Question

An excess of zinc metal is added to \(50.0 \mathrm{~mL}\) of a \(0.100 \mathrm{M} \mathrm{AgNO}_{3}\) solution in a constant-pressure calorimeter like the one pictured in Figure 5.8 . As a result of the reaction \(\mathrm{Zn}(s)+2 \mathrm{Ag}^{+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+2 \mathrm{Ag}(s)\) the temperature rises from \(19.25^{\circ} \mathrm{C}\) to \(22.17^{\circ} \mathrm{C}\). If the heat capacity of the calorimeter is \(98.6 \mathrm{~J} /{ }^{\circ} \mathrm{C},\) calculate the enthalpy change for the given reaction on a molar basis. Assume that the density and specific heat of the solution are the same as those for water, and ignore the specific heats of the metals.

Short answer

The molar enthalpy change is approximately \(-359.28 \text{ kJ/mol}\).

Step-by-step solution

Calculate the Total Heat Absorbed

First, calculate the heat absorbed by the whole system (solution and calorimeter). The temperature change is given as \(\Delta T = 22.17^{\circ} \mathrm{C} - 19.25^{\circ} \mathrm{C} = 2.92^{\circ} \mathrm{C}\). Using the formula \( q = C_{\text{cal}} \times \Delta T \), where \( C_{\text{cal}} = 98.6 \text{ J/}^{\circ} \mathrm{C} \) (calorimeter's heat capacity), we find \( q_{\text{cal}} = 98.6 \times 2.92 = 288.712 \text{ J} \).

Calculate Heat Released by Reaction

Since the density and specific heat are the same as water, we calculate the heat change for the solution. Use \( q = m \cdot c \cdot \Delta T \), where \( m = 50.0 \text{ mL} = 50.0 \text{ g} \), \( c = 4.18 \text{ J/g}^{\circ}\mathrm{C} \) (specific heat capacity of water), and \( \Delta T = 2.92^{\circ} \mathrm{C} \). So, \( q_{\text{solution}} = 50.0 \times 4.18 \times 2.92 = 609.48 \text{ J} \).

Determine Total Heat

Add the heat absorbed by the calorimeter to the heat change of the solution to find the total heat released by the reaction: \( q_{\text{total}} = q_{\text{solution}} + q_{\text{cal}} = 609.48 + 288.712 = 898.192 \text{ J} \).

Find Moles of Silver Nitrate

Determine the amount in moles of \( \mathrm{AgNO}_{3} \) using its concentration. Since \( \text{Volume} = 50.0 \text{ mL} = 0.0500 \text{ L} \) and \( \text{Molarity} = 0.100 \text{ M} \), the moles of \( \mathrm{AgNO}_{3} \) are \( 0.0500 \times 0.100 = 0.00500 \text{ mol} \).

Calculate Enthalpy Change per Mole

Since 2 moles of \( \mathrm{Ag}^{+} \) are involved per mole of reaction according to the stoichiometry, only half the moles are considered for the reaction: \( 0.00250 \text{ mol of } \mathrm{AgNO}_3 \) participate. Calculate \( \Delta H \) using the formula \( \Delta H = \frac{q_{\text{total}}}{\text{moles reacted}} = \frac{898.192}{0.00250} \approx 359276.8 \text{ J/mol} \).

Convert Units to kJ/mol

Convert the enthalpy from Joules to kilojoules: \( 359276.8 \text{ J/mol} \) to \( 359.2768 \text{ kJ/mol} \). Therefore, the molar enthalpy change is \(-359.28 \text{ kJ/mol} \) after considering the reaction releases heat (exothermic).

Key concepts

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. This particular exercise uses a constant-pressure calorimeter to observe the enthalpy change of the reaction between zinc and silver nitrate. In this setup, the calorimeter measures the heat resulting from the chemical reaction as the temperature of the solution rises.
This heat measurement is indirect and relies on the assumption that all the heat evolved from the reaction is absorbed by both the solution and the calorimeter.

• The equation used to calculate the absorbed heat is: \(q = C_{\text{cal}} \times \Delta T\)
• Here, \(C_{\text{cal}}\) is the heat capacity of the calorimeter and \(\Delta T\) represents the change in temperature of the system.

For accurate results, it's important to ensure that the calorimeter is well-insulated to avoid heat loss to the surroundings. The heat change measured by calorimetry is usually then related to enthalpy change, which can provide valuable insights into the energy changes during reactions.

Molarity

Molarity is one of the primary ways to express concentration. It signifies the number of moles of solute present in one liter of solution. In this exercise, the molarity of the \(\mathrm{AgNO}_3\) solution is given as \(0.100 \mathrm{M}\).
To find the number of moles of \(\mathrm{AgNO}_3\), you use the formula:

• \(\text{moles} = \text{molarity} \times \text{volume (L)}\)

The volume must be converted from milliliters to liters by dividing by 1000. So, for \(50.0 \text{ mL}\), it becomes \(0.0500 \text{ L}\). Calculating gives:

• \(0.100 \text{ M} \times 0.0500 \text{ L} = 0.00500 \text{ mol}\)

Molarity allows chemists to accurately explore reactions by considering amounts of reactants and predicting theoretical yields.

Reaction Stoichiometry

Reaction stoichiometry involves the quantitative relationships between the reactants and products in a chemical reaction. It relies heavily on balanced chemical equations to determine these relationships. In this reaction:
\(\mathrm{Zn}(s) + 2 \mathrm{Ag}^+(aq) \rightarrow \mathrm{Zn}^{2+}(aq) + 2 \mathrm{Ag}(s)\),
stoichiometry plays a critical role in finding how much of each reactant is used and how much product is formed.

• From the equation, one mole of zinc reacts with two moles of silver ions.
• This implies for every 0.00500 moles of \(\mathrm{AgNO}_3\), which provides two moles \(\mathrm{Ag}^+\), half of that, or 0.00250 moles, participate in the reaction with zinc.

Understanding stoichiometry is crucial because it allows the calculation of moles of products formed and can be used to calculate theoretical yields from reactions.

Specific Heat Capacity

Specific heat capacity is a property that tells us how much heat energy is needed to change the temperature of a given substance. It is typically expressed as Joules per gram per degree Celsius (\(\text{J/g}^{\circ} \text{C}\)). In this exercise, water's specific heat capacity is utilized since it is often used as an approximation for solutions.
The specific heat of water is approximately \(4.18 \text{ J/g}^{\circ} \text{C}\). This value means that \(4.18 \text{ Joules}\) of energy are required to raise the temperature of \(1\text{ gram}\) of water by \(1^{\circ} \text{C}\).

• In calculating the heat absorbed by the solution, this specific heat value is multiplied by the mass of the solution and the temperature change \(\Delta T\).
• The formula used is: \(q = m \cdot c \cdot \Delta T\)
• Where \(m\) is the mass of the solution and \(c\) is the specific heat capacity of the solvent, in this case, water.

Knowing the specific heat allows scientists to determine how much heat energy is involved in heating or cooling a substance.