Problem 98 - Chemistry and Chemical Reactivity

Question

A bomb calorimetric experiment was run to determine the enthalpy of combustion of ethanol. The reaction is $$ \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\ell)+3 \mathrm{O}_{2}(\mathrm{g}) \rightarrow 2 \mathrm{CO}_{2}(\mathrm{g})+3 \mathrm{H}_{2} \mathrm{O}(\ell) $$ The bomb had a heat capacity of \(550 \mathrm{J} / \mathrm{K},\) and the calorimeter contained \(650 \mathrm{g}\) of water. Burning \(4.20 \mathrm{g}\) of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\ell)\) resulted in a rise in temperature from \(18.5^{\circ} \mathrm{C}\) to \(22.3^{\circ} \mathrm{C}\). Calculate \(\Delta U\) for the combustion of ethanol, in \(\mathrm{kJ} / \mathrm{mol}\).

Step-by-Step Solution

Answer

\(\Delta U = -136.15\, \mathrm{kJ/mol}\).

Step 1: Calculate the Temperature Change

First, determine the temperature change in the calorimeter. The initial temperature was \(18.5^{\circ} \mathrm{C}\), and the final temperature was \(22.3^{\circ} \mathrm{C}\). So, the change in temperature, \(\Delta T\), is:\[\Delta T = 22.3 - 18.5 = 3.8^{\circ} \mathrm{C}.\]

Step 2: Calculate the Heat Absorbed by the Calorimeter and Water

Calculate the total heat absorbed using the heat capacities of the bomb and the water. First, use the formula for the bomb: \[q_{bomb} = C_{bomb} \times \Delta T = 550\, \mathrm{J/K} \times 3.8\, \mathrm{K} = 2090\, \mathrm{J}.\]Then, calculate the heat absorbed by the water (using specific heat capacity of water, \(4.18\, \mathrm{J/g\,K}\)): \[q_{water} = m_{water} \times c \times \Delta T = 650\, \mathrm{g} \times 4.18\, \mathrm{J/g\,K} \times 3.8\, \mathrm{K} = 10327\, \mathrm{J}.\]The total heat absorbed by the system is:\[q_{total} = q_{bomb} + q_{water} = 2090 + 10327 = 12417\, \mathrm{J}.\]

Step 3: Convert Total Heat to Kilojoules

Since the units of energy are usually expressed in kilojoules for such problems, convert \(q_{total}\) from joules to kilojoules: \[q_{total} = 12417\, \mathrm{J} = 12.417\, \mathrm{kJ}.\]

Step 4: Calculate Moles of Ethanol Combusted

To find \(\Delta U\) per mole, determine the number of moles of ethanol combusted. The molar mass of ethanol \((\mathrm{C}_2\mathrm{H}_5\mathrm{OH})\) is \(46.08\, \mathrm{g/mol}\). Using the mass given, \(4.20\, \mathrm{g}\): \[n = \frac{4.20\, \mathrm{g}}{46.08\, \mathrm{g/mol}} = 0.0912\, \mathrm{mol}.\]

Step 5: Calculate \(\Delta U\) per Mole of Ethanol

Now calculate \(\Delta U\) per mole of ethanol by dividing the total heat by the number of moles:\[\Delta U = \frac{12.417\, \mathrm{kJ}}{0.0912\, \mathrm{mol}} = -136.15\, \mathrm{kJ/mol}.\]The negative sign indicates that the reaction is exothermic.

Key Concepts

Understanding Bomb Calorimetry

Bomb calorimetry is a technique used to measure the heat of combustion of a substance. In this method, the substance is placed in a strong container called a bomb, which is then submerged in a water-filled calorimeter. When the substance burns, it releases heat, raising the temperature of the water. By knowing the mass of the water and its specific heat capacity, we can determine the heat released. This experiment allows us to precisely calculate the enthalpy change of the combustion reaction.

• The heat absorbed by the calorimeter and water is measured in joules.
• Through calibration, bomb calorimeters can account for their own heat capacity.
• This method provides a controlled environment to ensure accurate results.

By carefully following the temperature changes and heat absorbed, bomb calorimetry gives us a way to understand the energy aspects of reactions through precise measurements.

Thermodynamic Calculations Behind Combustion

The process of determining the enthalpy of combustion involves several thermodynamic calculations. First, we measure the temperature change to find the heat absorbed by the calorimeter, considering both the calorimeter's and water's heat capacities.
Then, from this heat, we calculate its equivalent in kilojoules since most chemical energy data is reported in this unit.
Following that, it's essential to know the amount of substance burned to compute the enthalpy per mole. This involves converting the mass of the compound into moles using its molar mass. The calculations allow us to link physical measurements to chemical energy changes:

• Heat absorbed is determined through the equation: \[q = C \times \Delta T\] for both the calorimeter and the water.
• Conversion from joules to kilojoules modernizes the data for broader application.
• The molar calculation provides an understanding of energy changes relative to a specific amount.

Completing these calculations helps in interpreting the thermal energy changes involved in the combustion process.

Exothermic Reactions and Their Implications

Exothermic reactions are those that release energy in the form of heat into their surroundings. The combustion of ethanol in a bomb calorimeter is a quintessential example. This released energy makes the reaction exothermic, as indicated by a negative enthalpy change.
When analyzing these reactions:

• The negative sign of \(\Delta U\) shows energy is being released.
• Identifying whether a reaction is exothermic can help predict its spontaneity and effect on the environment.
• Knowing the energy release helps in energy balance calculations for industrial processes.

Understanding exothermic reactions is essential in many fields as it helps predict energy management and efficiency aspects of chemical processes. These reactions are not only important in laboratories but play a crucial role in environmental and industrial applications as well.