Question

Insoluble \(\mathrm{AgCl}(\mathrm{s})\) precipitates when solutions of \(\mathrm{AgNO}_{3}(\mathrm{aq})\) and \(\mathrm{NaCl}(\mathrm{aq})\) are mixed. \(\mathrm{AgNO}_{3}(\mathrm{aq})+\mathrm{NaCl}(\mathrm{aq}) \rightarrow \mathrm{AgCl}(\mathrm{s})+\mathrm{NaNO}_{3}(\mathrm{aq})\) $$ \Delta_{\mathrm{r}} H^{\circ}=? $$ To measure the energy evolved in this reaction, \(250 . \mathrm{mL}\) of \(0.16 \mathrm{M} \mathrm{AgNO}_{3}(\mathrm{aq})\) and \(125 \mathrm{mL}\) of \(0.32 \mathrm{M} \mathrm{NaCl}(\mathrm{aq})\) are mixed in a coffee-cup calorimeter. The temperature of the mixture rises from \(21.15^{\circ} \mathrm{C}\) to \(22.90^{\circ} \mathrm{C} .\) Calculate the enthalpy change for the precipitation of AgCl(s), in kJ/mol. (Assume the density of the solution is \(1.0 \mathrm{g} / \mathrm{mL}\) and its specific heat capacity is \(4.2 \mathrm{J} / \mathrm{g} \cdot \mathrm{K}\) )

Short answer

\(-68.91\; \mathrm{kJ/mol}\)

Step-by-step solution

Calculate the Total Volume of the Mixture

Firstly, calculate the total volume by adding the volumes of the two solutions mixed. Since \(250 \mathrm{mL}\) of \(\mathrm{AgNO}_3\) is mixed with \(125 \mathrm{mL}\) of \(\mathrm{NaCl}\), the total volume is \(250 + 125 = 375 \mathrm{mL}\). This is equivalent to \(375 \mathrm{g}\) since the density is \(1.0 \mathrm{g/mL}\).

Calculate the Temperature Change

Determine the temperature change \(\Delta T\) of the mixture. It is calculated as the final temperature minus the initial temperature: \(22.90^{\circ}\mathrm{C} - 21.15^{\circ}\mathrm{C} = 1.75^{\circ}\mathrm{C}\).

Calculate the Heat Absorbed by the Solution

Using the specific heat capacity formula, \(q = mc\Delta T\), calculate the heat absorbed by the solution. Here, \(m = 375 \mathrm{g}\), \(c = 4.2 \mathrm{J/g\cdot K}\), and \(\Delta T = 1.75^{\circ}\mathrm{C}\). Thus, \(q = 375 \cdot 4.2 \cdot 1.75 = 2756.25 \mathrm{J}\).

Convert Heat to Kilojoules

Convert the heat from joules to kilojoules by dividing by 1000. Therefore, \(2756.25 \mathrm{J}\) is equal to \(2.75625 \mathrm{kJ}\).

Determine the Limiting Reactant

Calculate moles of each reactant to find the limiting reactant. Moles of \(\mathrm{AgNO}_3 = 0.16 \mathrm{M} \times 0.250 \mathrm{L} = 0.040 \mathrm{mol}\). Moles of \(\mathrm{NaCl} = 0.32 \mathrm{M} \times 0.125 \mathrm{L} = 0.040 \mathrm{mol}\). Both have equal moles, hence there is no limiting reactant, the moles of \(\mathrm{AgCl}\) formed is \(0.040 \mathrm{mol}\).

Calculate Enthalpy Change per Mole

Calculate the enthalpy change \(\Delta H^{\circ}\) in kJ/mol by dividing the total heat change by the moles of \(\mathrm{AgCl}\) formed. \(\Delta H^{\circ} = \frac{-2.75625\mathrm{kJ}}{0.040\mathrm{mol}} = -68.91 \mathrm{kJ/mol}\). The negative sign indicates that the reaction is exothermic.

Key concepts

Precipitation Reaction

A precipitation reaction occurs when two soluble ionic compounds in aqueous solution are mixed and an insoluble solid, called a precipitate, forms. The equation for the reaction is:

• \(\text{AgNO}_3(\text{aq}) + \text{NaCl}(\text{aq}) \rightarrow \text{AgCl}(\text{s}) + \text{NaNO}_3(\text{aq})\)

The solid \(\text{AgCl}\) is the precipitate because it does not dissolve in water.
Precipitation reactions are a type of double displacement reaction, where the cations and anions swap partners. As the mixed solutions combine, the ions reorganize into a solid. This reaction is used to remove unwanted ions in a solution or to obtain a desired product, like \(\text{AgCl}\) in this scenario.
Identifying whether a reaction will result in a precipitate mainly relies on understanding the solubility rules. These rules indicate which combinations of ions will yield an insoluble product. In this example, \(\text{AgCl}\) is known to be insoluble when formed in solution, thus precipitating out as a solid.

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. It allows us to determine the enthalpy change of a reaction, which is the heat absorbed or released. In the context of our scenario, a coffee-cup calorimeter is used to assess the energy change when \(\text{AgCl}\) precipitates from the reaction.
Use the calorimetry equation:

• \(q = mc\Delta T\) where
  • \(q\) is the heat absorbed or released,
  • \(m\) is the mass of the solution,
  • \(c\) is the specific heat capacity of the solution, and
  • \(\Delta T\) is the change in temperature.

Calorimetry captures the temperature rise as the reaction occurs in the calorimeter, providing data on how much heat is involved in the process. In the exercise, the temperature of the solution rises by \(1.75^{\circ}\)C, indicating that heat is being released from the reaction.
By using calorimetry, we convert this temperature change into a measurable heat exchange that allows us to calculate the enthalpy change for the reaction. This makes calorimetry a crucial method in physical chemistry for analyzing reaction energetics.

Exothermic Reaction

Exothermic reactions are chemical reactions that release energy in the form of heat. In an exothermic reaction, the products are more stable and have lower energy than the reactants.

• \(\Delta H\) (enthalpy change) is negative, indicating energy is released.

In the given reaction, when solutions of \(\text{AgNO}_3\) and \(\text{NaCl}\) are mixed, \(\text{AgCl}\) precipitates, and the temperature of the surrounding solution increases from \(21.15^{\circ}\)C to \(22.90^{\circ}\)C. This rise in temperature is a clear indicator of an exothermic reaction.
Exothermic reactions can often be seen in everyday activities, such as burning wood or the reaction in hand warmers. These reactions release energy, making them useful for generating warmth or power. It's the energy released during the formation of bonds in the product that contributes to the overall heat release. Understanding this concept helps in comprehensively utilizing reactions like precipitation in practical applications.