To determine the mass of natural gas [tex]\((\text{CH}_4)\)[/tex] that must be burned to emit 345 kJ of heat, we need to follow these steps:
1. Identify the given reaction and its enthalpy change ([tex]\(\Delta H\)[/tex]):
[tex]\[
\text{CH}_4(g) + 2 O_2(g) \rightarrow \text{CO}_2(g) + 2 \text{H}_2O(g)
\][/tex]
The enthalpy change for the reaction, [tex]\(\Delta H_{\text{rxn}}^{\circ}\)[/tex], is [tex]\(-802.3 \text{ kJ}\)[/tex]. This value means 802.3 kJ of energy is released (since it's negative) for every mole of [tex]\(\text{CH}_4\)[/tex] burned.
2. Calculate the number of moles of [tex]\(\text{CH}_4\)[/tex] required to emit 345 kJ of heat:
[tex]\[
\text{Moles of } \text{CH}_4 = \frac{\text{Heat emitted}}{|\Delta H_{\text{rxn}}|}
\][/tex]
[tex]\[
\text{Moles of } \text{CH}_4 = \frac{345 \text{ kJ}}{802.3 \text{ kJ/mol}}
\][/tex]
[tex]\[
\text{Moles of } \text{CH}_4 \approx 0.430014 \text{ moles}
\][/tex]
3. Determine the molar mass of [tex]\(\text{CH}_4\)[/tex]:
[tex]\[
\text{Molar mass of } \text{CH}_4 = 12.01 \text{ (for C)} + 4 \times 1.01 \text{ (for H)} = 16.04 \text{ g/mol}
\][/tex]
4. Calculate the mass of [tex]\(\text{CH}_4\)[/tex]:
[tex]\[
\text{Mass of } \text{CH}_4 = \text{Moles of } \text{CH}_4 \times \text{Molar mass of } \text{CH}_4
\][/tex]
[tex]\[
\text{Mass of } \text{CH}_4 = 0.430014 \text{ moles} \times 16.04 \text{ g/mol}
\][/tex]
[tex]\[
\text{Mass of } \text{CH}_4 \approx 6.90 \text{ grams}
\][/tex]
To summarize, to emit 345 kJ of heat, approximately 6.90 grams of natural gas ([tex]\(\text{CH}_4\)[/tex]) must be burned.
