. Convert this thickness into centimeters for compatibility with the given density, recognizing that 1 millimeter equals 0.1 centimeters. Calculating the mass becomes essential to further analyze how long it will take for dust to accumulate over an area.

Mass and Volume Relationship

The relationship between mass and volume is crucial in understanding how substances behave. Mass refers to the amount of material within an object, while volume quantifies the space the object occupies. In calculations involving dust deposition, we use this relationship to estimate how long it will take for a measurable layer of dust to accumulate.Convert the layer's thickness from millimeters to centimeters, giving us a volume of \(1000\, \text{cm}^3\) per square meter, assuming a 1mm thick layer is uniformly distributed across a square meter.Once the volume is calculated, multiply it by the dust density to find the total mass of this layer: \[Mass = \text{Density} \times \text{Volume} = 2 \, \text{g/cm}^3 \times 1000 \, \text{cm}^3 = 2000 \, \text{g}\]This result allows us to use the dust deposition rate in mg/m\(^2\) per hour to estimate the time period for deposition. This calculation is essential in environmental science and helps to predict the effects of air quality on air and surface pollution over time.