Question

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

Short answer

\(\Delta U \approx 2801.493\, \mathrm{kJ/mol}\) for glucose.

Step-by-step solution

Calculate Temperature Change

First, determine the change in temperature of the calorimeter. The formula to use is:\[\Delta T = T_{\text{final}} - T_{\text{initial}}\]Substitute the given values:\[\Delta T = 25.22^{\circ} \mathrm{C} - 21.70^{\circ} \mathrm{C} = 3.52^{\circ} \mathrm{C}\]

Calculate Heat Absorbed by Water

Use the formula for heat absorbed by the water, \( q = m c \Delta T \), where \( m \) is the mass of the water, \( c \) is the specific heat capacity of water \( (4.184\, \mathrm{J/g^{\circ}C}) \), and \( \Delta T \) is the temperature change.\[q_{\text{water}} = 575 \times 4.184 \times 3.52 = 8473.696\, \mathrm{J}\]

Calculate Heat Absorbed by Bomb Calorimeter

The heat absorbed by the calorimeter is given by the calorimeter's heat capacity multiplied by the temperature change:\[q_{\text{bomb}} = 650 \times 3.52 = 2288\, \mathrm{J}\]

Total Heat Released by Combustion Reaction

Add the heat absorbed by both the water and the calorimeter to find the total heat released by the glucose:\[q_{\text{total}} = q_{\text{water}} + q_{\text{bomb}} = 8473.696 + 2288 = 10761.696\, \mathrm{J}\]

Convert Heat to Molar Basis

To find \( \Delta U \) per mole of glucose burned, calculate the number of moles of glucose:The molar mass of glucose \( \mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6 \) is 180.16 g/mol.\[\text{moles of glucose} = \frac{0.692}{180.16} = 0.003841\, \text{mol}\]Now, calculate \( \Delta U \) per mole:\[\Delta U = \frac{10761.696}{0.003841} = 2801493.64\, \mathrm{J/mol} \approx 2801.493\, \mathrm{kJ/mol}\]

Provide Final Expression for \(\Delta U\)

Thus, the change in internal energy \( \Delta U \) per mole of glucose is determined to be approximately:\[\Delta U \approx 2801.493\, \mathrm{kJ/mol}\]

Key concepts

Enthalpy Change

Enthalpy change represents the heat absorbed or released during a chemical reaction at constant pressure. In calorimetry, however, experiments are often performed at constant volume, leading to a measurement of the change in internal energy, \( \Delta U \), instead of the enthalpy change (\( \Delta H \)). Still, these changes are closely related, particularly when considering reactions in a bomb calorimeter, as the pressure-volume work is often negligible. This makes \( \Delta U \) a good estimate for the heat change under constant pressure conditions when calculated per mole of a substance. Understanding the connection between enthalpy and internal energy is critical for interpreting calorimetry data correctly.

Specific Heat Capacity

Specific heat capacity is a property of a material and is defined as the amount of heat required to raise the temperature of one gram of the substance by one degree Celsius. For water, a common substance used in calorimeters, this value is \( 4.184 \, \text{J/g}^{\circ} \text{C} \). Knowing the specific heat capacity is essential when calculating the heat exchanged with a substance, as it allows us to determine how much heat the substance either absorbs or releases during a temperature change.
In the exercise, water in the calorimeter absorbs heat from the reaction, and its specific heat capacity plays a critical role in determining the total heat change. This allows us to quantify the energy released by the glucose combustion.

Molar Mass

Molar mass is a fundamental concept in chemistry, indicating the mass of one mole of a particular substance. It provides a bridge between the mass of a substance and the moles, which is crucial when dealing with chemical equations. In the example problem, the molar mass of glucose \( \mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6 \) is 180.16 g/mol. This value is used to convert the mass of burned glucose into moles, enabling the calculation of the energy change per mole of glucose.
When solving calorimetry problems, accurately knowing the molar mass ensures the correct interpretation of energy changes on a molar level, which is often the standard unit of measurement in energy calculations.

Temperature Change

Temperature change in calorimetry is a direct indication of energy exchange. It measures how much the temperature of a system (or surroundings) shifts as a result of a chemical or physical process. The formula used is \( \Delta T = T_{\text{final}} - T_{\text{initial}} \). In the calorimetric setup, the temperature increase reflects the exothermic nature of the glucose combustion.
This increase in temperature is recorded and used to calculate the total heat absorbed by both the water and the calorimeter. The overall heat change is split into contributions from these two sources to precisely determine how much energy was released per mole of glucose. This process highlights the importance of precise temperature measurements in energy calculations.