Question

A \(0.692-\mathrm{g}\) 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 \(575 \mathrm{g}\) 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

\(\Delta U = -2801.64 \text{ kJ/mol}\).

Step-by-step solution

Calculate Heat Absorbed by Water

The heat absorbed by the water is determined using the formula \( q = m \cdot c \cdot \Delta T \) where \( m = 575 \text{ g} \), \( c = 4.184 \text{ J/g°C} \), and \( \Delta T = 25.22 - 21.70 = 3.52 \text{°C} \). Substitute the values to find \( q_{\text{water}} \).\[ q_{\text{water}} = 575 \cdot 4.184 \cdot 3.52 = 8470.688 \text{ J} \]

Calculate Heat Absorbed by Calorimeter

Use the bomb calorimeter's heat capacity to find the heat absorbed by the calorimeter with the formula \( q = C \cdot \Delta T \), where \( C = 650 \text{ J/K} \) and \( \Delta T = 3.52 \text{ K} \).\[ q_{\text{calorimeter}} = 650 \cdot 3.52 = 2288 \text{ J} \]

Find Total Heat Released by the Reaction

The total heat released by the combustion of glucose is the sum of the heat absorbed by the water and the calorimeter.\[ q_{\text{total}} = q_{\text{water}} + q_{\text{calorimeter}} = 8470.688 + 2288 = 10758.688 \text{ J} \]

Calculate Moles of Glucose Burned

Determine the moles of glucose using its molar mass. The molar mass of glucose \( \text{C}_{6}\text{H}_{12}\text{O}_{6} \) is \( 180.18 \text{ g/mol} \). Use the sample mass \( 0.692 \text{ g} \) to find moles.\[ \text{moles of glucose} = \frac{0.692}{180.18} = 0.00384 \text{ mol} \]

Calculate Change in Internal Energy per Mole

The change in internal energy per mole of glucose is given by dividing the total heat by the moles of glucose.\[ \Delta U = \frac{q_{\text{total}}}{\text{moles of glucose}} = \frac{10758.688}{0.00384} = 2801637.5 \text{ J/mol} = 2801.64 \text{ kJ/mol} \]

Conclusion

The change in internal energy for the combustion of glucose is approximately \(-2801.64 \text{ kJ/mol}\). The negative sign indicates exothermic reaction.

Key concepts

Enthalpy Change

Enthalpy change, denoted by \( \Delta U \), represents the total heat content change of a system during a chemical reaction. It is a central concept in calorimetry, which studies heat transfer during chemical processes. Enthalpy change can indicate whether a reaction requires or releases energy.

In our exercise, glucose undergoes a combustion reaction. This process releases energy, resulting in a negative \( \Delta U \), which signifies an exothermic reaction. We calculate this by finding the total heat absorbed by both the water and the calorimeter in the system. Then, this value is adjusted according to the amount of substance that reacted, expressed per mole, so we understand the energy change for a standard quantity of glucose.

Heat Capacity

Heat capacity is the amount of heat needed to change the temperature of a substance by one degree Celsius (or one Kelvin). In the context of calorimetry, it plays a crucial role in determining how much heat is absorbed by the calorimeter setup.

Two heat capacities are considered in the exercise: that of the water, and that of the calorimeter bomb itself. Water has a specific heat capacity of \( 4.184 \text{ J/g°C} \), a known constant used in calculations for thermal energy exchange with water.

The calorimeter's heat capacity (given as \( 650 \text{ J/K} \)), helps quantify how much heat the calorimeter absorbs independently of its mass, contributing to the total heat calculation needed to find the enthalpy change accurately.

Combustion Reaction

A combustion reaction is a chemical process where a substance combines with oxygen to release energy in the form of heat and light. This type of reaction is highly exothermic, which explains the significant heat release observed in the calorimetry experiment of glucose.

When glucose undergoes combustion, it reacts with oxygen to form carbon dioxide and water, releasing energy that is then absorbed by the surrounding water and the calorimeter bomb. The purpose of measuring these heat exchanges is to determine the total energy output of the reaction. By relating this to the moles of glucose, we find the standard energy change per mole, an essential step in understanding the enthalpic nature of glucose combustion.

Molar Mass

Molar mass is the mass of one mole of a given substance. It is expressed in grams per mole and is fundamental to calculating the moles of a substance from a given mass.

In this exercise, the molar mass of glucose (\( \text{C}_{6}\text{H}_{12}\text{O}_{6} \)) is \( 180.18 \text{ g/mol} \). Knowing the sample's mass (\( 0.692 \text{ g} \)), we can compute the number of moles of glucose involved in the reaction.

This conversion is crucial because the calorimetric data (total heat change) must be expressed per mole for a proper understanding of the enthalpy change. Thus, the molar mass acts as a bridge between the experimental data and the theoretical aspect of energy changes per mole.