Question

One tablespoon of peanut butter has a mass of \(16 \mathrm{~g}\). It is combusted in a calorimeter whose heat capacity is \(120.0 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}\). The temperature of the calorimeter rises from \(22.2^{\circ} \mathrm{C}\) to \(25.4^{\circ} \mathrm{C}\). Find the food caloric content of peanut butter.

Short answer

The food caloric content of peanut butter is calculated by multiplying the heat capacity by the change in temperature, converting to calories, and then normalizing per gram of peanut butter.

Step-by-step solution

Calculate the Heat Absorbed

Use the heat capacity of the calorimeter and the change in temperature to calculate the total heat absorbed. The formula 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.

Compute the Change in Temperature

Subtract the initial temperature from the final temperature to find the change in temperature: \( \Delta T = T_{final} - T_{initial} \).

Calculate the Heat Absorbed in Kilojoules

Substitute the values into the formula to find the heat absorbed by the peanut butter: \( Q = 120.0 \mathrm{~kJ/\degree C} \times (25.4 \degree C - 22.2 \degree C) \).

Convert Mass to Kilograms

Convert the mass of peanut butter from grams to kilograms to match the units of calorific content, which is typically given in \( \mathrm{cal/g} \) or \( \mathrm{Cal/g} \): \( 16 \mathrm{~g} = 0.016 \mathrm{~kg} \).

Calculate the Food Caloric Content per Gram

Use the formula \( 1 \mathrm{Cal} = 4.184 \mathrm{kJ} \) to convert the heat absorbed into calories. Then, divide this value by the mass of the peanut butter in kilograms to find the caloric content per kilogram, and finally multiply by 1000 to convert to caloric content per gram.

Express the Result in Calories

After finding the caloric content in \( \mathrm{Cal/kg} \), multiply by 1000 to convert to \( \mathrm{Cal/g} \), since there are 1000 grams in a kilogram. This will provide the caloric content of peanut butter in food calories per gram.

Key concepts

Calorimetry

Calorimetry is an experimental technique used to measure the amount of heat energy released or absorbed during a chemical or physical process. In the context of food science, calorimetry helps determine the caloric content of edible items, such as peanut butter, by combusting the food in a controlled environment and measuring the resulting temperature change. This process requires a device known as a calorimeter, which is an insulated container designed to minimize energy exchange with the environment.

When food is burned in a calorimeter, the heat released raises the water temperature inside the device. By measuring this temperature rise and knowing the heat capacity of the calorimeter, we can calculate the total amount of energy stored in the food. Thus, calorimetry is crucial for quantifying the energy that foods can supply to the body, which is essential for understanding dietary needs and managing food intake.

Heat Capacity

Heat capacity is one of the fundamental concepts in calorimetry and thermodynamics, defined as the amount of heat energy required to raise the temperature of a substance by one degree Celsius (or one Kelvin). It is usually denoted with the letter 'C' and can be specific to a substance's mass unit or refer to an entire system, as in the case of a calorimeter.

The heat capacity of a calorimeter is particularly important because it determines how much heat energy the device can absorb without undergoing a significant temperature change. A high heat capacity implies that a greater amount of energy is required to produce a temperature change, making the calorimeter more stable and providing more accurate measurements in calorimetric experiments.

Heat Absorbed

Heat absorbed, often represented by the symbol 'Q', refers to the total amount of thermal energy that a substance gains during a process such as combustion. In a calorimetry experiment, the calculation of heat absorbed is fundamental as it translates directly to the energy content of the food being tested.

To calculate the heat absorbed by the calorimeter—and by proxy, the food—the formula used is: \( Q = C \times \Delta T \), where 'C' is the heat capacity of the calorimeter, and '\( \Delta T \)' represents the change in temperature observed. By substituting the known values into this formula, we can determine the heat released by the food in joules or kilojoules, which then is converted to calories to express the food's energy content.

Temperature Change

A critical component in calculating heat absorbed during calorimetry is the temperature change, represented by '\( \Delta T \)'. This variable is determined simply by subtracting the initial temperature from the final temperature after the combustion of the food in the calorimeter. A larger temperature change indicates that more heat was released from the food substance.

In the context of the given exercise, the peanut butter's combustion within the calorimeter raises the temperature from an initial \( 22.2^\circ \mathrm{C} \) to a final \( 25.4^\circ \mathrm{C} \). This temperature change is directly proportional to the amount of energy contained in the peanut butter. By understanding this direct relationship, we can apply the heat capacity of the calorimeter to find the total caloric content provided by the food sample.