When \(1.0 \mathrm{g}\) of fructose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s)\), a sugar commonly found in fruits, is bumed in oxygen in a bomb calorimeter, the temperature of the calorimeter increases by \(1.58^{\circ} \mathrm{C}\). If the heat capacity of the calorimeter and its contents is \(9.90 \mathrm{kJ} /^{\circ} \mathrm{C}\), what is \(q\) for this combustion?

Understanding a bomb calorimeter is key to grasping the way we measure the energy changes in chemical reactions. A bomb calorimeter is a type of constant-volume calorimeter used in measuring the heat of combustion of a particular reaction. When a sample like fructose combusts in the presence of oxygen, it releases energy in the form of heat, which causes a temperature increase within the calorimeter. The peculiar name 'bomb' comes from the fact that the reaction takes place in a sealed, bomb-like container that withstands the high pressure produced during the combustion process.

Within the calorimeter, the energy released by the combustion of the sample is absorbed by the surrounding water and the calorimeter itself. A temperature sensor tracks the change in temperature, which is directly related to the energy released. Since no heat escapes, the bomb calorimeter allows for an accurate measurement of the heat involved in the reaction.

Heat capacity is a vital concept when working with calorimetry. It refers to the amount of heat required to raise the temperature of a substance by a specified amount, often one degree Celsius or Kelvin. This property is intrinsic to the substance; each material will have its own unique heat capacity. In the context of calorimetry, the heat capacity of the calorimeter setup is crucial because it is used to calculate the total amount of heat absorbed or released during the chemical reaction.

For instance, in the given problem, the heat capacity of the calorimeter and its contents is given as 9.90 kJ/°C. It means that for every degree Celsius increase in temperature, the calorimeter absorbs 9.90 kJ of energy. Understanding this allows us to use the formula: \(q = C \times \Delta T\), where \(q\) is the heat released or absorbed, \(C\) represents the heat capacity, and \(\Delta T\) is the change in temperature, to calculate the heat involved in the reaction.

Enthalpy of combustion is a term that often comes up in chemical thermodynamics. It describes the heat released when one mole of a substance burns completely in oxygen. It's an essential quantity as it reflects the energy content of fuels and materials. In the calorimetric context, it is obtained by burning a known quantity of the material and measuring the heat output using a device like a bomb calorimeter.

To relate this to the example of fructose combustion, by measuring the heat released from burning a known mass of fructose, we can calculate the enthalpy of combustion on a per mole basis. In practice, the measured heat (\(q\)) divided by the number of moles of fructose combusted will give us the enthalpy of combustion in units of kJ/mol. These values are of immense importance in not only chemistry but also in industries where energy production and management are critical.