Question

In a coffee-cup calorimeter, 50.0 \(\mathrm{mL}\) of 0.100\(M \mathrm{AgNO}_{3}\) and 50.0 \(\mathrm{mL}\) of 0.100 \(\mathrm{M} \mathrm{HCl}\) 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 reacture in kJ/mol of AgCl formed. Assume that the combined solution has a mass of 100.0 \(\mathrm{g}\) and a specific heat capacity of 4.18 \(\mathrm{J} / \rho \mathrm{C} \cdot \mathrm{g} .\)

Short answer

The heat that accompanies this reaction in the formation of AgCl is 66.88 kJ/mol.

Step-by-step solution

Calculate the moles of reactants

First, we need to find the moles of both Ag⁺ and Cl⁻ in the solutions using the volume and concentration. Moles of Ag⁺: Moles = Molarity × Volume Moles of Ag⁺ = 0.100 M × 50.0 mL = 0.100 mol/L × 0.050 L = 0.005 mol Moles of Cl⁻: Moles of Cl⁻ = 0.100 M × 50.0 mL = 0.100 mol/L × 0.050 L = 0.005 mol

Determine the limiting reactant

In this case, there are equal moles of Ag⁺ and Cl⁻, and the stoichiometry of the reaction is 1:1. Therefore, both reactants are limiting, and there will be 0.005 mol of AgCl formed.

Calculate the heat released

The heat (q) released during the reaction can be calculated using the formula: q = mcΔT where m is the mass, c is the specific heat capacity, and ΔT is the change in temperature. In our case, the mass of the combined solution is 100.0 g, the specific heat capacity is 4.18 J/g°C, and the change in temperature is the final temperature minus the initial temperature (ΔT = 23.40°C - 22.60°C). q = (100.0 g)(4.18 J/g°C)(23.40 °C - 22.60 °C) q = (100.0 g)(4.18 J/g°C)(0.80 °C) q = 334.4 J

Calculate the heat released per mole of AgCl formed

Now we need to calculate the heat released in kJ/mol of AgCl. We have found that 0.005 moles of AgCl are formed, and the heat released is 334.4 J. We can convert the heat into kJ and then divide by the moles of AgCl formed. q = 334.4 J × (1 kJ / 1000 J) = 0.3344 kJ Heat released per mole of AgCl formed = Heat released / Moles of AgCl formed Heat released per mole of AgCl formed = 0.3344 kJ / 0.005 mol = 66.88 kJ/mol Solution: The heat that accompanies this reaction in the formation of AgCl is 66.88 kJ/mol.

Key concepts

Enthalpy Change

In a chemical reaction, **enthalpy change** (\( \Delta H \)) represents the heat energy absorbed or released when the reaction occurs at constant pressure. In calorimetry, we often measure the enthalpy change of a reaction by mixing reactants in a calorimeter and observing the temperature change that occurs. This provides a hands-on approach to understanding how much heat is exchanged with the surroundings during a reaction.

In our exercise, the reaction between \( \mathrm{AgNO}_{3} \) and \( \mathrm{HCl} \) in a coffee-cup calorimeter produces solid \( \mathrm{AgCl} \). We observe a change in temperature from 22.60°C to 23.40°C. This temperature rise indicates that the reaction is **exothermic**, releasing heat into the surroundings. By calculating the heat exchanged (using the formula \( q = mc\Delta T \)) and knowing the number of moles of product formed, we can determine the enthalpy change per mole of product. In this case:

• Mass (\( m \)) of solution = 100 g
• Specific heat capacity (\( c \)) = 4.18 J/g°C
• Temperature change (\( \Delta T \)) = 0.80°C
• Total heat (\( q \)) = 334.4 J

This energy is then distributed per mole of \( \mathrm{AgCl} \) formed, leading us to conclude an enthalpy change of **66.88 kJ/mol**.

Limiting Reactant

The concept of a **limiting reactant** is crucial in determining how much product a chemical reaction can produce. The limiting reactant is the substance that is completely consumed first, thus stopping the reaction from proceeding further and determining the maximum amount of product formed.

In the given reaction between \( \mathrm{AgNO}_{3} \) and \( \mathrm{HCl} \), both reactants are mixed in equal molar amounts (0.005 mol each). Because the reaction stoichiometry is 1:1 (one mole of \( \mathrm{Ag}^{+} \) reacts with one mole of \( \mathrm{Cl}^{-} \) to form one mole of \( \mathrm{AgCl} \)), neither reagent is in excess. This means both are consumed completely, and both act as the limiting reactant.

Hence, every mole of \( \mathrm{AgNO}_{3} \) reacts with every mole of \( \mathrm{HCl} \) available, leading to the formation of 0.005 mol of \( \mathrm{AgCl} \), making the calculation of the heat per mole straightforward. Understanding limiting reactants helps in efficiently using reactants in industrial and laboratory settings.

Specific Heat Capacity

**Specific heat capacity** is a property that indicates how much heat energy a substance can absorb or release per unit mass to produce a temperature change of one degree Celsius (or Kelvin). It is denoted by \( c \) and typically measured in units of J/g°C.

For our calorimetry exercise, the given specific heat capacity for the solution is 4.18 J/g°C. This indicates that for every gram of solution, 4.18 Joules are required to raise the temperature by 1°C. Knowing the specific heat capacity is indispensable for calculating the heat change in calorimetry experiments. It allows us to relate the amount of heat exchanged with the environment when a substance undergoes a temperature change.

In the experiment discussed, the specific heat capacity helped determine the total energy change using the formula \( q = mc\Delta T \). With a mass of 100 g and a temperature change of 0.80°C, the heat exchanged was calculated to be 334.4 J. Without precise specific heat capacity values, enthalpy calculations could lead to significant errors. This concept highlights how energy interactions occur in substances and is essential in disciplines like thermodynamics and physical chemistry.