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 is C12H22O11(s) + 12O2(g) -> 12CO2(g) + 11H2O(l), and the change in internal energy per mole of sucrose combusted is 5,637.09 kJ/mol.

Step-by-step solution

Write the unbalanced combustion equation for sucrose

The combustion reaction involves the reaction of a substance with oxygen to produce carbon dioxide and water. The unbalanced chemical equation for the combustion of sucrose is: C12H22O11(s) + O2(g) -> CO2(g) + H2O(l)

Balance the combustion equation of sucrose

To balance the equation, we need to ensure that the number of atoms of each element is the same on both sides of the equation. We can do this by adjusting the coefficients in front of the compounds: C12H22O11(s) + 12O2(g) -> 12CO2(g) + 11H2O(l) Now the equation is balanced.

Determine the number of moles of sucrose combusted

We are given that 1.46 g of table sugar (sucrose) has been combusted. To determine the number of moles, we can use the molecular weight of sucrose, which is 342.3 g/mol: Moles of sucrose = (mass of sucrose) / (molecular weight of sucrose) Moles of sucrose = 1.46 g / 342.3 g/mol Moles of sucrose = 0.00426 mol

Calculate the change in internal energy per mole of sucrose combusted

We are given that 24.00 kJ of heat was liberated during the combustion of 1.46 g of sucrose. We also know that the number of moles of sucrose combusted is 0.00426 mol. ΔE = (heat liberated during combustion) / (number of moles of sucrose combusted) ΔE = 24.00 kJ / 0.00426 mol ΔE = 5637.09 kJ/mol Hence, the change in internal energy per mole of sucrose combusted is 5,637.09 kJ/mol.

Key concepts

Enthalpy Change

Enthalpy change is a key concept in thermodynamics when it comes to understanding chemical reactions, like combustion. Enthalpy (\(\Delta H\)) is the heat absorbed or released at constant pressure during a reaction.
For combustion reactions, such as that of sucrose, you're observing an exothermic reaction—where heat is released. The enthalpy change can tell us how much heat energy is produced when a certain amount of a reactant burns completely.
In constant-volume calorimetry, the heat measured corresponds to the change in internal energy (\(\Delta E\)), not exactly the enthalpy change. However, for reactions involving gases, the difference between \(\Delta H\) and \(\Delta E\) is often minimal.
This is because the work done by the system, often negligible at constant volume, involves no significant pressure-volume work (because, at constant volume, pressure changes don't do work).

• Enthalpy change symbolized as \(\Delta H\) represents heat absorbed/released at constant pressure.
• It's crucial for determining the thermal nature of reactions—endothermic or exothermic.

Sucrose Combustion

Sucrose combustion involves the reactivity of sucrose with oxygen resulting in carbon dioxide and water. It's a classic example of a **combustion reaction**. In the combustion of sucrose, the reactants and conditions usually produce large amounts of energy in the form of heat.
Since sucrose is a carbohydrate, its combustion is similar to the burning of other hydrocarbons, which also produce carbon dioxide and water.
This specific combustion equation represents quite a common type of reaction in our everyday environment and is critical in both biological and industrial chemistry.
The combustion of sucrose can be summarized by the reaction:

• The process involves the burning of sucrose in the presence of oxygen.
• The end products are carbon dioxide (\(\mathrm{CO}_2\)) and water (\(\mathrm{H}_2\mathrm{O}\)).

Balanced Chemical Equation

A balanced chemical equation is an essential aspect of chemistry, ensuring that the law of conservation of mass is upheld. This means the number of atoms for each element in the reactants side matches those in the product's side.
For the combustion of sucrose, the equation initially written is unbalanced and needs adjustments. By balancing, we ensure that none of the atoms are lost, just rearranged.
The balanced equation for sucrose combustion is:\[\mathrm{C}_{12}\mathrm{H}_{22}\mathrm{O}_{11}(s) + 12\mathrm{O}_2(g) \rightarrow 12\mathrm{CO}_2(g) + 11\mathrm{H}_2\mathrm{O}(l)\]In this method, we equalized each type of atom (carbon, hydrogen, and oxygen) by adjusting coefficients. This balancing procedure follows strict stoichiometric rules to ensure all atoms from reactants become part of the products.

• Balances both sides by changing coefficients, not subscripts.
• Critical for accurate calculation of reactants and products in reactions.

Internal Energy Calculation

Internal energy measurements are essential for understanding the energy changes in a closed system. \(\Delta E\), the change in internal energy, is a calculated metric in thermochemistry that reflects the heat energy change when no work is done by or on the system.
When sucrose combusts in a bomb calorimeter, we consider it a constant-volume process. Here, the heat released (24.00 \mathrm{kJ}) becomes imperative for calculating \(\Delta E\) because it's equal to the heat measured in a constant-volume process.
The formula used is:\[\Delta E = \frac{\text{heat liberated}}{\text{moles of sucrose}}\]By applying this formula, the internal energy change per mole of sucrose combusted is calculated to be 5,637.09 \mathrm{kJ/mol}. Understanding these metrics helps us define the energy efficiency and output of chemical reactions.

• Internal energy change ties intimately with the heat changes in a reaction.
• In constant-volume calorimetry, \(\Delta E\) directly relates to measurable heat such as liberated or absorbed.