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-L sample of drinking water that just meets the standard? Parts per billion is defined on a mass basis as $$ \mathrm{Ppb}=\frac{\mathrm{g} \text { solute }}{\mathrm{g} \text { solution }} \times 10^{9} $$

Short answer

In a 1.00 L sample of drinking water that just meets the U.S. standard for arsenate, the mass of sodium arsenate (\(\mathrm{Na}_{3}\mathrm{AsO}_{4}\)) is \(2.79 \times 10^{-5}\,\mathrm{g}\).

Step-by-step solution

Calculate mass of arsenic

According to the U.S. standard, the drinking water should contain no greater than 10 ppb of arsenic. Here, we consider 1.00 L of water, which is equivalent to 1.00 kg. Using the given definition of ppb, we can find the mass of arsenic in the water sample: $$ \mathrm{Ppb}=\frac{\mathrm{g} \text { solute }}{\mathrm{g} \text { solution }}\times 10^{9}, $$ Rearrange the equation to find the mass of arsenic solute: $$ \mathrm{g} \text { solute }=\frac{\mathrm{Ppb} \times \mathrm{g} \text { solution }}{10^{9}}. $$ Insert the values for ppb and mass of solution: $$ \mathrm{g} \text { arsenic }=\frac{10 \times 1,000}{10^{9}}=10^{-5} \,\mathrm{g}. $$

Calculate moles of arsenic

Next, we need to convert the mass of arsenic into moles, using the molar mass of arsenic (As): $$ \mathrm{moles} =\frac{\mathrm{mass}}{\mathrm{molar \, mass}}. $$ The molar mass of arsenic is \(\mathrm{74.92\,g/mol}\), so: $$ \mathrm{moles\, As} =\frac{10^{-5}\,\mathrm{g}}{74.92\,\mathrm{g/mol}}=1.34 \times 10^{-7}\,\mathrm{mol}. $$

Calculate the moles of sodium arsenate

Since each molecule of sodium arsenate contains one arsenic atom, the number of moles of sodium arsenate in the sample is equal to the number of moles of arsenic: $$ \mathrm{moles\, Na}_{3}\mathrm{AsO}_{4} = 1.34 \times 10^{-7} \,\mathrm{mol}. $$

Calculate the mass of sodium arsenate

To obtain the mass of sodium arsenate in the sample, we will multiply the number of moles by the molar mass of sodium arsenate. The molar mass of sodium arsenate is: $$ \mathrm{3 \times (molar \, mass \, of \, Na) + molar \, mass\, of\, As+4 \times (molar\, mass\,of\, O) }. $$ The molar mass of sodium (Na), arsenic (As), and oxygen (O) are \(\mathrm{22.99\,g/mol}\), \(\mathrm{74.92\,g/mol}\), and \(\mathrm{16.00\,g/mol}\), respectively. Thus, $$ \mathrm{molar \, mass\, of\, Na}_{3}\mathrm{AsO}_{4} =\mathrm{3 \times 22.99 + 74.92 + 4 \times 16.00}=207.89\, \mathrm{g/mol}. $$ Now, we can find the mass of sodium arsenate in the sample: $$ \mathrm{mass\, of\, Na}_{3}\mathrm{AsO}_{4} = (\mathrm{moles}\,\mathrm{of\,Na}_{3}\mathrm{AsO}_{4}) \times (\mathrm{molar\,mass\,of\,Na}_{3}\mathrm{AsO}_{4})= (1.34 \times 10^{-7}\,\mathrm{mol}) \times (207.89\,\mathrm{g/mol})= 2.79\times 10^{-5}\,\mathrm{g}. $$ So, the mass of sodium arsenate in a 1.00 L sample of drinking water that just meets the U.S. standard is \(2.79 \times 10^{-5}\, \mathrm{g}\).

Key concepts

Parts Per Billion (PPB)

Understanding Parts Per Billion (PPB) is crucial when dealing with trace amounts of substances in solutions. PPB is a unit of measurement that indicates the concentration of a substance in a solution. It is used to express extremely small concentrations and is particularly useful in environmental science applications, such as measuring pollutants in water.
PPB describes one part of a substance for every billion parts of the solution. In our problem, the water can only contain a maximum of 10 parts of arsenic per billion parts of water, ensuring it meets safety standards.
To calculate PPB, the mass of the solute (in grams) is divided by the total mass of the solution (in grams), then multiplied by ten to the ninth power (\[ \text{Ppb} = \frac{\text{mass of solute in g}}{\text{mass of solution in g}} \times 10^9 \]).
This calculation helps ensure only safe amounts of harmful substances like arsenic are present in water, complying with environmental and health regulations.

Molar Mass Calculation

Molar mass calculation is a fundamental concept in chemistry, used to convert between the mass of a substance and the amount in moles. The molar mass is the mass of one mole of a given substance, expressed in grams per mole (g/mol).
To determine the molar mass of a compound, one must add up the atomic masses of all the atoms present in the molecule. For sodium arsenate, the chemical formula is \(\text{Na}_3\text{AsO}_4\). Let's break it down:

• 3 sodium (Na) atoms: Each with a molar mass of 22.99 g/mol, contributing \(3 \times 22.99 \,\text{g/mol}\).
• 1 arsenic (As) atom: Molar mass of 74.92 g/mol.
• 4 oxygen (O) atoms: Each with a molar mass of 16.00 g/mol, contributing \(4 \times 16.00\,\text{g/mol}\).

Adding these together, the molar mass of sodium arsenate is found to be 207.89 g/mol. This information is essential for converting the moles of a substance back to its mass, aiding in practical applications like the preparation of chemical solutions.

Water Safety Standards

Water safety standards are set by authorities like the U.S. Environmental Protection Agency (EPA) to ensure that water is safe to drink. These standards specify the maximum allowable concentrations of various contaminants, including heavy metals such as arsenic.
Arsenic is highly toxic and poses significant health risks even at low concentrations. The EPA has set a limit of 10 parts per billion (ppb) for arsenic in public water supplies. This strict regulation helps minimize health risks associated with exposure to arsenic, such as cancer and skin lesions.
Water safety standards reflect an incredible amount of research and are designed to protect public health. Tools like PPB are vital for monitoring contaminant levels and ensuring that water treatment processes effectively maintain these safety standards.

Chemical Solutions

Chemical solutions are homogeneous mixtures consisting of two or more substances. In these mixtures, a solute is dissolved in a solvent. Chemical solutions can vary widely in concentration, from very dilute to very concentrated forms.
In our case, the solute is sodium arsenate, and the solvent is water. Concentrations are often expressed in units like parts per million (ppm), parts per billion (ppb), or molarity, depending on the level of precision required.
Sodium arsenate in drinking water as an example illustrates the importance of limiting toxic substances to safe concentrations. It is important to understand the concepts around these units to calculate how much of a given compound is present in the water. This knowledge is especially critical for industries involved in water treatment and environmental protection, as it helps keep contaminants within regulated allowable limits.