A 0.1375-g sample of solid magnesium is burned in a constant-volume bomb calorimeter that has a heat capacity of \(3024 \mathrm{~J} /{ }^{\circ} \mathrm{C}\). The temperature increases by \(1.126^{\circ} \mathrm{C} .\) Calculate the heat given off by the burning \(\mathrm{Mg},\) in \(\mathrm{kJ} / \mathrm{g}\) and in \(\mathrm{kJ} / \mathrm{mol} .\)

Understanding heat capacity is crucial when analyzing calorimetry data. It's a term that defines how much heat energy is required to raise the temperature of a given substance by one degree Celsius (or one Kelvin, which has the same increment size). More formally, it is expressed in units of joules per degree Celsius (\( J/^\text{o}C \)).

For instance, in the exercise provided, the bomb calorimeter has a heat capacity of 3024 J/°C, meaning it takes 3024 joules of energy to raise the temperature of the entire calorimeter by one degree. This property simplifies our calculations because we regard the calorimeter as an isolated system where no heat is lost to the environment. Instead, all the heat of the reaction is absorbed by the calorimeter, leading to a measurable temperature change—a core aspect of calorimeter calculations.

The term enthalpy of combustion is a concept from thermodynamics referring to the heat released when one mole of a substance is completely burned in excess oxygen under standard conditions. It is a crucial parameter for understanding energy content of fuels and is measured in kilojoules per mole (\( kJ/mol \) ).

In our exercise, the magnesium's enthalpy of combustion is unknown, but we can calculate it indirectly by using the heat absorbed by the calorimeter, which is reflected by its temperature increase. By knowing this, along with the mass and molar mass of magnesium, we can then convert the absorbed heat into the enthalpy of combustion for the sample provided.

The process of calorimeter calculations involves finding the relationship between the substance's temperature change and the amount of heat absorbed or released. The basic approach follows these four steps:

• Firstly, calculate the total heat absorbed by the calorimeter using the heat capacity and the observed temperature change.
• Secondly, convert that heat from joules to kilojoules if necessary, as scientific convention often uses kilojoules for enthalpy values.
• Then, find the heat given off by the reaction per gram of substance by dividing the total heat by the sample's mass—all to tailor the value to a standard measurement unit.
• Finally, to relate this to common chemical quantities, convert this per-gram value to per mole by multiplying by the molar mass of the substance.

In the context of the exercise, these steps yield the energy released by the burning magnesium sample in kilojoules per gram and per mole. Different substances will have different enthalpies of combustion, meaning the calorimetry calculations can help determine the energy efficiency of various fuels or reactants.