Question

A quantity of \(85.0 \mathrm{~mL}\) of \(0.600 \mathrm{M} \mathrm{HCl}\) is mixed with \(85.0 \mathrm{~mL}\) of \(0.600 \mathrm{M} \mathrm{KOH}\) in a constant- pressure calorimeter. The initial temperature of both solutions is the same at \(17.35^{\circ} \mathrm{C}\), and the final temperature of the mixed solution is \(19.02^{\circ} \mathrm{C}\). What is the heat capacity of the calorimeter? Assume that the specific heat of the solutions is the same as that of water and the molar heat of neutralization is \(-56.2 \mathrm{~kJ} / \mathrm{mol}\).

Short answer

The heat capacity of the calorimeter is approximately \(1.007 \text{ kJ/°C}.\)

Step-by-step solution

Calculate the Heat Absorbed by the Solution

First, find the total volume of the solution after mixing 85.0 mL of HCl and 85.0 mL of KOH. The total volume is 170.0 mL. Assume the density of the solution is the same as water (1 g/mL), so the mass of the solution is 170.0 g. The specific heat capacity of water is 4.18 J/g°C. Use the formula \( q = mc\Delta T \), where \( q \) is the heat absorbed, \( m \) is mass, \( c \) is specific heat, and \( \Delta T \) is the change in temperature. Calculate: \[ \Delta T = 19.02^{\circ} \mathrm{C} - 17.35^{\circ} \mathrm{C} = 1.67^{\circ} \mathrm{C} \]\[ q = 170.0 \mathrm{~g} \times 4.18 \mathrm{~J/g^{\circ}C} \times 1.67^{\circ} \mathrm{C} = 1185.19 \mathrm{~J} \]

Convert Heat from J to kJ

Convert the heat absorbed by the solution from Joules to kilojoules:\[ 1185.19 \text{ J} \times \frac{1 \text{ kJ}}{1000 \text{ J}} = 1.185 \text{ kJ} \]

Calculate Moles of Water Produced

Find the moles of HCl or KOH used, as both have the same concentration and volume. Using the formula \( ext{moles} = ext{volume (L)} \times ext{concentration (M)} \):\[ ext{Moles of water} = 0.085 ext{ L} \times 0.600 ext{ M} = 0.051 ext{ mol} \]

Calculate Heat Released by Neutralization

The heat released by the neutralization reaction is calculated using the given molar heat of neutralization:\[ q_{\text{reaction}} = 0.051 \text{ mol} \times (-56.2 \text{ kJ/mol}) = -2.8662 \text{ kJ} \]

Determine Heat Capacity of the Calorimeter

The heat absorbed by the calorimeter, \( q_{\text{cal}} \), can be calculated as:\[ q_{\text{cal}} = -q_{\text{reaction}} - q_{\text{solution}} \]Insert the values:\[ q_{\text{cal}} = -(-2.8662 \text{ kJ}) - 1.185 \text{ kJ} = 1.6812 \text{ kJ} \]

Calculate the Heat Capacity of the Calorimeter

Since the heat change of the calorimeter is related to its heat capacity, and the temperature change experienced by the calorimeter is the same as the solution's, use:\[ q_{\text{cal}} = C_{\text{cal}} \times \Delta T \]Solve for \( C_{\text{cal}} \):\[ C_{\text{cal}} = \frac{q_{\text{cal}}}{\Delta T} = \frac{1.6812 \text{ kJ}}{1.67^{\circ} \mathrm{C}} = 1.007 \text{ kJ/}\degree \text{C} \]

Key concepts

Heat Capacity

Heat capacity is a fundamental property of materials that defines the amount of heat required to change the temperature of an entire object by one degree Celsius. It is an extensive property, meaning it depends on the sample size or amount of material. In the context of calorimetry, the heat capacity of a calorimeter is crucial as it determines how effectively the calorimeter can absorb and measure heat changes during a reaction. By using the formula \[ q = C \times \Delta T \]where \( q \) is the heat change and \( \Delta T \) is the temperature change, you can calculate the calorimeter's heat capacity \( C \). This allows us to understand how much heat is absorbed or lost by the calorimeter during a chemical reaction.
The heat capacity can be expressed in different units, typically Joules per degree Celsius (J/°C) or kilojoules per degree Celsius (kJ/°C), and it is critical for accurately interpreting calorimetry experiments.

Neutralization Reaction

A neutralization reaction involves the reaction between an acid and a base to produce water and a salt. In the exercise, the neutralization occurs between hydrochloric acid (HCl) and potassium hydroxide (KOH).
The balanced chemical equation is:\[\text{HCl}_{(aq)} + \text{KOH}_{(aq)} \rightarrow \text{KCl}_{(aq)} + \text{H}_2\text{O}_{(l)}\]During a neutralization reaction, an exothermic reaction typically occurs, releasing energy in the form of heat. This release is due to the formation of strong water bonds when hydrogen ions \((\text{H}^+)\) from the acid combine with hydroxide ions \((\text{OH}^-)\) from the base.

The heat released in neutralization reactions is characterized by the molar heat of neutralization, often measured in kJ/mol. In the given problem, it is \(-56.2 \text{ kJ/mol}\), indicating that the process is exothermic and releases 56.2 kJ of energy per mole of water formed.

Specific Heat Capacity

Specific heat capacity is an intrinsic property of a substance that indicates the amount of heat required to raise the temperature of one gram of the substance by one degree Celsius. It is a vital concept in calorimetry, as it helps calculate how much heat is absorbed or released by a substance. For the solution in this problem, it's assumed to have the same specific heat capacity as water, which is typically \(4.18 \text{ J/g°C}\).
This assumption simplifies calculations by allowing us to apply the specific heat formula \[q = mc\Delta T\]where \( m \) is mass, \( c \) is specific heat capacity, and \( \Delta T \) is the change in temperature. Specific heat capacity is a critical factor in identifying how materials respond to heat changes, and it varies for different substances. Understanding it enables accurate prediction of temperature changes in reaction mixtures.

Enthalpy Change

The enthalpy change is a measure of the total heat content change in a system during a chemical reaction or physical process. It represents the heat exchanged under constant pressure and is often denoted by \( \Delta H \). In many reactions, especially neutralization reactions, enthalpy change helps determine whether the reaction is endothermic or exothermic.

• Endothermic reactions absorb heat (\( \Delta H > 0 \)).
• Exothermic reactions release heat (\( \Delta H < 0 \)).

In our specific exercise, the enthalpy change for the neutralization reaction is represented as the molar heat of neutralization. The given value of \(-56.2\text{ kJ/mol}\) indicates it is exothermic, meaning heat is released as the reaction proceeds.
Calculating enthalpy changes is essential in calorimetry as it allows chemists to understand energy dynamics in reactions. It provides insights into energy efficiency and the feasibility of industrial chemical processes.