Question

The enthalpy of combustion of benzoic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right)\) is commonly used as the standard for calibrating constant-volume bomb calorimeters; its value has been accurately determined to be \(-3226.7 \mathrm{~kJ} / \mathrm{mol}\). When \(1.9862 \mathrm{~g}\) of benzoic acid are burned in a calorimeter, the temperature rises from \(21.84^{\circ} \mathrm{C}\) to \(25.67^{\circ} \mathrm{C}\). What is the heat capacity of the bomb? (Assume that the quantity of water surrounding the bomb is exactly \(2000 \mathrm{~g} .\) )

Short answer

The heat capacity of the bomb calorimeter is approximately 13688.24 J/°C.

Step-by-step solution

Calculate Moles of Benzoic Acid

Find the molar mass of benzoic acid \( \mathrm{C}_{6}\mathrm{H}_{5}\mathrm{COOH} \): \( 7 \times 12.01 + 6 \times 1.01 + 2 \times 16.00 = 122.12 \, \mathrm{g/mol} \). Calculate the moles of benzoic acid by dividing the mass by its molar mass: \[ \text{moles} = \frac{1.9862}{122.12} \approx 0.01626 \, \text{mol} \].

Determine Heat Released

Use the enthalpy of combustion to find the heat released. The enthalpy of combustion is \(-3226.7 \, \mathrm{kJ/mol}\). Thus, the heat released \( q \) is calculated by: \[ q = 0.01626 \, \text{mol} \times (-3226.7 \, \mathrm{kJ/mol}) = -52.45 \, \mathrm{kJ} \].

Convert Heat Released to Joules

Convert the heat from kJ to J (1 kJ = 1000 J): \[ q = -52.45 \, \mathrm{kJ} \times 1000 = -52450 \, \mathrm{J} \].

Determine Temperature Change

Find the temperature change \( \Delta T \) of the calorimeter: \[ \Delta T = 25.67^{\circ} \mathrm{C} - 21.84^{\circ} \mathrm{C} = 3.83^{\circ} \mathrm{C} \].

Calculate Heat Capacity of the Bomb

Using the formula \( q = C \Delta T \), solve for heat capacity \( C \): \[ C = \frac{|q|}{\Delta T} = \frac{52450}{3.83} \approx 13688.24 \, \mathrm{J/^{\circ}C} \].

Key concepts

Benzoic Acid

Benzoic acid, with the chemical formula \(\mathrm{C}_{6}\mathrm{H}_{5}\mathrm{COOH}\), is an aromatic carboxylic acid that is often used in the calibration of calorimeters.
Calorimeters are devices used to measure the heat of chemical reactions, and benzoic acid provides a reliable standard due to its well-known and consistent enthalpy of combustion value. The enthalpy of combustion refers to the amount of heat released when one mole of a substance is burned in excess oxygen. For benzoic acid, this value is accurately determined to be \(-3226.7 \, \mathrm{kJ/mol}\).
This consistency is what makes benzoic acid a common choice for ensuring calorimeters give predictable and accurate readings.

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes.
The main tool used in calorimetry is the calorimeter, a device designed to measure the amount of heat involved in these processes. In this exercise, a bomb calorimeter is used, which is particularly suited for reactions that occur at constant volume, such as combustion reactions.
The calorimeter must be calibrated to provide meaningful results, and this is where substances like benzoic acid play a crucial role. By using a substance with a known enthalpy of combustion, one can deduce the heat capacity of the calorimeter, which is an essential part of making accurate heat measurements.

• Forms of calorimetry also include differential scanning calorimetry and isothermal titration calorimetry.

Heat Capacity

Heat capacity is a measure of the amount of heat energy required to change the temperature of a substance by a given amount.
In calorimetry, understanding the heat capacity of the calorimeter is essential to determine the heat changes in reactions accurately. The heat capacity of an object depends on its mass and the specific heat, which is a property of the material.
To find the heat capacity of the bomb calorimeter in this exercise, the heat released by the combustion of benzoic acid is divided by the change in temperature observed. This calculation gives the heat capacity, allowing us to determine how much heat is absorbed by the calorimeter during the reaction. The formula to calculate it is \( C = \frac{|q|}{\Delta T} \), where \( q \) is the amount of heat exchanged, and \( \Delta T \) is the change in temperature.

Temperature Change

Temperature change, denoted as \( \Delta T \), is a critical measurement in calorimetry for identifying the heat changes that occur during reactions.
In the context of the provided exercise, the temperature change is the difference between the final and initial temperature readings. This was calculated as \( 3.83^{\circ} \mathrm{C} \), marking how much the temperature increased due to the combustion of benzoic acid.
This change in temperature is directly related to the heat released by the reaction. When combined with the heat capacity of the calorimeter, it allows us to determine the precise amount of heat generated, or absorbed, by the system.

• In scientific studies, accurate temperature measurements are essential, as even small errors can lead to incorrect conclusions about the enthalpy change of a reaction.