Lesson 6 - End of Unit SAQ

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A mixed gas stream (70% by volume of nitrogen and 30% by volume of carbon dioxide) is used to pressurise and dispense a keg of beer. If the beer temperature was 5°C, what gas pressure would you choose to yield an equilibrium dissolved CO₂ concentration of 1.5 v/v (at STP)?

The Henry’s law constant for N₂ and CO₂ in beer is 54,000 bar a/mol fr and 900 bar a/ mol fr respectively, both at 5°C. Assume that the molecular mass for beer is the same as for water and the beer density is 1,000 kg/m³.

Response:

We can use Henry’s law for CO₂ to determine the partial pressure of CO₂ p, which would result in an equilibrium concentration of CO₂ x in solution:

First let’s determine the mol fraction x corresponding to the required 1.5 v/v at STP.

At STP (101,325 Pa and 273.15 K), the density of CO₂ is given by an application of the concept of molar volume. We know that the molar volume at STP is 22.4 m³ and we also know that the molar mass of CO₂ is 1 x 12 + 2 x 16 = 44 kg/kmol. Hence the density of CO₂ is given by:

Using 1 L of beer (approximated as water) as a basis, a dissolved CO₂ concentration of 1.5 v/v suggests that 1.5 L of CO₂ would be dissolved. Hence the mass of 1.5 L of CO₂ is given by:

Expressing this as a molar quantity by dividing the mass of CO₂ by the molar mass of CO₂:

Considering beer to be approximated as water with a density of 1,000 kg/m³, the mass of 1 L of water is 1 kg. Remember that the molar mass of water H₂O is 2 x 1 + 16 = 18 kg/kmol. The molar quantity is given by:

Hence the molar fraction x of CO₂ in water (beer) is given by:

We know from the problem statement that the Henry’s law constant is 900 bar a/mol fr, hence the partial pressure of CO₂ is given by:

Stating Dalton’s law as the following:

and introducing the molar fraction of CO₂ in the gas mixture (don’t get confused with the molar fractions in Dalton’s and Henry’s laws – we emphasised this in the learning lesson):

 is the molar fraction of CO₂ in the gas mixture. We know this to be equivalent to the volume fraction of 0.3. Hence the total gas pressure P_T required is given by: