Question

When \(0.514 \mathrm{~g}\) of biphenyl \(\left(\mathrm{C}_{12} \mathrm{H}_{10}\right)\) undergoes combustion in a bomb calorimeter, the temperature rises from \(25.8^{\circ} \mathrm{C}\) to \(29.4^{\circ} \mathrm{C}\). Find \(\Delta E_{\mathrm{rxn}}\) for the combustion of biphenyl in \(\mathrm{kJ} / \mathrm{mol}\) biphenyl. The heat capacity of the bomb calorimeter, determined in a separate experiment, is \(5.86 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}\).

Short answer

The change in energy for the combustion of biphenyl (\triangle E_{rxn}) is calculated to be the heat absorbed by the calorimeter divided by the number of moles of biphenyl.

Step-by-step solution

Calculate the heat absorbed by the calorimeter

Determine the change in temperature and use the heat capacity of the calorimeter to find the total heat absorbed. The change in temperature is the final temperature minus the initial temperature: \(29.4^\circ C - 25.8^\circ C\). Multiply this value by the heat capacity of the calorimeter to find the heat absorbed.

Calculate the moles of biphenyl

Calculate the number of moles of biphenyl that underwent combustion. To do this, divide the mass of biphenyl by its molar mass. The molar mass of biphenyl (C12H10) is calculated as: \(12 \cdot 12.01 \text{g/mol} + 10 \cdot 1.01 \text{g/mol}\).

Calculate the enthalpy of reaction per mole

The enthalpy change of the reaction per mole of biphenyl is the heat absorbed by the calorimeter divided by the number of moles of biphenyl. Keep in mind that the heat absorbed should be in kJ to match the units requested for the answer.

Key concepts

Bomb Calorimeter

A bomb calorimeter is an essential device in thermochemistry used to measure the heat of combustion of a particular reaction. It consists of a strong, sealed container known as a 'bomb,' which is filled with oxygen and placed in an insulated water bath. The sample of interest, such as biphenyl in our exercise, is ignited in the bomb, and the resulting temperature change in the water bath is measured. The heat capacity of the calorimeter, which is determined through a calibration process with a substance of known enthalpy change, allows us to calculate the total heat absorbed or released during the reaction.

Due to its ability to withstand high pressures, the bomb calorimeter can accurately measure the heat of combustion without the interference of the atmosphere. This makes it possible to completely ascertain the amount of heat evolved during combustion reactions, providing us with the capability to perform precise energy calculations.

Combustion Analysis

Combustion analysis is a method commonly used in chemistry to determine the composition of a substance by burning it and analyzing the resulting products. In the context of the exercise, biphenyl is subjected to combustion in the presence of excess oxygen within the bomb calorimeter. The combustion process breaks down the compound into simpler molecules, usually carbon dioxide and water. By measuring the temperature change, we can infer the amount of heat released during the reaction. This is critical for understanding the energy content of fuels and other materials.

Combustion analysis not only allows us to derive energy values but also provides insight into the empirical formulae and molar amounts of unknown samples. It's particularly useful in the field of organic chemistry, where determining the structure of hydrocarbons and other carbon-based compounds is essential.

Enthalpy Change

Enthalpy change, denoted as \(\Delta H\) or \(\Delta E_{rxn}\) in the context of bomb calorimetry, reflects the amount of heat absorbed or released by a system at constant pressure during a chemical reaction. The measurement of enthalpy change is fundamental in understanding energy transfer in reactions. For the case of biphenyl combustion, the temperature rise in the calorimeter's water bath is used to calculate the heat involved in the reaction, and thus the enthalpy change.

The enthalpy change for a reaction can be reported in joules or kilojoules per mole (kJ/mol), indicating the energy change associated with the combustion of one mole of a substance. In the exercise, after calculating the heat absorbed using the change in temperature and the calorimeter's heat capacity, we can determine the \(\Delta E_{rxn}\) per mole of biphenyl, which is crucial for scientists and engineers when designing energy-related processes.

Molar Mass Calculation

The molar mass of a substance is the weight of one mole (6.022 \(\times\) 10\(^{23}\) entities) of that substance and is expressed in grams per mole (g/mol). To calculate the molar mass of biphenyl (C\(_{12}\)H\(_{10}\)), as shown in the exercise, we sum the molar masses of each element in the compound, multiplied by the number of atoms of that element present in the formula unit. For carbon (C), with a molar mass of approximately 12.01 g/mol, and hydrogen (H), with a molar mass of about 1.01 g/mol, the molar mass calculation is as follows:

\(12 \cdot 12.01 \text{ g/mol} + 10 \cdot 1.01 \text{ g/mol} = 154.12 \text{ g/mol}\) for biphenyl.

This allows us to convert the mass of biphenyl used in the combustion reaction into moles, a step that is crucial in determining the enthalpy change per mole of the substance. Understanding how to calculate molar mass is vital not only for performing energy calculations but also in a plethora of chemical analyses and quantitative studies.