Problem 69 - Chemistry The Molecular Science

Question

Benzoic acid, \(\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{2},\) occurs naturally in many berries. Suppose you burn \(1.500 \mathrm{~g}\) of the compound in a combustion calorimeter and find that the temperature of the calorimeter increases from \(22.50^{\circ} \mathrm{C}\) to \(31.69^{\circ} \mathrm{C}\). The calorimeter contains \(775 \mathrm{~g}\) water, and the bomb has a heat capacity of \(893 \mathrm{~J}^{\circ} \mathrm{C}^{-1}\). Calculate \(\Delta_{\mathrm{r}} E\) per mole of benzoic acid.

Step-by-Step Solution

Answer

\(\Delta_r E \approx -3224 \text{ J/mol}\), considering significant figures and units.

Step 1: Calculate Heat Absorbed by Water

The heat absorbed by the water can be calculated using the formula \[ q = m \cdot c \cdot \Delta T \]where \( m \) is the mass of the water (775 g), \( c \) is the specific heat capacity of water (4.18 J/g°C), and \( \Delta T \) is the change in temperature. Calculate\[q = 775 \cdot 4.18 \cdot (31.69 - 22.50) \].

Step 2: Calculate Heat Absorbed by the Bomb

For the bomb, the heat absorbed can be calculated using:\[ q = C \cdot \Delta T \]where \( C \) is the heat capacity of the bomb (893 J/°C) and \( \Delta T \) is the change in temperature (31.69°C - 22.50°C). Calculate\[ q = 893 \cdot (31.69 - 22.50) \].

Step 3: Calculate Total Heat Released by Combustion

The total heat released by the combustion reaction is the sum of the heat absorbed by both the water and the bomb:\[ q_{total} = q_{water} + q_{bomb} \].

Step 4: Calculate Moles of Benzoic Acid

Use the molar mass of benzoic acid (\( C_7H_6O_2 \)) to find the moles of benzoic acid burnt. The molar mass is calculated as follows: \[ (7 \times 12.01) + (6 \times 1.01) + (2 \times 16.00) \].For 1.500 g of benzoic acid, calculate the moles using:\[ \text{moles} = \frac{1.500}{\text{molar mass}} \].

Step 5: Calculate \(\Delta_r E\) per Mole of Benzoic Acid

With the total heat released (in terms of energy \( q_{total} \)) and the moles of benzoic acid known from previous steps, calculate \( \Delta_r E \) using:\[ \Delta_r E = \frac{q_{total}}{\text{moles}} \].Make sure the result is in Joules per mole.

Key Concepts

Benzoic Acid

Benzoic acid is an important organic compound with the chemical formula \( \mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{2} \). It is commonly found in various berries and is known for its role in everyday applications, particularly as a precursor for the synthesis of numerous other chemicals.
Benzoic acid features a benzene ring attached to a carboxyl group \( (-\mathrm{COOH}) \), making it a carboxylic acid. This type of compound is characterized by its acidic properties, which allow it to participate in various reactions. One of these is combustion, where it acts as a fuel source.
In the context of calorimetry, burning benzoic acid provides a measurable release of energy, crucial for understanding enthalpic changes during chemical reactions. This energy release can be precisely quantified using a combustion calorimeter in experiments like the one described, allowing for deeper insights into the thermodynamic behavior of this compound.

Combustion Reaction

A combustion reaction is a chemical process where a substance combines with oxygen, releasing energy in the form of heat and light. In the case of benzoic acid, the combustion reaction can be represented as:
\[ \mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{2} + 15/2 \mathrm{O}_2 \rightarrow 7 \mathrm{CO}_2 + 3 \mathrm{H}_2\mathrm{O} \]
This reaction is exothermic, meaning it releases energy. The energy released during this reaction is captured by the calorimeter setup and can be measured to determine the energy change, \( \Delta_r E \), of the reaction.
Understanding combustion reactions is key in calorimetry because it involves the complete oxidation of a compound. In the described experiment, the heat produced by burning 1.500 g of benzoic acid is absorbed by the surrounding water and the calorimeter's bomb, allowing the calculation of the reaction's energy dynamics.

Heat Capacity

Heat capacity is a property that describes the amount of heat required to change a substance's temperature by a certain amount. It is a crucial parameter in calorimetry because it helps quantify the heat transfer during a reaction.
In a calorimeter, both the water and the bomb have specific heat capacities. For water, it is given as \( 4.18 \mathrm{~J/g}^\circ \mathrm{C} \), and for the calorimeter bomb, it is reported as \( 893 \mathrm{~J/^\circ C} \). These values help determine how much energy is absorbed by each during the combustion process.
By calculating the total heat absorbed by the water and the bomb, the total energy released by the combustion reaction can be inferred. In this experiment, conducting these calculations allows us to assess the energy output of the benzoic acid combustion and understand its thermodynamic profile.

Molar Mass Calculation

Calculating molar mass involves summing the atomic masses of all atoms in a molecule. For benzoic acid, \( \mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{2} \), the molar mass is determined by:
- Carbon (C): \( 7 \times 12.01 \mathrm{~g/mol} \)
- Hydrogen (H): \( 6 \times 1.01 \mathrm{~g/mol} \)
- Oxygen (O): \( 2 \times 16.00 \mathrm{~g/mol} \)
Adding these values together gives the molar mass of benzoic acid, which is essential for converting mass to moles in calorimetry experiments.
Knowing the molar mass allows us to determine how many moles of benzoic acid were combusted, which is a necessary step in finding the reaction energy per mole (\( \Delta_r E \)). This process is integral to understanding the energetic implications of the compound's combustion, as it bridges the measured heat with the chemical amount involved in the reaction.