To determine the fraction of CH4 in the mixture, you need to calculate the mass of CH4 in the mixture and then divide it by the total mass of the mixture.
First, you need to calculate the molar mass (mass of one mole) of CH4 and C2H6. The molar mass of CH4 is calculated as follows:
1 carbon atom (C) * atomic mass of carbon (12.01 g/mol) + 4 hydrogen atoms (H) * atomic mass of hydrogen (1.01 g/mol) = 16.04 g/mol
Now, calculate the number of moles of CH4 in the mixture by dividing the mass of CH4 by its molar mass:
moles of CH4 = mass of CH4 / molar mass of CH4
moles of CH4 = 13.43 g / 16.04 g/mol ≈ 0.836 mol
Since CH4 and C2H6 react completely, the moles of CO2 and H2O produced will also be equal to the moles of CH4 and C2H6 in the mixture.
Next, calculate the mass of CO2 and H2O produced by adding their respective masses:
mass of CO2 + mass of H2O = 64.84 g
The molar masses of CO2 and H2O are calculated as follows:
CO2: 1 carbon atom (C) * atomic mass of carbon (12.01 g/mol) + 2 oxygen atoms (O) * atomic mass of oxygen (16.00 g/mol) = 44.01 g/mol
H2O: 2 hydrogen atoms (H) * atomic mass of hydrogen (1.01 g/mol) + 1 oxygen atom (O) * atomic mass of oxygen (16.00 g/mol) = 18.02 g/mol
Now, let's calculate the moles of CO2 and H2O produced:
moles of CO2 = mass of CO2 / molar mass of CO2
moles of CO2 = 44.01 g/mol
moles of H2O = mass of H2O / molar mass of H2O
moles of H2O = 18.02 g / 18.02 g/mol ≈ 3 mol
Since the moles of CO2 and H2O are equal to the moles of CH4 and C2H6, we can calculate the moles of C2H6:
moles of C2H6 = moles of CO2 - moles of CH4
moles of C2H6 = 3 mol - 0.836 mol ≈ 2.164 mol
Next, calculate the molar mass of C2H6:
2 carbon atoms (C) * atomic mass of carbon (12.01 g/mol) + 6 hydrogen atoms (H) * atomic mass of hydrogen (1.01 g/mol) = 30.07 g/mol
Now, calculate the mass of C2H6 in the mixture by multiplying the moles of C2H6 by its molar mass:
mass of C2H6 = moles of C2H6 * molar mass of C2H6
mass of C2H6 = 2.164 mol * 30.07 g/mol ≈ 65.05 g
Finally, calculate the fraction of CH4 in the mixture by dividing the mass of CH4 by the total mass of the mixture:
fraction of CH4 = mass of CH4 / (mass of CH4 + mass of C2H6)
fraction of CH4 = 13.43 g / (13.43 g + 65.05 g) ≈ 0.171
Therefore, the fraction of CH4 in the mixture is approximately 0.171, or 17.1%.