Question

Methane burns with oxygen to produce carbon dioxide and water as a gas. The balanced thermochemical equation is $$ \mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) \atop \Delta H^{\circ}=-802 \mathrm{~kJ} $$ How much methane, in grams, must be burned to release \(432 \mathrm{~kJ}\) of heat?

Short answer

Approximately 8.64 grams of methane must be burned to release 432 kJ of heat.

Step-by-step solution

Understanding the Thermochemical Equation

Recognize that the given thermochemical equation represents the burning of methane in the presence of oxygen to yield carbon dioxide and water vapor, along with the release of heat. The \( \Delta H^\circ \) value of -802 kJ indicates that 802 kJ of heat is released when 1 mole of methane (\text{CH}_4\text{(g)}) reacts with 2 moles of oxygen (\text{O}_2\text{(g)}). Note that the negative sign indicates the reaction is exothermic (releases heat).

Calculate Moles of Methane to Release 432 kJ

Set up a proportion based on the thermochemical equation, using the fact that 1 mole of methane releases 802 kJ to find out how many moles are needed to release 432 kJ. \[ \frac{1 \text{ mole } \text{CH}_4}{802 \text{ kJ}} = \frac{x \text{ moles } \text{CH}_4}{432 \text{ kJ}} \] Solve for \(x\text{(moles of CH}_4\text{)}\) by cross-multiplying: \[ x \times 802 \text{ kJ} = 1 \text{ mole } \text{CH}_4 \times 432 \text{ kJ} \] \[ x = \frac{1 \text{ mole } \text{CH}_4 \times 432 \text{ kJ}}{802 \text{ kJ}} \] \[ x \approx 0.5387 \text{ moles } \text{CH}_4 \]

Convert Moles of Methane to Grams

Use the molar mass of methane (\text{CH}_4\text{)} to convert the amount in moles to grams. Methane has a molar mass of approximately 16.04 g/mol (12.01 g/mol for carbon and 1.01 g/mol for each hydrogen, with 4 hydrogens in the molecule). \[ \text{Mass in grams} = \text{Moles of } \text{CH}_4 \times \text{Molar mass of } \text{CH}_4 \] \[ \text{Mass in grams} = 0.5387 \text{ moles } \times 16.04 \text{ g/mol} \] \[ \text{Mass in grams} \approx 8.64 \text{ g } \]

Key concepts

Enthalpy Change

Enthalpy change is a measure of the heat content in a system and is central to thermochemistry and thermodynamics. It's denoted by ΔH, and it represents the amount of energy absorbed or released by a system during a reaction at constant pressure. In a thermochemical equation, a negative ΔH value, such as -802 kJ, indicates an exothermic reaction, where the system releases heat to its surroundings. Conversely, a positive ΔH would signify an endothermic reaction, where the system absorbs heat.

Understanding the role of enthalpy change in chemical equations helps in predicting the heat flow during reactions. It is valuable for various applications, such as calculating the energy required for heating or cooling in industrial processes, or the energy released during combustion, as seen in the burning of methane to release heat.

Stoichiometry

Stoichiometry is the section of chemistry that involves calculating the relative quantities of reactants and products in chemical reactions. It's a foundational concept for scientists and engineers, as it facilitates the prediction of yield and the optimization of reactions. In the case of the methane combustion reaction, stoichiometry is utilized to relate the mole quantities of methane and oxygen to the energy produced during the reaction.

By setting up proportionate ratios, as seen in the step-by-step solution, we can determine exactly how many moles of methane are required to produce a certain amount of energy. This precise calculation is critical to ensure that reactions are conducted efficiently and resources are not wasted, which is especially important in large-scale industrial manufacturing and energy production processes.

Mole Concept

The mole concept is a bridge between the atomic scale and the macroscopic world, allowing chemists to count particles by weighing them. A mole is defined as the amount of substance containing as many entities (atoms, molecules, ions, etc.) as there are atoms in 12 grams of carbon-12. This number is Avogadro's number, approximately 6.022 x 10²³.

In practical terms, relating moles to grams using the molar mass of a substance, which is the mass of one mole of that substance, is essential. For methane (CH₄), with a molar mass of 16.04 g/mol, we use the mole concept to convert given grams of methane to moles or vice versa. This conversion is crucial in understanding how much of a material is needed to participate in a reaction, enabling precise planning and execution of chemical processes.

Exothermic Reactions

Exothermic reactions are chemical reactions that release energy, usually in the form of heat, to their surroundings. These reactions are characterized by a negative enthalpy change (ΔH). Common examples include combustion reactions, like the burning of methane mentioned in our exercise. When chemical bonds are formed in the products (CO₂ and H₂O), more energy is released than is consumed to break the bonds in the reactants (CH₄ and O₂).

The energy released in exothermic reactions is often harnessed for practical purposes such as heating homes or fueling engines. It's crucial to understand and control exothermic reactions to manage the energy release, ensuring safe and efficient processes, especially in industrial and environmental settings where control of heat emissions is paramount.