Partial Pressure and Mole Fraction

Partial Pressure and Mole Fraction

Mole fraction is the ratio of the number of moles of one component in a mixture to the total number of moles in the mixture. In the case of a mixture of gases, the mole fraction of one particular gas would equal the number of moles of that gas divided by the total number of moles in the mixture.

gas-mixture-mole-fraction

The mole fraction of gas A in the mixture above would be:

mole-fraction-A

 

where

X = mole fraction

n = number of moles

Mole fraction can be used to determine partial pressure of a gas in a mixture of gases. The partial pressure of a gas in a mixture of gases is equal to the total pressure of the mixture times that mole fraction of that particular gas:

partial-pressure-A

 

where

Pgas A = partial pressure of gas A

XA = mole fraction of gas A

Ptotal = total pressure of the mixture of gases

 

Let’s look at some sample problems.

 

Sample Problem

A mixture of helium (8.00 g) and argon (40.0 g) in a container at 300. K has a total gas pressure of 0.906 atmosphere. What is the partial pressure of helium in the mixture?

We are given the total gas pressure of 0.906. We simply need to multiply this total pressure by the mole fraction of helium to get the partial pressure of helium.

First, we need to convert our masses to moles:

mol-He

 

mol-Ar

 

The mole fraction of helium is:

mole-fraction-He

 

Notice that there are no units associated with mole fraction, because it is a ratio and the unit mol cancels out.

The partial pressure of helium is equal to the product of the mole fraction of helium and the total gas pressure:

partial-pressure-He

 

The partial pressure of helium in this mixture is 0.604 atm

Sample Problem

A mixture of 4.00 g of O2 and 6.00 g of CH4 is placed in a 20.0 L vessel at 0 °C. What is the partial pressure of each gas, and what is the total pressure in the vessel?

First, we can convert the 4.00 g of O2 and 6.00 g CH4 to their respective number of moles by dividing by their molar masses. We can then plug the total number of moles of gases into the Ideal Gas Law to get the total pressure. Once we have the total pressure, we can multiply that by the mole fraction of each gas to get the partial pressure of each gas.

Converting mass to moles:

mol-O2

 

mol-CH4

 

The total number of moles of gas in the mixture is the sum of the number of moles of these two gases:

 

499mol

 

We will plug this value for ntotal into the ideal gas law:

ideal-gas-law-small

 

And solve for P (which in this case is the total pressure because we we will substitute the total number of moles of gas for n).

ideal-gas-law-P

 

The problem provided us with the following additional variables:

Given:

V = 20.0 L

T = 0 °C

We need to convert temperature into Kelvin:

27315K

 

Now we plug our values into the above equation:

560atm

 

The total pressure of the mixture is 0.560 atm. To solve for the partial pressure of each gas, we will multiply this total pressure by the mole fraction of each gas. Recall that mole fraction of one gas in a mixture is equal to the number of moles of that gas in the mixture divided by the total number of moles of gas in the mixture:

mole-fraction-A

 

The mole fraction of O2 would be:

251

 

The mole fraction of CH4 is:

749

 

Notice that the sum of the mole fractions of each gas in the mixture equals 1.

We will now multiply the total pressure obtained above by the mole fraction of each gas to get the partial pressure of each gas.

partial-pressure-A

 

mole-fraction-O2

 

mole-fraction-CH4

 

The sum of the partial pressures of each gas in the mixture should be equal to the total pressure. 0.141 atm + 0.419 atm = 0.560 atm.

PO2 = 0.141 atm, PCH4 = .0419 atm