Problem 148 - Chemistry The Molecular Science

Question

A \(60.0-\mathrm{mL}\) sample of \(2.00-\mathrm{M} \mathrm{NaCl}\) and a \(40.0-\mathrm{mL}\) sample of \(0.500-\mathrm{M} \mathrm{NaCl}\) are mixed. Then, additional distilled water is added until the total volume is \(500 . \mathrm{mL}\). Calculate the molarity of the \(\mathrm{NaCl}\) in the final solution.

Step-by-Step Solution

Answer

The final molarity of the NaCl solution is 0.280 M.

Step 1: Calculate Moles of NaCl in Each Sample

First, calculate the number of moles in the first solution (60.0 mL of 2.00 M NaCl). The formula to use is: \[ \text{moles} = \text{molarity} \times \text{volume in liters} \]. Thus, for the first sample: \[ \text{moles} = 2.00 \frac{\mathrm{mol}}{\mathrm{L}} \times 0.0600 \mathrm{L} = 0.120 \text{ moles of NaCl} \].Next, repeat the calculation for the second sample (40.0 mL of 0.500 M NaCl). \[ \text{moles} = 0.500 \frac{\mathrm{mol}}{\mathrm{L}} \times 0.0400 \mathrm{L} = 0.0200 \text{ moles of NaCl} \].

Step 2: Sum Up Total Moles of NaCl

Add the moles from both samples to get the total moles present after mixing. \[ \text{Total moles of NaCl} = 0.120 + 0.0200 = 0.140 \text{ moles} \]

Step 3: Calculate Total Volume

The total volume after mixing and adding water is given as 500 mL. Convert this to liters for molarity calculation:\[ \text{Total volume} = 500 \text{ mL} = 0.500 \text{ L} \]

Step 4: Calculate the Final Molarity

To find the molarity of the NaCl in the final solution, use the formula:\[ \text{Molarity (M)} = \frac{\text{Total moles of NaCl}}{\text{Total volume in liters}} \]Substitute the calculated values:\[ \text{Molarity (M)} = \frac{0.140 \text{ moles}}{0.500 \text{ L}} = 0.280 \text{ M} \]

Key Concepts

Moles Calculation

Calculating the moles of a substance is a fundamental step in chemistry for understanding the quantity of material present. This requires knowledge of the solution's molarity and volume. Molarity is the number of moles of solute per liter of solution. To find the moles, the formula is: \[\text{moles} = \text{molarity} \times \text{volume in liters}\]. When provided with volume in milliliters, it's essential to convert it to liters by dividing by 1000. For example, a 60.0 mL solution converted to liters is 0.0600 L. By using the molarity of the solution, like 2.00 M for NaCl, you multiply by the volume in liters to calculate the moles. This conversion is crucial for precision, especially when preparing solutions or calculating reactions.

Volume Conversion

Volume conversion is a simple yet vital process in solutions chemistry. It ensures that units are consistent, which is important for accurate calculations. Often, volumes are provided in milliliters (mL), but calculations typically require liters (L).

• Convert mL to L: Divide by 1000.
• Convert L to mL: Multiply by 1000.

In the exercise, you have solutions measured in milliliters: 60.0 mL and 40.0 mL. The total desired volume is 500 mL. Converting these to liters before proceeding with calculations is essential: 60.0 mL becomes 0.0600 L, and 40.0 mL becomes 0.0400 L. Similarly, 500 mL is converted to 0.500 L. This consistency helps when applying the molarity formula later in the calculation.

Molarity Formula

The molarity formula is essential for determining the concentration of solutions in chemistry. Molarity is defined as the number of moles of solute per liter of solution, expressed as \( M = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \).
This formula helps in understanding how concentrated a solution is. It is crucial for predicting reaction yields or diluting solutions. For the given problem, once you have calculated the total moles after mixing both samples, you determine the molarity by dividing total moles by the total volume in liters.
In this scenario, using a final volume of 0.500 L and finding the total moles of NaCl to be 0.140, the molarity is calculated as 0.280 M. Therefore, understanding and applying this formula helps paint the big picture of what's happening chemically in the solution.

Concentration Dilution

Concentration and dilution are key parts of preparing solutions in chemistry. When substances are mixed and diluted, the overall concentration changes. The principle of dilution can be explained by the formula: \[ C_1V_1 = C_2V_2 \] where \( C \) represents concentration and \( V \) represents volume. When mixing two solutions of different concentrations, as in this exercise, the total number of moles remains consistent if you then dilute the solution with water.
It's important to note that while the concentration changes, the total amount of solute (in moles) does not. After mixing, the original moles in solutions are combined, and when more water is added, the molarity decreases. Understanding this principle is crucial in lab settings, as it influences how solutions behave and react with other chemicals.