Question

An alternative approach to bomb calorimetry is to establish the heat capacity of the calorimeter, exclusive of the water it contains. The heat absorbed by the water and by the rest of the calorimeter must be calculated separately and then added together. A bomb calorimeter assembly containing $983.5\mathrm{g}$ water is calibrated by the combustion of $1.354\mathrm{g}$ anthracene. The temperature of the calorimeter rises from 24.87 to ${35.63}^{\circ }\mathrm{C}.$ When $1.053\mathrm{g}$ citric acid is burned in the same assembly, but with 968.6 g water, the temperature increases from 25.01 to ${27.19}^{\circ }\mathrm{C}$. The heat of combustion of anthracene, ${\mathrm{C}}_{14}{\mathrm{H}}_{10}(\mathrm{s}),$ is $-7067\mathrm{kJ}/\mathrm{mol}$ ${\mathrm{C}}_{14}{\mathrm{H}}_{10}\cdot$ What is the heat of combustion of citric acid, ${\mathrm{C}}_6{\mathrm{H}}_8{\mathrm{O}}_7,$ expressed in $\mathrm{kJ}/\mathrm{mol}?$

Short answer

The heat of combustion of citric acid is $-2335.70kJ/mol$.

Step-by-step solution

Calculate the heat capacity of the calorimeter

For anthracene combustion, the total heat calculated as how much kcal/mole (or kJ/mole) it emits, multiplied by how many moles of anthracene is combusted, obtaining $-7067\mathrm{kJ}/\mathrm{mol}\times \frac{1.354g}{178.23g/mol}=-53.73kJ$, where $178.23g/mol$ is the molecular weight of anthracene. The heat capacity ($C$) of the calorimeter is calculated as follows: $C=-\frac{Q}{\mathrm{\Delta }T}$ where $Q$ is the total heat produced and $\mathrm{\Delta }T$ is the change in temperature. Plugging in the values: $C=-\frac{-53.73kJ}{{35.63}^{\circ }\mathrm{C}-{24.87}^{\circ }\mathrm{C}}=5.66kJ{/}^{\circ }\mathrm{C}$.

Calculate the heat produced from the combustion of citric acid

Using the heat capacity of the calorimeter, calculate the heat produced from the combustion of citric acid $\mathrm{\Delta }H=C\mathrm{\Delta }T=(5.66kJ{/}^{\circ }\mathrm{C})({27.19}^{\circ }\mathrm{C}-{25.01}^{\circ }\mathrm{C})=-12.35kJ$. $\mathrm{\Delta }H$ is the heat absorbed by the calorimeter and the water it contains.

Calculate the heat of combustion of citric acid (Q)

Now, we know the total heat produced and the mass of citric acid that was burned. To find the heat of combustion of citric acid per mole, we first calculate the amount of citric acid in moles, which is given by $\frac{1.053g}{M}$, where M is the molar mass of citric acid $=210.14g/mol$. Finally, to obtain the heat of combustion, the heat produced is divided by the mole of citric acid burned. $Q=-\frac{-12.35kJ}{\frac{1.053g}{210.14g/mol}}=-2335.70kJ/mol$.

Key concepts

Heat Capacity

Heat capacity is a fundamental concept in the study of thermodynamics, especially in the context of bomb calorimetry. It represents the amount of heat needed to raise the temperature of an object or substance by one degree Celsius. In bomb calorimetry, understanding heat capacity is crucial because it allows us to determine how much heat the calorimeter itself absorbs during a reaction.

To calculate the heat capacity of a calorimeter, we use the formula:

• $C=-\frac{Q}{\mathrm{\Delta }T}$

Here, $Q$ is the total heat produced by the combustion reaction, and $\mathrm{\Delta }T$ is the change in temperature observed. This equation helps isolate the calorimeter's contribution to the heat absorbed, excluding the effect of water or other components.

In a practical sense, this is done by using a known reaction, like the combustion of anthracene, to calibrate the calorimeter. As detailed in the solution, by knowing the specific change in temperature and the amount of heat released by a known mass of anthracene, the heat capacity of the calorimeter can be understood and later used to determine the heat changes in other reactions.

Combustion Reaction

A combustion reaction is a chemical process where a substance (typically a hydrocarbon) reacts with oxygen to produce heat and light. This is exothermic, meaning it releases energy. These reactions are essential for bomb calorimetry because they provide a controlled way to measure a substance's heat of combustion.

The heat released in a combustion reaction can be used to calculate the energy content of a substance. For example, when anthracene burns, it releases a significant amount of energy, measured in kilojoules per mole (kJ/mol). This known value helps calibrate the calorimeter so that unknown samples like citric acid can be analyzed under the same conditions.

During these reactions in a calorimeter setup, the goal is to capture all the heat released to accurately determine a substance's energy content. By comparing the initial and final temperatures of the calorimeter, one can calculate how much heat the reaction released. This forms the basis for the further calculation of unknown substances like citric acid in the exercise.

Calorimeter Assembly

A calorimeter assembly is a setup designed to measure the heat of chemical reactions or physical changes, as well as heat capacity. A bomb calorimeter is a specific type used for measuring high-temperature combustion reactions, making it ideal for analyzing energy content in substances like fuels.

A bomb calorimeter assembly typically includes:

• A strong, sealed container (called a bomb) where the reaction takes place.
• A surrounding water bath to absorb the heat released by the reaction.
• Thermometers or sensors to measure the temperature change.

The assembly ensures that the reaction happens in an isolated environment where the energy release can be measured accurately. Any heat absorbed by the calorimeter is calculated to extract the data from the reaction alone.

In this exercise, the bomb calorimeter was used to measure the heat released when burning a known sample, anthracene, to find the calorimeter's heat capacity. The same setup was then used to determine the heat of combustion for citric acid, ensuring consistent results and accurate measurements.