Problem 37 - General Chemistry: Principles and Modern Applications

Question

The Environmental Protection Agency has a limit of 15 ppm for the amount of lead in drinking water. If a \(1.000 \mathrm{mL}\) sample of water at \(20^{\circ} \mathrm{C}\) contains \(15 \mathrm{ppm}\) of lead, how many lead ions are there in this sample of water? What is the mole fraction of lead ion in solution?

Step-by-Step Solution

Answer

There are approximately \(2.91 \times 10^{18}\) lead ions in a water sample of 1.000 ml with 15 ppm of lead, and the mole fraction of lead ion in the solution is \(8.7 \times 10^{-8}\).

Step 1: Convert ppm to moles

ppm means parts per million. In this case, it says there are 15 parts of lead in 1 million parts of water. Since there are 1000 mL in one liter, there is 1 mg of lead in this sample. Since the molar mass of lead is approximately 207 g/mol, to convert mg to g, divide by 1000. So there are approximately \(\frac{1}{1000} \times \(\frac{1}{207} = 4.83 \times 10^{-6}\) mol of lead.

Step 2: Compute the number of ions

Using Avogadro's number, which states that one mole of a substance contains \(6.022 \times 10^{23}\) particles, the number of lead ions can be computed as \(4.83 \times 10^{-6}\) mol \(\times 6.022\times10^{23}\) mol\(^{-1}\) = \(2.91 \times 10^{18}\) ions.

Step 3: Calculate mole fraction

The mole fraction of a substance is the number of moles of the substance divided by the total number of moles in the solution. Here, the total moles of the solution need to be calculated first. Given the density of water to be 1g/mL at \(20^{\circ} \mathrm{C}\), then the mass of one liter of water is 1000g. Thus, its moles would be \(\frac{1000}{18.02} = 55.5\) moles (molar mass of water is 18.02). Hence, the mole fraction of lead would be \(\frac{4.83 \times 10^{-6}}{55.5} = 8.7 \times 10^{-8}\).

Key Concepts

ppm conversion

When we talk about **ppm**, which stands for parts per million, we're essentially discussing how many parts of a substance are present in one million parts of the solution. It's a handy way to express extremely small concentrations, especially in environmental contexts like lead concentration in water.
Imagine you have a swimming pool filled with one million little balls, and 15 of these balls are red, representing lead in our context. That's what 15 ppm means!

• This ratio technique allows chemists and environmental scientists to describe very dilute concentrations easily.

In the context of our water example, since we have a 1 mL sample and water's density is 1 g/mL, it equates to 1 g or 1000 mg of water. Thus, 15 ppm means 15 mg of lead per kg of water. However, since we are dealing with 1 mL (or 1g) water, it simplifies to 1 mg of lead in 1 mL of water.
To convert this to the number of moles, use the molar mass of lead (207 g/mol). First, convert milligrams to grams: 1 mg = 0.001 g. Then find moles by \( \frac{0.001}{207} \) which gives approximately \( 4.83 \times 10^{-6} \) moles of lead.

mole fraction

The **mole fraction** allows us to understand the proportion of a solute (like lead) within a solution relative to the total number of moles present. It’s expressed without units and helps in various calculations, such as determining phase equilibria and colligative properties.
In our example, we calculated that there are approximately \( 4.83 \times 10^{-6} \) moles of lead in the sample.

• We also determine the moles of water by using its density. Since 1 liter equals 1000 grams and the molar mass of water is approximately 18.02 g/mol, the moles of water becomes about 55.5 moles.

Next, the mole fraction of lead (\( X_{lead} \)) is calculated using: \( X_{lead} = \frac{\text{moles of lead}}{\text{total moles in solution}} = \frac{4.83 \times 10^{-6}}{55.5} \)
This results in a mole fraction of approximately \( 8.7 \times 10^{-8} \). Even though this number is very small, it accurately reflects the minuscule presence of lead compared to water.

Avogadro's number calculation

**Avogadro's number** is a constant, \( 6.022 \times 10^{23} \), which tells us how many particles are in one mole of a substance. Think of it as a bridge between the macroscopic world of grams and liters, and the microscopic realm of atoms and molecules.
In chemistry, this constant is immensely helpful as it allows us to convert moles into actual countable numbers of atoms or molecules.
For the lead in our water sample, once we know there are \( 4.83 \times 10^{-6} \) moles of lead, we can easily determine the number of lead ions because each mole contains Avogadro's number of particles.\(\text{Number of lead ions} = 4.83 \times 10^{-6} \times 6.022 \times 10^{23} \approx 2.91 \times 10^{18}\)
So, despite appearing in small quantities (\( ppm \)), the actual count of these particles is extraordinarily large!