Question

A \(1.800-g\) sample of phenol \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}\right)\) was burned in a bomb calorimeter whose total heat capacity is \(11.66 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}\). The temperature of the calorimeter plus contents increased from \(21.36\) to \(26.37^{\circ} \mathrm{C}\). (a) Write a balanced chemical equation for the bomb calorimeter reaction. (b) What is the heat of combustion per gram of phenol? Per mole of phenol?

Short answer

The heat of combustion per gram of phenol is \(32.43\, \mathrm{kJ/g}\), and the heat of combustion per mole of phenol is approximately \(3051.54\, \mathrm{kJ/mol}\).

Step-by-step solution

Write the balanced chemical equation

Phenol has molecular formula \(\mathrm{C}_6 \mathrm{H}_5 \mathrm{OH}\). When it burns, it will react with oxygen (\(\mathrm{O}_2\)) to produce carbon dioxide (\(\mathrm{CO}_2\)) and water (\(\mathrm{H}_2\mathrm{O}\)). The balanced chemical equation for phenol combustion can be written as: \(\mathrm{C}_6 \mathrm{H}_5 \mathrm{OH} + \dfrac{15}{2}\,\mathrm{O}_2 \rightarrow 6\,\mathrm{CO}_2 + 3\,\mathrm{H}_2\mathrm{O}\) Step 2: Calculate the heat generated in the reaction

Find the heat generated

The total heat capacity of the calorimeter is given as \(11.66 \, \mathrm{kJ}/ { }^{\circ} \mathrm{C}\). The temperature of the calorimeter plus contents increased from \(21.36^{\circ} \mathrm{C}\) to \(26.37^{\circ} \mathrm{C}\). Therefore, the heat generated in the reaction, denoted by \(q\), can be calculated using the formula: \(q = C\Delta T\) Where \(C\) is the heat capacity of the calorimeter and \(\Delta T\) is the change in temperature. \(q = 11.66\, \mathrm{kJ}/{ }^{\circ}\mathrm{C} \times (26.37 - 21.36)\,^{\circ}\mathrm{C}\) \(q = 11.66\, \mathrm{kJ}/{ }^{\circ}\mathrm{C} \times 5.01\,^{\circ}\mathrm{C}\) \(q = 58.37\, \mathrm{kJ}\) Step 3: Calculate heat of combustion per gram of phenol

Heat of combustion per gram of phenol

Given the mass of phenol as \(1.800\,g\), we can calculate the heat of combustion per gram of phenol by dividing the heat generated by the mass of phenol: \(\frac{\text{Heat of combustion}}{\text{gram of phenol}} = \frac{58.37\, \mathrm{kJ}}{1.800\,g}\) \(\frac{\text{Heat of combustion}}{\text{gram of phenol}} = 32.43\, \mathrm{kJ/g}\) Step 4: Calculate heat of combustion per mole of phenol

Heat of combustion per mole of phenol

To calculate heat of combustion per mole of phenol, first find the molar mass of phenol: \(\text{Molar mass of phenol} = 6 \times 12.01\, \text{g/mol (C)} + 6 \times 1.01\, \text{g/mol (H)} + 1 \times 16.00\, \text{g/mol (O)} = 94.11\, \text{g/mol\) Next, divide the heat of combustion per gram of phenol by the molar mass of phenol: \(\frac{\text{Heat of combustion}}{\text{mole of phenol}} = 32.43\, \mathrm{kJ/g} \times 94.11\, \mathrm{g/mol}\) \(\frac{\text{Heat of combustion}}{\text{mole of phenol}} = 3051.54\, \mathrm{kJ/mol}\) In summary, the heat of combustion per gram of phenol is \(32.43\, \mathrm{kJ/g}\), and the heat of combustion per mole of phenol is approximately \(3051.54\, \mathrm{kJ/mol}\).

Key concepts

Calorimetry

Calorimetry is a branch of science focusing on measuring the heat of chemical reactions or physical changes. This technique is crucial in understanding the heat of combustion, which is the amount of heat generated when a substance fully reacts with oxygen. A calorimeter, the device used in those measurements, is designed to insulate the reaction from its surroundings to ensure accurate heat measurement.

In the case of a bomb calorimeter, the substance is burned in a sealed container, and the heat released causes an increase in temperature measured within the system. Calculating the energy change involves multiplying the calorimeter's heat capacity by the observed temperature rise, a clear example of how calorimetry depends on precise temperature and heat capacity measurements. By accounting for the calorimeter's heat capacity, one can determine the total heat released by the reaction.

Enthalpy Change

Enthalpy change, denoted by \(\Delta H\), is the amount of heat absorbed or released by a system at constant pressure. It is a central concept in thermochemistry and is used to quantify the energy changes during chemical reactions, like combustion. Positive \(\Delta H\) values indicate endothermic reactions where the system absorbs heat, while negative \(\Delta H\) values represent exothermic reactions, with heat being released.

For combustion reactions, the enthalpy change is typically negative, reflecting the exothermic nature of these reactions. By dividing the total heat released (q) by the number of moles of the substance burned, we obtain the molar enthalpy change. This value is key to understanding the energy content of fuels and other substances and can be used to predict the energy produced in similar reactions.

Stoichiometry

Stoichiometry is essentially the math of chemistry; it entails calculating the relative quantities of reactants and products in chemical reactions. In the context of heat of combustion, stoichiometry allows us to relate the mass of a substance to the amount of heat released upon its complete combustion.

By using the balanced chemical equation and the molar masses of the compounds involved, one can convert between grams and moles, thus bridging the gap between laboratory measurements and theoretical calculations. For combustion reactions, stoichiometry is pivotal to determine how the mass of a substance relates to the amount of heat energy that can be obtained from it. The accurate stoichiometric calculations are fundamental for practical applications such as determining fuel efficiency or synthesizing the desired end-products in industry.