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 sucrose is burned in a bomb calorimeter. The temperature of the calorimeter increases from 21.94 to \(24.62^{\circ} \mathrm{C} .(\mathbf{a})\) 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?

Short answer

(a) The total heat capacity of the calorimeter is 18.44 kJ/°C. (b) If the size of the sucrose sample had been exactly twice as large, the temperature change of the calorimeter would have been 5.36°C.

Step-by-step solution

(a) Calculate heat released by combustion of sucrose

: We start by finding how much heat is released by the combustion of the sucrose sample. We know that the heat of combustion of sucrose is 16.49 kJ/g and the mass of the sucrose is 3.00 g. We can use this information to calculate the total heat released: Total heat released (q) = heat of combustion × mass of sucrose \(q = 16.49 \, \text{kJ/g} \times 3.00 \, \text{g} = 49.47 \, \text{kJ}\)

(a) Calculate the change in temperature

: The temperature change of the calorimeter is given by the difference of the initial and final temperatures: ΔT = T_final - T_initial ΔT = 24.62°C - 21.94°C ΔT = 2.68°C

(a) Calculate the heat capacity of the calorimeter

: To find the heat capacity of the calorimeter, C_cal, we need to use the formula: C_cal = q / ΔT Plugging in the values, we get: C_cal = \(49.47 \, \text{kJ} / 2.68 \, ^{\circ}\text{C} = 18.44 \, \text{kJ/^{\circ}C}\) So, the total heat capacity of the calorimeter is 18.44 kJ/°C.

(b) Calculate the heat released for twice the mass of sucrose

: Now we need to check the temperature change if the mass of the sucrose sample were twice as large. We first find the heat released by combustion of twice the mass of sucrose: Total heat released (q') = heat of combustion × 2 × mass of sucrose \(q' = 16.49 \, \text{kJ/g} \times 2 \times 3.00 \, \text{g} = 98.94 \, \text{kJ}\)

(b) Calculate the new temperature change

: Next, we determine the temperature change associated with double the amount of sucrose, using the heat capacity of the calorimeter that we calculated earlier: ΔT' = q' / C_cal ΔT' = \(98.94 \, \text{kJ} / 18.44 \, \text{kJ/^{\circ}C} = 5.36 \,^{\circ}\text{C}\) So if the amount of sucrose burned was exactly twice as large, the temperature change of the calorimeter would have been 5.36°C.

Key concepts

Heat of Combustion

The heat of combustion is a critical concept in calorimetry that refers to the amount of heat released when a specific amount of a substance is burned in oxygen. For sucrose, or common table sugar, the heat of combustion is measured in kilojoules per gram (kJ/g). This value indicates the energy content of the substance.

• In the exercise, the heat of combustion of sucrose is given as 16.49 kJ/g.
• This value helps in calculating how much heat is produced when a known quantity of sucrose is burned.

Imagine when you burn a piece of wood in a fireplace to warm a room. Similarly, in a laboratory setting, the heat of combustion tells us how efficient a fuel is at releasing energy, which is key to understanding caloric content and its potential uses.

Bomb Calorimeter

A bomb calorimeter is a device used to measure the heat of combustion of a sample. It provides a constant-volume environment for the reaction to occur.

• This setup ensures that the energy change can be accurately measured without losing heat to the surroundings.
• The calorimeter consists of a robust "bomb" where the sample is combusted in the presence of oxygen and surrounded by water.
• The overall apparatus includes a thermometer to track temperature changes and, hence, calculate energy changes.

In our exercise, the bomb calorimeter records the temperature change from burning sucrose. This change helps compute the energy released, allowing the calculation of the calorimeter's heat capacity. Understanding the bomb calorimeter's operation is essential for accurate measurement of combustion processes, making it invaluable in both research and industrial applications.

Heat Capacity

Heat capacity is a vital property in calorimetry, defining how much heat is needed to change the temperature of an object. It is expressed in units like kJ/°C.

• The heat capacity of a calorimeter is crucial because it determines how much the temperature will rise when a specific amount of heat is absorbed.
• In our example, the calorimeter's heat capacity was found using the formula: \[ C_{\text{cal}} = \frac{q}{\Delta T} \]where \(q\) is the heat absorbed, and \(\Delta T\) is the change in temperature.
• Here, the calculated heat capacity was 18.44 kJ/°C.

Heat capacity provides insight into how efficiently a calorimeter absorbs energy, which is necessary for precise measurements. By knowing how much energy is needed to increase the temperature, you can predict the effects of varying sample sizes. This is demonstrated in the exercise by predicting changes in temperature for different amounts of sucrose combusted.