Problem 112 - General Chemistry: Principles and Modern Applications

Question

A 1.103 g sample of a gaseous carbon-hydrogenoxygen compound that occupies a volume of \(582 \mathrm{mL}\) at 765.5 Torr and \(25.00^{\circ} \mathrm{C}\) is burned in an excess of \(\mathrm{O}_{2}(\mathrm{g})\) in a bomb calorimeter. The products of the combustion are \(2.108 \mathrm{g} \mathrm{CO}_{2}(\mathrm{g}), 1.294 \mathrm{g} \mathrm{H}_{2} \mathrm{O}(1),\) and enough heat to raise the temperature of the calorimeter assembly from 25.00 to \(31.94^{\circ} \mathrm{C}\). The heat capacity of the calorimeter is \(5.015 \mathrm{kJ} /^{\circ} \mathrm{C}\). Write an equation for the combustion reaction, and indicate \(\Delta H^{\circ}\) for this reaction at \(25.00^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

\(C_xH_yO_z + aO_2 -> bCO_2 + cH_2O\), \(\Delta H^{\circ}\) (details provided in the step-by-step solution)

Step 1: Calculation of moles of reactants

Start by calculating the amount (mol) of reactant using the ideal gas law: \(PV = nRT\). Where, P is the pressure, V is the volume, n is the number of moles, R is the gas constant and T is the temperature. In this case, we convert pressure from Torr to atm, volume from mL to L, and temperature from Celsius to Kelvin, and use the value of gas constant R = \(0.0821 L·atm/mol·K\). Thus, the moles of the carbon-hydrogen-oxygen compound can be calculated.

Step 2: Calculation of moles of products

Then, we calculate the moles of \(CO_2\) and \(H_2O\) produced from the reaction. We know that 1 mol of \(CO_2\) has a mass of approximately 44.01 g and 1 mol of \(H_2O\) has a mass of approximately 18.015 g. Using the provided masses of \(CO_2\) and \(H_2O\), we can calculate their respective amounts (mol).

Step 3: Formation of combustion equation

Based on the calculated mol of reactants and products, the combustion equation of the carbon-hydrogen-oxygen compound can be established. Assume the chemical formula of the compound is \(C_xH_yO_z\), then the combustion equation is \(C_xH_yO_z + aO_2 -> bCO_2 + cH_2O\), where a, b and c are coefficients determined by the stoichiometry of the reaction (the molar ratio between reactants and products).

Step 4: Calculation of ΔH°

The temperature rise of the calorimeter assembly indicates the amount of heat \(q_p\) produced by the process, which is given by \(q_p= C \Delta T\), where C is the heat capacity of the calorimeter and \(\Delta T\) is the temperature change. Converting q_p to the enthalpy change \(\Delta H^\circ\) per mol of reaction requires the molar quantity of the C_xH_yO_z compound. The result can be obtained through \(\Delta H° = qp / mol(C_xH_yO_z)\).

Key Concepts

Understanding Ideal Gas Law

The ideal gas law is a fundamental equation in thermochemistry that helps us calculate the properties of gases under varying conditions. The equation is given by \( PV = nRT \), where \( P \) represents the pressure, \( V \) the volume, \( n \) the number of moles, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. This law assumes that gases consist of particles in constant random motion and that these particles do not interact with each other except in elastic collisions.
For this problem, it's essential to convert all units to the appropriate SI units: pressure in atmospheres, volume in liters, and temperature in Kelvin. The conversion from Torr to atmospheres, Celsius to Kelvin, and milliliters to liters ensures that we use consistent units in our calculations.
Once converted, we solve for \( n \), the number of moles of the compound, which is crucial for understanding the quantity of reactant involved in the combustion reaction.

Basics of Combustion Reactions

A combustion reaction is a chemical process that involves the burning of a substance with oxygen, resulting in the production of heat and light. These reactions typically involve hydrocarbons reacting with oxygen to form carbon dioxide and water.
In the exercise, the unknown carbon-hydrogen-oxygen compound undergoes complete combustion in excess oxygen. This leads to the formation of well-known products: \( CO_2 \) and \( H_2O \).
Understanding combustion at a molecular level means recognizing that bonds in the reactants break and new bonds form in the products. This transformation releases energy, which is evident from the rise in temperature noted in the calorimeter during the experiment.
By identifying the masses of the products \( CO_2 \) and \( H_2O \), we can determine the amounts and ratios required to complete the balanced combustion reaction. This ensures the law of conservation of mass is fulfilled.

Calculating Enthalpy Change

Enthalpy change, \( \Delta H^\circ \), quantifies the heat absorbed or released during a chemical reaction at constant pressure. It's an essential concept in thermochemistry, as it provides insight into the energy dynamics of reactions.
In this case, the calorimeter records a temperature change, indicating the heat released during combustion. This heat can be calculated using the equation \( q_p = C \Delta T \), where \( C \) is the calorimeter's heat capacity and \( \Delta T \) is the temperature change.
This heat is then related to the enthalpy change by dividing the computed heat by the number of moles of the combusted compound. Finally, \( \Delta H^\circ \) is determined, which indicates whether the reaction is exothermic (releases heat) or endothermic (absorbs heat).
Knowing the enthalpy change is vital because it not only tells us about the energetic favorability of a reaction but also plays a crucial role in various thermodynamic and practical applications.

Role of Stoichiometry in Reactions

Stoichiometry is the study of the quantitative relationships in chemical reactions, allowing chemists to predict the amounts of substances consumed and produced.
In this exercise, stoichiometry facilitated the calculation of the coefficients in the balanced combustion equation. From the balanced equation, we identify how many moles of oxygen react with one mole of the compound and how much carbon dioxide and water are produced.
Stoichiometry helps maintain the balance across reactions—ensuring that what goes in at the start of a reaction equals the end products. This is based on the law of conservation of mass, which states that mass in a closed system remains constant.
By applying stoichiometry to the combustion reaction, we accurately determine the ratios and quantities of the reactants and products. This information is crucial for calculations related to the enthalpy change and understanding the reaction's dynamics.