Problem 113 - 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 \(\left(1.0 \mathrm{g} \text { of methylene blue, } \mathrm{C}_{16} \mathrm{H}_{18} \mathrm{Cl} \mathrm{N}_{3} \mathrm{S}\right.\) in 50.0 mL of water). After the dye has mixed with the water in the pool, you take a sample of the water. Using an instrument such as a spectrophotometer, you determine that the concentration of the dye in the pool is 4.1 \(\times 10^{-8}\) M. What is the volume of water in the pool?

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

You need to find the volume of a pool using a dye solution. A dye solution containing 1.0 g of methylene blue dissolved in 50.0 mL of water is added to the pool. The final concentration of dye in the pool water is given as 4.1 \( \times 10^{-8} \) M.

Step 2: Convert Mass of Dye to Moles

Calculate the number of moles of methylene blue in the dye solution. First, determine its molar mass. The molar mass of methylene blue (\( \mathrm{C}_{16} \mathrm{H}_{18} \mathrm{Cl} \mathrm{N}_{3} \mathrm{S} \)) is calculated as follows:- \( 16 \times 12.01 \) (C) = 192.16 g/mol- \( 18 \times 1.01 \) (H) = 18.18 g/mol- \( 1 \times 35.45 \) (Cl) = 35.45 g/mol- \( 3 \times 14.01 \) (N) = 42.03 g/mol- \( 1 \times 32.07 \) (S) = 32.07 g/molTotal molar mass = 319.79 g/mol.Convert mass to moles: \( \frac{1.0 \text{ g}}{319.79 \text{ g/mol}} = 3.13 \times 10^{-3} \text{ moles} \).

Step 3: Relate Moles of Dye to Pool Volume

The final molarity (concentration) after the dye fully disperses in the pool is known: \( 4.1 \times 10^{-8} \) M. Use the formula for molarity: \( C = \frac{n}{V} \), where \( C \) is concentration, \( n \) is moles of solute, and \( V \) is volume of solution in liters.Rearrange to solve for \( V \): \( V = \frac{n}{C} = \frac{3.13 \times 10^{-3} \text{ moles}}{4.1 \times 10^{-8} \text{ M}} \).

Step 4: Calculate the Pool Volume

Substitute the moles and concentration values into the formula to find the volume of the pool:\[ V = \frac{3.13 \times 10^{-3}}{4.1 \times 10^{-8}} \approx 7.63 \times 10^{4} \text{ L} \]This result means the volume of the pool is approximately 76,300 liters.

Key Concepts

Dye Concentration

Dye concentration plays a crucial role in solution chemistry when determining the amount of solute in a solution. Concentration is defined as the amount of a substance within a certain volume of a solution. In this context, methylene blue is added to the pool water, and a sample is taken to measure its concentration. The dye concentration is given as 4.1 \( \times 10^{-8} \) M, which is a measure of molarity, or moles per liter. This tells us the number of moles of methylene blue per liter of water in the pool. Knowing the concentration allows us to connect the amount of dye added to the pool with the overall volume of the pool. This value indicates how dilute the methylene blue is after being mixed in such a large volume of water.

Volume Calculation

Calculating the volume of a pool can be tricky, especially if it's irregularly shaped. However, using solution chemistry, we can determine the volume by measuring the dye concentration. Once the dye is sufficiently mixed, we take a sample of the pool water to measure how concentrated the dye now is. We then apply the formula for concentration, \( C = \frac{n}{V} \), where \( C \) is the concentration, \( n \) is the number of moles of the dye, and \( V \) is the volume of the pool in liters.

By rearranging the formula to solve for volume, we get \( V = \frac{n}{C} \). This step is essential because it transforms the problem into a manageable mathematical equation. Despite the pool's irregular shape, we can calculate its volume precisely just by knowing the concentration of methylene blue after mixing. Therefore, solution chemistry offers us a practical solution without needing to measure physical dimensions.

Molarity

Molarity is a fundamental concept in chemistry, representing the concentration of a solute in a solution. It is denoted by the symbol \( M \) and is measured in moles per liter. For methylene blue in this exercise, the known molarity of the dye solution after mixing is 4.1 \( \times 10^{-8} \) M. This value is crucial for calculating the volume of the swimming pool.

To understand molarity in practice, imagine dissolving methylene blue in a container of water. Molarity tells us how many moles of the dye exist in each liter of that water. The whole process is summarized in the formula \( C = \frac{n}{V} \), where \( C \) is molarity, \( n \) is the number of moles, and \( V \) is volume in liters. This relationship makes molarity a powerful tool for determining other properties of the solution, like the total volume when the solution spreads through a larger body of water, such as a swimming pool.