To calculate the mole fraction of each component, we first need to find the total moles of the gas mixture.
Total moles = moles of methane + moles of ethane
Total moles = 0.5 moles CH4 + 0.2 moles C2H6
Total moles = 0.7 moles
Now we can calculate the mole fraction of each component:
Mole fraction of methane CH4 = moles of methane / total moles
Mole fraction of CH4 = 0.5 moles / 0.7 moles
Mole fraction of CH4 = 0.714 rounded to three decimal places
Mole fraction of ethane C2H6 = moles of ethane / total moles
Mole fraction of C2H6 = 0.2 moles / 0.7 moles
Mole fraction of C2H6 = 0.286 rounded to three decimal places
Now we can calculate the partial pressures of each component using Dalton's Law of Partial Pressures:
Partial pressure of methane P_CH4 = mole fraction of CH4 × total pressure
P_CH4 = 0.714 × 2 atm
P_CH4 = 1.428 atm
Partial pressure of ethane P_C2H6 = mole fraction of C2H6 × total pressure
P_C2H6 = 0.286 × 2 atm
P_C2H6 = 0.572 atm
The total pressure of the gas mixture is the sum of the partial pressures of each component:
Total pressure = P_CH4 + P_C2H6
Total pressure = 1.428 atm + 0.572 atm
Total pressure = 2 atm
So, the mole fractions of methane and ethane are 0.714 and 0.286, respectively, and the total pressure of the gas mixture is 2 atm.