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To determine how much heat is released when [tex]\(10.0 \, \text{g}\)[/tex] of iron ([tex]\(Fe\)[/tex]) and [tex]\(2.00 \, \text{g}\)[/tex] of oxygen ([tex]\(O_2\)[/tex]) are reacted, follow these steps:
1. Find the molar mass of iron and oxygen:
[tex]\[ \text{Molar mass of } Fe = 55.845 \, \text{g/mol} \][/tex]
[tex]\[ \text{Molar mass of } O_2 = 32.00 \, \text{g/mol} \][/tex]
2. Calculate the moles of [tex]\(Fe\)[/tex] and [tex]\(O_2\)[/tex] given their masses:
[tex]\[ \text{Moles of } Fe = \frac{\text{mass of } Fe}{\text{molar mass of } Fe} = \frac{10.0 \, \text{g}}{55.845 \, \text{g/mol}} \approx 0.179 \, \text{mol} \][/tex]
[tex]\[ \text{Moles of } O_2 = \frac{\text{mass of } O_2}{\text{molar mass of } O_2} = \frac{2.00 \, \text{g}}{32.00 \, \text{g/mol}} = 0.0625 \, \text{mol} \][/tex]
3. Determine the heat released per mole of each reactant for the given reaction:
The reaction releases [tex]\(1652 \, \text{kJ}\)[/tex] per [tex]\(4\)[/tex] moles of [tex]\(Fe\)[/tex] and [tex]\(3\)[/tex] moles of [tex]\(O_2\)[/tex]:
[tex]\[ \text{Heat released per mole of } Fe = \frac{1652 \, \text{kJ}}{4} \approx 413 \, \text{kJ/mol} \][/tex]
[tex]\[ \text{Heat released per mole of } O_2 = \frac{1652 \, \text{kJ}}{3} \approx 550.67 \, \text{kJ/mol} \][/tex]
4. Calculate the heat released based on the amount of each reactant:
[tex]\[ \text{Heat released based on } Fe = \text{moles of } Fe \times \text{heat released per mole of } Fe = 0.179 \, \text{mol} \times 413 \, \text{kJ/mol} \approx 73.95 \, \text{kJ} \][/tex]
[tex]\[ \text{Heat released based on } O_2 = \text{moles of } O_2 \times \text{heat released per mole of } O_2 = 0.0625 \, \text{mol} \times 550.67 \, \text{kJ/mol} \approx 34.42 \, \text{kJ} \][/tex]
5. Determine the actual heat released by identifying the limiting reagent:
The actual heat released will be determined by the limiting reagent, which is the reactant that produces the lesser amount of heat. In this case:
[tex]\[ \text{Heat released} = \min(73.95 \, \text{kJ}, 34.42 \, \text{kJ}) = 34.42 \, \text{kJ} \][/tex]
Therefore, when [tex]\(10.0 \, \text{g}\)[/tex] of iron and [tex]\(2.00 \, \text{g}\)[/tex] of oxygen are reacted, the heat released is [tex]\(34.42 \, \text{kJ}\)[/tex].
1. Find the molar mass of iron and oxygen:
[tex]\[ \text{Molar mass of } Fe = 55.845 \, \text{g/mol} \][/tex]
[tex]\[ \text{Molar mass of } O_2 = 32.00 \, \text{g/mol} \][/tex]
2. Calculate the moles of [tex]\(Fe\)[/tex] and [tex]\(O_2\)[/tex] given their masses:
[tex]\[ \text{Moles of } Fe = \frac{\text{mass of } Fe}{\text{molar mass of } Fe} = \frac{10.0 \, \text{g}}{55.845 \, \text{g/mol}} \approx 0.179 \, \text{mol} \][/tex]
[tex]\[ \text{Moles of } O_2 = \frac{\text{mass of } O_2}{\text{molar mass of } O_2} = \frac{2.00 \, \text{g}}{32.00 \, \text{g/mol}} = 0.0625 \, \text{mol} \][/tex]
3. Determine the heat released per mole of each reactant for the given reaction:
The reaction releases [tex]\(1652 \, \text{kJ}\)[/tex] per [tex]\(4\)[/tex] moles of [tex]\(Fe\)[/tex] and [tex]\(3\)[/tex] moles of [tex]\(O_2\)[/tex]:
[tex]\[ \text{Heat released per mole of } Fe = \frac{1652 \, \text{kJ}}{4} \approx 413 \, \text{kJ/mol} \][/tex]
[tex]\[ \text{Heat released per mole of } O_2 = \frac{1652 \, \text{kJ}}{3} \approx 550.67 \, \text{kJ/mol} \][/tex]
4. Calculate the heat released based on the amount of each reactant:
[tex]\[ \text{Heat released based on } Fe = \text{moles of } Fe \times \text{heat released per mole of } Fe = 0.179 \, \text{mol} \times 413 \, \text{kJ/mol} \approx 73.95 \, \text{kJ} \][/tex]
[tex]\[ \text{Heat released based on } O_2 = \text{moles of } O_2 \times \text{heat released per mole of } O_2 = 0.0625 \, \text{mol} \times 550.67 \, \text{kJ/mol} \approx 34.42 \, \text{kJ} \][/tex]
5. Determine the actual heat released by identifying the limiting reagent:
The actual heat released will be determined by the limiting reagent, which is the reactant that produces the lesser amount of heat. In this case:
[tex]\[ \text{Heat released} = \min(73.95 \, \text{kJ}, 34.42 \, \text{kJ}) = 34.42 \, \text{kJ} \][/tex]
Therefore, when [tex]\(10.0 \, \text{g}\)[/tex] of iron and [tex]\(2.00 \, \text{g}\)[/tex] of oxygen are reacted, the heat released is [tex]\(34.42 \, \text{kJ}\)[/tex].
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