To determine the percentage by mass of CH4 in the mixture, we need to calculate the moles of methane (CH4) and ethane (C2H6) present in the mixture.
Step 1: Calculate the moles of CH4 and C2H6:
To calculate the moles, we need to use the molar mass of each compound:
Molar mass of CH4 = 12.01 g/mol (C) + 4(1.01 g/mol) = 16.04 g/mol
Molar mass of C2H6 = 2(12.01 g/mol) + 6(1.01 g/mol) = 30.07 g/mol
Now we can calculate the moles using the given mass of the mixture:
Moles of CH4 = (mass of CH4 in the mixture) / (molar mass of CH4)
Moles of C2H6 = (mass of C2H6 in the mixture) / (molar mass of C2H6)
However, we don't know the individual masses of CH4 and C2H6 in the mixture, so we need to find them.
Step 2: Assume a total mass for the mixture:
Let's assume the total mass of the mixture is 100 g.
Step 3: Calculate the masses of CH4 and C2H6 in the mixture:
To calculate the masses, we can assume that the percent composition of CH4 is x%, and the rest (100-x)% is C2H6.
Mass of CH4 = (x/100) * 100 g
Mass of C2H6 = ((100 - x)/100) * 100 g
Step 4: Calculate the moles of CH4 and C2H6 using the assumed masses:
Moles of CH4 = (mass of CH4) / (molar mass of CH4)
Moles of C2H6 = (mass of C2H6) / (molar mass of C2H6)
Step 5: Use the combustion reactions and enthalpies to calculate heat released by CH4 and C2H6:
The balanced combustion reactions for CH4 and C2H6 are:
CH4 + 2O2 → CO2 + 2H2O
C2H6 + 7/2O2 → 2CO2 + 3H2O
The heat released by CH4 can be calculated using its standard enthalpy of combustion:
Heat released by CH4 = (moles of CH4) * (standard enthalpy of combustion of CH4)
The heat released by C2H6 can be calculated using its standard enthalpy of combustion:
Heat released by C2H6 = (moles of C2H6) * (standard enthalpy of combustion of C2H6)
The total heat released by the mixture will be the sum of the heat released by CH4 and C2H6:
Total heat released = Heat released by CH4 + Heat released by C2H6
Step 6: Set up the equation using the given total heat released and solve for x:
Total heat released = (x/100) * (moles of CH4) * (standard enthalpy of combustion of CH4) + ((100 - x)/100) * (moles of C2H6) * (standard enthalpy of combustion of C2H6)
Given: Total heat released = -369 kJ
Standard enthalpy of combustion of CH4 = -890.3 kJ/mol
Standard enthalpy of combustion of C2H6 = -1569.7 kJ/mol
Solve the equation for x to find the percentage of CH4 in the mixture.
For a more automated approach, you could use a software tool like a spreadsheet or a programming language to plug in the values and calculate the percentage.