Sure! Let's solve this step-by-step.
Step 1: Understand the given reaction and heat change (ΔH).
The balanced chemical reaction is:
[tex]\[ CH_4(g) + 2 O_2(g) \rightarrow CO_2(g) + 2 H_2O(g) \][/tex]
The heat change (ΔH_rxn) for the combustion reaction of methane (CH_4) is -802.3 kJ, which means that burning 1 mole of CH_4 releases 802.3 kJ of heat.
Step 2: Determine the amount of heat we want to emit.
We want to emit 345 kJ of heat.
Step 3: Calculate the number of moles of CH_4 required to emit 345 kJ of heat.
Since 1 mole of CH_4 releases 802.3 kJ of heat, we can set up the following proportion:
[tex]\[ \text{Number of moles of CH_4} = \frac{\text{Heat emitted}}{\left|\Delta H_{rxn}\right|} = \frac{345 \text{ kJ}}{802.3 \text{ kJ/mol}} \][/tex]
This simplifies to:
[tex]\[ \text{Number of moles of CH_4} = 0.43001371058207655 \text{ mol} \][/tex]
Step 4: Calculate the molar mass of CH_4.
The molar mass of methane (CH_4) is calculated as follows:
[tex]\[ \text{Molar mass of CH_4} = 12.01 \left(\text{for C}\right) + 4 \times 1.008 \left(\text{for H}\right) = 16.042 \text{ g/mol} \][/tex]
Step 5: Calculate the mass of CH_4 needed in grams.
With the number of moles of CH_4 known, we can calculate the mass:
[tex]\[ \text{Mass of CH_4} = \text{Number of moles of CH_4} \times \text{Molar mass of CH_4} \][/tex]
[tex]\[ \text{Mass of CH_4} = 0.43001371058207655 \text{ mol} \times 16.042 \text{ g/mol} = 6.898279945157673 \text{ g} \][/tex]
Step 6: Round the mass to three significant figures.
The mass of CH_4 needed is then rounded to three significant figures:
[tex]\[ \text{Mass of CH_4} \approx 6.898 \text{ g} \][/tex]
Therefore, the mass of natural gas (CH_4) that you must burn to emit 345 kJ of heat is approximately [tex]\( \boxed{6.898 \text{ g}} \)[/tex].
