Question

A 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 \((1.0 \mathrm{g}\) 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?

Short answer

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

Step-by-step solution

Identify the Given Information

We have 1.0 g of methylene blue dye mixed in 50.0 mL of water, and after mixing, the concentration of dye in the pool is measured as \(4.1 \times 10^{-8} \text{ M}\).

Determine the Moles of Dye Used

First, we need to calculate the number of moles of methylene blue initially used. The molar mass of methylene blue \(\text{C}_{16} \text{H}_{18} \text{ClN}_{3} \text{S}\) can be calculated as follows:\[16(12.01) + 18(1.01) + 1(35.45) + 3(14.01) + 1(32.07) = 319.85 \, \text{g/mol}\]Next, convert grams to moles:\[\text{Moles of dye} = \frac{1.0 \, \text{g}}{319.85 \, \text{g/mol}} \approx 0.00313 \, \text{mol}\]

Use Dye Concentration to Find Pool Volume

The concentration of the dye in the pool is given as \(4.1 \times 10^{-8} \, \text{M}\). Use the formula \(\text{Concentration} = \frac{\text{moles}}{\text{volume}}\) to find the volume of the pool:\[\text{Volume of pool} = \frac{0.00313 \, \text{mol}}{4.1 \times 10^{-8} \, \text{M}} \approx 7.63 \times 10^4 \, \text{L}\]

Convert Volume to Desired Units

The volume is already in liters, which is a suitable unit for large volumes like a swimming pool. Thus, the calculated volume of the swimming pool is approximately \(76,300 \, \text{L}\).

Key concepts

Molarity

Molarity is a way to express the concentration of a solution. It's defined as the number of moles of a solute dissolved in one liter of solution. This is represented by the formula \( M = \frac{n}{V} \), where \( M \) is the molarity, \( n \) is the number of moles of solute, and \( V \) is the volume of the solution in liters.
In dilution calculations like the one in this exercise, molarity helps us determine how concentrated the dye is after it's been mixed into the pool water. The initial concentration from the solution provides a basis for determining the final volume when the concentration changes, due to the dilution across the larger volume of pool water.
To calculate moles from mass and molarity, first find the molar mass of the compound to convert grams to moles. This helps to align the units with molarity calculations.

• Mentally link molarity with measuring concentration, just like measuring teaspoons per cup in cooking.
• Consider it a conversion factor that bridges between the mass of a substance and its distribution in a liquid.

Spectrophotometry

Spectrophotometry is a technique used to measure how much a chemical substance absorbs light. By measuring the intensity of light before and after it passes through the solution, the spectrophotometer can gauge the concentration of the dye in the pool. This method is particularly useful for solutions where concentration changes down to low ranges, like the dye in this pool.
The spectrophotometer uses unique wavelengths that certain chemicals absorb or transmit. Different substances absorb varying amounts of light at different wavelengths, giving each its signature. This property allows us to determine very precise concentrations.

• Imagine a spectrophotometer like a person checking how dark tea is by the amount of color transmitted through it.
• It is efficient for very diluted solutions, which can be challenging to measure by volume or mass alone.

Molecular Weight

Molecular weight, also referred to as the molar mass, is how we express the "weight" of a molecule. It's calculated by summing the atomic weights of each atom that makes up the molecule based on its chemical formula. In the case of methylene blue, the given formula \(\text{C}_{16} \text{H}_{18} \text{ClN}_{3} \text{S}\) leads to a molar mass of 319.85 g/mol.
This molar mass is crucial for converting between grams and moles, a necessary step to use in molarity calculations. If you know the mass of substance used, calculating the number of moles gives insight into the behavior of the solution within the dilution.

• Molecular weight bridges elements of the periodic table into actionable measurements for practical uses.
• Think of it as the molecular scale that helps predict how many particles are present in a given mass of substance.

Volume of a Solution

When we talk about volume in the context of solutions, we're discussing the space that the solution occupies. This space is typically measured in liters for larger volumes, such as a swimming pool. In this exercise, once the molarity of the pool water is known, we can find the volume of the pool by rearranging the concentration formula: \(\text{Concentration} = \frac{\text{moles}}{\text{volume}}\).
Understanding volumes is key to many practical tasks in chemistry. In this example, calculating the pool's volume based on dye concentration involved spectrophotometry for an accurate measure. By finding the moles of dye and its concentration after dilution, you can back calculate to estimate how much water the dye has spread through.

• Think of large volumes in terms of everyday objects, such as a pool fuller than a bathtub but less than an entire house.
• Volume ties together with concentration to discern just how spread out a solution is, offering insights into its uniformity and extent.