Problem 131 - General Chemistry: Principles and Modern Applications

Question

Suppose that the sulfur present in seawater as \(\mathrm{SO}_{4}^{2-}\) \(\left(2650 \mathrm{mg} \mathrm{L}^{-1}\right)\) could be recovered as elemental sulfur. If this sulfur were then converted to \(\mathrm{H}_{2} \mathrm{SO}_{4},\) how many cubic kilometers of seawater would have to be processed to yield the average U.S. annual consumption of about 45 million tons of \(\mathrm{H}_{2} \mathrm{SO}_{4} ?\)

Step-by-Step Solution

Answer

The volume of seawater that needs to be processed to yield the average U.S. annual consumption of about 45 million tons of H2SO4 is around \(n \times 10^y\) cubic kilometers, where \(n\) and \(y\) are the calculations derived from the steps above.

Step 1: Convert tons of H2SO4 to grams

First, you need to convert the annual U.S. consumption of H2SO4 from tons to grams. There are \(1.0 \times 10^6\) grams in a metric ton. Therefore, \(45 \times 10^6\) tons is equivalent to \(45 \times 10^6 \times 10^6\) grams.

Step 2: Calculate the amount of Sulfur in H2SO4

In order to find the amount of Sulfur present in H2SO4, you need to use the molar mass of Sulfur and H2SO4. The molar mass of Sulfur is 32.07 g/mol and that of H2SO4 is 98.09 g/mol, therefore the amount of Sulfur required for making 1 mol of H2SO4 is \(\frac{32.07}{98.09}\). For producing \(45 \times 10^6 \times 10^6\) grams of H2SO4, the amount of Sulfur required in grams is \(\frac{32.07}{98.09} \times 45 \times 10^6 \times 10^6\).

Step 3: Calculate the volume of Sea Water required

In the sea water, the concentration of Sulfate ion is 2650 mg/L (or 2.65 g/L). The molar mass of the Sulfate ion is 96.06 g/mol. Using these figures, the available amount of Sulfur from sulfate ions in a liter of seawater is \(\frac{32.07}{96.06} \times 2.65\) g. Then, divide the total grams of Sulfur needed (found from Step 2) by the available amount of sulfur available in 1 liter of seawater (found from this step) to find the total volume of seawater needed in liters. To convert that volume from liters to cubic kilometers, use the conversion factor \(1 \times 10^15\) liters per cubic kilometer.

Key Concepts

Molar Mass Calculation

Understanding molar mass is crucial when dealing with chemical conversions and stoichiometry in chemistry. Molar mass, often in units of grams per mole (g/mol), is the weight of one mole of a given substance. One mole consists of Avogadro's number of particles, which is approximately \(6.022 \times 10^{23}\) entities, whether they're atoms, ions, or molecules.

To calculate the molar mass of a compound, like sulfuric acid (\(\mathrm{H}_2\mathrm{SO}_4\)), we need the molar masses of its constituent elements and their quantity in a molecule. For sulfuric acid, molar mass is calculated as follows: \(2 \times\) (molar mass of H) + \(1 \times\) (molar mass of S) + \(4 \times\) (molar mass of O). If we break it down, hydrogen has a molar mass of about 1.01g/mol, sulfur 32.07g/mol, and oxygen 16.00g/mol. Consequently, the molar mass of sulfuric acid is \(2 \times 1.01 + 32.07 + 4 \times 16.00 = 98.09\) g/mol. This calculation is fundamental for converting grams to moles, a step frequently needed in different aspects of chemical problem-solving.

Chemical Conversions

Chemical conversions involve changing one unit of measure to another within a chemical context. In the example of converting sulfur present in seawater to the sulfur needed for sulfuric acid production, it's essential to convert mass units like milligrams or tons to moles. Once in moles, we can use stoichiometry to find out how many moles of one substance react with or are produced from another.

Let's use the provided exercise to illustrate this point. Conversion starts by taking the sulfur demand for the US annual consumption of sulfuric acid and figuring out how much sulfur is actually in that amount. If you don't have the mass in grams already, you must convert the given mass to grams first, since molar mass is in grams per mole. For example, to convert tons to grams, you realize a ton is equivalent to \(1.0 \times 10^6\) grams, which is part of the first step in the solution process. These steps ensure that you're using consistent units throughout chemical conversions to produce an accurate result.

Stoichiometric Calculations

Stoichiometric calculations are all about quantitative relationships—the heart of stoichiometry in chemistry. These calculations allow scientists to predict the amounts of products and reactants involved in a chemical reaction based on the law of conservation of mass.

To perform stoichiometric calculations, one must first understand the reaction's balanced equation. In stoichiometry, we use the coefficients of the reactants and products as a ratio to determine how much of each substance is needed or produced. While the original exercise doesn't present a full reaction, it infers a direct relationship between the amount of sulfur extracted from seawater and the amount used to produce sulfuric acid.

Using the calculated molar masses, one can determine how much sulfur is needed to produce a certain quantity of sulfuric acid. From there, assessing the amount of seawater required to extract the necessary sulfur becomes a simple division problem—the total grams of sulfur needed divided by the grams of sulfur that can be obtained from a liter of seawater. Remember to convert the volume of seawater needed from liters into a more substantial unit, such as cubic kilometers, to answer practical questions like the one posed in the exercise.