Lesson 4 - Gas Transfer

Let’s do a worked example of Henry’s law in action.

Deoxygenated and carbonated blending water is stored in a tank at 2°C. Prior to entering the tank, the water is saturated with CO₂ to a level of 4 g/L. What would the required (100% purity) CO₂ headspace pressure be to ensure the dissolved CO₂ concentration doesn’t change?

The Henry’s law constant, H, for CO₂ and water at 2°C is 80,000 kPa a/mol fr. Water density can be taken as 1,000 kg/m³.

All CO₂ involved can be considered to be 100% pure.

So, restating Henry’s law:

Where

p is the partial pressure of the gas above the solution

H is the Henry’s law constant

x is the molar fraction of the gas dissolved in the liquid

As we are dealing with a pure headspace gas, we know that the headspace pressure and partial pressure are the same, so our unknown variable is p!

We were given the Henry’s law constant, H, (with units kPa a/mol fr) so we just need to make sure our units of x, the molar fraction, match. So, what do we know about the amount of CO₂ dissolved in the water? We were told that the concentration is 4 g/L. (Read this back to yourself and make sure you understand it’s 4 g of CO₂ dissolved in 1 litre of water.)

We know that a molar fraction is the ratio of the number of moles (or kmoles) of substance A, to the total number of moles (or kmoles) in the mixture. We’ve considered molar fractions in gas mixtures previously, however, in this case, we’re applying the same concept of a ratio of the number of moles, but we’re considering a dissolved gas in a liquid as our mixture. Don’t get confused! As a result, we need our molar fraction to be given by:

This can also be represented as:

So, let’s look at converting a dissolved gas concentration of 4 g/L into a molar fraction:

Converting 4 g of CO₂ into kmoles:

We’ve previously worked out that the molecular mass of CO₂ is 12 + 2×16 = 44 kg/kmol. So,

Now, let’s move on to how many kmoles there are in 1 litre of carbonated water? Well, we know that carbonated water would be composed of water and dissolved CO₂, so there are two components. We’ve just worked out how many kmoles of CO₂ there are, so now we just need to calculate the number of kmoles of 1 litre of water.

Mass of 1 L of water:

And the molecular mass of water is 2×1 + 16 = 18 kg/kmole.

Hence

Hence the molar fraction of dissolved CO₂ in water is,

Using Henry’s law to solve for the required headspace pressure:

Are you wondering why is this pressure in kPa a and not Pa? Check the units of the Henry’s law constant! (kPa a/mol fr)

So, we’d need a headspace pressure of pure CO₂ of 1.3 bar a (or 0.3 bar g) to hold the dissolved CO₂ level at saturation.