Problem 79 - Chemistry Matter and Change

Question

If you dilute 20.0 \(\mathrm{mL}\) of a 3.5 \(\mathrm{M}\) solution to make 100.0 \(\mathrm{mL}\) of solution, what is the molarity of the dilute solution?

Step-by-Step Solution

Answer

The molarity of the diluted solution is \(0.7\,\text{M}\).

Step 1: Write down the given information

We are given the following information: Initial volume (\(V_1\)) = 20.0 mL Initial concentration (\(C_1\)) = 3.5 M Final volume (\(V_2\)) = 100.0 mL

Step 2: Convert volume to liters

As the molarity is expressed in moles per liter, we need to convert the volumes from mL to L: \(V_1 = 20.0\,\text{mL} \times \frac{1\,\text{L}}{1000\,\text{mL}} = 0.020\,\text{L}\) \(V_2 = 100.0\,\text{mL} \times \frac{1\,\text{L}}{1000\,\text{mL}} = 0.100\,\text{L}\)

Step 3: Apply the dilution formula

Use the dilution formula \(C_1V_1 = C_2V_2\) to find the final concentration (\(C_2\)): \(C_2 = \frac{C_1V_1}{V_2}\)

Step 4: Plug in the values and solve for final concentration

Substitute the given values and solve for \(C_2\): \(C_2 = \frac{3.5\,\text{M} \times 0.020\,\text{L}}{0.100\,\text{L}}\) \(C_2 = \frac{0.07\,\text{M}\,\text{L}}{0.100\,\text{L}}\) \(C_2 = 0.7\,\text{M}\)

Step 5: State the final answer

The molarity of the diluted solution is 0.7 M.

Key Concepts

Molarity Concentration

Understanding molarity is fundamental in chemistry for solving various problems involving solutions. Molarity, denoted as M, is a measure of concentration that tells us how many moles of a solute are present in one liter of solution. It is calculated using the formula:

\[\begin{equation}M = \frac{moles\text{ of solute}}{liters\text{ of solution}}\end{equation}\]
When dealing with molarity, we typically deal with solutions where the solute is evenly dispersed within the solvent. A higher molarity means the solution is more concentrated, whereas a lower molarity indicates a more diluted solution. In our exercise, we're adjusting the concentration of a solution from a higher molarity to a lower one by adding solvent, thereby increasing the total volume of the solution without changing the amount of solute.

• Molarity is a way to quantify the concentration of a solution.
• It is calculated as moles of solute per liter of solution.
• Changing the volume of the solution alters the molarity.

Dilution Formula

The dilution formula is a useful tool for scientists and students alike, enabling the calculation of the new concentration of a solution after it has been diluted with more solvent. This formula is expressed as:

\[\begin{equation}C_1V_1 = C_2V_2\end{equation}\]
where

• \texttt{C}_{1} is the initial molarity,
• \texttt{V}_{1} is the initial volume,
• \texttt{C}_{2} is the final molarity, and
• \texttt{V}_{2} is the final volume of the solution after dilution.

This equation shows that the product of the initial concentration and volume equals the product of the final concentration and volume. It is a direct reflection of the conservation of mass, as the amount of substance in the solution remains unchanged after dilution. In the exercise, by applying this formula, one can calculate the molarity of the diluted solution, given the initial molarity, the initial volume, and the final volume.

To use this formula effectively, it is imperative to ensure all volumes are in the same unit, typically liters, and that the concept of conservation of solute is understood.

• The dilution formula is a reflection of the conservation of mass in a solution.
• It is key to volume and concentration calculations in dilution problems.
• Understanding the formula aids in accurately predicting the outcome of dilution.

Volume Conversion

Volume conversion is an essential skill in chemistry, particularly when working with measurements in different units. Most commonly, we convert between milliliters (\texttt{mL}) and liters (\texttt{L}), remembering that 1000 mL equal 1 L. This conversion is necessary because molarity is expressed in terms of liters.

To perform volume conversions, use the following relationship:

\[\begin{equation}1\text{ L} = 1000\text{ mL}\end{equation}\]
This means that to convert from milliliters to liters, you divide the volume in milliliters by 1000. Conversely, to convert from liters to milliliters, you multiply the volume in liters by 1000. In the provided exercise, the volumes were given in milliliters and needed to be converted to liters before applying the dilution formula. Ignoring this step can lead to incorrect calculations and results.

Having a firm grasp of volume conversions is vital for accurately solving problems in chemistry that involve different units of measure.

• Conversion is critical when dealing with various measurement units.
• Liters and milliliters are the common units used in volumetric analysis.
• Understanding these basic conversions prevents errors in calculating concentrations.