Problem 115 - Chemistry and Chemical Reactivity

Question

You need to know the volume of water in a small swimming pool, but, owing to the pool's irregular shape, it is not a simple matter to determine its dimensions and calculate the volume. To solve the problem, you stir in a solution of a dye (1.0 g of methylene blue, \(\mathrm{C}_{16} \mathrm{H}_{18} \mathrm{ClN}_{3} \mathrm{S}\), in \(50.0 \mathrm{mL}\) of water). After the dye has mixed with the water in the pool, you take a sample of the water. Using a spectrophotometer, you determine that the concentration of the dye in the pool is \(4.1 \times 10^{-8} \mathrm{M}\) What is the volume of water in the pool?

Step-by-Step Solution

Answer

The volume of water in the pool is approximately 76,300 liters.

Step 1: Calculate Moles of Dye

Start by determining the moles of methylene blue in the dye solution. The given mass of methylene blue is 1.0 g. Use the molar mass of methylene blue (C_{16}H_{18}ClN_{3}S), which is approximately 319.85 g/mol.\[ \text{Moles of methylene blue} = \frac{1.0 \text{ g}}{319.85 \text{ g/mol}} \approx 3.13 \times 10^{-3} \text{ mol} \]

Step 2: Use Concentration to Find Pool Volume

The concentration of methylene blue in the pool is given as \(4.1 \times 10^{-8} \text{ M}\). This means there are \(4.1 \times 10^{-8} \text{ moles}\) of dye per liter of water. Using the moles of dye calculated in the previous step, the total volume \(V\) of the pool can be calculated using the formula:\[ \text{Concentration} = \frac{\text{Moles of solute}}{\text{Volume of solution}} \]Solving for \(V\):\[ V = \frac{3.13 \times 10^{-3} \text{ mol}}{4.1 \times 10^{-8} \text{ mol/L}} \approx 7.63 \times 10^{4} \text{ L} \]

Step 3: Verify Units and Reasonability

Verify that the units used are correct and the answer makes sense. In this problem, the units of concentration are moles per liter, which are consistent with our calculation approach for volume (liters). The order of magnitude for pool volume seems reasonable for a small swimming pool.

Key Concepts

Molarity

Molarity is a way to express concentration. It tells us how much solute is present in a given volume of solution. When dealing with chemical solutions, understanding molarity helps us know the strength of a solution.
Molarity is represented as \({\text{M}}\), defined as the number of moles of solute per liter of solution. This is an important concept when you need to predict how a solution will react in various chemical processes.

• Formula: \( \text{Molarity} (M) = \frac{\text{Moles of Solute}}{\text{Volume of Solution in Liters}} \)
• Unit: Moles per liter (mol/L)

In our pool example, the molarity of methylene blue after mixing is given, which is pivotal for calculating the pool's volume.

Moles

The concept of moles is central in chemistry as it allows us to link the microscopic world of atoms to the macroscopic world we can measure. A mole is a quantity that represents \(6.022 \times 10^{23}\) particles of a substance, and this number is known as Avogadro's number.
Understanding moles helps to convert between atoms or molecules and grams that we can measure.

• Formula: \( \text{Moles} = \frac{\text{Mass of Substance}}{\text{Molar Mass}} \)

In the exercise, the moles of methylene blue were calculated by dividing its mass by its molar mass. This value was then used to find the volume of the pool.

Volume Calculation

To find the volume of the pool, we need to understand the relationship between moles, molarity, and volume. Once we have the moles of the solute and the molarity of the solution, we can calculate the volume of the solution that would contain those moles at the given concentration.
In the pool scenario, using the formula \( \text{Volume} (V) = \frac{\text{Moles of Solute}}{\text{Concentration}} \), we calculated how large the pool must be to achieve the observed concentration.

• Total Volume \(V\) was solved using: \( V = \frac{3.13 \times 10^{-3}\, \text{mol}}{4.1 \times 10^{-8}\, \text{mol/L}} \)

The volume obtained was about \(7.63 \times 10^4\, \text{L}\), making sense for a small pool.

Methylene Blue

Methylene blue is a synthetic dye with the chemical formula \({\text{C}}_{16} {\text{H}}_{18} {\text{ClN}}_{3} {\text{S}}\). It is commonly used in scientific experiments as a staining agent due to its vivid color and properties.
Methylene blue is soluble in water, making it ideal for mixing evenly throughout a solution or in a swimming pool scenario.

• Molecular Weight: Approximately 319.85 g/mol
• Use in Experiment: Acts as a marker to determine concentration and volume in the pool

Its properties make methylene blue suitable for spectrophotometry, as its concentration can be accurately measured using this technique.