Question

A solution is prepared by mixing \(75.0 \mathrm{mL}\) of \(0.020\) \(M\) \(\mathrm{BaCl}_{2}\) and \(125 \mathrm{mL}\) of \(0.040\) \(M\) \(\mathrm{K}_{2} \mathrm{SO}_{4}\). What are the concentrations of barium and sulfate ions in this solution? Assume only \(\mathrm{SO}_{4}^{2-}\) ions \(\left(\text { no } \mathrm{HSO}_{4}^{-}\right)\) are present.

Short answer

The concentrations of barium and sulfate ions in the final solution are \(0.00750 \mathrm{M}\) and \(0.0250 \mathrm{M}\), respectively.

Step-by-step solution

Calculate moles of Barium ions (Ba^2+) in BaCl2 solution

First, we will calculate the moles of Ba^2+ ions in the given 75.0 mL of 0.020 M BaCl2 solution using the formula: Moles of Ba^2+ = Molarity of BaCl2 * Volume of BaCl2 solution (in liters) moles of Ba^2+ = (0.020 mol/L) * (0.075 L) moles of Ba^2+ = 0.00150 mol

Calculate moles of Sulfate ions (SO_4^2-) in K2SO4 solution

Next, we will calculate the moles of SO_4^2- ions in the given 125 mL of 0.040 M K2SO4 solution using the same formula: Moles of SO_4^2- = Molarity of K2SO4 * Volume of K2SO4 solution (in liters) moles of SO_4^2- = (0.040 mol/L) * (0.125 L) moles of SO_4^2- = 0.00500 mol

Calculate the final volume of the solution

The final volume of the solution is the total volume of both solutions combined. Total Volume = Volume of BaCl2 solution + Volume of K2SO4 solution Total Volume = 75.0 mL + 125 mL Total Volume = 200.0 mL = 0.200 L

Calculate the concentration of Barium ions (Ba^2+) in the final solution

Now, we will calculate the concentration of Ba^2+ ions in the final solution using the formula: Concentration (Molarity) = Moles of Ba^2+ / Total Volume (in liters) Concentration of Ba^2+ = (0.00150 mol) / (0.200 L) Concentration of Ba^2+ = 0.00750 M

Calculate the concentration of Sulfate ions (SO4^2-) in the final solution

Finally, calculate the concentration of SO_4^2- ions in the final solution using the same formula: Concentration (Molarity) = Moles of SO_4^2- / Total Volume (in liters) Concentration of SO_4^2- = (0.00500 mol) / (0.200 L) Concentration of SO_4^2- = 0.0250 M Thus, the concentrations of barium and sulfate ions in the final solution are 0.00750 M and 0.0250 M, respectively.

Key concepts

Molarity

Molarity is a measure of the concentration of a solute in a solution, defined as the number of moles of solute per liter of solution. It is often denoted by the symbol 'M' and is expressed in units of moles per liter (mol/L). For students grappling with this concept, think of molarity as a way to quantify how 'strong' or 'concentrated' a solution is.

To calculate the molarity of a solution, one can use the following formula:
\[ \text{Molarity (M)} = \frac{\text{Number of moles of solute}}{\text{Volume of solution in liters}} \]

When mixing different solutions, molarity helps to predict the final concentration of ions in the mixture. For example, in the provided exercise, the molarity is used to determine the concentrations of barium and sulfate ions after mixing two solutions of known concentrations. It's essential to articulate each step in calculating molarities, as understanding the concept of molarity is fundamental for mastering topics in chemistry such as dilutions, chemical reactions, and stoichiometry.

Mixing Solutions

When mixing solutions of different concentrations, it is crucial to understand that the total volume of the mixed solution affects the final concentration of the solutes. In the context of our exercise, we mix a bacl2 solution with a K2SO4 solution, each with its own initial volumes and molarities.

Mixing Rule

One key principle to remember is that the number of moles of solute remains unchanged when solutions are mixed; only the total volume changes. Therefore, to find the final concentration, one must first calculate the moles of solute present in each individual solution before they are combined. After this, you can determine the total volume of the resulting mixture and then apply the molarity formula to find the concentrations of the relevant ions.
This process illustrates the conservation principle in mixing solutions: the sum of solute particles (moles) is maintained, providing a logical pathway for students to find final concentrations after mixing solutions of different molarity values.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the relative quantities of reactants and products in chemical reactions. In the context of solution concentration calculations, stoichiometry can inform us about the ratios in which different ions combine, or in this case, dissociate in water.

Understanding stoichiometry is crucial for correctly calculating the number of moles of ions in a solution. For example, when a salt like \( BaCl_2 \) dissolves in water, it separates into its respective ions, Ba+2 and Cl-, and the same goes for \( K_2SO_4 \) separating into K+ and SO4^2- ions.

In our exercise, the stoichiometry concept is thinly veiled, but it's there when determining the number of moles of Ba^2+ and SO_4^2- ions released in solution. Making certain to reference the balanced chemical equations that define the dissociation can also aid in a deeper understanding of the reactions taking place and their relevance to the moles of ions present in a solution. When improving upon the exercise solution, a stoichiometric perspective ensures accurate calculations and a more profound comprehension of the chemical system at hand.