Stearic acid \(\left[\mathrm{CH}_{3}\left(\mathrm{CH}_{2}\right)_{16} \mathrm{CO}_{2} \mathrm{H}\right]\) is a fatty acid, the part of fat that stores most of the energy. \(1.0 \mathrm{~g}\) of stearic acid was burned in a bomb calorimeter. The bomb had a heat capacity of 652 \(\mathrm{~J} /{ }^{\circ} \mathrm{C} .\) If the temperature of \(500 \mathrm{~g}\) water \(\left(c=4.18 \mathrm{~J} / \mathrm{g}^{\circ} \mathrm{C}\right)\) rose from \(25.0\) to \(39.3^{\circ} \mathrm{C}\), how much heat was released when the stearic acid was burned? \(\left[\right.\) Given \(\left.C_{p}\left(\mathrm{H}_{2} \mathrm{O}\right)=4.18 \mathrm{~J} / \mathrm{g}^{\circ} \mathrm{C}\right]\) (a) \(39.21 \mathrm{~kJ}\) (b) \(29.91 \mathrm{~kJ}\) (c) \(108 \mathrm{~kJ}\) \(\begin{array}{llll}\text { (d) } 9.32 \mathrm{~kJ} & \therefore \therefore\end{array}\)

Calorimetry is an essential laboratory technique used to measure the amount of heat energy released or absorbed in chemical reactions and physical changes. Central to this technique is the calorimeter, a device designed to minimize energy exchange with the environment so accurate measurements can be made. In our exercise, a bomb calorimeter—a type of constant-volume calorimeter—is used to determine the enthalpy of combustion of stearic acid.

The calorimeter consists of an insulated container fitted with a thermometer or temperature sensor. The substance being tested is placed in the calorimeter and subjected to a reaction, such as combustion. The heat produced by this reaction causes a temperature change in the surrounding water, and, using the specific heat capacity of water, we can calculate the amount of heat released by the substance.

The enthalpy of combustion is a crucial value in thermochemistry that measures the heat released when one mole of a substance burns completely with oxygen under standard conditions. It is a form of enthalpy change, denoted \(\Delta H_{\text{comb}}\), and is typically expressed in joules per mole (J/mol).

In the context of the given exercise, the enthalpy of combustion of stearic acid can be determined using the bomb calorimeter. This is calculated by burning a known mass of stearic acid and measuring the temperature change in the system. The resulting value represents the energy density or the energy per gram, of the compound, which is an essential parameter for understanding fuel efficiency and nutritional content of foods.

Specific heat capacity, often simply called specific heat, is the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius (or one Kelvin). It is an intensive property of matter and is unique for different substances. The symbol for specific heat is \(c\), and it is measured in joules per gram per degree Celsius (J/g°C).

The formula \(Q = mc\Delta T\) highlights how we use the specific heat in calorimetric calculations to find the heat \(Q\) involved in a temperature change \(\Delta T\) of a mass \(m\) of substance. In our problem, we apply the specific heat of water, which is high compared to many other substances, to calculate the amount of heat absorbed when stearic acid is burned.

Thermochemistry is the study of the heat energy associated with chemical reactions and physical transformations. A key concept of thermochemistry is the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred or transformed. This foundational principle is leveraged in calorimetry, where the heat released by a substance during a reaction is absorbed by its surroundings, and we assume no heat is lost to the environment.

Using thermochemical principles, we can understand the exercise at hand. When the stearic acid burns in the bomb calorimeter, the heat released by this exothermic reaction is absorbed by the water and the bomb itself. By knowing the specific heat capacities and the resultant temperature change, we can calculate the total energy released during combustion. This total energy is reflective of the chemical potential energy stored within the stearic acid, which is transformed into thermal energy.