Example 3.1 Problem Statement:
A mixture of gases has the following composition by mass:
O_{2}  16.0% 
CO  4.0% 
CO_{2}  17.0% 
N_{2}  63.0% 
To reach mass fractions from mass percent composition, we would move the
decimal place over two places. For example, the mixture is 16%
O_{2}, or we could say the mass fraction of O_{2} in the
mixture is 0.16. Determine the molar composition, or mole fractions, of this mixture. (hint:
you first need to find the total moles of the mixture)
To solve Example 3.1:
To simplify our calculations, we will begin by assuming a basis.
Both the initial conditions and the final answer are fractions or percents, so we are allowed to assume a basis.
Mass fractions are provided in the problem statement, so we will assume a basis of 100 g.
This means that we are assuming that there are 16.0 g O_{2}, 4.0 g CO,
17.0 g CO_{2}, and 63.0 g N_{2} in the system.
As the hint suggested, the first step is to determine the total moles in the system.
To find this, we will calculate the amount of moles of each compound in the system and add them together.
First, the amount of moles of each compound can be calculated using their molecular weights:
n_{O2} = 16.0 g O_{2} x (1 mol O_{2} / 32.0 g O_{2}) = 0.5 mole O_{2}
n_{CO} = 4.0 g CO x (1 mol CO / 30.0 g CO) = 0.125 mol CO
n_{CO2} = 17.0 g CO_{2} x (1 mol CO_{2} / 44.0 g CO_{2}) = 0.3864 mole CO_{2}
n_{N2} = 63.0 g N_{2} x (1 mol N_{2} / 34.0 g N_{2}) = 1.8529 mole N_{2}
Now, the total moles can be calculated:
n_{TOTAL} = 0.5 moles + 0.125 moles + 0.3864 moles + 1.8529 moles
n_{TOTAL} = 2.8643 moles
Remember, the mole fraction, x_{A}, is defined in Equation 2 as:
x_{A} = n_{A} / n_{TOTAL}
Finally, to determine the mole fractions, we can use this equation with the calculated values for the amount of moles for each compound and the total moles:
x_{O2} = (0.5 moles O_{2}) / (2.8643 moles Total)
x_{O2} = 0.1746
x_{CO} = (0.125 moles O_{2}) / (2.8643 moles Total)
x_{CO} = 0.0436
x_{CO2} = (0.3864 moles O_{2}) / (2.8643 moles Total)
x_{CO2} = 0.1349
x_{N2} = (1.8529 moles O_{2}) / (2.8643 moles Total)
x_{N2} = 0.6469
To make sure our answer is reasonable, add the fractions together to see if we get 1.0:
x_{TOTAL} = 0.1746 + 0.0436 + 0.1349 + 0.6469
x_{TOTAL} = 1.0
The fractions add up to 1.0, so our answer is reasonable.
