Question

The heat capacity of a bomb calorimeter was determined by burning 6.79 g methane (energy of combustion \(=-802 \mathrm{kJ} / \mathrm{mol}\) \(\mathrm{CH}_{4}\)) in the bomb. The temperature changed by \(10.8^{\circ} \mathrm{C}\). a. What is the heat capacity of the bomb? b. \(A 12.6-g\) sample of acetylene, \(C_{2} H_{2},\) produced a temperature increase of \(16.9^{\circ} \mathrm{C}\) in the same calorimeter. What is the energy of combustion of acetylene (in \(\mathrm{kJ} / \mathrm{mol}\) )?

Short answer

a. The heat capacity of the bomb calorimeter is approximately -31.4 kJ/°C. b. The energy of combustion of acetylene is approximately -1098 kJ/mol.

Step-by-step solution

Calculate the amount of heat released by methane combustion

In this step, we calculate the total heat (q) released by the combustion of 6.79 g of methane, using its energy of combustion (-802 kJ/mol). First, we need to know the molar mass of methane, CH₄. Molar mass of CH₄ = 12.01 (C) + 1.01 × 4 (H) = 16.05 g/mol Now, we can calculate the moles of methane burned in the experiment: Moles of CH₄ = 6.79 g / 16.05 g/mol ≈ 0.423 mol With the moles of CH₄ and the energy of combustion, we can calculate the total heat released: q = moles × energy of combustion = 0.423 mol × (-802 kJ/mol) = -339.5 kJ

Calculate the heat capacity of the calorimeter

Since we know the temperature change and the total heat released, we can now determine the heat capacity (C) of the bomb calorimeter: C = q / ΔT = -339.5 kJ / 10.8 °C ≈ -31.4 kJ/°C b. Calculate the energy of combustion of acetylene

Calculate the heat generated by acetylene combustion

We know that a 12.6 g sample of acetylene (C₂H₂) generated a temperature increase of 16.9 °C in the calorimeter. With the heat capacity of the calorimeter we calculated in part a, we can determine the total heat generated by the acetylene combustion: q = C × ΔT = -31.4 kJ/°C × 16.9 °C = -531.6 kJ

Calculate the energy of combustion of acetylene

To calculate the energy of combustion of acetylene per mole, we first need to know the molar mass of C₂H₂: Molar mass of C₂H₂ = 12.01 × 2 (C) + 1.01 × 2 (H) = 26.04 g/mol Now, we can calculate the moles of acetylene burned in the experiment: Moles of C₂H₂ = 12.6 g / 26.04 g/mol ≈ 0.484 mol Finally, we determine the energy of combustion of acetylene: Energy of combustion = q / moles = -531.6 kJ / 0.484 mol ≈ -1098 kJ/mol So, the energy of combustion of acetylene is approximately -1098 kJ/mol.

Key concepts

Combustion Analysis

Combustion analysis is a vital process used in chemistry to determine the composition of a substance by burning it and analyzing the resulting products. During combustion, the substance reacts with an excess of oxygen to produce carbon dioxide, water, and other gases depending on the components in the substance. In a bomb calorimeter, the heat generated from the combustion process is trapped and measured, allowing for precise determinations of a compound's energy content.

By understanding the masses of the reactants and the products, we can calculate the amount of substance combusted. For example, if we know the mass of methane and its molar mass, we can find the number of moles combusted as in the original exercise. This information is an essential first step in determining the total energy released, which is essential in various fields such as environmental science and energy production.

Heat Capacity Calculation

The heat capacity of an object or substance is a measure of how much heat energy it takes to raise the temperature of that substance by one degree Celsius. In the field of calorimetry, the heat capacity calculation is crucial for determining the heat change in chemical reactions or physical changes. A bomb calorimeter has a known and constant heat capacity, allowing it to absorb heat without undergoing a phase change or chemical reaction.

By using the formula for heat capacity (C = q / ΔT), where q is the quantity of heat supplied and ΔT is the temperature change, we can determine the calorimeter's heat capacity. The calculated heat capacity is a critical factor in subsequent experiments, ensuring accurate results when measuring the energy content of other substances as illustrated by our example with methane combustion.

Calorimetry

Calorimetry is the science of measuring the change in energy of a system by observing the temperature change of a calorimeter. A bomb calorimeter, one of the most accurate types, is used to determine the energy of combustion of substances. It is a sealed container capable of withstanding the high pressure of a combustion reaction. Inside the calorimeter, a known mass of a substance is burned in the presence of oxygen, and the resulting temperature change is measured.

The heat of the reaction is directly proportional to the temperature change, assuming no heat loss to the environment. For substances with unknown energies of combustion, the bomb calorimeter is instrumental, as we illustrated with the acetylene sample in the exercise solution. By relating the observed temperature change to the known heat capacity of the calorimeter, we can calculate the energy released or absorbed during reactions.

Energy of Combustion

The energy of combustion is the amount of heat released when a compound undergoes complete combustion with oxygen to form water, carbon dioxide, and other gases. It is a fundamental thermodynamic quantity indicative of a substance's potential as a fuel. The higher the energy of combustion, the more energy is available for use, such as powering engines or generating electricity.

To calculate the energy of combustion per mole of substance, the total heat released during combustion (found through calorimetry) is divided by the number of moles of the substance that was burned. As displayed in the solution for acetylene, this provides the specific energy content necessary for comparative fuel efficiency studies and other practical applications where energy production or requirement is a concern.