Problem 114 - Chemistry The Central Science In Si Units

Question

The U.S. standard for arsenate in drinking water requires that public water supplies must contain no greater than 10 parts per billion (ppb) arsenic. If this arsenic is present as arsenate, \(\mathrm{AsO}_{4}^{3-},\) what mass of sodium arsenate would be present in a \(1.00-\mathrm{L}\) sample of drinking water that just meets the standard? Parts per billion is defined on a mass basis as $$ \mathrm{ppb}=\frac{\mathrm{g} \text { solute }}{\mathrm{g} \text { solution }} \times 10^{9} $$

Step-by-Step Solution

Answer

The mass of sodium arsenate needed is approximately 0.0000278 grams.

Step 1: Understand the Units and Definitions

The problem is asking for the mass of sodium arsenate in a 1 L sample of water if it contains 10 ppb of arsenic as arsenate. Parts per billion (ppb) is given by \(\text{ppb} = \frac{\text{g solute}}{\text{g solution}} \times 10^{9}\). This means 10 ppb signifies \(10 \times 10^{-9}\) grams of arsenic per gram of water solution.

Step 2: Calculate Arsenic Mass in 1 L of Water

Since 1 L of water has a mass of approximately 1000 g, the mass of arsenic in 1 L of water at 10 ppb is \(10 \times 10^{-9} \times 1000 = 10 \times 10^{-6}\) grams of arsenic.

Step 3: Find Molar Masses and Moles of Arsenic

Find the molar mass of arsenic (As) which is approximately 74.92 g/mol. Calculate the moles of arsenic by using the formula \(\text{moles of As} = \frac{10 \times 10^{-6} \text{ g}}{74.92 \text{ g/mol}}\).

Step 4: Calculate Moles of Sodium Arsenate

The chemical formula for sodium arsenate is \(\text{Na}_3\text{AsO}_4\). The stoichiometric ratio of arsenic to sodium arsenate is 1:1. Therefore, the moles of sodium arsenate is equal to the moles of arsenic calculated.

Step 5: Calculate Mass of Sodium Arsenate

Find the molar mass of sodium arsenate \(\text{Na}_3\text{AsO}_4\). The molar masses are: Na (22.99 g/mol), As (74.92 g/mol), O (16.00 g/mol), so the molar mass of \(\text{Na}_3\text{AsO}_4\) is \(3 \times 22.99 + 74.92 + 4 \times 16.00 = 207.87\) g/mol. Thus, the mass of sodium arsenate in the water is \(\text{moles of \(\text{Na}_3\text{AsO}_4\)} \times 207.87 \text{ g/mol}\).

Key Concepts

Sodium Arsenate

Sodium arsenate is a chemical compound with the formula \(\text{Na}_3\text{AsO}_4\). It consists of three sodium ions \(\text{Na}^+\) and one arsenate ion \(\text{AsO}_4^{3-}\). This compound is significant in water chemistry due to its toxicity. Arsenate ions can replace phosphate ions in biochemistry processes, disrupting cellular activities.
In terms of structure, the arsenate ion is similar to phosphate, having a tetrahedral shape. Each oxygen atom is connected to the central arsenic atom, sharing electrons to form a stable ion. Because of its chemical nature, sodium arsenate is often employed in laboratory contexts and industrial applications.
Understanding the role of sodium arsenate in water supply is crucial for managing its levels, as high concentrations can have adverse effects on health. Therefore, water chemistry involves strict regulations to ensure sodium arsenate remains within safe limits.

Parts Per Billion

Parts per billion (ppb) is a measurement unit used to describe the concentration of a solute within a solution. PPB is particularly useful when dealing with very low concentrations, like those found in trace chemical analysis. In this context, it defines how many parts of a solute are present in a billion parts of the total solution by mass.
The formula for ppb is given by:

• \(\text{ppb} = \frac{\text{g solute}}{\text{g solution}} \times 10^9\)

To visualize it, if a solution contains 10 ppb of a solute, this implies that 10 grams of solute is mixed with one billion grams of the solution, which is practically immeasurable in everyday contexts. Therefore, understanding and calculating ppb is essential in fields like environmental science and water quality management, ensuring harmful compounds like sodium arsenate remain within safe thresholds.

Molar Mass

Molar mass is the mass of one mole of a given substance and is measured in grams per mole (g/mol). It is essentially the sum of the atomic masses of all the atoms in a molecule. Understanding molar mass is fundamental in chemical calculations, as it allows for the conversion between moles and grams, facilitating various stoichiometric calculations.
For instance, to determine the molar mass of sodium arsenate \(\text{Na}_3\text{AsO}_4\), you would add together the molar masses of three sodium atoms \(3 \times 22.99\), one arsenic atom \(74.92\), and four oxygen atoms \(4 \times 16.00\), resulting in a total of approximately 207.87 g/mol.
Accurately calculating molar masses likewise aids in determining how many moles of a compound are present in a given mass and is critical when calculating reactants or products in a chemical reaction.

Chemical Calculations

Chemical calculations involve several key steps that form the backbone of solving problems in water chemistry and beyond. These calculations often require understanding units, converting between masses, and working through stoichiometry.
In our example, to find the mass of sodium arsenate in a liter of water solution, we first determined the mass of arsenic using the definition of parts per billion. By converting this mass to moles using arsenic's molar mass, we could then seamlessly calculate the number of moles of sodium arsenate, leveraging the stoichiometric ratio from its chemical formula.
Finally, by applying the molar mass of sodium arsenate, we convert the calculated moles back into mass. These calculations are repeated steps in chemistry, enabling scientists to predict product yields and understand reactant consumption in chemical equations. Mastering these skills is fundamental for anyone studying chemical processes or engaging with environmental sciences.