Problem 129 - General Chemistry

Question

The enthalpy of combustion, \(\Delta H\), for benzoic acid, \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\), is \(-3226 \mathrm{~kJ} / \mathrm{mol}\). When a sample of benzoic acid was burned in a calorimeter (at constant pressure), the temperature of the calorimeter and contents rose from \(23.44^{\circ} \mathrm{C}\) to \(27.65^{\circ} \mathrm{C}\). The heat capacity of the calorimeter and contents was \(12.41 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}\). What mass of benzoic acid was burned?

Step-by-Step Solution

Answer

The mass of benzoic acid burned is 1.98 g.

Step 1: Determine the Temperature Change

Calculate the change in temperature of the calorimeter using the initial and final temperatures.\[\Delta T = T_{\text{final}} - T_{\text{initial}} = 27.65^\circ C - 23.44^\circ C = 4.21^\circ C\]

Step 2: Calculate the Heat Absorbed by the Calorimeter

Use the heat capacity of the calorimeter to calculate the total heat absorbed.\[q = C \times \Delta T = 12.41 \, \text{kJ/}^\circ C \times 4.21^\circ C = 52.24 \, \text{kJ}\]

Step 3: Relate Heat Absorbed to Moles of Benzoic Acid Combusted

Since the calorimeter process occurs at constant pressure, the heat absorbed is equal to the enthalpy change for the combustion of benzoic acid.\[\Delta H = -q = -52.24 \, \text{kJ}\]

Step 4: Calculate Moles of Benzoic Acid Burned

Use the known enthalpy of combustion to find the number of moles of benzoic acid burned.\[\text{Moles of benzoic acid} = \frac{q}{\Delta H_{\text{combustion}}} = \frac{52.24 \, \text{kJ}}{3226 \, \text{kJ/mol}} = 0.0162 \, \text{mol}\]

Step 5: Convert Moles to Mass

Calculate the mass of benzoic acid burned using the molar mass.- The molecular formula is \( \text{C}_6 \text{H}_5 \text{COOH} \), and the molar mass is computed as follows: - \( 6(12.01) + 5(1.01) + 12.01 + 16.00 \times 2 + 1.01 = 122.12 \, g/mol \)\[\text{Mass of benzoic acid} = 0.0162 \, \text{mol} \times 122.12 \, \text{g/mol} = 1.98 \, \text{g}\]

Key Concepts

Calorimetry

Calorimetry is the science of measuring heat transfer during chemical reactions or physical changes. It is essential in studying thermodynamic processes in a controlled environment. In a calorimetric experiment, such as the one involving benzoic acid, an insulated device called a calorimeter helps measure the heat absorbed or released by a substance.
At constant pressure, like in this exercise, the calorimeter captures the enthalpy change (\(\Delta H\)) when benzoic acid combusts. When the sample burned, it released heat, causing the temperature of the calorimeter and its contents to rise.
A calorimeter measures heat using its known heat capacity, which is the amount of heat needed to raise its temperature by one degree Celsius.

• Heat capacity (\(C\)) of the calorimeter was given as 12.41 kJ/\(^\circ\)C.
• Temperature change (\(\Delta T\)) was calculated to be 4.21\(^\circ\)C, which helps in determining the heat (\(q\)) absorbed.
• Using \(q = C \times \Delta T\), we find that the calorimeter absorbed 52.24 kJ of heat.

This value is crucial because it helps establish the connection between the experiment's data and the theoretical enthalpy of combustion.

Benzoic Acid

Benzoic acid (\(\text{C}_6\text{H}_5\text{COOH}\)) is an aromatic carboxylic acid. It is used as a standard substance in calorimetry due to its stable and well-known enthalpy of combustion. Having a relatively simple structure, benzoic acid is a perfect candidate for calculating precise combustion energy.
When benzoic acid combusts, it undergoes a chemical reaction that releases energy. This reaction can be described as:\[\text{C}_6\text{H}_5\text{COOH} + \text{excess } \text{O}_2 \rightarrow \text{CO}_2 + \text{H}_2\text{O} + \text{energy (heat)}\]The process releases 3226 kJ/mol of benzoic acid, as shown in this exercise.

• Enthalpy (\(\Delta H\)) tells us how much energy is released per mole.
• In the lab, this enthalpy change is tracked via the temperature increase in the calorimeter setup.
• Using this known enthalpy of combustion allows researchers to quantify other energetic aspects of reactions efficiently.

Understanding this foundational reaction is critical for further studying chemical thermodynamics and energy transformations.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance expressing the mass relationship between individual atoms in molecules. It is crucial for converting between grams and moles, enabling quantitative analysis of reactions and substances.
In the case of benzoic acid (\(\text{C}_6\text{H}_5\text{COOH}\)), each element's atomic mass contributes to the total molar mass:

• Carbon (\(C\)) has an atomic mass of 12.01 g/mol, and there are 7 carbon atoms, contributing 84.07 g/mol.
• Hydrogen (\(H\)) has an atomic mass of 1.01 g/mol, and there are 6 hydrogen atoms, adding another 6.06 g/mol.
• Oxygen (\(O\)) has an atomic mass of 16.00 g/mol, with 2 oxygen atoms present, summing up to 32.00 g/mol.

Adding these up, benzoic acid has a molar mass of 122.12 g/mol.
In the exercise solution, this molar mass allows for the conversion from moles of benzoic acid to grams, allowing us to solve for the mass of benzoic acid burned. Understanding molar mass helps in accurately describing the amount of substances participated in reactions and their stoichiometry.