Problem 28 - Analytical Chemistry

Question

Calculate the concentration of barium in the solution at equilibrium when \(15.0 \mathrm{~mL}\) of \(0.200 \mathrm{M}\) \(\mathrm{K}_{2} \mathrm{CrO}_{4}\) is added to \(25.0 \mathrm{~mL}\) of \(0.100 \mathrm{M} \mathrm{BaCl}_{2}\)

Step-by-Step Solution

Answer

The concentration of barium in the solution at equilibrium is 0 M.

Step 1: Determine the Initial Moles

Calculate the moles of each reactant initially present before any reaction occurs. Moles of \(K_2CrO_4 = 0.200 \, M \times 0.015 \, L = 0.003 \: moles\) Moles of \(BaCl_2 = 0.100 \, M \times 0.025 \, L = 0.0025 \: moles\)

Step 2: Write the Balanced Reaction

Here is the balanced chemical equation for the reaction between barium chloride and potassium chromate:\[ Ba^{2+} + CrO_4^{2-} \rightarrow BaCrO_4(s) \]This indicates a 1:1 mole ratio between \(Ba^{2+}\) and \(CrO_4^{2-}\) forming the insoluble salt \(BaCrO_4\).

Step 3: Identify the Limiting Reactant

Since the reaction ratio is 1:1, the reactant with fewer moles will be the limiting reactant. Here, \(Ba^{2+}\) is the limiting reactant because there are only 0.0025 moles of \(Ba^{2+}\), compared to 0.003 moles of \(CrO_4^{2-}\).

Step 4: Calculate Moles at Equilibrium

All of the limiting reactant \(Ba^{2+}\) will react, so 0.0025 moles of \(BaCrO_4\) will precipitate. Remaining moles of \(CrO_4^{2-} = 0.003 - 0.0025 = 0.0005 \: moles\).

Step 5: Determine Total Volume of Solution

Calculate the new total volume of the solution after mixing: Total Volume = 15 mL + 25 mL = 40 mL = 0.040 L.

Step 6: Calculate Concentration of Barium at Equilibrium

Since all \(Ba^{2+}\) ions precipitate out, the concentration of barium ions \([Ba^{2+}]\) in the solution at equilibrium is 0 M.

Key Concepts

Limiting Reactant

In any chemical reaction, it is crucial to determine which reactant will be completely used up first. This reactant is known as the limiting reactant. It determines the maximum amount of product that can be formed and effectively limits the extent of the reaction.

When we look at the original exercise involving \[ Ba^{2+} + CrO_4^{2-} \rightarrow BaCrO_4(s) \]we find that barium ions \( Ba^{2+} \) is the limiting reactant. This is decided by comparing the initial moles of each reactant:

• 0.0025 moles of \( Ba^{2+} \)
• 0.003 moles of \( CrO_4^{2-} \)

Since \( Ba^{2+} \) has fewer moles than \( CrO_4^{2-} \), it entirely reacts, forming \( BaCrO_4 \) and defining the endpoint of the reaction.

Recognizing the limiting reactant is essential. It tells us precisely how much product will be formed and which reactants, if any, will remain. Understanding this can help in predicting the result of any mixture of chemicals, saving both time and resources.

Precipitation Reaction

A precipitation reaction occurs when two solutions combine to form an insoluble solid, known as a precipitate. In the given exercise, \( BaCrO_4 \) is the insoluble product of the reaction between \( Ba^{2+} \) and \( CrO_4^{2-} \).

The balanced chemical equation for this precipitation is:\[ Ba^{2+} + CrO_4^{2-} \rightarrow BaCrO_4(s) \]As \( Ba^{2+} \) ions from \( BaCl_2 \) interact with \( CrO_4^{2-} \) ions from \( K_2CrO_4 \), they form a solid precipitate, which will settle out of the solution.

Precipitation reactions are vital because they are used in:

• Purifying compounds
• Removing ions from solutions
• Identifying compounds through visual evidence

When analyzing solutions, noting the formation of precipitates can also provide information about the ions present in the solution.

Balanced Chemical Equation

Understanding a balanced chemical equation is fundamental to comprehending any chemical reaction. It indicates the proportion of reactants and products involved in the reaction, expressed through their molecular formulas.

For the reaction in our exercise, the equation is:\[ Ba^{2+} + CrO_4^{2-} \rightarrow BaCrO_4(s) \]This equation tells us several crucial details:

• The stoichiometry is 1:1 for \( Ba^{2+} \) and \( CrO_4^{2-} \)
• It illustrates that one mole of barium ion reacts with one mole of chromate ion to produce one mole of barium chromate

A balanced equation ensures that the law of conservation of mass is upheld during the reaction, meaning the number of atoms of each element is the same on both sides.

In practice, creating and balancing chemical equations is crucial for analyzing quantitative aspects of reactions, such as predicting the amounts of products formed or reactants needed, as illustrated in this exercise.