Problem 59 - Chemistry and Chemical Reactivity

Question

If you want a solution that is \(0.100 \mathrm{m}\) in ions, what mass of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) must you dissolve in \(125 \mathrm{g}\) of water? (Assume total dissociation of the ionic solid.)

Step-by-Step Solution

Answer

Dissolve 0.592 g of \( \mathrm{Na}_2 \mathrm{SO}_4 \) in 125 g of water.

Step 1: Calculate the Total Ion Molarity

Since total dissociation occurs, determine the ion concentration produced by \( \mathrm{Na}_2 \mathrm{SO}_4 \). \( \mathrm{Na}_2 \mathrm{SO}_4 \) dissociates in water into 2 \( \mathrm{Na}^+ \) ions and 1 \( \mathrm{SO}_4^{2-} \) ion. Thus, for each mole of \( \mathrm{Na}_2 \mathrm{SO}_4 \), 3 moles of ions are produced.

Step 2: Determine Moles of \( \mathrm{Na}_2 \mathrm{SO}_4 \) Needed

The target molality is \(0.100\, \mathrm{m}\), which means 0.100 moles of ions per kg of water. With 125 g of water (which is 0.125 kg), the required moles of ions is \(0.100 \times 0.125 = 0.0125 \) moles. To achieve this with \( \mathrm{Na}_2 \mathrm{SO}_4 \), calculate the moles of \( \mathrm{Na}_2 \mathrm{SO}_4 \) needed: \( \frac{0.0125}{3} = 0.00417 \) moles.

Step 3: Convert Moles to Mass

Convert the moles of \( \mathrm{Na}_2 \mathrm{SO}_4 \) (0.00417 moles) into grams using its molar mass. The molar mass of \( \mathrm{Na}_2 \mathrm{SO}_4 \) is approximately \(142.04 \mathrm{g/mol}\). Therefore, the required mass is \(0.00417 \times 142.04 = 0.592\,\mathrm{g}.\)

Key Concepts

Ion Concentration

Ion concentration refers to the number of ions present in a solution, which is crucial for understanding the behavior and properties of the solution. For ionic compounds, like sodium sulfate (\(\text{Na}_2\text{SO}_4\)), ion concentration is determined by how the compound dissociates in water.
When you dissolve \(\text{Na}_2\text{SO}_4\), it fully dissociates into ions: two sodium ions (\(\text{Na}^+\)) and one sulfate ion (\(\text{SO}_4^{2-}\)). This means each mole of \(\text{Na}_2\text{SO}_4\) gives you three moles of ions. Therefore, the ion concentration is essentially linked to the number of moles the compound dissociates into.
Knowing how to calculate ion concentration from a compound is important because it helps predict how the solution behaves chemically. For instance, the concentration of ions affects the solution's electrical conductivity and its reactivity. It plays a key role in many chemical and biological processes. To calculate the total ion concentration, multiply the molarity of the solution by the number of ions each formula unit produces when dissolved.

Molarity

Molarity is a measure of concentration that expresses the number of moles of a solute per liter of solution. However, in this exercise, we come across a different term: molality, which is slightly different yet equally important to understand.
Molality (\(m\)), unlike molarity, is defined as the moles of solute per kilogram of solvent. In the context of our problem, a target molality of \(0.100\, m\) is aimed at indicating that there should be \(0.100\,\text{moles}\) of ions per kilogram of water.
A clear understanding of both molarity and molality is crucial for performing successful chemical calculations. It helps in analyzing reactions more precisely, especially under varying temperature conditions, as molality is not affected by temperature changes. When dealing with solutions in lab settings, knowing the difference can hugely impact the outcome of experiments and reactions.

Moles to Mass Conversion

Converting moles to mass is an essential calculation in chemistry that allows us to transition from a conceptual understanding of chemical quantities to tangible, practical measurements. This is crucial when preparing solutions for chemical reactions, like dissolving \(\text{Na}_2\text{SO}_4\) in water.
The process involves using the molar mass of a substance, which is the mass of one mole of its particles. For \(\text{Na}_2\text{SO}_4\), the molar mass is approximately \(142.04\,\text{g/mol}\). To find out how much mass you need for a certain number of moles, simply multiply the moles by the molar mass. For example, to dissolve \(0.00417\) moles of \(\text{Na}_2\text{SO}_4\), you calculate: \(0.00417\, \text{moles} \times 142.04\,\text{g/mol}\), resulting in \(0.592\,\text{g}\).
This conversion is fundamental for ensuring that the correct amounts of substances are used in reactions, thus ensuring accurate and desirable outcomes in laboratory and industrial settings.