Question

The energy content of food is typically determined using a bomb calorimeter. Consider the combustion of a \(0.30-\mathrm{g}\) sample of butter in a bomb calorimeter having a heat capacity of 2.67 \(\mathrm{kJ}^{\prime} \mathrm{C}\) . If the temperature of the calorimeter increases from \(23.5^{\circ} \mathrm{C}\) to \(27.3^{\circ} \mathrm{C}\) , calculate the energy of combustion per gram of butter.

Short answer

To calculate the energy of combustion per gram of butter, first find the change in temperature (ΔT) by subtracting the initial temperature from the final temperature: \(ΔT = 27.3°C - 23.5°C = 3.8°C\). Next, calculate the heat released (q) during combustion using the formula \(q = C × ΔT\), where C is the heat capacity of the calorimeter (2.67 kJ/°C): \(q = (2.67 kJ/°C) × (3.8°C) = 10.146 kJ\). Finally, find the energy of combustion per gram of butter by dividing the heat released (q) by the mass of the butter sample: \(Energy\; of\; combustion\; per\; gram\; of\; butter = (10.146 kJ) / (0.30 g) = 33.82 kJ/g\). Therefore, the energy of combustion per gram of butter is approximately 33.82 kJ/g.

Step-by-step solution

Calculate the change in temperature

We need to calculate the change in temperature (ΔT). We can find this by subtracting the initial temperature from the final temperature. ΔT = T_final - T_initial ΔT = 27.3°C - 23.5°C ΔT = 3.8°C

Calculate the heat released (q)

Now, we will calculate the heat released during combustion using the formula: q = C × ΔT Where C is the heat capacity of the calorimeter (2.67 kJ/°C) and ΔT is the change in temperature (3.8°C) calculated in step 1. q = (2.67 kJ/°C) × (3.8°C) q = 10.146 kJ

Calculate the energy of combustion per gram of butter

Finally, we will find the energy of combustion per gram of butter by dividing the heat released (q) by the mass of the butter sample. Energy of combustion per gram of butter = q / mass of the butter sample Energy of combustion per gram of butter = (10.146 kJ) / (0.30 g) Energy of combustion per gram of butter = 33.82 kJ/g The energy of combustion per gram of butter is approximately 33.82 kJ/g.

Key concepts

Bomb Calorimeter

A bomb calorimeter is an essential tool in thermochemistry used to measure the heat of combustion of a sample. It consists of a strong container where a reaction occurs, and the container is submerged in water which absorbs the heat produced.
This setup ensures that all the heat transfer happens in a closed system, making it possible to accurately measure energy changes. The term "bomb" relates to the sealed, pressure-resistant environment inside the calorimeter. In practice, when a sample is burned in this device, it allows scientists to measure the exact amount of heat generated during combustion.
This device is crucial in fields such as nutrition and material sciences, as it helps determine the energy content of substances.

Heat Capacity

Heat capacity is another key concept in calorimetry and refers to the amount of heat required to change a substance's temperature by one degree Celsius. In our example, the bomb calorimeter has a specified heat capacity of 2.67 kJ/°C.
This value indicates how much energy is needed to raise the entire calorimeter's temperature by one degree. Heat capacity is different for every material and setup, depending on factors like mass and specific heat (which is the heat capacity per unit mass).

• A higher heat capacity means more energy is required to alter the temperature.
• A lower heat capacity indicates less energy is needed.

Precisely knowing the heat capacity is crucial for accurate energy calculations.

Energy of Combustion

The energy of combustion is the total energy released as heat when a substance combusts with oxygen under standard conditions. It helps us understand how much energy can be extracted from a specific amount of substance.
In the exercise, the energy of combustion of butter is calculated by measuring how much heat was absorbed by the calorimeter. By using the heat capacity and the temperature change, we can determine the total heat released. The formula used is:
\( q = C \times \Delta T \)
Where \( q \) is the heat released, \( C \) is the heat capacity, and \( \Delta T \) is the temperature change. Calculating the energy per gram gives insights into the efficiency and energy content of the substance being analyzed.

Temperature Change

Temperature change plays a vital role in calorimetry as it directly relates to the energy changes during a reaction. The exercise outlines how the initial and final temperatures of the calorimeter are measured to find the temperature change (\( \Delta T \)).
By subtracting the initial temperature from the final temperature, we obtain \( \Delta T \):

• Initial temperature: 23.5°C
• Final temperature: 27.3°C
• \( \Delta T = 27.3°C - 23.5°C = 3.8°C \)

This change is crucial to determine the heat absorbed by the bomb calorimeter. Understanding temperature change helps visualize the extent of energy involved in the combustion process.