Question

A \(8.50 \mathrm{~g}\) sample of \(\mathrm{KCl}\) is dissolved in \(66.0 \mathrm{~mL}\) of water. The resulting solution is then added to \(72.0 \mathrm{~mL}\) of a \(0.280 \mathrm{M} \mathrm{CaCl}_{2}(a q)\) solution. Assuming that the volumes are additive, calculate the concentrations of each ion present in the final solution.

Short answer

K⁺: 0.826 M, Cl⁻: 1.119 M, Ca²⁺: 0.146 M

Step-by-step solution

Calculate moles of KCl

First, determine the number of moles of potassium chloride (KCl) in the initial sample. The molar mass of KCl is approximately 74.55 g/mol. Use the formula: \[ \text{Moles of KCl} = \frac{8.50 \text{ g}}{74.55 \text{ g/mol}} \]Calculating this gives:\[ \text{Moles of KCl} \approx 0.114 \text{ moles} \]

Determine moles of CaCl2 and ions in solution

Next, determine the moles of calcium chloride (CaCl2) in the solution. We know the concentration of the CaCl2 solution is 0.280 M (moles per liter) and the volume is 72.0 mL or 0.072 L:\[ \text{Moles of CaCl2} = 0.280 \text{ M} \times 0.072 \text{ L} \]Calculating this, we get:\[ \text{Moles of CaCl2} \approx 0.02016 \text{ moles} \]Since CaCl2 dissociates into one calcium ion (Ca²⁺) and two chloride ions (Cl⁻) for each formula unit, the moles of ions are:- Moles of Ca²⁺ = 0.02016 moles- Moles of Cl⁻ (from CaCl2) = 2 \times 0.02016 moles \approx 0.04032 moles

Calculate total volume of the solution

Add the initial volumes of water and both solutions together, assuming they are additive:\[ \text{Total volume} = 66.0 \text{ mL} + 72.0 \text{ mL} = 138.0 \text{ mL} = 0.138 \text{ L} \]

Calculate concentration of K⁺ ions

KCl dissociates into one potassium ion (K⁺) and one chloride ion (Cl⁻). Since the moles of KCl are 0.114, the moles of ions will be:- Moles of K⁺ = 0.114 moles- Moles of Cl⁻ (from KCl) = 0.114 molesUsing the total volume, the concentration of potassium ions (K⁺) is:\[ \text{Concentration of K⁺} = \frac{0.114 \text{ moles}}{0.138 \text{ L}} \approx 0.826 \text{ M} \]

Calculate total concentration of Cl⁻ ions

Add chloride ions from both KCl and CaCl2:- Total moles of Cl⁻ = 0.114 moles (from KCl) + 0.04032 moles (from CaCl2) = 0.15432 molesThe concentration of Cl⁻ ions is:\[ \text{Concentration of Cl⁻} = \frac{0.15432 \text{ moles}}{0.138 \text{ L}} \approx 1.119 \text{ M} \]

Calculate concentration of Ca²⁺ ions

Finally, since the concentration is required for calcium ions, we use the already calculated moles of calcium ions:\[ \text{Concentration of Ca²⁺} = \frac{0.02016 \text{ moles}}{0.138 \text{ L}} \approx 0.146 \text{ M} \]

Key concepts

Understanding Molarity

Molarity is a way of expressing the concentration of a solution. It refers to the number of moles of solute per liter of solution and is denoted by the symbol 'M'. For example, a 1 M solution contains 1 mole of solute in every liter of solution. This unit is particularly useful because it allows chemists to calculate how much solute is dissolved in a solution given its volume and molarity.

To calculate molarity, you can use the formula:

• \(M = \frac{n}{V}\)

where \(n\) is the number of moles of solute and \(V\) is the volume of the solution in liters.

In our original problem, we calculated the molarity of potassium ions \((K^+)\) by determining the moles of KCl, and then dividing that by the total volume of the solution in liters. Similarly, molarity can be applied to find the concentration of any ion or compound in a solution by using this straightforward process.

Calculating Ionic Concentration

Ionic concentration pertains to the concentration of individual ions in a solution. When a compound dissolves in water, it dissociates into its constituent ions. Understanding this process is crucial, as it determines how each ion contributes to the solution's properties.

For ionic concentration calculations, focus on how the compound dissociates:

• For \(\text{KCl}\), it dissociates into one \(\text{K}^+\) ion and one \(\text{Cl}^-\) ion.
• For \(\text{CaCl}_2\), it dissociates into one \(\text{Ca}^{2+}\) ion and two \(\text{Cl}^-\) ions.

To find each ion's concentration, multiply the moles of compound by the number of ions produced, and then divide by the total solution volume.

This method was applied in the exercise to calculate the concentration of \(\text{Cl}^-\) ions by considering contributions from both \(\text{KCl}\) and \(\text{CaCl}_2\). Remember to add up all moles of \(\text{Cl}^-\) ions from different sources before determining the final concentration.

Exploring the Dissolution Process

The dissolution process involves a solute (like salt) dissolving in a solvent (like water). This action results in the solute particles spreading evenly throughout the solvent, forming a solution. Famous for its ability to dissolve numerous substances, water acts as a solvent in our example problem.

The key steps in the dissolution process include:

• The breakdown of interactions holding the ions together in the solid form of a compound, like \(\text{KCl}\), when added to water.
• The ions being surrounded by water molecules, which helps stabilize the solution. This is known as solvation.

In the case of \(\text{CaCl}_2\) and \(\text{KCl}\), both salts dissolve and dissociate into their respective ions, creating an aqueous solution that contains \(\text{K}^+\), \(\text{Ca}^{2+}\), and \(\text{Cl}^-\) ions. Understanding this process is key to predicting how solutions behave chemically and physically.