Problem 85 - Chemistry The Central Science In Si Units

Question

A 40-lb container of peat moss measures \(14 \times 20 \times 30\) in. A 40-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? (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.?

Step-by-Step Solution

Answer

(a) Peat moss density: 0.132 g/cm³, topsoil density: 2.522 g/cm³. Peat moss is "lighter." (b) 16 bags of peat moss are needed.

Step 1: Convert Units for Peat Moss Volume

First, convert the volume of peat moss from cubic inches to cubic centimeters. The dimensions of the container are given as \(14 \times 20 \times 30\) inches. Calculate the volume in cubic inches first, then convert to cubic centimeters using the conversion factor \(1 \text{ in}^3 = 16.387 \text{ cm}^3\):\[\text{Volume in cubic inches} = 14 \times 20 \times 30 = 8400 \text{ in}^3\]\[\text{Volume in } \mathrm{cm}^3 = 8400 \times 16.387 = 137650.8 \mathrm{cm}^3\]

Step 2: Calculate Density of Peat Moss

Density is mass divided by volume. The mass of the peat moss is 40 lb, which can be converted to grams (1 lb = 453.592 g):\[\text{Mass in grams} = 40 \times 453.592 = 18143.68 \text{ g} \]Now, calculate the density using the volume in \(\mathrm{cm}^3\):\[\text{Density of peat moss} = \frac{18143.68}{137650.8} = 0.132 \text{ g/cm}^3\]

Step 3: Convert Units for Topsoil Volume

Convert the volume of topsoil from gallons to cubic centimeters. Given the volume of topsoil is 1.9 gallons, and using the conversion factor \(1 \text{ gallon} = 3785.41 \text{ cm}^3\):\[\text{Volume in } \mathrm{cm}^3 = 1.9 \times 3785.41 = 7192.279 \mathrm{cm}^3\]

Step 4: Calculate Density of Topsoil

Similar to peat moss, the mass of the topsoil is 40 lb, convert it to grams:\[\text{Mass in grams} = 40 \times 453.592 = 18143.68 \text{ g}\]Now, calculate the density of the topsoil:\[\text{Density of topsoil} = \frac{18143.68}{7192.279} = 2.522 \text{ g/cm}^3\]

Step 5: Compare Densities

Compare the densities of peat moss and topsoil to determine which one is lighter. The density of peat moss is \(0.132 \text{ g/cm}^3\) and the density of topsoil is \(2.522 \text{ g/cm}^3\). Since the density of peat moss is much lower, it is correct to say that peat moss is "lighter" than topsoil.

Step 6: Calculate Area Coverage of Peat Moss

To cover an area of \(15.0 \text{ ft} \times 20.0 \text{ ft}\) to a depth of \(3.0 \text{ inches}\), first convert the area to square inches and depth to inches:\[\text{Area in } \text{ft}^2 = 15.0 \times 20.0 = 300 \text{ ft}^2\]Convert the area to square inches using \(1 \text{ ft} = 12 \text{ in}\):\[\text{Area in } \text{in}^2 = 300 \times 144 = 43200 \text{ in}^2\]Now calculate the volume required for the depth of \(3\) inches:\[\text{Volume in cubic inches} = 43200 \times 3 = 129600 \text{ in}^3\]Convert the volume to cubic centimeters:\[\text{Volume in } \mathrm{cm}^3 = 129600 \times 16.387 \approx 2122615.2 \mathrm{cm}^3\]

Step 7: Calculate Number of Bags Required

Finally, use the volume of one bag of peat moss ( \(137650.8 \text{ cm}^3\)) to determine the number of bags needed:\[\text{Number of bags} = \frac{2122615.2}{137650.8} \approx 15.42\]Since bags cannot be fractional, we round up to \(16\) bags.

Key Concepts

Unit Conversion

Understanding unit conversion is important for accurately measuring and expressing quantities in different units. It allows us to switch between systems, such as from inches to centimeters or gallons to cubic centimeters, which is essential when working on density calculations.

• To convert from inches to centimeters, multiply by 2.54 because 1 inch equals 2.54 centimeters.
• For volume, cubic inches can be converted to cubic centimeters using the conversion factor 1 cubic inch is approximately equal to 16.387 cubic centimeters.
• To convert gallons to cubic centimeters, use the conversion factor where 1 gallon equals 3785.41 cubic centimeters.

When converting mass, the standard conversion from pounds to grams is applied. Since 1 pound is equal to 453.592 grams, this conversion is straightforward. By understanding these conversions, you ensure precision in calculations, leading to accurate density values.

Volume Measurement

Volume measurement is about determining the space that a substance occupies, which is crucial in density calculations. To calculate the volume of rectangular objects like the peat moss container, multiply the length, width, and height. In this case, the volume of peat moss in cubic inches is calculated as:\[ \text{Volume} = 14 \times 20 \times 30 = 8400 \text{ in}^3 \]For liquids like topsoil given in gallons, you'll convert to cubic centimeters for consistency in units:\[ 1.9 \text{ gallons} \times 3785.41 \text{ cm}^3/\text{gallon} = 7192.279 \text{ cm}^3 \]Ensuring the volume is in cubic centimeters allows for more straightforward density calculations, as the mass is also in grams.

Mass and Weight

Mass and weight are often used interchangeably, but they're different concepts. Mass measures the amount of matter in an object and is expressed in grams or kilograms, whereas weight is the gravitational force on an object. We use mass when calculating density because it remains constant regardless of location.For instance, the mass of the peat moss and the topsoil, each with a weight of 40 pounds, converts to grams as:\[ 40 \text{ lb} \times 453.592 \frac{\text{g}}{\text{lb}} = 18143.68 \text{ g} \]Mass conversion is crucial for density calculations, as density requires comparing mass to volume in the same unit system, such as grams and cubic centimeters.

Density Comparison

Density is defined as mass per unit volume. It helps compare how compact the substance is in different materials. A lower density indicates a lighter substance when comparing similar masses.For instance, to find the density of peat moss:\[ \text{Density of peat moss} = \frac{18143.68 \text{ g}}{137650.8 \text{ cm}^3} = 0.132 \text{ g/cm}^3 \]And for topsoil:\[ \text{Density of topsoil} = \frac{18143.68 \text{ g}}{7192.279 \text{ cm}^3} = 2.522 \text{ g/cm}^3 \]Comparing these densities, peat moss has a lower density than topsoil, indicating that, at the same weight, peat moss occupies more space. This comparison confirms that peat moss is indeed "lighter" than topsoil, referring to its lesser density.