Problem 43 - General Chemistry: Principles and Modern Applications

Question

Refer to Example \(7-3 .\) Based on the heat of combustion of sucrose established in the example, what should be the temperature change \((\Delta T)\) produced by the combustion of \(1.227 \mathrm{g} \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\) in a bomb calorimeter assembly with a heat capacity of \(3.87 \mathrm{kJ} /^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The temperature change \(\Delta T\) produced by the combustion of 1.227g sucrose in a bomb calorimeter assembly with a heat capacity of 3.87kJ/degC is approximately 4.26 degrees Celsius.

Step 1: Analyze the Information Given

We know the heat capacity \(C\) of the bomb calorimeter is \(3.87 \, \mathrm{kJ/ ^{\circ}C}\). We also know the mass \(m\) of the sucrose (\(\mathrm{C}_{12}\mathrm{H}_{22}\mathrm{O}_{11}\)) combusted is \(1.227 \, \mathrm{g}\). The molar mass of sucrose is approximately 342.30 g/mol. The heat of combustion is the heat released when one mole of a substance is completely burned in excess oxygen. From Example 7-3 we know that the heat of combustion \(q_p\) for sucrose is -5648 kJ/mol.

Step 2: Calculate the Amount of Sucrose Combusted

We calculate the number of moles \(n\) of sucrose that has been combusted using the formula: \(n = \frac{m_i}{MM} = \frac{1.227\, \mathrm{g}}{342.3\, \mathrm{g/mol}}\), where \(m_i\) is the initial mass of the sucrose and \(MM\) is the molar mass.

Step 3: Calculate the Heat of Combustion for the Given Mass

We calculate the heat \(q_p\) resulting from the combustion of the given amount of sucrose using the formula \(-q_p = n \cdot \Delta H = 1.227\, \mathrm{g} \cdot \frac{-5648\, \mathrm{kJ/mol}}{342.3\, \mathrm{g/mol}}\). This is the heat that is transferred to the calorimeter.

Step 4: Calculate Change in Temperature

Finally, calculate the temperature change \(ΔT\) produced by the combustion. This is gotten by dividing the calculated heat of combustion by the heat capacity of the calorimeter assembly. Use the formula \(\Delta T = \frac{|q_p|}{C} = \frac{|-16.50\, \mathrm{kJ}|}{3.87\, \mathrm{kJ/ ^{\circ}C}}\). This will give you the temperature change of the bomb calorimeter.

Key Concepts

Heat of Combustion

The heat of combustion is an important concept in chemistry. It is the total amount of energy released as heat when a substance undergoes complete combustion with oxygen. For substances like sucrose, the heat of combustion can be expressed in terms of energy per mole, such as kilojoules per mole (kJ/mol).
The heat of combustion can vary greatly depending on the type of substance being burned:

• *High heat of combustion:* These substances release a lot of energy, often used as efficient fuels.
• *Low heat of combustion:* These substances release less energy, not ideal for fuel purposes.

In the given problem, the heat of combustion for sucrose is provided as \(-5648 \, \mathrm{kJ/mol}\). This indicates the energy change involved in burning one mole of sucrose. The negative sign indicates an exothermic reaction, meaning heat is released.

Bomb Calorimeter

The bomb calorimeter is a crucial device used in calorimetry to measure the heat of combustion. It allows scientists to analyze how much heat is released during chemical reactions.
Here's how a bomb calorimeter works:

• *Sealed chamber:* The substance to be combusted is placed in a sealed bomb, which is submerged in water or another fluid with a known heat capacity.
• *Controlled ignition:* The sample is ignited in a controlled manner, allowing it to burn completely.
• *Heat measurement:* The heat released by the combustion is absorbed by the surrounding fluid, causing a rise in temperature. This rise in temperature is measured to calculate the heat of the reaction.

The heat capacity of the bomb calorimeter, given as \(3.87 \, \mathrm{kJ/ ^{\circ}C}\) in this problem, is important. It tells us how much heat is needed to raise the temperature of the system by one degree Celsius. This value is critical in accurately measuring the heat exchange during the combustion process.

Temperature Change Calculation

Temperature change is vital to understanding energetic processes in calorimetry. Once you measure the heat exchanged, you can calculate how it affects temperature. In chemistry problems like this one, you often have to find the temperature change caused by a chemical reaction inside a calorimeter.
To calculate this change, follow these steps:

• *Determine total heat released:* First, calculate the heat produced by the combustion reaction (\(-q_p\)). Use the number of moles of the substance and multiply by the heat of combustion.
• *Use the calorimeter's heat capacity:* Once you have the total heat, divide it by the calorimeter’s heat capacity. This will show the change in temperature.
• *Formula usage:* The change in temperature can be calculated as \(\Delta T = \frac{|q_p|}{C}\), where \(\Delta T\) is the temperature change, \(q_p\) is the absolute heat change, and \(C\) is the heat capacity of the calorimeter.

Understanding this calculation helps you see how much a substance’s combustion can impact the system’s temperature, allowing for deeper insights into reaction energetics.