Question

Describe how you would prepare \(2.50 \times 10^{2} \mathrm{~mL}\) of \(0.50 \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}\). What mass (in grams) of sodium sulfate, \(\mathrm{Na}_{2} \mathrm{SO}_{4},\) is needed?

Short answer

18 grams of sodium sulfate is needed.

Step-by-step solution

Identify the Desired Molarity and Volume

We know we need a solution with a molarity of 0.50 M and a volume of 250 mL (or 0.250 L).

Use the Molarity Formula

The molarity formula is given by \( M = \frac{n}{V} \), where \( n \) is the number of moles and \( V \) is the volume in liters.

Calculate the Number of Moles Required

Rearranging the molarity formula to find the number of moles: \( n = M \times V = 0.50 \text{ M} \times 0.250 \text{ L} = 0.125 \text{ moles} \text{ of } \mathrm{Na}_2 \mathrm{SO}_4 \).

Determine the Molar Mass of Sodium Sulfate

The molar mass of \( \mathrm{Na}_2 \mathrm{SO}_4 \) is calculated as follows: \( 2(23.0) + 32.1 + 4(16.0) = 142.1 \text{ g/mol} \).

Calculate the Mass of Sodium Sulfate Needed

Use the formula \( \text{Mass} = n \times \text{Molar Mass} \). Thus, the mass required is \( 0.125 \text{ moles} \times 142.1 \text{ g/mol} = 17.7625 \text{ g} \).

Round the Mass to the Appropriate Significant Figures

Given the significant figures from the initial data, round the mass of sodium sulfate to 18 grams.

Key concepts

Molarity Calculation

Molarity is a simple yet fundamental concept in solution chemistry. It represents the concentration of a solution and is expressed as moles of solute per liter of solution (mol/L or M). To calculate the molarity, use 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 this exercise, we need a solution with a molarity of 0.50 M and a volume of 250 mL, which converts to 0.250 L. This conversion from milliliters to liters is crucial as molarity is typically a measure of how many moles exist in a liter of solution.
Understanding how to calculate and use molarity is vital in many real-world applications, such as preparing solutions in chemistry labs, pharmaceutical preparations, and understanding chemical reactivity.

Stoichiometry

Stoichiometry involves the quantitative relationships between reactants and products in a chemical reaction. Although no reaction is detailed in the exercise, stoichiometry principles apply when calculating how much of a substance is needed for a specific molarity.
When you know the amount of one reactant (or solute in this case) you can determine the required amount of another by using the concept of moles. The molarity formula implicitly uses stoichiometry because it balances the amount of solute (in moles) with the volume of solvent to achieve the desired solution concentration.
Stoichiometry is like a recipe for a chemical reaction. It ensures that you have just enough of each component to proceed without waste, making it an essential tool for laboratory and industrial processes.

Molar Mass

Molar mass is the mass of one mole of a substance and is measured in grams per mole (g/mol). It is calculated by summing the atomic masses of all atoms in a molecular formula. For sodium sulfate, \( \mathrm{Na}_2 \mathrm{SO}_4 \), the molar mass calculation goes as follows:

• Two sodium atoms: \( 2 \times 23.0 = 46.0 \text{ g/mol} \)
• One sulfur atom: \( 32.1 \text{ g/mol} \)
• Four oxygen atoms: \( 4 \times 16.0 = 64.0 \text{ g/mol} \)

This adds up to a total molar mass of \( 142.1 \text{ g/mol} \).
Knowing the molar mass is key to converting between the mass of a substance and the number of moles, which directly ties into many calculations in chemistry. In this exercise, it allowed us to determine how much sodium sulfate is needed to achieve a specific molarity.

Significant Figures

Significant figures are the digits in a number that are reliable and contribute to its precision. In scientific calculations, they reflect the accuracy of the experimental data. When rounding, it’s essential to consider the number of significant figures from the initial measurements or data.
In the given exercise, the required solution concentration and volume led us to initial data with two significant figures. Therefore, when calculating the mass of sodium sulfate needed, we corrected the final answer to match this level of precision, resulting in a rounded mass of 18 grams.
Mastering significant figures in calculations ensures consistent and accurate scientific measurements, prevents overstatement of data precision, and aligns results with the inherent accuracy of the given data.