Problem 102 - Chemistry

Question

Answer the following questions: (a) An old rule of thumb in cooking says: "A pint's a pound the world around." What is the density in \(\mathrm{g} / \mathrm{mL}\) of a substance for which \(1 \mathrm{pt}=1\) lb exactly? (b) There are exactly 640 acres in 1 square mile. How many square meters are in 1 acre? (c) A certain type of wood has a density of \(0.40 \mathrm{~g} / \mathrm{cm}^{3} .\) What is the mass of \(1.0\) cord of this wood in kg, where 1 cord is 128 cubic feet of wood?

Step-by-Step Solution

Answer

(a) Density: 0.958 g/mL; (b) 1 acre: 4,046.86 m²; (c) Cord mass: 1,449.50 kg.

Step 1: Understanding Density Conversion

First, we need to understand the conversion between pounds and grams, and between pints and milliliters. 1 pound (lb) is equal to 453.592 grams (g), and 1 pint (pt) is equal to 473.176 milliliters (mL).

Step 2: Calculating Density in g/mL

Using the rule of thumb 'a pint's a pound', the density is defined by: \[ \text{Density} = \frac{1\,\text{pound}}{1\,\text{pint}} = \frac{453.592 \,\text{g}}{473.176\,\text{mL}} \approx 0.958 \, \text{g/mL} \]

Step 3: Acre to Square Meter Conversion

There are 640 acres in 1 square mile. Use the fact that 1 square mile = 2,589,988.11 square meters. Therefore, the area of 1 acre is: \[ \text{1 acre} = \frac{2,589,988.11 \, \text{m}^2}{640} \approx 4,046.86 \, \text{m}^2 \]

Step 4: Wood Density Calculation

Start by converting cubic feet to cubic centimeters. 1 cubic foot = 28,316.8466 cubic centimeters (cm³), thus 128 cubic feet is:\[ 128 \,\text{ft}^3 \times 28,316.8466\,\text{cm}^3/\text{ft}^3 = 3,623,755.565\,\text{cm}^3 \] Given the density is 0.40 g/cm³:\[ \text{Mass} = \text{Density} \times \text{Volume} = 0.40\,\text{g/cm}^3 \times 3,623,755.565\,\text{cm}^3 = 1,449,502.226\,\text{g} \] Finally, convert this mass to kilograms by dividing by 1,000:\[ 1,449,502.226\,\text{g} = 1,449.502\,\text{kg} \]

Step 5: Conclusion: Compile Answers

(a) The density of the given substance is approximately 0.958 g/mL. (b) There are approximately 4,046.86 square meters in 1 acre. (c) The mass of 1 cord of this wood is approximately 1,449.50 kg.

Key Concepts

Unit Conversion

Understanding unit conversion is essential in solving many scientific problems, especially those related to density and area. In our first exercise, we need to convert pounds (lbs) to grams (g) and pints (pt) to milliliters (mL). Here are the facts you need:

• 1 pound is equivalent to 453.592 grams.
• 1 pint is equivalent to 473.176 milliliters.

When doing conversions, always write down the conversion factor and perform the calculation step by step. Multiply the original quantity by the conversion factor to switch between units. Ensuring accurate conversions is the key to getting the correct density calculation.

Volume Measurement

Volume measurement is crucial when calculating the density of substances. In the exercise, you encounter the volume conversion from cubic feet (ft³) to cubic centimeters (cm³). Let’s go through what's given:

• 1 cubic foot is equal to 28,316.8466 cubic centimeters.

For the specific exercise, we multiplied this conversion factor by the given volume of 128 cubic feet:Calculate the total volume in cubic centimeters:\[128 \, ext{ft}^3 \times 28,316.8466\, \text{cm}^3/\text{ft}^3 = 3,623,755.565\, \text{cm}^3\]This accurate volume measurement is necessary for further calculations involving density.

Density of Wood

The density of wood helps us determine its mass when its volume is known. Here, we are given a wood density of 0.40 g/cm³ and need to find the mass of 1.0 cord of wood with a volume of 128 cubic feet:1. Calculate the volume in cubic centimeters, as shown in the previous section.2. Use the formula to compute mass:\[\text{Mass} = \text{Density} \times \text{Volume} = 0.40\,\text{g/cm}^3 \times 3,623,755.565\,\text{cm}^3 = 1,449,502.226\,\text{g}\]Finally, convert this mass to kilograms by dividing by 1,000 to get:\[1,449,502.226\,\text{g} = 1,449.502\,\text{kg}\]This calculation tells you the total mass of the wood based on its density and volume.

Area Conversion

Converting units of area can often be a bit tricky because they involve squared units. For example, when a conversion is needed from acres to square meters, different conversion factors apply than those used for linear measurements. Here’s what we need to do:

• Recall that there are 640 acres in 1 square mile.
• 1 square mile is equal to 2,589,988.11 square meters.

To find square meters per acre, divide the total area in square meters of a square mile by 640:\[\text{1 acre} = \frac{2,589,988.11 \, \text{m}^2}{640} \approx 4,046.86 \, \text{m}^2\]This calculation helps you understand the size of an acre in terms of square meters, a common unit used globally.