#### Learning Objective

- Demonstrate an understanding of partial pressures and mole fractions.

#### Key Points

- The total pressure of a mixture of gases can be defined as the sum of the pressures of each individual gas: [latex]P_{total}=P_1+P_2+…+\;P_n[/latex] .
- The partial pressure of an individual gas is equal to the total pressure multiplied by the mole fraction of that gas.
- Boyle’s Law and the Ideal Gas Law tell us the total pressure of a mixture depends solely on the number of moles of gas, and not the kinds of molecules; Dalton’s Law allows us to calculate the total pressure in a system from each gas’ individual contribution.

#### Terms

- mole fractionnumber of moles of one particular gas divided by the total moles of gas in the mixture
- Dalton’s Law of Partial Pressuresthe total pressure exerted by the mixture of non-reactive gases is equal to the sum of the partial pressures of each individual gas; also known as Dalton’s Law of Partial Pressures

Because it is dependent solely the number of particles and not the identity of the gas, the Ideal Gas Equation applies just as well to mixtures of gases is does to pure gases. In fact, it was with a gas mixtureâ€”ordinary airâ€”that Boyle, Gay-Lussac, and Charles performed their early experiments. The only new concept we need to deal with gas mixtures is partial pressure, a concept invented by the famous English chemist John Dalton (1766-1844). Dalton correctly reasoned that the low density and high compressibility of gases were indicative of the fact that they consisted mostly of empty space; from this, it Dalton concluded that when two or more different gases occupy the same volume, they behave entirely independently of one another.

Dalton’s Law (also called Dalton’s Law of Partial Pressures) states that the total pressure exerted by the mixture of non-reactive gases is equal to the sum of the partial pressures of individual gases. Mathematically, this can be stated as follows:

[latex]{P}_{total} = {P}_{1}+{P}_{2}+…+\;{P}_{n}[/latex]

where P_{1}, P_{2} and P_{n} represent the partial pressures of each compound. It is assumed that the gases do not react with each other.

## Example

A 2.0 L container is pressurized with 0.25 atm of oxygen gas and 0.60 atm of nitrogen gas. What is the total pressure inside the container?

[latex]P_{total}=P_{O_2}+P_{N_2}=0.25+0.60=0.85\;\text{atm}[/latex]

The total pressure inside the contain is 0.85 atm.

## Calculating the Mole Fraction

The mole fraction is a way of expressing the relative proportion of one particular gas within a mixture of gases. We do this by dividing the number of moles of a particular gas *i* by the total number of moles in the mixture:

[latex]x_i=\frac{\text{number of moles }i}{\text{total number moles of gas}}[/latex]

## Example

A 3.0 L container contains 4 mol He, 2 mol Ne, and 1 mol Ar. What is the mole fraction of neon gas?

[latex]x_{Ne}=\frac{\text{number of moles Ne}}{\text{total number moles of gas}}=\frac{2}{4+2+1}=\frac{2}{7}[/latex]

The mole fraction of neon gas is 2/7 or 0.28.

## Calculating Partial Pressure

The partial pressure of one individual gas within the overall mixtures, *pi*, can be expressed as follows:

[latex]{P}_{i}={P}_{total}{x}_{i}[/latex]

where *xi* is the mole fraction.

## Example

A mixture of 2 mol H_{2} and 3 mol He exerts a total pressure of 3 atm. What is the partial pressure of He?

[latex]{P}_{He}={P}_{total}{x}_{He}=(3)\left(\frac{3}{5}\right)=\frac{9}{5}\text{atm}[/latex]

## Calculating Total Pressure

We know from Boyle’s Law that the total pressure of the mixture depends solely on the number of moles of gas, regardless of the types and amounts of gases in the mixture; the Ideal Gas Law reveals that the pressure exerted by a mole of molecules does not depend on the identity of those particular molecules; Dalton’s Law now allows us to calculate the total pressure in a system when we know each gas individual contribution.

## Example

Consider a container of fixed volume 25.0 L. We inject into that container 0.78 moles of N_{2} gas at 298 K. From the Ideal Gas Law, we can easily calculate the measured pressure of the nitrogen gas to be 0.763 atm.

We now take an identical container of fixed volume 25.0 L, and we inject into that container 0.22 moles of O_{2} gas at 298K. The measured pressure of the oxygen gas is 0.215 atm.

As a third measurement, we inject 0.22 moles of O_{2} gas at 298K into the first container, which already has 0.78 moles of N_{2}. (Note that the mixture of gases we have prepared is very similar to that of air. ) The measured pressure in this container is now found to be 0.975 atm.

Our data show that the total pressure of the mixture of N_{2} and O_{2} in the container is equal to the sum of the pressures of the N_{2} and O_{2} samples taken separately. We now define the partial pressure of each gas in the mixture to be the pressure of each gas as if it were the only gas present. Our measurements demonstrate that the partial pressure of N_{2 }as part of the gas PN_{2} is 0.763 atm, and the partial pressure of O_{2 }as part of the gas PO_{2}, is 0.215 atm.