Question

The bomb calorimeter in Exercise 108 is filled with \(987 \mathrm{~g}\) water. The initial temperature of the calorimeter contents is \(23.32^{\circ} \mathrm{C}\). A \(1.056-\mathrm{g}\) sample of benzoic acid \(\left(\Delta E_{\text {comb }}=-26.42 \mathrm{~kJ} / \mathrm{g}\right)\) is combusted in the calorimeter. What is the final temperature of the calorimeter contents?

Short answer

The final temperature of the calorimeter contents is approximately \(16.91^{\circ}C\).

Step-by-step solution

Calculate the total heat released by the combustion of benzoic acid

To calculate the total heat released, we will multiply the mass of the benzoic acid sample and the heat of combustion per gram: \(q = m_{BA} \Delta E_{comb}\) where \(m_{BA}\) is the mass of benzoic acid sample and \(\Delta E_{comb}\) is the heat of combustion per gram. \(q = (1.056 g)\left(-26.42 \frac{kJ}{g}\right)\) First, convert kJ to J: \(q = (1.056 g)\left(-26.42 \frac{1000 J}{g}\right)\) Now, calculate q: \(q = -27,932.32 J\) The negative sign indicates that the heat is released during the combustion process.

Calculate the final temperature of the calorimeter contents using the heat transfer formula

Now, we will use the heat transfer formula to find the final temperature of the calorimeter contents. The heat released by the combustion of benzoic acid will be absorbed by the water in the calorimeter: \(q = m_{H2O}c\Delta T\) where \(m_{H2O}\) is the mass of water in the calorimeter and c is specific heat capacity of water. Substitute the values: \(-27,932.32 J = (987g)\left(4.184 \frac{J}{g^{\circ}C}\right)(T_f - 23.32^{\circ}C)\) Now, divide by the product of the mass of water and the specific heat capacity of water: \(\frac{-27,932.32 J}{(987g)(4.184 \frac{J}{g^{\circ}C})} = T_f - 23.32^{\circ}C\) Calculate the value on the left side: \(-6.41^{\circ}C = T_f - 23.32^{\circ}C\) Now, add the initial temperature to both sides: \(-6.41^{\circ}C + 23.32^{\circ}C = T_f\) Finally, calculate the final temperature: \(T_f = 16.91^{\circ}C\) The final temperature of the calorimeter contents is approximately \(16.91^{\circ}C\).

Key concepts

Heat of Combustion

The heat of combustion refers to the energy released as heat when a substance undergoes complete combustion with oxygen. In our case, benzoic acid is being combusted. This is a crucial concept in bomb calorimetry, as it's the primary data used to determine thermal changes in the system.
The heat of combustion is typically expressed in terms of energy per gram or mole. For benzoic acid, it's given as \(\Delta E_{comb} = -26.42 \text{kJ/g}\). The negative sign indicates that energy is released into the surroundings — a common characteristic of exothermic reactions.
To calculate the total heat produced from burning a specific amount of a substance, we multiply its mass by the heat of combustion:

• \(q = m \Delta E_{comb}\)

This gives us the total energy released as heat in the reaction.

Specific Heat Capacity

Specific heat capacity is a measure of the amount of heat energy required to raise the temperature of a given quantity of a substance by one degree Celsius. In most calorimetry problems, including this one, specific heat capacity is crucial because it helps us relate energy change to temperature change.
For water, which is often used in calorimeters due to its high specific heat capacity, this value is \(4.184 \frac{\text{J}}{\text{g} ^{\circ} C}\). This means water can absorb a lot of heat with only a small change in temperature, making it a very efficient medium for measuring energy changes during reactions.
In calorimetry calculations, specific heat is denoted as \(c\), and it appears in the equation that determines temperature changes:

• \(q = mc\Delta T\)

This indicates that the heat absorbed or released by the substance is proportional to its mass, specific heat, and temperature change.

Temperature Change

When we talk about temperature change in a calorimeter, we're usually interested in understanding how much the temperature of water changes due to the heat released from a reaction. This is calculated using the formula:

• \(\Delta T = \frac{q}{mc}\)

where \(q\) is the heat energy transferred, \(m\) is the mass of the substance experiencing the temperature change (like water in the calorimeter), and \(c\) is the specific heat capacity.
In our problem, the temperature change is calculated as the heat absorbed by the water over the product of the water's mass and specific heat capacity.
Once you find \(\Delta T\), you can add this change to the initial temperature of the system to find the final temperature, which reflects how much the energy from combustion affected the temperature.

Energy Transfer

Energy transfer is a fundamental concept in bomb calorimetry, as it explains the passage of energy from the chemical reaction to the calorimeter. This transfer causes the temperature changes we measure.
In the combustion of benzoic acid within the calorimeter, energy is transferred from the breaking and forming of bonds in the reactants to the surrounding water. This is indicative of the first law of thermodynamics, stating that energy cannot be created or destroyed, only transferred.
The reaction releases energy, measured as \(q\), which is then absorbed by the water:

• \(q_{\text{reaction}} = -q_{\text{water}}\)

This equation highlights the energy conservation, as the energy lost by the reaction equals the energy gained by the water. By observing how much the water's temperature changes, we determine the extent of energy transfer.