Problem 60 - Chemistry The Science in Context

Question

If the concentration of mercury in the water of a polluted lake is 0.33 micrograms per liter, what is the total mass of mercury in the lake, in kilograms, if the lake has a surface area of \(10.0 \mathrm{km}^{2}\) and an average depth of \(15 \mathrm{m} ?\)

Step-by-Step Solution

Answer

Answer: The total mass of mercury in the lake is approximately 0.0495 kilograms.

Step 1: Convert the concentration of mercury to a more suitable unit

First, we need to convert the concentration of mercury from micrograms per liter to grams per cubic meter. To do this, we can use these conversion factors: 1 microgram= 10^{-6} grams and 1 liter = 0.001 cubic meters. Based on this, the conversion factor is: (10^{-6} grams/microgram) * (0.001 cubic meters/liter). Now, let's convert the concentration of mercury to grams per cubic meter: 0.33 micrograms/liter * (10^{-6} grams/microgram) * (0.001 cubic meters/liter) 0.33 * 10^{-6} grams * 0.001 cubic meters = 3.3 * 10^{-7} grams/cubic meter

Step 2: Find the volume of the lake

The volume of the lake can be calculated by multiplying its surface area and average depth. Since the surface area is given in square kilometers, we need to convert it to square meters. We can use this conversion factor: 1 square kilometer = 10^{6} square meters. So, the volume of the lake (in cubic meters) can be calculated as: (surface area of the lake) * (average depth of the lake) = (10.0 km^2) * (15 m) = 10.0 * (10^{6} m^2) * 15 m = 1.5 * 10^{8} m^3

Step 3: Calculate the total mass of mercury in the lake

Now, we can use the converted concentration (3.3 * 10^{-7} grams/m^3) and the volume of the lake (1.5 * 10^{8} m^3) to calculate the total mass of mercury in grams in the lake: (total mass of mercury in the lake) = (concentration) * (volume) = (3.3 * 10^{-7} grams/m^3) * (1.5 * 10^{8} m^3) = 4.95 * 10^{1} grams Finally, let's convert this value into kilograms by using the conversion factor 1 kg = 1000 grams: (total mass of mercury in the lake) = (4.95 * 10^{1} grams) / (1000 grams/kg) = 4.95 * 10^{-2} kg So, the total mass of mercury in the lake is approximately 0.0495 kilograms.

Key Concepts

Unit Conversion

When working with measurements in different unit systems, it's essential to perform unit conversions to ensure that all the values are compatible for calculations. In environmental studies, for example, mercury concentration might be measured in micrograms per liter, but calculations could require the measurements in grams per cubic meter.

To perform these conversions, one must use conversion factors, which are ratios that represent how one unit of measure relates to another. For instance, knowing that 1 microgram equals 10^{-6} grams allows you to convert micrograms to grams. Similarly, understanding that there are 10^{6} square meters in a square kilometer helps in converting surface areas. Accurate unit conversion is critical in performing correct calculations and obtaining meaningful results.

In our exercise, the conversion factors are neatly employed to transform the concentration of mercury (from micrograms per liter to grams per cubic meter) and the lake's surface area (from square kilometers to square meters). Remembering that these conversions are vital helps prevent errors in scientific assessments, environmental planning, and in making informed decisions based on the data.

Environmental Chemistry

Environmental chemistry focuses on chemical phenomena within the environment, including the effects of human activities on natural systems and the behavior of pollutants like mercury. The accurate measurement of contaminant concentrations is a cornerstone of environmental chemistry and is indispensable for assessing the health of ecosystems and risks to public health.

Mercury, for example, can have severe consequences for both aquatic life and the humans who consume contaminated water or fish. By calculating its concentration in a lake, as demonstrated in the given exercise, scientists can make inferences about potential sources and effects of pollution and devise strategies for remediation. Understanding the concepts of concentration and mass—and their relationship to the environment—allows for a quantifiable assessment of pollution, which is essential for regulatory compliance, cleanup efforts, and public awareness.

Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. However, its principles also extend to calculating the quantities of substances in non-reactive contexts, such as in the measurement of pollutant concentrations in an environment.

This concept becomes crucial when determining the total mass of a substance within a particular volume; it's the stoichiometric relationship between concentration and volume that allows us to calculate the overall quantity of the substance. In our exercise, once the concentration of mercury was converted to a usable unit, stoichiometry was used to determine the total mass of mercury in the lake by multiplying the concentration by the volume of water.

Grasping this relationship is not only fundamental in chemistry but is also widely applicable in environmental studies, engineering, and any field requiring precise measurement of material quantities.