Problem 80 - Chemistry The Central Science

Question

Methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) is used as a fuel in race cars. (a) Write a balanced equation for the combustion of liquid methanol in air. (b) Calculate the standard enthalpy change for the reaction, assuming \(\mathrm{H}_{2} \mathrm{O}(g)\) as a product. (c) Calculate the heat produced by combustion per liter of methanol. Methanol has a density of \(0.791 \mathrm{~g} / \mathrm{mL}\). (d) Calculate the mass of \(\mathrm{CO}_{2}\) produced per kJ of heat emitted.

Step-by-Step Solution

Answer

(a) The balanced equation for the combustion of methanol is: \[ \mathrm{CH}_{3} \mathrm{OH}(l) + \frac{3}{2}\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) + 2\mathrm{H}_{2}\mathrm{O}(g)\] (b) The standard enthalpy change for the reaction is -725.6 kJ/mol. (c) The heat produced per liter of methanol is approximately -17920 kJ. (d) The mass of CO2 produced per kJ of heat emitted is 0.0607 g CO2/kJ.

Step 1: (a) Balanced Combustion Equation

Combustion reactions involve the reaction of a compound (in this case, methanol) with oxygen to produce water and carbon dioxide. The balanced equation for the combustion of liquid methanol (CH3OH) in air is: \[ \mathrm{CH}_{3} \mathrm{OH}(l) + \frac{3}{2}\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) + 2\mathrm{H}_{2}\mathrm{O}(g)\]

Step 2: (b) Standard Enthalpy Change

To find the standard enthalpy change for this reaction, we will use the following formula: ΔH = ΣΔHf(products) - ΣΔHf(reactants) We need the standard enthalpies of formation (ΔHf) for all the substances in the balanced equation. These values are often found in tables and are given as: ΔHf(CH3OH) = -238.7 kJ/mol ΔHf(O2) = 0 kJ/mol (since O2 is in its standard state) ΔHf(CO2) = -393.5 kJ/mol ΔHf(H2O) = -241.8 kJ/mol Now, we can calculate the standard enthalpy change for the reaction: ΔH = [1 * (-393.5) + 2 * (-241.8)] - [1 * (-238.7) + 3/2 * 0] ΔH = -725.6 kJ/mol

Step 3: (c) Heat Produced per Liter

Methanol has a density of 0.791 g/mL. To calculate the heat produced by the combustion of one liter of methanol, we need to convert the mass of one liter of methanol to moles and multiply it by the heat produced per mole. First, we find the mass of one liter of methanol: Mass of methanol = (1000 mL) * (0.791 g/mL) = 791 g Now, we convert the mass to moles: Moles of methanol = mass/(molar mass) Moles of methanol = 791 g / (32.04 g/mol) ≈ 24.7 moles Heat produced per liter = moles of methanol * ΔH Heat produced per liter = 24.7 moles * (-725.6 kJ/mol) ≈ -17920 kJ

Step 4: (d) Mass of CO2 produced per kJ

From the balanced chemical equation, 1 mol of methanol produces 1 mol of CO2. To find the mass of CO2, we need to determine the mass of CO2 produced per mole of methanol reacted. Mass of CO2 per mole = (1 mol CO2 / 1 mol CH3OH) * (44.01 g/mol CO2) Mass of CO2 per mole = 44.01 g Now, we need to find the mass of CO2 produced per kJ of heat emitted. Mass of CO2 per kJ = (44.01 g/mol)/(725.6 kJ/mol) = 0.0607 g CO2 / kJ

Key Concepts

Balanced Chemical Equation

Understanding the balanced chemical equation is crucial for comprehending the stoichiometry of any chemical reaction, including combustion. The equation represents the reactants and products in their molecular or molar proportions. For example, the combustion of methanol ((CH₃OH)) involves its reaction with oxygen (O₂), resulting in carbon dioxide (CO₂) and water (H₂O). Balancing this equation is a matter of ensuring that the number of atoms for each element is equal on both sides of the equation.

To arrive at a balanced equation for the combustion of methanol, one must account for each hydrogen, carbon, and oxygen atom. The balanced equation, as provided in the exercise solution, is as follows:
\[ CH_3OH(l) + \frac{3}{2}O_2(g) \rightarrow CO_2(g) + 2H_2O(g)\]
This indicates that one molecule of liquid methanol reacts with one and a half molecules of gaseous oxygen to produce one molecule of gaseous carbon dioxide and two molecules of gaseous water. Balancing chemical equations is a foundational skill in chemistry that ensures the law of conservation of mass is upheld, and it sets the stage for further calculation in stoichiometry and thermochemistry.

Standard Enthalpy Change

The standard enthalpy change (cap{H}) of a reaction quantifies the heat absorbed or released under standard conditions, which is typically measured in kilojoules per mole (kJ/mol). In an exothermic reaction like combustion, cap{H} is negative, indicating that heat is released to the surroundings. To calculate the standard enthalpy change, one must use the standard enthalpies of formation (cap{H}_fdelta H_f) for all the reactants and products involved in the balanced equation.

The formula to determine cap{H} is given by the sum of the enthalpies of formation of the products minus the sum of the enthalpies of formation of the reactants:
cap{H} = cap{H}_f(products) - cap{H}_f(reactants). In this context, 'formation' refers to the creation of a compound from its constituent elements in their standard states. The standard enthalpy change provides crucial information about the energy efficiency of a fuel, which is especially important in applications like race cars, where methanol is commonly used.

Heat of Combustion

The heat of combustion is the amount of heat released when a substance fully oxidizes during combustion. It is an essential factor in the energy content of fuels. The heat of combustion can be experimentally determined, but it can also be calculated using the standard enthalpy change of the reaction and the stoichiometry of the involved species.

In the example of methanol combustion, we calculate the heat produced by one liter of methanol by finding its mass and converting it to moles. The density of methanol is given as 0.791 g/mL, so one liter (1000 mL) would weigh 791 g. After converting the mass to moles using the molar mass of methanol (32.04 g/mol), we multiply the number of moles by the standard enthalpy change of the reaction to obtain the total heat produced per liter of methanol. This heat quantity is an important parameter for understanding the energy efficiency and power output when methanol is used as a fuel.

Stoichiometry

Stoichiometry is the aspect of chemistry that involves calculating the quantities of reactants and products in a chemical reaction. It relies on the balanced chemical equation and applies the laws of conservation of mass and matter. Stoichiometry is a fundamental concept that allows chemists to predict the outcomes of reactions, quantify the amounts needed or produced, and make conversions between mass and moles.

Using stoichiometry, we can determine the mass of CO₂ produced per kJ of heat emitted by using the molar mass of CO₂ and the heat of combustion calculated earlier. From the methanol combustion equation, it’s clear that each mole of methanol yields one mole of CO₂. By dividing the molar mass of CO₂ by the heat of combustion (in kJ/mol), we can find the mass of CO₂ produced per unit of energy. This type of stoichiometric calculation is essential for evaluating the environmental impact of fuel combustion, as in the case of race cars that use methanol as a fuel.