Question

The concentration of \(\mathrm{Cu}^{2+}\) ions in the water (which also contains sulfate ions) discharged from a certain industrial plant is determined by adding excess sodium sulfide \(\left(\mathrm{Na}_{2} \mathrm{~S}\right)\) solution to \(0.800 \mathrm{~L}\) of the water. The molecular equation is $$ \mathrm{Na}_{2} \mathrm{~S}(a q)+\mathrm{CuSO}_{4}(a q) \longrightarrow \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)+\operatorname{CuS}(s) $$ Write the net ionic equation and calculate the molar concentration of \(\mathrm{Cu}^{2+}\) in the water sample if \(0.0177 \mathrm{~g}\) of solid CuS is formed.

Short answer

The molar concentration of \( \mathrm{Cu}^{2+} \) is \( 2.31 \times 10^{-4} \text{ M} \).

Step-by-step solution

Identify the Net Ionic Equation

To find the net ionic equation, identify the participating ions in the reaction. The overall chemical equation is \( \mathrm{Na}_{2} \mathrm{S}(aq) + \mathrm{CuSO}_{4}(aq) \rightarrow \mathrm{Na}_{2} \mathrm{SO}_{4}(aq) + \mathrm{CuS}(s) \). Since \( \mathrm{Na}_{2} \mathrm{SO}_{4} \) is soluble and dissociates into ions, the net ionic equation focuses only on the formation of the solid: \( \mathrm{Cu}^{2+}(aq) + \mathrm{S}^{2-}(aq) \rightarrow \mathrm{CuS}(s) \).

Determine the Molar Mass of CuS

Calculate the molar mass of \( \mathrm{CuS} \). The atomic mass of \( \mathrm{Cu} \) is approximately 63.55 g/mol and that of \( \mathrm{S} \) is 32.07 g/mol. Therefore, the molar mass of \( \mathrm{CuS} \) is \( 63.55 + 32.07 = 95.62 \) g/mol.

Calculate Moles of CuS Formed

Use the mass and molar mass of \( \mathrm{CuS} \) to find the moles. Given that 0.0177 g of \( \mathrm{CuS} \) is formed, the moles of \( \mathrm{CuS} \) is \( \frac{0.0177 \text{ g}}{95.62 \text{ g/mol}} = 1.85 \times 10^{-4} \text{ mol} \).

Find Concentration of Cu^{2+} Ions

The net ionic equation shows \( 1:1 \) molar ratio between \( \mathrm{Cu}^{2+} \) and \( \mathrm{CuS} \). Therefore, the moles of \( \mathrm{Cu}^{2+} \) is the same as \( \mathrm{CuS} \), which is \( 1.85 \times 10^{-4} \text{ mol} \). To find the molar concentration, divide the moles of \( \mathrm{Cu}^{2+} \) by the volume of water in liters: \( \frac{1.85 \times 10^{-4} \text{ mol}}{0.800 \text{ L}} = 2.31 \times 10^{-4} \text{ M} \).

Key concepts

Molar Concentration

Molar concentration is a measure used to express how many moles of a solute are present in one liter of solution. It is a fundamental concept in chemistry that helps in determining the concentration of ions or molecules, facilitating chemical analysis and reactions. To calculate molar concentration, you divide the number of moles of solute by the volume of the solution in liters.
For example, if we found that 0.0177 g of solid CuS was formed during a reaction and we calculated 1.85 x 10^-4 moles of CuS, we can determine the concentration of Cu^{2+} ions in the 0.800 L of solution.
By using the formula:

• Molar Concentration = \( \frac{\text{moles of solute}}{\text{volume of solution in liters}} \)

We get \( \frac{1.85 \times 10^{-4} \text{ mol}}{0.800 \text{ L}} = 2.31 \times 10^{-4} \text{ M} \).

Chemistry Problem Solving

Chemistry problem solving often involves breaking down complex chemical reactions into simpler steps to analyze and understand the changes occurring at the molecular level. This requires understanding stoichiometry, the balancing of chemical equations, and the recognition of ion dissociations in aqueous solutions.
In our exercise, we tackled the problem by writing the net ionic equation and calculating the molar concentration of specific ions. Problem-solving started with identifying reactants and products, then focusing on ions that participate directly in forming a precipitate. This emphasized observing molecular interactions.
The method involves:

• Writing balanced molecular and net ionic equations.
• Calculating molar mass and number of moles formed from the reactants.
• Using stoichiometry to relate moles of reactants and products.

Feasibility of Chemical Reactions

The feasibility of chemical reactions depends on whether and how efficiently reactants convert into products. This is influenced by factors like solubility, reaction kinetics, and thermodynamic stability.
In the given example, the reaction involves the precipitation of CuS from aqueous copper sulfate and sodium sulfide. Understanding solubility rules is crucial, as the reaction depends on forming an insoluble solid (CuS) that can precipitate out of the solution. The net ionic equation \( \mathrm{Cu}^{2+}(aq) + \mathrm{S}^{2-}(aq) \rightarrow \mathrm{CuS}(s) \) elucidates this, focusing on the formation of the solid product.
To judge feasibility, observe reactants' conditions and products' formation:

• Address solubility and concentration of ions involved.
• Assess if reaction conditions favor product formation.
• Ensure stoichiometric ratios support complete reaction.

Concentration Calculation

Calculating concentration is integral to assessing reaction progress and final product yield. It requires precise measurements of reactant quantities and volumes.
For the calculation of Cu^{2+} ion concentration in our example, knowing the mass of CuS formed was essential. By converting this mass into moles using the molar mass of CuS and then relating these moles back to the initial aqueous solution, accurate concentration derivation was possible.
Steps include:

• Determining molar mass: CuS has a molar mass = 95.62 g/mol.
• Calculating moles: \( \frac{0.0177 \text{ g}}{95.62 \text{ g/mol}} = 1.85 \times 10^{-4} \text{ mol} \).
• Finding concentration using: \( \frac{ \text{moles} }{0.800 \text{ L}} = 2.31 \times 10^{-4} \text{ M} \).

These steps help ensure precise determination of the substance's concentration in a given solution.