Problem 59 - Chemistry An Atoms First Approach

Question

In a coffee-cup calorimeter, \(50.0 \mathrm{mL}\) of \(0.100 \mathrm{M} \mathrm{AgNO}_{3}\) and \(50.0 \mathrm{mL}\) of \(0.100 \mathrm{M}\) HCl are mixed to yield the following reaction: $$\mathrm{Ag}^{+}(a q)+\mathrm{Cl}^{-}(a q) \longrightarrow \mathrm{AgCl}(s)$$ The two solutions were initially at \(22.60^{\circ} \mathrm{C},\) and the final temperature is \(23.40^{\circ} \mathrm{C}\). Calculate the heat that accompanies this reaction in \(\mathrm{kJ} / \mathrm{mol}\) of \(\mathrm{AgCl}\) formed. Assume that the combined solution has a mass of \(100.0 \mathrm{g}\) and a specific heat capacity of \(4.18 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\).

Step-by-Step Solution

Answer

The heat that accompanies this reaction is \(66.88 \: \text{kJ/mol}\) of AgCl formed.

Step 1: Calculate the temperature change

Find the difference between the final temperature and the initial temperature. ΔT = T_final - T_initial = 23.4°C - 22.6°C = 0.8°C

Step 2: Calculate the heat gained by the solution

Use the formula Q = mcΔT, where Q is the heat gained by the solution, m is the mass of the combined solution, c is the specific heat capacity, and ΔT is the temperature change we found in step 1. Q = (100.0 g)(4.18 J/°C·g)(0.8°C) Q = 334.4 J

Step 3: Calculate the moles of AgCl formed

From the balanced equation, we know that 1 mole of AgNO3 reacts with 1 mole of HCl to produce 1 mole of AgCl. Both reactants have the same concentration (0.100 M) and volume (50.0 mL), so the limiting reactant will be completely consumed. Since the reaction is 1:1, the number of moles formed will also be 0.100 mol/L * 0.050 L = 0.005 moles of AgCl.

Step 4: Calculate the heat released per mole of AgCl

Since the heat gained by the solution is equal to the heat released during the reaction, divide the heat (in Joules) by the number of moles of AgCl to obtain the heat in J/mol. Heat released = (334.4 J) / (0.005 mol) Heat released = 66880 J/mol

Step 5: Convert the heat to kJ/mol

Finally, convert the heat released from J/mol to kJ/mol by dividing by 1000: Heat released = (66880 J/mol) / 1000 Heat released = 66.88 kJ/mol The heat that accompanies this reaction is 66.88 kJ/mol of AgCl formed.

Key Concepts

Calorimetry

Calorimetry is a technique used to measure the heat of chemical reactions or physical changes as well as heat capacity. The process involves using a calorimeter, which is an insulated container designed to minimize energy exchange with the surroundings. When a reaction occurs inside the calorimeter, such as the dissolution and reaction of AgNO₃ and HCl to form AgCl, the heat change is confined to the container, allowing for accurate measurements.
In our example, the coffee-cup calorimeter holds both solutions at a known initial temperature. When they react, the temperature change is a direct indication of the heat involved in the reaction. The calorimeter's simplicity makes it ideal for such experiments, though more sophisticated versions can offer even greater precision.

Specific Heat Capacity

Specific heat capacity (c) is a property that indicates how much energy is required to raise the temperature of 1 gram of a substance by 1 degree Celsius. The value of specific heat capacity varies among substances; for water, it is approximately 4.18 J/g°C, which is what we assume for our mixed solution in the calorimetry exercise. This concept is crucial because it affects how substances respond to heat.
In practice, knowing the specific heat capacity allows us to calculate the amount of heat energy (Q) absorbed or released by a substance during a temperature change (ΔT). This calculation is fundamental when determining the heat of reaction in a calorimeter.

Stoichiometry

Stoichiometry refers to the quantitative relationship between reactants and products in a chemical reaction. It is derived from the balanced chemical equation, and it allows us to predict the amounts of substances consumed and produced during a reaction. In this exercise, the stoichiometry of the reaction between AgNO₃ and HCl is 1:1, meaning one mole of silver nitrate reacts with one mole of hydrochloric acid to produce one mole of silver chloride (AgCl).
By applying stoichiometry, we identified that equal volumes and concentrations of AgNO₃ and HCl will produce an equal amount of AgCl. It enabled us to calculate the moles of AgCl formed, which is necessary for determining the heat per mole released in the reaction.

Heat Transfer

Heat transfer is the process of thermal energy moving from a warmer object to a cooler one. In calorimetry, when a reaction takes place, the transfer of heat occurs between the reaction and the solution in which it is contained. This transfer continues until thermal equilibrium is reached—that is, until both the reaction and the solution are at the same temperature.

Understanding Heat Transfer in Calorimetry

When AgNO₃ reacts with HCl to form AgCl, the temperature change we observe is the result of heat transfer between the reaction and the calorimeter's solution. By knowing the specific heat capacity, mass, and temperature change of the solution, we can calculate the total amount of heat transferred using the formula Q = mcΔT. This heat corresponds to the energy released or absorbed by the reaction.