Question

A \(40-\mathrm{lb}\) container of peat moss measures \(14 \times 20 \times 30 \mathrm{in}\). A \(40-\mathrm{lb}\) container of topsoil has a volume of 1.9 gal. (a) Calculate the average densities of peat moss and topsoil in units of \(\mathrm{g} / \mathrm{cm}^{3}\). Would it be correct to say that peat moss is "lighter" than topsoil? Explain. (b) How many bags of peat moss are needed to cover an area measuring \(15.0 \mathrm{ft} \times 20.0 \mathrm{ft}\) to a depth of 3.0 in.?

Short answer

The average densities of peat moss and topsoil are 0.0851 g/cm³ and 0.2514 g/cm³, respectively. Since the density of peat moss is less than the density of topsoil, it is correct to say that peat moss is "lighter" than topsoil. To cover an area of 15.0 ft × 20.0 ft to a depth of 3.0 inches, approximately 4 bags of peat moss are required.

Step-by-step solution

Convert units to match

We need to convert the volumes of peat moss and topsoil to the same unit to find their densities. Let's convert the volume of the peat moss container from inches to centimeters and the volume of the topsoil container from gallons to cubic centimeters. 1 inch = 2.54 cm 1 gallon = 3,785.41 cm³

Calculate the volume of the peat moss container

Using the given dimensions for the peat moss container, we can find the volume by multiplying the length, width, and height. Volume = Length × Width × Height Volume = \(14\:\text{in}\) × \(20\:\text{in}\) × \(30\:\text{in}\) Volume = \(14\:\text{in}\) × \(20\:\text{in}\) × \(30\:\text{in}\) × \((2.54\:\mathrm{cm/in})^3\) Volume = 21282.24 cm³

Calculate the volume of the topsoil container

We will now convert the given volume of the topsoil container to cubic centimeters. Volume = 1.9 gal × 3,785.41 cm³/gal Volume = 7192.279 cm³

Calculate the densities of peat moss and topsoil

To calculate the densities of peat moss and topsoil, we will use the formula: Density = Mass / Volume (a) Peat moss: Density = \(40\:\text{lb}\) × 453.592 g/lb ÷ 21282.24 cm³ Density = 0.0851 g/cm³ (b) Topsoil: Density = \(40\:\text{lb}\) × 453.592 g/lb ÷ 7192.279 cm³ Density = 0.2514 g/cm³

Compare the densities

Now, we can compare the densities and determine if it is correct to say that peat moss is lighter than topsoil: Density of peat moss = 0.0851 g/cm³ Density of topsoil = 0.2514 g/cm³ Since the density of peat moss is less than the density of topsoil, it is correct to say that peat moss is "lighter" than topsoil.

Calculate the number of bags of peat moss required

To find the number of bags of peat moss needed to cover the given area to a certain depth, we first need to calculate the volume of peat moss needed. Area = Length × Width = \(15.0\:\text{ft}\) × \(20.0\:\text{ft} = 300\:\text{ft}^2\) Depth = 3.0 in = 3.0 in × 2.54 cm/in = 7.62 cm Volume_needed = Area × Depth Volume_needed = 300 ft² × 7.62 cm × \((30.48\:\mathrm{cm/ft})^2\) Volume_needed = 65776.32 cm³ Now, we will divide the volume needed by the volume of one bag of peat moss to obtain the number of bags required. Number_of_bags = Volume_needed / Volume_of_one_bag Number_of_bags = 65776.32 cm³ ÷ 21282.24 cm³ Number_of_bags ≈ 3.09 As we can't use a fraction of a bag, we will round up to find that 4 bags of peat moss are required to cover the specified area to a depth of 3.0 inches.

Key concepts

Unit Conversion in Chemistry

In chemistry, unit conversion is a fundamental skill that enables scientists and students to compare measurements and perform calculations with different units. It often involves converting units of mass, volume, and density to a common system so that they can be compared accurately. For instance, the exercise presented required converting the volume of peat moss from inches to centimeters and the volume of topsoil from gallons to cubic centimeters.

To make such conversions accurate and easy to understand, it's essential to know the conversion factors, such as:

• 1 inch = 2.54 cm
• 1 gallon = 3,785.41 cm³

Additionally, remembering that conversions can be performed by multiplying the initial measurement by the conversion factor helps ensure that units will be consistent across different measurements, allowing for correct density calculations.

Volume Calculation

Volume is the amount of space that a substance or object occupies, and it is a crucial parameter in density calculations. To calculate the volume of solid objects, like the container of peat moss, we typically use the formula for the volume of a rectangular prism:
Volume = Length × Width × Height
Using the exercise's measurements, the volume was calculated by multiplying the container's length, width, and height. Converting these measurements from inches to the standard unit of centimeters in chemistry (cm³) involved raising the conversion factor (2.54 cm/in) to the third power, since volume is a three-dimensional measurement.

For liquids, such as the topsoil in gallons, we use conversion factors to translate gallons into cubic centimeters, as different materials are often compared in a common unit of measure in scientific calculations.

Mass to Volume Ratio

The mass to volume ratio, more commonly known as density, is a characteristic property of a material and a critical concept in many fields of science, including chemistry. Density is defined as the mass of the material divided by its volume, and it is usually expressed in grams per cubic centimeter (g/cm³) in the metric system.

In the given exercise, the density of peat moss and topsoil was calculated using the formula:
Density = Mass / Volume
First, the mass of each container was converted to grams (g), and then the volume was calculated in cubic centimeters (cm³) as described in the previous sections. These values were then used to find the densities. The comparison of the densities of peat moss and topsoil allowed us to determine that peat moss is 'lighter' since it has a lower density, meaning less mass per unit of volume.