Question

Consider the combustion of one mole of methyl alcohol, \(\mathrm{CH}_{3} \mathrm{OH}(l),\) which yields carbon dioxide gas and steam. (a) Write a thermochemical equation for the reaction. (Use Table 8.2.) (b) Calculate \(\Delta E\) at \(25^{\circ} \mathrm{C}\).

Short answer

Question: Calculate the change in internal energy (∆E) at 25°C for the combustion of 1 mole of methyl alcohol. Answer: The change in internal energy (∆E) at 25°C for the combustion of 1 mole of methyl alcohol is approximately -639641 J/mol.

Step-by-step solution

(a) Writing a thermochemical equation for the reaction

: First, let's write the balanced chemical equation for the combustion of 1 mole of methyl alcohol: $$\mathrm{CH}_3\mathrm{OH}(l) + \dfrac{3}{2}\mathrm{O}_2(g) \rightarrow \mathrm{CO}_2(g) + 2\mathrm{H}_2\mathrm{O}(g)$$ Now, using Table 8.2, we can find the standard enthalpies of formation (\(\Delta H_f^\circ\)) for each molecule involved in the reaction: $$\Delta H_f^\circ(\mathrm{CH}_3\mathrm{OH}(l)) = -238.7\ \text{kJ/mol}$$ $$\Delta H_f^\circ(\mathrm{O}_2(g)) = 0\ \text{kJ/mol}\ (\text{since oxygen is in its elemental form})$$ $$\Delta H_f^\circ(\mathrm{CO}_2(g)) = -393.5\ \text{kJ/mol}$$ $$\Delta H_f^\circ(2\mathrm{H}_2\mathrm{O}(g)) = 2(-241.8)\ \text{kJ/mol} = -483.6\ \text{kJ/mol}$$ To find the enthalpy change \(\Delta H\) for the reaction, we can use the following formula: $$\Delta H = \sum \Delta H_f^\circ(\text{products}) - \sum \Delta H_f^\circ(\text{reactants})$$

(b) Calculate the enthalpy change for the reaction

: We can now substitute the values we found above: $$\Delta H = (-393.5\ \text{kJ/mol} - 483.6\ \text{kJ/mol}) - (-238.7\ \text{kJ/mol})$$ $$\Delta H = -638.4\ \text{kJ/mol}$$

Calculate the change in internal energy for the reaction

: To find the change in internal energy (∆E), we can use the following relationship: $$\Delta E = \Delta H - nRT$$ Where n is the change in the number of moles of gases in the reaction, R is the ideal gas constant (8.314 J/mol·K), and T is the temperature in Kelvin. First, let's find the change in the number of moles of gases in the reaction: $$n = \text{(number of moles of gaseous products)} - \text{(number of moles of gaseous reactants)}$$ $$n = (1 + 2) - \left(\dfrac{3}{2}\right) = \dfrac{1}{2}$$ Now, convert the temperature from Celsius to Kelvin: $$T = 25^{\circ} \mathrm{C} + 273.15 = 298.15\ \text{K}$$ Now, we can plug in the values into the ∆E equation: $$\Delta E = \Delta H - nRT$$ $$\Delta E = -638.4\ \text{kJ/mol} - \left(\dfrac{1}{2}\mathrm{mol}\right)(8.314\ \mathrm{J/mol·K})(298.15\ \mathrm{K})$$ (Note that we need to convert kJ to J for the units to match up, by multiplying by 1000) $$\Delta E = -638400\ \mathrm{J/mol} - \left(\dfrac{1}{2}\mathrm{mol}\right)(8.314\ \mathrm{J/mol·K})(298.15\ \mathrm{K})$$ $$\Delta E = -638400\ \mathrm{J/mol} - 1240.63\ \mathrm{J/mol}$$ $$\Delta E = -639640.63\ \mathrm{J/mol}$$ The change in internal energy (∆E) at \(25^{\circ} \mathrm{C}\) is approximately -639641 J/mol.

Key concepts

Enthalpy Change

Enthalpy change, denoted as (ΔH), is a critical concept in thermodynamics and describes the heat absorbed or released during a chemical reaction occurring at constant pressure. It's directly related to the energy involved in breaking and forming chemical bonds.

In the context of a combustion reaction, such as the burning of methyl alcohol ((CH_3)(OH)), enthalpy change can be calculated by subtracting the sum of the standard enthalpies of formation ((ΔH_f^∘)) of the reactants from that of the products. The standard enthalpy of formation for a compound is the change in enthalpy when one mole of the substance is formed from its elements under standard conditions (i.e., 298 K and 1 atm).

To illustrate, when you write a thermochemical equation for a reaction, you should also include the enthalpy change, showing whether heat was absorbed ((ΔH) > 0, endothermic) or released ((ΔH) < 0, exothermic) in the course of the reaction. For our methyl alcohol combustion, the enthalpy change calculated from given values shows it to be exothermic, to the tune of -638.4 kJ/mol, indicating that energy is released as heat to the surroundings during the reaction.

An understanding of enthalpy change is not only fundamental to predicting the energetic outcome of reactions but also to various applications such as designing energy-efficient processes in industries or understanding environmental impact through heat release.

Combustion Reaction

A combustion reaction is a type of chemical reaction where a substance combines with oxygen to release heat and light, typically resulting in the production of a flame. Common examples include the burning of wood, coal, or hydrocarbons like methyl alcohol.In this exercise, the combustion of methyl alcohol is explored. The general equation for the combustion of a hydrocarbon or alcohol can be represented as:Hydrocarbon/Alcohol + Oxygen → Carbon Dioxide + WaterThe balanced thermochemical equation for the combustion of one mole of methyl alcohol to yield carbon dioxide and steam is expressed as:CH_3OH(l) + 3/2 O_2(g) → CO_2(g) + 2 H_2O(g)With the correct balance of reactants, such a combustion reaction can be complete, meaning it will convert all available fuel into CO_2 and H_2O. This type of reaction is of particular importance in understanding the energy transformations involved in the burning of fuels and has immense implications in fields ranging from material science to environmental engineering, where energy output and emissions need to be managed.

Internal Energy

In thermodynamics, internal energy is the total of all forms of energy contained within a system. This encompasses the kinetic energy of particles, which is related to their temperature, as well as the potential energy resulting from molecular interactions. The symbol (ΔE) often represents the change in internal energy of a system during a reaction.Internal energy is particularly noteworthy because it reflects the energy changes excluding work done by or on the system at constant volume. This distinction is crucial when examining energy change in a chemical reaction, especially if gas expansion or compression occurs during the process. By combining the concepts of enthalpy change ((ΔH)) with the equation (ΔE = ΔH - nRT), where 'n' is the change in moles of gas, 'R' the ideal gas constant, and 'T' the absolute temperature, we gain insight into how the internal energy changes during a reaction.

For the given exercise, the combustion of methyl alcohol, we can calculate the change in internal energy by considering the enthalpy change and the work associated with gas expansion or compression. This is because, in a system at constant pressure, part of the energy released during combustion is spent on work done by the system as it expands. Here, the change in internal energy for the reaction ((ΔE)) comes out to be approximately -639641 J/mol at 25°C, indicating the energy released as heat after accounting for work done by gases expanding.