Problem 103 - Chemistry An Atoms First Approach

Question

The bomb calorimeter in Exercise 102 is filled with 987 g water. The initial temperature of the calorimeter contents is \(23.32^{\circ} \mathrm{C} . \mathrm{A}\) \(1.056-\mathrm{g}\) 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?

Step-by-Step Solution

Answer

The final temperature of the calorimeter contents is \(16.53^\circ \mathrm{C}\).

Step 1: Identify the given information

We are given the following information: - Initial temperature of calorimeter contents: \(23.32^\circ \mathrm{C}\) - Mass of water: \(987\,\mathrm{g}\) - Mass of benzoic acid sample: \(1.056 \, \mathrm{g}\) - Combustion energy of benzoic acid: \(-26.42\, \mathrm{kJ/g}\)

Step 2: Calculate the energy released during combustion of benzoic acid

To calculate the energy released during combustion of benzoic acid, multiply the mass of benzoic acid by its combustion energy. \[q_\text{combusted} = m_\mathrm{BA} \times \Delta {E}_\mathrm{comb}\] \[q_\text{combusted} = (1.056\,\mathrm{g})\times(-26.42\,\mathrm{kJ/g})\] \[q_\text{combusted} = -27.92\,\mathrm{kJ}\]

Step 3: Convert the energy released to joules

Since the specific heat of water is given in joules, we convert the energy in kJ to J. \[q_\text{combusted}= -27.92\, \mathrm{ kJ} \times \frac{1000\, \mathrm{J}}{1\, \mathrm{ kJ}}\] \[q_\text{combusted} = -27920\, \mathrm{J}\]

Step 4: Equate the heat lost by benzoic acid to the heat gained by water

The heat lost by the benzoic acid is equal to the heat gained by the water. The specific heat capacity of water is \(4.18\,\mathrm{J/g^{\circ} C}\). We can write the equation as: \[q_\text{gained} = q_\text{combusted}\] \[m_\mathrm{w} \times c_\mathrm{w} \times \Delta T = q_\text{combusted}\]

Step 5: Solve for the temperature change

Rearrange the equation above to find the temperature change \(\Delta T\). \[\Delta T = \frac{q_\text{combusted}}{m_\mathrm{w} \times c_\mathrm{w}}\] \[\Delta T = \frac{-27920\,\mathrm{J}}{(987\,\mathrm{g}) \times (4.18\,\mathrm{J/g^{\circ} C})}\] \[\Delta T = -6.79\, ^\circ \mathrm{C}\]

Step 6: Calculate the final temperature

To find the final temperature of the calorimeter contents, simply add the temperature change to the initial temperature. \[T_\text{final} = T_\text{initial} + \Delta T\] \[T_\text{final} = 23.32^\circ \mathrm{C} - 6.79^\circ \mathrm{C}\] \[T_\text{final} = 16.53^\circ \mathrm{C}\] The final temperature of the calorimeter contents is \(16.53^\circ \mathrm{C}\).

Key Concepts

Heat of Combustion

Understanding the concept of heat of combustion is fundamental when examining thermochemical processes like the combustion of substances in a bomb calorimeter. Heat of combustion, symbolized as \( \Delta H_c \), is eminently the amount of heat released when a substance is burned completely in the presence of oxygen under standard conditions. It is often measured in kilojoules per gram (kJ/g), indicating the energy released by burning one gram of the substance.

The heat of combustion is intrinsic to a substance and can be determined experimentally using caloric measurements, such as in a bomb calorimeter. This property is utilized when we wish to gauge the energy content of fuels or food. In our exercise, the heat of combustion of benzoic acid is given, which allows us to calculate the thermal energy released upon its combustion.

Specific Heat Capacity

The specific heat capacity of a substance is a measure of the amount of heat required to raise the temperature of one gram of the substance by one degree Celsius (or one Kelvin). It's a property that varies from material to material, represented by the symbol \(c\) and typically measured in joules per gram degree Celsius \(\mathrm{J/g^\circ C}\).

In practical terms, substances with high specific heat capacities, like water, can absorb a lot of heat with only a minimal change in temperature. This is important in calorimetry, as the specific heat capacity of the calorimeter's water plays a crucial part in determining the temperature change induced by a reaction. This concept was vital in the exercise to relate the amount of energy released by the benzoic acid to the observable change in temperature of the water in the calorimeter.

Enthalpy Change

Enthalpy change, symbolized by \( \Delta H \), represents the difference in enthalpy, or the heat content of a system at constant pressure, between the products and the reactants of a chemical reaction. For exothermic reactions—like combustion—the enthalpy change is negative because heat is released to the surroundings, manifesting as an increase in the temperature of the surrounding environment or, in our case, the water within the calorimeter.

In thermodynamics, \( \Delta H \), or the change in enthalpy, is equivalently the amount of heat transferred at a constant pressure. Since our benzoic acid was combusted within the calorimeter, a vessel designed to withstand high pressure, the enthalpy change could be directly associated with the heat of combustion provided for benzoic acid (\( \Delta H_c = -26.42 \mathrm{kJ/g}\)).

Thermal Energy Calculations

Thermal energy calculations involve determining the amount of heat transferred in a process, and it is essential for understanding energy flows in physical and chemical contexts. These calculations are often utilized to find out changes in temperature, heat capacities, phase changes, and chemical reactions.

In the given problem, thermal energy calculations enabled us to connect the heat of combustion of benzoic acid with the specific heat capacity of water and the mass of water to determine the final temperature change. We used the formula \[q = mc\Delta T\] where \(q\) is the heat transferred, \(m\) is the mass, \(c\) is specific heat capacity, and \(\Delta T\) is the temperature change. By following a systematic approach in solving the problem, the relation between the thermal energy released from the substance and the resultant temperature change of the water was elucidated. The negative sign in the enthalpy change indicates heat being released to the surroundings, a convention which neatly fits the context of the benzoic acid’s exothermic combustion reaction.