Problem 59 - Chemistry The Central Science In Si Units

Question

Under constant-volume conditions, the heat of combustion of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) is \(16.49 \mathrm{~kJ} / \mathrm{g}\). A \(3.00-\mathrm{g}\) sample of su- crose is burned in a bomb calorimeter. The temperature of the calorimeter increases from 21.94 to \(24.62^{\circ} \mathrm{C}\). (a) What is the total heat capacity of the calorimeter? (b) If the size of the sucrose sample had been exactly twice as large, what would the temperature change of the calorimeter have been?

Step-by-Step Solution

Answer

(a) The total heat capacity of the calorimeter is 18.46 kJ/°C. (b) The temperature change would be 5.36°C with double the sucrose sample.

Step 1: Calculate Total Heat Released by 3.00g of Sucrose

To find the total heat released, we multiply the heat of combustion per gram of sucrose by the mass of the sucrose sample. The heat of combustion is given as 16.49 kJ/g. Thus, for 3.00 g of sucrose:\[ Q = 16.49 \, \text{kJ/g} \times 3.00 \, \text{g} = 49.47 \, \text{kJ} \]

Step 2: Calculate the Temperature Change of Calorimeter

The temperature increases from 21.94°C to 24.62°C. The change in temperature, \( \Delta T \), is:\[ \Delta T = 24.62^{\circ} \text{C} - 21.94^{\circ} \text{C} = 2.68^{\circ} \text{C} \]

Step 3: Determine Total Heat Capacity of the Calorimeter

The total heat capacity, \( C \), of the calorimeter can be determined using the formula:\[ Q = C \times \Delta T \]Thus, rearranging for \( C \), we have:\[ C = \frac{Q}{\Delta T} = \frac{49.47 \, \text{kJ}}{2.68^{\circ} \text{C}} = 18.46 \, \text{kJ/}^{\circ} \text{C} \]

Step 4: Calculate New Heat Released for Double the Sucrose Mass

If the sucrose sample is doubled to 6.00 g, the total heat released becomes:\[ Q_{\text{new}} = 16.49 \, \text{kJ/g} \times 6.00 \, \text{g} = 98.94 \, \text{kJ} \]

Step 5: Determine New Temperature Change of Calorimeter

Using the total heat capacity of the calorimeter found in Step 3, find the new temperature change, \( \Delta T_{\text{new}} \), with twice the sucrose mass:\[ \Delta T_{\text{new}} = \frac{Q_{\text{new}}}{C} = \frac{98.94 \, \text{kJ}}{18.46 \, \text{kJ/}^{\circ} \text{C}} = 5.36^{\circ} \text{C} \]

Key Concepts

Heat of Combustion

The heat of combustion is a critical concept in thermochemistry, representing the energy released when a substance is burned in the presence of oxygen. It is crucial for understanding how much energy fuels can provide. In our exercise, the heat of combustion for sucrose is given as 16.49 kJ/g. This means that burning 1 gram of sucrose releases 16.49 kilojoules of energy.

• This heat is a result of the breaking of chemical bonds in the reactants and the formation of new bonds in the products.
• Typically, the heat of combustion is measured under constant-volume conditions to ensure accuracy, as it is in our scenario.

Knowing the heat of combustion helps predict how much energy a specific amount of a substance will release, which is essential in energy calculations and applications.

Bomb Calorimeter

A bomb calorimeter is a device used to measure the heat of combustion of a sample accurately. It operates at constant volume, making it perfect for determining how much energy substances like sucrose release when burned.

• The calorimeter contains a strong, sealed "bomb" where the combustion reaction occurs. This prevents gases from escaping, ensuring accurate energy measurement.
• Heat from the reaction is transferred to the water and materials surrounding the bomb, causing a measurable temperature change.

The principle is simple: by knowing the heat transferred and the resulting temperature change, you can calculate the total heat capacity of the calorimeter, as was done in our exercise.

Heat Capacity

Heat capacity is a measure of how much heat a system can absorb before its temperature rises. It plays a crucial role in calorimetry as it helps determine how energy input translates to temperature changes.

• In our scenario, the total heat capacity of the calorimeter indicates how much heat is needed to raise its temperature by one degree Celsius.
• The formula used is: \[ Q = C \times \Delta T \] where \( Q \) is the heat absorbed, \( C \) is the heat capacity, and \( \Delta T \) is the change in temperature.

Understanding heat capacity allows you to predict the temperature change for different amounts of heat input, crucial for experimental and practical applications.

Temperature Change

Temperature change is an essential part of the calorimetry process. It indicates the energy transfer within the system after a reaction occurs.

• In our exercise, when 3.00 g of sucrose is burned, the temperature of the calorimeter changes by 2.68°C. This change results from the transferred heat energy of 49.47 kJ.
• For twice the sucrose, the temperature change increases to 5.36°C, showing a proportional relationship between the amount of substance and the energy released.

The ability to measure and predict temperature changes is vital in controlling chemical processes and ensuring safety, efficiency, and optimal energy usage in various applications.