Lesson 2 - Gas Laws

As we defined above, a molar fraction (sometimes called a mol or mole fraction) is defined as the ratio of the number of moles of a specific gas to the total number of moles of gas in the mixture. So, in a CO₂/N₂ mixture of 100 kmols, a CO₂ mole fraction of 0.25 would mean that there were 0.25 x 100 = 25 kmols of CO₂ and 75 kmols of N₂. This seems simple enough, but what if the gas mixture composition is given as a volume fraction, and not as a molar fraction? This is actually quite common, so how do we convert to a molar fraction to use Dalton’s law?

In order to convert from a molar fraction to a volume fraction, and vice versa, we use the concept of molar volume, which we described previously as:

The molar volume suggests that the volume of 1 kmol of an ideal gas at STP is 22.4 m³.

The implication of the concept of molar volume is that in a gas mixture at any pressure and temperature, the volume fraction (v_i/v_T) of any gas in the mixture is equal to the molar fraction x_i. So, we have an easy conversion:

Example:

10 m³ of a gas mixture of CO₂ and N₂ used for dispensing draught lagers has an N₂ mole fraction of 0.3. What are the respective volumes of CO₂ and N₂ in the mixture?

We know that the molar fractions and volume fractions of an ideal gas mixture are equal:

Consequently:

Therefore:

And similarly, for CO₂:

Now let us consider a worked example of Dalton’s law in action.

Example:

Two differing dispense gases are being considered to dispense a lager: 100% CO₂ and a mixture of 30/70% (v/v) N₂/CO₂. If we require a CO₂ partial pressure of 0.8 bar g irrespective of which gas is used, would we use the same gas pressure for each gas? If not, which pressures would we use? Atmospheric pressure is 100 kPa.

The first part of the example is asking whether we would use the same gas pressure for either dispense gas to achieve a CO₂ partial pressure of 0.8 bar g. For the 100% CO₂ dispense gas, as there is only one gas present, the concept of partial pressure may be a little confusing but as the molar fraction of CO₂ in this gas is 1.0, then the total gas pressure is the CO₂ partial pressure.

If we applied the same gas pressure for both dispense gas options, then the CO₂ partial pressure would be less for the 30/70 gas mixture (as the volume/molar fraction of CO₂ is 0.7). We would have to apply a higher gas pressure for the 30/70 gas mixture to achieve the same partial pressure as the 100% CO₂ gas option.

For the 100% CO₂ dispense gas option, a gas pressure of 0.8 bar g would give an effective partial pressure of 0.8 bar g. This would be equivalent to an absolute pressure of 0.8 + 1 = 1.8 bar a.

For the 30/70% (v/v) gas, we can express Dalton’s law as the following:

And introducing the molar fraction of CO₂:

Remember that the molar fraction is equivalent to the volume fraction so:

We should note here that we have expressed the required CO₂ partial pressure as an absolute pressure.

Therefore, P_T is:

We can see that the required total gas pressures, to achieve the same CO₂ partial pressures are quite different (1.8 bar a vs 2.57 bar a).

We should keep this concept of partial pressure and how it is influenced by the composition of a gas mixture at the front of our mind, as we will see in a later lesson, how important it is in getting a gas to dissolve into a liquid.