Under constant-volume conditions, the heat of combustion of benzoic acid (C₆H₅O₆) is 15.57 kJ/g. A 3.500-g sample of sucrose is burned in a bomb calorimeter. The temperature of the calorimeter increases from 20.94 to 24.72 °C. (b) If the size of the sucrose sample had been exactly twice as large, what would the temperature change of the calorimeter have been?

Verified step by step guidance

1. Identify the relationship between the mass of the sample and the temperature change in a bomb calorimeter. The heat released is directly proportional to the mass of the sample burned.
2. Recognize that the heat released by the combustion of the sucrose sample is absorbed by the calorimeter, causing the temperature to rise. This is described by the equation: \( q = C_{cal} \Delta T \), where \( q \) is the heat absorbed, \( C_{cal} \) is the calorimeter's heat capacity, and \( \Delta T \) is the temperature change.
3. Calculate the heat released by the original 3.500-g sucrose sample using the heat of combustion of benzoic acid as a reference. Since the problem does not provide the heat of combustion for sucrose, assume it is known or provided in a complete problem statement.
4. Determine the heat capacity of the calorimeter using the initial temperature change and the heat released by the original sample: \( C_{cal} = \frac{q}{\Delta T} \).
5. Use the calculated heat capacity and the doubled mass of sucrose to find the new temperature change: \( \Delta T_{new} = \frac{2q}{C_{cal}} \).

Key Concepts

Heat of Combustion

The heat of combustion is the amount of energy released when a substance undergoes complete combustion with oxygen. It is typically expressed in kJ/g or kJ/mol. In this context, the heat of combustion of benzoic acid is given, which serves as a reference for understanding the energy changes involved when burning other substances, such as sucrose.

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. In a bomb calorimeter, the heat released from a reaction is absorbed by the surrounding water, leading to a temperature change that can be measured. This temperature change is directly related to the amount of substance burned and the heat of combustion, allowing for calculations of energy release.

Temperature Change and Mass Relationship

The temperature change in a calorimeter is proportional to the mass of the substance being burned and its heat of combustion. If the mass of the sample is doubled, the energy released will also double, leading to a corresponding increase in temperature change. This relationship is crucial for predicting how changes in sample size affect the calorimeter's temperature response.