Problem 110 - Chemistry The Central Science In Si Units

Question

In 2014 , a major chemical leak at a facility in West Virginia released \(28,390 \mathrm{~L}\) of MCHM (4-methylcyclohexylmethanol, \(\mathrm{C}_{8} \mathrm{H}_{16} \mathrm{O}\) ) into the Elk River. The density of MCHM is 0.9074 \(\mathrm{g} / \mathrm{mL}\). (a) Calculate the initial molarity of MCHM in the river, assuming that the first part of the river is \(2.00 \mathrm{~m}\) deep, \(90.0 \mathrm{~m}\) wide, and \(90.0 \mathrm{~m}\) long. (b) How much farther down the river would the spill have to spread in order to achieve a "safe" MCHM concentration of \(1.00 \times 10^{-4} \mathrm{M}\) ? Assume the depth and width of the river are constant and the concentration of MCHM is uniform along the length of the spill.

Step-by-Step Solution

Answer

The initial molarity is 0.01240 M. The spill must spread 11,071 meters more downriver to reach a safe concentration.

Step 1: Calculate Volume of the River Section

To find the initial molarity, we need the volume of the section of the river where the MCHM was initially released. The volume \( V \) is given by multiplying the depth, width, and length of the river section:\[V = 2.00 \text{ m} \times 90.0 \text{ m} \times 90.0 \text{ m}\]Converting cubic meters to liters (since \(1 \text{ m}^3 = 1000 \text{ L}\)), we find:\[V = 2.00 \times 90.0 \times 90.0 \times 1000 \text{ L}\]\[V = 16,200,000 \text{ L}\]

Step 2: Calculate Mass of MCHM

The next step is to calculate the mass of MCHM released. First, convert the volume from liters to milliliters (since 1 L = 1000 mL):\[28,390 \text{ L} \times 1000 = 28,390,000 \text{ mL}\]Using the density of MCHM, we can find the mass using the formula \( \text{mass} = \text{density} \times \text{volume} \):\[\text{mass} = 0.9074 \text{ g/mL} \times 28,390,000 \text{ mL}\]\[\text{mass} = 25,756,586 \text{ g}\]

Step 3: Calculate Moles of MCHM

Convert mass to moles using the molar mass of MCHM \( \mathrm{C}_8\mathrm{H}_{16}\mathrm{O} \). The molar mass is approximately 128.2 g/mol (obtained by summing the atomic masses: \(8 \times 12.01 + 16 \times 1.008 + 16.00\)): \[\text{moles} = \frac{25,756,586 \text{ g}}{128.2 \text{ g/mol}}\]\[\text{moles} = 200,890.7 \text{ moles}\]

Step 4: Calculate Initial Molarity of MCHM

Molarity is defined as moles of solute per liter of solution. Using the moles calculated and the volume of the river section:\[M = \frac{200,890.7 \text{ moles}}{16,200,000 \text{ L}}\]\[M = 0.01240 \text{ M}\]

Step 5: Determine New Volume for Safe MCHM Concentration

To achieve a safe concentration of \(1.00 \times 10^{-4} \text{ M}\), we need to determine the new volume (\(V'\)) of the river through which the MCHM must be spread:\[1.00 \times 10^{-4} \text{ M} = \frac{200,890.7 \text{ moles}}{V'}\]Rearruating gives:\[V' = \frac{200,890.7 \text{ moles}}{1.00 \times 10^{-4} \text{ M}}\]\[V' = 2,008,907,000 \text{ L}\]

Step 6: Calculate Additional Length of River Needed

The additional length \( L' \) of the river needed, while keeping width and depth constant, is found by:\[L' = \frac{V' - V}{\text{width} \times \text{depth}}\]\[L' = \frac{2,008,907,000 \text{ L} - 16,200,000 \text{ L}}{90 \text{ m} \times 2 \text{ m}}\]\[L' = \frac{1,992,707,000 \text{ L}}{180 \text{ m}^2}\]\[L' = 11,070,594.4 \text{ m} \approx 11,071 \text{ m}\]

Step 7: Total Length Down the River

The total distance down the river where the concentration will be safe, includes the initial 90 meters:\[L_{\text{total}} = 90 \text{ m} + 11,071 \text{ m} = 11,161 \text{ m}\]

Key Concepts

Density and Volume

The concepts of density and volume are foundational in chemistry, helping us understand how substances occupy space and how their mass relates to volume.
Density is a measure of how much mass is contained in a given volume. It's expressed in units such as grams per milliliter (g/mL) for liquids. The formula for density is:

• \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)

In this problem, the density of MCHM was given as 0.9074 g/mL. This means each milliliter of MCHM weighs 0.9074 grams.
To calculate volume in this exercise, we needed the dimensions of the river section where the chemical spill occurred. The river’s volume was determined by multiplying its depth, width, and length.

• \(V = \text{depth} \times \text{width} \times \text{length} \)

After finding the volume in cubic meters, it was converted to liters for consistency with the chemical spill volume, using the conversion: \( 1 \text{ m}^3 = 1000 \text{ L} \).
Understanding these conversions is crucial in ensuring correct calculations in chemical problems.

Mole Concept

The mole concept is a central idea in chemistry that relates the mass of a substance to the number of molecules or atoms it contains. One mole is defined as \(6.022 \times 10^{23}\) entities (Avogadro's number), whether they are atoms, molecules, or formula units.
In this exercise, we used molar mass, which is the mass of one mole of a substance measured in grams/mole, to convert the mass of MCHM to moles. The molar mass of MCHM (\(\text{C}_8\text{H}_{16}\text{O}\)) was calculated by summing the atomic masses of its constituent elements:

• Carbon (C): 12.01 g/mol
• Hydrogen (H): 1.008 g/mol
• Oxygen (O): 16.00 g/mol

This results in 128.2 g/mol for MCHM.Using this, the moles of MCHM were calculated:

• \[ \text{moles} = \frac{\text{mass of MCHM}}{\text{molar mass of MCHM}} \]

Such calculations are pivotal in determining the chemical quantity in reactions and environmental assessments.

River Pollution Calculation

Calculating pollution concentration in a river involves determining the molarity of the pollutant, which is the moles of solute per liter of solution. Molarity is crucial for understanding the concentration levels and potential impact of the pollutant, considering environmental safety standards.
In this scenario, the initial molarity of MCHM in the river was computed from the total moles of MCHM and the section's volume:

• \[ M = \frac{\text{moles of MCHM}}{\text{Volume of river section in L}} \]

To achieve a safe concentration level (\(1.00 \times 10^{-4} \text{ M}\)), the spread of the pollutant had to be expanded. By equating the desired molarity with the known moles of MCHM, the new total volume of the river was calculated, and subsequently, the additional length downriver was determined.
These calculations are essential in environmental chemistry to forecast the flow and concentration of pollutants, ensuring water safety and regulatory compliance.