Problem 18 - Analytical Chemistry

Question

Calculate the molar concentration of \(\mathrm{NaCl}\), to the correct number of significant figures, if \(1.917 \mathrm{~g}\) of \(\mathrm{NaCl}\) is placed in a beaker and dissolved in \(50 \mathrm{~mL}\) of water measured with a graduated cylinder. If this solution is quantitatively transferred to a 250 -mL volumetric flask and diluted to volume, what is its concentration to the correct number of significant figures?

Step-by-Step Solution

Answer

The final concentration of the NaCl solution is 0.13 M.

Step 1: Calculate the number of moles of NaCl

First, we need to find the number of moles of NaCl in the sample. Use the formula \(\text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\). The molar mass of NaCl is \(58.44 \text{ g/mol}\). Thus, the number of moles is \(\frac{1.917 \text{ g}}{58.44 \text{ g/mol}} = 0.03280 \text{ mol}\).

Step 2: Calculate the initial concentration in the 50 mL solution

Now, we calculate the initial molar concentration of NaCl in the beaker before dilution. Use the formula \(C = \frac{n}{V}\), where \(n\) is the number of moles and \(V\) is the volume in liters. Hence, the concentration \(C\) is \(\frac{0.03280 \text{ mol}}{0.050 \text{ L}} = 0.656 \text{ M}\).

Step 3: Calculate the final concentration after dilution to 250 mL

Dilution involves adding more solvent to achieve a specific concentration. Use the dilution formula \(C_1V_1 = C_2V_2\). The initial concentration \(C_1 = 0.656 \text{ M}\) and the initial volume \(V_1 = 0.050 \text{ L}\). The final volume \(V_2 = 0.250 \text{ L}\). Substitute into the formula: \(0.656 \times 0.050 = C_2 \times 0.250\). Solving, \(C_2 = \frac{0.656 \times 0.050}{0.250} = 0.1312 \text{ M}\).

Step 4: Adjust to significant figures

Look at the significant figures in the given values: the number of significant figures in 1.917 is 4, 50 mL is 2, and 250 mL is 3. The limiting value is the initial volume of 50 mL with 2 significant figures. Thus, we round 0.1312 M to 0.13 M, maintaining 2 significant figures.

Key Concepts

Significant Figures

Understanding significant figures is essential when performing calculations in chemistry. Significant figures pertain to the number of meaningful digits used to express a value. They convey the precision of a measurement. When dealing with measurements, it's crucial to consider significant figures to ensure accurate and reliable results.

• Rules for significant figures: Non-zero digits are always significant. Zeros between non-zero digits are significant. Leading zeros are not significant. Trailing zeros in a decimal point are significant.
• Application: When you multiply or divide numbers, the result should have as many significant figures as the number with the fewest significant figures in the calculation.

This principle was applied when calculating the final concentration of NaCl. We identified that the measurement of 50 mL limited us to 2 significant figures. Thus, the final result, which was initially 0.1312 M, was rounded to 0.13 M to convey the most reliable measurement.

Dilution Formula

Dilution is a method used to decrease the concentration of a solution by adding more solvent. The dilution formula is a go-to tool for this and is mathematically expressed as:
\[ C_1V_1 = C_2V_2 \]

• \(C_1\) represents the initial concentration of the solution.
• \(V_1\) is the initial volume of the solution.
• \(C_2\) is the final concentration after dilution.
• \(V_2\) is the final volume of the solution after dilution.

In our exercise, the dilution process took a concentrated 50 mL solution of NaCl and increased its volume to 250 mL, thus reducing its concentration from 0.656 M to 0.1312 M, before rounding based on significant figures. This process is a straightforward application of how diluting a solution impacts its molar concentration.

Moles and Molar Mass

A fundamental concept in chemistry is that the mole is a unit for counting particles, much like a dozen pertains to twelve of something.
Molar mass, expressed in g/mol, indicates the mass of one mole of a substance, thereby acting as a bridge from lab measurements to the theoretical understanding of substances.

• To calculate moles, use the formula: \( \text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \).
• Moles provide a link between the mass of a substance and the amount of particles it contains physically, such as atoms or molecules.

In this example, to find out how many moles of NaCl were present, we divided its mass, 1.917 g, by its molar mass, 58.44 g/mol, to yield 0.03280 moles. This process is crucial for finding concentrations as it translates tangible measurements into actionable chemical data.