Question

The heat capacity of a bomb calorimeter was determined by burning \(6.79 \mathrm{~g}\) 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 \(12.6-\mathrm{g}\) sample of acetylene, \(\mathrm{C}_{2} \mathrm{H}_{2}\), 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

The heat capacity of the bomb is approximately \(31.437 \mathrm{~kJ/}^{\circ}\mathrm{C}\), and the energy of combustion of acetylene is approximately \(1098.74 \mathrm{~kJ/mol}\).

Step-by-step solution

Calculate the molar mass of methane

First, we need to calculate the molar mass of CH4: C = 12.01g/mol H = 1.008g/mol Molar mass of CH4 = C + 4 * H = 12.01 + (4 * 1.008) = 16.042 g/mol

Convert grams of methane to moles

Next, we'll convert the given mass of methane (6.79 g) to moles using the molar mass calculated above: moles of methane = (6.79 g) / (16.042 g/mol) = 0.423 mol

Calculate energy released

Now, calculate the energy released during the burning of methane: Energy released = moles of methane * energy of combustion per mole Energy released = (0.423 mol) * (-802 kJ/mol) = -339.546 kJ

Determine the heat capacity of the bomb

Using the formula q = CΔT: Heat capacity of bomb (C) = -Energy released / ΔT C = 339.546 kJ / 10.8 °C ≈ 31.437 kJ/°C #Part b - Find the energy of combustion of acetylene#

Calculate the molar mass of acetylene

First, we need to calculate the molar mass of C2H2: Molar mass of C2H2 = 2 * C + 2 * H = (2 * 12.01) + (2 * 1.008) = 26.036 g/mol

Convert grams of acetylene to moles

Next, we'll convert the given mass of acetylene (12.6 g) to moles using the molar mass calculated above: moles of acetylene = (12.6 g) / (26.036 g/mol) = 0.484 mol

Calculate energy released by acetylene

Using the heat capacity of the bomb found in part (a) and the temperature change given for acetylene, we can calculate the energy released by acetylene: Energy released by acetylene = (31.437 kJ/°C) * (16.9 °C) = 531.593 kJ

Determine the energy of combustion per mole of acetylene

Finally, divide the energy released by the number of moles of acetylene to get the energy of combustion per mole of acetylene: Energy of combustion per mole = Energy released / moles of acetylene Energy of combustion per mole of acetylene = (531.593 kJ) / (0.484 mol) ≈ 1098.74 kJ/mol The heat capacity of the bomb is approximately 31.437 kJ/°C, and the energy of combustion of acetylene is approximately 1098.74 kJ/mol.

Key concepts

Understanding Heat Capacity in Bomb Calorimetry

Heat capacity is a critical concept in bomb calorimetry, which is a method used to measure the heat of combustion of a substance. Heat capacity (\( C \)) refers to the amount of heat (\( q \)) required to raise the temperature of the calorimeter by one degree Celsius (\( \text{°C} \)). In other words, it measures the calorimeter's ability to absorb heat without undergoing a significant change in temperature. \(
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\) To calculate the heat capacity of the bomb calorimeter, you would use the formula: \[ C = \frac{-q}{\Delta T} \( \(1\) \) \]. The negative sign indicates that the combustion process releases heat, causing the temperature to rise. The specific heat capacity for the calorimeter is constant; it's like a fingerprint unique to the device. Understanding this concept is crucial because it helps us to correctly interpret how much energy is generated from a chemical reaction, which is essential for fields such as material science, thermodynamics, and chemical engineering. \(
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\) In our exercise, we utilized the known energy of combustion of methane to determine the heat capacity of the bomb calorimeter. By burning a known amount of methane and measuring the temperature change, we applied the above formula to find the calorimeter's heat capacity. This value is then integral in determining the energy of combustion of other substances, as shown in part B of the exercise with acetylene.

Energy of Combustion Explored

The energy of combustion is the amount of energy released when a substance combusts or burns in the presence of oxygen. It is a measure of the potential energy stored within chemical bonds of the substance. In bomb calorimetry, the energy of combustion is used to calculate the amount of heat produced during the burning process. \(
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\) \[ \text{Energy of combustion} = \frac{\text{Energy released}}{\text{moles of substance}} \(\(2\)\) \] This energy is usually expressed in kilojoules per mole (\( kJ/mol \)), which signifies the amount of energy that one mole of a substance can release upon complete combustion. Knowing this value is critical for a variety of applications, including fuel efficiency, nutritional energy content of food, and the synthesis of chemicals. \(
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\) In the case of our acetylene example, we found the energy of combustion by using the calorimeter's heat capacity and the measured temperature increase. The result gives us a quantifiable understanding of how much energy acetylene can provide, which can have practical implications in industrial processes where acetylene is used as a fuel source.

Navigating Molar Mass Calculation

Molar mass is a foundational concept in chemistry that refers to the mass of one mole of a substance, expressed in grams per mole (\( g/mol \)). It's calculated by summing the masses of the individual atoms in a molecule based on the periodic table. \(
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\) The formula to calculate the molar mass of a chemical compound is given by: \[ \text{Molar Mass} = \sum (\text{n}_{\text{element}} \times \text{atomic weight of the element}) \(\(3\)\) \] where \( \text{n}_{\text{element}} \) stands for the number of atoms of each element in the molecule. \(
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\) For instance, in the exercise, calculating the molar mass of methane (\( CH_4 \)) and acetylene (\( C_2H_2 \)) allowed us to convert grams of the substances to moles, linking the macroscopic mass of the sample with its molecular composition. This step is vital when working with the stoichiometry of reactions or when interpreting the results of calorimetric experiments, as it provides a bridge between the measured physical quantity and the underlying molecular entities.