Question

Combustion of table sugar produces \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(l) .\) When \(1.46 \mathrm{g}\) table sugar is combusted in a constant-volume (bomb) calorimeter, \(24.00 \mathrm{kJ}\) of heat is liberated. a. Assuming that table sugar is pure sucrose, \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}(s)\) write the balanced equation for the combustion reaction. b. Calculate \(\Delta E\) in \(\mathrm{kJ} / \mathrm{mol} \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\) for the combustion reaction of sucrose.

Short answer

The balanced equation for the combustion of sucrose (table sugar) is \(C_{12}H_{22}O_{11}(s) + 12O_2(g) → 12CO_2(g) + 11H_2O(l)\). The change in internal energy per mole of sucrose for the combustion reaction is approximately -5635 kJ/mol.

Step-by-step solution

Write the balanced equation for the combustion of sucrose

To do this, we write the equation for the combustion of sucrose. During combustion, sucrose reacts with oxygen to produce carbon dioxide and water. The equation is: \(C_{12}H_{22}O_{11}(s) + O_2(g) → CO_2(g) + H_2O(l)\) Now, we have to balance the equation. There are 12 carbon atoms, 22 hydrogen atoms, and 11 oxygen atoms in a sucrose molecule. Therefore, the balanced equation is: \(C_{12}H_{22}O_{11}(s) + 12O_2(g) → 12CO_2(g) + 11H_2O(l)\)

Calculate the number of moles of sucrose combusted

To do this, we will use the given mass of sucrose and its molar mass. The molar mass of sucrose is: M(C) = 12.01 g/mol, M(H) = 1.01 g/mol, M(O) = 16.00 g/mol M(C_12H_22O_11) = 12(12.01) + 22(1.01) + 11(16.00) = 342.30 g/mol The number of moles of sucrose combusted is: n = mass / molar mass = 1.46 g / 342.30 g/mol ≈ 0.00426 mol

Calculate the change in internal energy per mole of sucrose combusted

We are given that 24.00 kJ of heat is liberated in the reaction. Since the reaction takes place in a constant-volume calorimeter, we know that the change in internal energy ΔE is equal to the heat liberated q: ΔE = q First, we calculate the internal energy change for the given mass of sucrose: ΔE = -24.00 kJ The negative sign indicates that the reaction is exothermic. Now, we need to find the change in internal energy per mole of sucrose. To do this, we divide the ΔE value we found by the number of moles of sucrose combusted: ΔE/mol = ΔE / n ≈ (-24.00 kJ) / 0.00426 mol ≈ -5635 kJ/mol The change in internal energy per mole of sucrose for the combustion reaction is approximately -5635 kJ/mol.

Key concepts

Enthalpy of Combustion

The enthalpy of combustion is a significant concept in chemical reactions, especially when we're looking at energy release during the process of burning a substance like sucrose, which is common table sugar. Enthalpy, represented by the symbol \(H\), is a measurement of the total heat content of a system. When we discuss the enthalpy of combustion, we are referring to the change in enthalpy (\(\Delta H\)) that occurs when one mole of a substance combusts completely with oxygen under standard conditions.

In such exothermic reactions (reactions that release energy), the enthalpy change is typically negative, indicating that energy is being released into the surroundings. Since we can't measure enthalpy directly, we often calculate it indirectly by using a calorimeter, such as in the given exercise where sucrose is combusted in a bomb calorimeter. The observed heat release helps to calculate the \(\Delta H\) for the reaction. For students, it's paramount to remember that the enthalpy of combustion is an extensive property; it depends on the amount of the substance combusted.

Stoichiometry

Stoichiometry is the aspect of chemistry that involves calculating the quantities of reactants and products in a chemical reaction. It is based on the conservation of mass, where the total mass of reactants equals the total mass of products. A balanced chemical equation is essential for these calculations because it tells us the exact ratio in which molecules react and the ratio in which they are produced.

In the combustion of sucrose, the balanced equation, \(C_{12}H_{22}O_{11}(s) + 12O_2(g) \rightarrow 12CO_2(g) + 11H_2O(l)\), shows the stoichiometry of the reaction. From this, we can deduce that one mole of sucrose reacts with twelve moles of oxygen to produce twelve moles of carbon dioxide and eleven moles of water. Knowing the molar mass of sucrose allows us to calculate how many moles were combusted, which is crucial for determining the enthalpy of combustion per mole.

Chemical Thermodynamics

Chemical thermodynamics deals with the study of energy and work of chemical processes. It's about understanding how energy is transformed during chemical reactions and how these changes affect the properties of substances. An important term here is internal energy, denoted by \(E\), which refers to the total of all kinetic and potential energies of all the particles in a system.

When a reaction occurs at a constant volume, such as in a bomb calorimeter, the change in internal energy (\(\Delta E\)) can be measured directly as heat (\(q\)). This measurement is extremely relevant as it is related to the enthalpy change of the reaction. The exercise involves calculating \(\Delta E\) per mole after a constant-volume combustion, which highlights the practical application of chemical thermodynamics principles. Always keep in mind, \(\Delta E\) is not always equal to \(\Delta H\); the relationship between them usually also considers work done by the system and can vary depending on constant pressure or constant volume conditions.