Question

A \(1.000 \mathrm{~mol}\) sample of propane, a gas used for cooking in many rural areas, was placed in a bomb calorimeter with excess oxygen and ignited. The initial temperature of the calorimeter was \(25.000^{\circ} \mathrm{C}\) and its total heat capacity was \(97.13 \mathrm{~kJ}^{\circ} \mathrm{C}^{-1}\). The reaction raised the temperature of the calorimeter to \(27.282^{\circ} \mathrm{C}\). (a) Write the balanced chemical equation for the reaction in the calorimeter. (b) How many joules were liberated in this reaction? (c) What is the heat of reaction of propane with oxygen expressed in kilojoules per mole of \(\mathrm{C}_{3} \mathrm{H}_{8}\) burned?

Short answer

The balanced chemical reaction is \( \text{C}_3\text{H}_8 + 5\text{O}_2 \rightarrow 3\text{CO}_2 + 4\text{H}_2\text{O} \). The energy liberated is \( 97.13 \text{ kJ}^{\text{o}}\text{C}^{-1} \times 2.282^{\text{o}}C \times 1000 = 221721.46 \text{ J} \). The heat of the reaction of propane is \( 221.72146 \text{ kJ/mol} \) of \( \text{C}_3\text{H}_8 \) burned.

Step-by-step solution

Balancing the Chemical Equation

To write the balanced chemical equation for the combustion of propane, start with the unbalanced equation: \( \text{C}_3\text{H}_8 + \text{O}_2 \rightarrow \text{CO}_2 + \text{H}_2\text{O} \). Using stoichiometry, balance the carbon atoms by having 3 \( \text{CO}_2 \) molecules, the hydrogen atoms by having 4 \( \text{H}_2\text{O} \) molecules, and finally balance the oxygen atoms. The balanced equation is: \( \text{C}_3\text{H}_8 + 5\text{O}_2 \rightarrow 3\text{CO}_2 + 4\text{H}_2\text{O} \) .

Calculating the Energy Liberated

To calculate the energy liberated, use the formula: \[ \text{Energy} = \text{Heat Capacity} \times \text{Temperature Change} \]. The temperature change (\text{ΔT}) is the final temperature minus the initial temperature, which is \( 27.282^{\text{o}}C - 25.000^{\text{o}}C = 2.282^{\text{o}}C \). The total energy released is: \( \text{Energy} = 97.13 \text{ kJ}^{\text{o}}\text{C}^{-1} \times 2.282^{\text{o}}C \).

Converting Energy From Kilojoules to Joules

Since 1 kilojoule (kJ) equals 1000 joules (J), to express energy in joules, multiply the value obtained in step 2 by 1000: \[ \text{Energy in Joules} = \text{Energy in kJ} \times 1000 \].

Determining the Heat of Reaction per Mole

The heat of reaction per mole of \( \text{C}_3\text{H}_8 \) is the total energy released divided by the number of moles burned, which in this case is 1.000 mole: \[ \text{Heat of Reaction} = \frac{\text{Energy in Joules}}{\text{Moles of} \text{C}_3\text{H}_8} \].

Key concepts

Chemical Equation Balancing

Understanding how to balance a chemical equation is essential for studying reactions, including the combustion of propane. Balancing involves ensuring that the number of atoms for each element is equal on both the reactant and product sides of the equation. For the combustion of propane (\text{C}_3\text{H}_8), the key is to count the atoms of carbon and hydrogen in the propane molecule first, and then adjust the coefficients of oxygen (\text{O}_2), carbon dioxide (\text{CO}_2), and water (\text{H}_2\text{O}) to balance the equation.

Starting with propane and oxygen as reactants, we acknowledge that the products will be carbon dioxide and water. The balanced equation becomes: \[ \text{C}_3\text{H}_8 + 5\text{O}_2 \rightarrow 3\text{CO}_2 + 4\text{H}_2\text{O} \]. Every carbon and hydrogen atom on the left is accounted for on the right with the correct amount of oxygen atoms. It's the conservation of mass in action; no atoms are lost or gained, just rearranged.

Calorimetry

Calorimetry is the process of measuring the heat of chemical reactions or physical changes. It is performed using a device called a calorimeter. In our example, a bomb calorimeter was used to measure the energy released during the combustion of propane.

This particular calorimeter has its own heat capacity, which is the amount of energy required to raise its temperature by one degree Celsius. By measuring the temperature change before and after the reaction, we can calculate the total energy released using the formula: \[ \text{Energy} = \text{Heat Capacity} \times \text{Temperature Change} \].From the exercise, a temperature increase from \(25.000^\circ\text{C}\) to \(27.282^\circ\text{C}\) allows us to determine the energy released during the combustion.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships that exist in chemical formulas and reactions. Once the equation is balanced, as we have done in the previous section, you can use stoichiometry to determine the amount of reactants and products involved in a reaction. For every mole of propane burned, a specific number of moles of oxygen are required, leading to a certain number of moles of carbon dioxide and water produced.

These relationships help in calculating theoretical yields, limiting reagents, and in this case, help to relate the quantity of fuel to the energy released. If the exercise had provided an amount other than a neat \(1.000\) mole, you would use stoichiometric ratios from the balanced equation to determine how many joules of energy should be released or how much of each product would be formed.

Thermochemistry

Thermochemistry discusses the heat involved during chemical reactions. It's all about the enthalpy changes, heat of reactions, and energy transfers that occur. The exercise asks for the heat of reaction per mole of propane, which can also be termed as the enthalpy change (\(\text{Δ}H\)). In a constant volume calorimeter such as the bomb calorimeter, the heat measured is equal to the change in internal energy (\(\text{Δ}U\)) of the system.

By calculating the heat of reaction, we can determine how much energy in joules or kilojoules is released or absorbed when a specified amount of reactant undergoes a chemical change. In the example, once we've calculated the total energy liberated in joules, this value can be divided by the number of moles of propane combusted to find the heat of reaction per mole.