Question

What mass of natural gas ( \(\mathrm{CH}_{4}\) ) must burn to emit \(267 \mathrm{~kJ}\) of heat? $$\begin{array}{r}\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) \\\\\Delta H_{\mathrm{rxn}}^{\circ}=-802.3 \mathrm{~kJ}\end{array}$$

Short answer

The mass of \(\mathrm{CH}_{4}\) needed is calculated as the product of the number of moles from Step 2 and its molar mass (16.04 g/mol).

Step-by-step solution

Understand the Chemical Equation

Review the balanced chemical equation for the combustion of methane \(\mathrm{CH}_{4}(g) + 2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g) + 2 \mathrm{H}_{2}O(g)\). The \(\Delta H_{\mathrm{rxn}}^{\circ} = -802.3 \mathrm{~kJ}\) indicates the amount of heat released (exothermic reaction) when 1 mole of methane is burned.

Calculate Moles of Methane

Use the provided heat energy (267 kJ) and the standard enthalpy change of the reaction to calculate the moles of methane burned. The equation to use is: \[\rfrac{-267\mathrm{~kJ}}{-802.3\mathrm{~kJ/mol}} = \mathrm{moles\ of\ CH_{4}}\]

Calculate the Mass of Methane

To find the mass of methane that produced the given amount of heat, multiply the moles of methane by its molar mass (16.04 g/mol for \(\mathrm{CH}_{4}\)).

Key concepts

Understanding Chemical Equations

Chemical equations are vital in explaining how substances react together to form new products. In the case of the combustion of methane, the equation \( \mathrm{CH}_{4}(g) + 2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g) + 2 \mathrm{H}_{2}O(g) \) is a balanced chemical equation.

It shows one molecule of methane (\(\mathrm{CH}_{4}\)) reacting with two molecules of oxygen (\(\mathrm{O}_{2}\)) to produce one molecule of carbon dioxide (\(\mathrm{CO}_{2}\)) and two molecules of water (\(\mathrm{H}_{2}O\)), both in gaseous form. Each side of the equation has the same number of atoms for each element, satisfying the Law of Conservation of Mass. The coefficient '2' in front of \(\mathrm{O}_{2}\) and \(\mathrm{H}_{2}O\) balances the equation, ensuring that there are equal numbers of elements on both sides of the reaction.

Combustion of Methane

Combustion is a high-temperature exothermic reaction that involves a fuel and an oxidant leading to the generation of heat and light.

In this case, methane, the primary component of natural gas, combusts in the presence of oxygen, releasing energy. The process can be represented through the reaction \( \mathrm{CH}_{4}(g) + 2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g) + 2 \mathrm{H}_{2}O(g) \) where \(\Delta H_{\mathrm{rxn}}^{\circ} = -802.3 \mathrm{~kJ}\) signifies the amount of energy released as heat when one mole of methane combusts.

Heat Released During Combustion

By quantifying the \(\Delta H_{\mathrm{rxn}}^{\circ}\), scientists can determine the energy yield of methane, which has practical applications in energy production and calculating the efficiency of fuel combustion.

Molar Mass

Molar mass is a critical concept in chemistry that relates the mass of a substance to its amount in moles. It is expressed in grams per mole (g/mol).

The molar mass of methane, \(\mathrm{CH}_{4}\), is calculated by summing the atomic masses of one carbon atom and four hydrogen atoms, essentially computed as \(12.01 g/mol\) (from carbon) plus \(4 \times 1.01 g/mol\) (from hydrogen), which totals \(16.04 g/mol\). This value allows chemists to convert between mass and moles of a substance, providing a bridge between the macroscopic and molecular worlds.

Exothermic Reactions

Exothermic reactions are chemical reactions that release energy, typically in the form of heat, to their surroundings. The combustion of methane is a classic example of such a reaction.

When an exothermic reaction occurs, like in the equation \( \mathrm{CH}_{4}(g) + 2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g) + 2 \mathrm{H}_{2}O(g) \), energy is released as the chemical bonds in the reactants break and new bonds form to create the products. The negative sign in the standard enthalpy change (\(\Delta H_{\mathrm{rxn}}^{\circ}\)) indicator, which in this case is \(-802.3 \mathrm{~kJ}\), signifies that the energy is exiting the system and being released into the surroundings.

Importance of Exothermicity

The concept of exothermicity is not only crucial in thermodynamics but also in practical applications such as heating homes or powering engines, where the released energy is harnessed for various purposes.