Problem 56 - Chemistry A Molecular Approach

Question

A dioxin-contaminated water source contains \(0.085 \%\) dioxin by mass. How much dioxin is present in \(2.5 \mathrm{~L}\) of this water? Assume a density of \(1.00 \mathrm{~g} / \mathrm{mL}\)

Step-by-Step Solution

Answer

There is approximately 2.125 grams of dioxin in 2.5 liters of the contaminated water.

Step 1: Convert liters to milliliters

Since the density is given in grams per milliliter, convert the volume of water from liters to milliliters. There are 1000 milliliters in 1 liter, so multiply the volume in liters by 1000 to get the volume in milliliters.

Step 2: Calculate the mass of the water

Use the density of water to calculate the mass of the 2.5 L of water. Since the density of water is approximately 1.00 g/mL, the mass of the water in grams is equal to the volume in milliliters.

Step 3: Calculate the mass of dioxin

Use the percentage by mass of dioxin (0.085g/mL) as a conversion factor to find the mass of dioxin in the 2.5 L of water by multiplying it with the total mass of the water.

Key Concepts

Percent by Mass

Understanding the concept of percent by mass is essential when dealing with solutions like the dioxin-contaminated water. It refers to the mass of a solute—in this case, dioxin—divided by the total mass of the solution, and then multiplied by 100 to get a percentage. In mathematical terms, it's expressed as:
\[\begin{equation} \text{Percent by mass} = \left( \frac{\text{Mass of solute}}{\text{Total mass of solution}} \right) \times 100 \end{equation}\]
In the provided exercise, dioxin represents a very small proportion, 0.085%, of the water's total mass. Accurately quantifying this small percentage is crucial in understanding the level of contamination and potential health risks.

Density Conversions

In scientific calculations, density conversions are necessary to relate mass and volume in different units. The density of a substance is its mass per unit volume. The water in our problem has a density of 1.00 gram per milliliter (g/mL). It's important to note that 1 mL corresponds to 1 cubic centimeter (1 mL = 1 cm^3). Understanding that the density can be applied across different volume units is key.
Converting from liters to milliliters is straightforward, as there are 1000 milliliters in 1 liter. This conversion is frequently used when densities are provided in g/mL, as in our exercise.

Mass-Volume Calculations

Mass-volume calculations involve finding the mass of a liquid from its volume using the formula:
\[\begin{equation} \text{Mass} = \text{Density} \times \text{Volume} \end{equation}\]
For the dioxin contamination problem, we know the volume of water (2.5 liters) and its density (1.00 g/mL), which allows us to find the mass directly. It's essential to ensure the volume is in the same unit as the density (milliliters in this case), requiring the initial conversion step from liters to milliliters.

Contaminant Quantification

Quantifying a contaminant in a sample involves determining the amount of the contaminant present in a given quantity of material or solution. In the context of the problem, we use the percent by mass of dioxin to calculate its mass in the water. By multiplying the total mass of the water by the percent by mass of dioxin (expressed as a decimal), we can find the amount of dioxin:
\[\begin{equation} \text{Mass of dioxin} = \text{Total mass of water} \times \text{(Percent by mass of dioxin / 100)} \end{equation}\]
This calculation provides us with a way to quantify the extent of contamination, which is vital for environmental assessment and health risk analysis.