Question

A 0.692 -g sample of glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6},\) is burned in a constant volume calorimeter. The temperature rises from \(21.70^{\circ} \mathrm{C}\) to \(25.22^{\circ} \mathrm{C} .\) The calorimeter contains \(575 \mathrm{g}\) of water and the bomb has a heat capacity of \(650 \mathrm{J} / \mathrm{K}\). What quantity of heat is evolved per mole of glucose?

Short answer

The heat evolved per mole of glucose is approximately 2798.24 kJ/mol.

Step-by-step solution

Calculate Temperature Change

Determine the change in temperature of the system by subtracting the initial temperature from the final temperature: \[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 25.22^\circ C - 21.70^\circ C = 3.52^\circ C \]

Calculate Heat Absorbed by Water

Use the formula for heat evolved: \[ q_{\text{water}} = m \cdot c \cdot \Delta T \] where \(m\) is the mass of the water, \(c = 4.18 \, \text{J/g}^\circ \text{C}\) is the specific heat capacity of water, and \(\Delta T\) is the temperature change.Calculate: \[ q_{\text{water}} = 575 \, \text{g} \times 4.18 \, \text{J/g}^\circ \text{C} \times 3.52^\circ C = 8459.84 \, \text{J} \]

Calculate Heat Absorbed by Calorimeter

Use the calorimeter's heat capacity and temperature change to calculate heat absorbed by the calorimeter: \[ q_{\text{calorimeter}} = C_\text{calorimeter} \cdot \Delta T \] where \(C_\text{calorimeter} = 650 \, \text{J/K}\).Calculate: \[ q_{\text{calorimeter}} = 650 \, \text{J/K} \times 3.52 \, K = 2288 \, \text{J} \]

Total Heat Evolved in Reaction

Sum the heat absorbed by the water and calorimeter to find the total heat evolved:\[ q_{\text{total}} = q_{\text{water}} + q_{\text{calorimeter}} = 8459.84 \, \text{J} + 2288 \, \text{J} = 10747.84 \, \text{J} \]

Calculate Moles of Glucose

Calculate the amount of moles of glucose combusted using the molar mass of glucose \(\mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6\), which is 180.16 g/mol:\[ n = \frac{0.692 \, \text{g}}{180.16 \, \text{g/mol}} = 0.003841 \, \text{mol} \]

Calculate Quantity of Heat Evolved per Mole of Glucose

Determine the heat evolved per mole of glucose:\[ q_\text{per mole} = \frac{q_{\text{total}}}{n} = \frac{10747.84 \, \text{J}}{0.003841 \, \text{mol}} = 2798242.45 \, \text{J/mol} \]Convert to kJ/mol:\[ q_\text{per mole} = 2798.24 \, \text{kJ/mol} \]

Key concepts

Heat Capacity

Heat capacity is a crucial property in calorimetry because it defines how much heat a substance can store as its temperature changes.
It is particularly important for understanding how a system in a calorimeter will evolve during a reaction. In calorimetry, two types of heat capacities are often noted: the specific heat capacity and the calorimeter's heat capacity.

• Specific heat capacity, denoted as \(c\), is the amount of heat required to raise the temperature of 1 gram of a substance by 1 degree Celsius.
• The calorimeter heat capacity refers to the heat needed to raise the entire calorimeter's temperature by 1 degree Celsius.

In many experiments, you’ll find the water's specific heat capacity used because water is often the medium in which reactions occur, due to its known and consistent heat capacity of \(4.18 \, \text{J/g}^\circ \text{C}\). This experiment also provides the bomb's calorimeter heat capacity, which is 650 \(\text{J/K}\).
Both water and the calorimeter contribute to the overall measurement of heat evolved during the glucose combustion process.

Glucose Combustion

Combustion refers to a chemical process that involves burning a substance in the presence of oxygen to release energy in the form of heat and light.
For glucose combustion, the chemical reaction can be represented as: \[\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s) + 6\mathrm{O}_2(g) \rightarrow 6\mathrm{CO}_2(g) + 6\mathrm{H}_2\mathrm{O}(g)\ + \, \text{energy}\] In this process, when glucose (\(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\)) is burned, it reacts with oxygen to produce carbon dioxide, water, and releases energy.
This experiment uses a calorimeter to keep this reaction confined and measure the heat released.
Understanding glucose combustion helps us know the amount of energy in foods, as glucose is a primary energy source in biological systems.

Enthalpy Change

Enthalpy change represents the total heat content change of a system when a chemical reaction occurs at constant pressure.
In constant volume calorimetry, such as a bomb calorimeter, we instead measure the \(\Delta U\), but it closely approximates enthalpy change (\(\Delta H\)) under constant pressure conditions because no PV work is performed.Bomb calorimeters directly measure the heat evolved (or absorbed) by the reaction.
Knowing the total heat evolved helps us to understand how much energy is involved in the overall reaction.
We calculated this heat as the sum of the heat absorbed by the water in the calorimeter and the calorimeter's own absorbed heat.For the glucose combustion, the calculated total heat released translates to a calorimeter reading whereby the rise in temperature helps us compute the energy change that happened within the calorimeter's enclosed system.

Molar Mass Calculation

The molar mass of a substance is the mass of one mole of its chemical units. It is an essential concept for converting measured mass into moles, a standard unit for quantifying chemical reactions.
For glucose, with the formula \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\), its molar mass is calculated as follows:

• 6 carbon atoms at \(12.01 \, \text{g/mol}\) = \(72.06 \, \text{g/mol}\)
• 12 hydrogen atoms at \(1.008 \, \text{g/mol}\) = \(12.096 \, \text{g/mol}\)
• 6 oxygen atoms at \(16.00 \, \text{g/mol}\) = \(96.00 \, \text{g/mol}\)
• Total molar mass = \(72.06 \, + 12.096 \, + 96.00 \, = 180.16 \, \text{g/mol}\)

To find how many moles of glucose were burned, the mass of glucose (\(0.692 \, \text{g}\)) is divided by this molar mass, providing a basis to calculate energy released per mole.
The precise conversion from mass to moles allows for the comparison of energy release on a per-mole basis, aiding in comprehensive understanding across various chemical reactions.