Lesson 2 - Gas Laws

The Gay-Lussac law relates the pressure and temperature of a fixed mass of gas at a constant volume. We owe Gay-Lussac a debt of thanks for his other contributions to brewing.

Joseph Gay-Lussac (1778-1850) was a French physicist and chemist. Our knowledge of brewing and the science behind it benefits from his work. This is not only because of his contribution to gas laws. He co-discovered the composition of water (the H₂O bit) and studied the properties of alcohol-water systems after which there is a unit named ‘degrees Gay-Lussac’. He also defined the overall chemical equation for fermentation. The latter we consider when we wish to define alcohol (and CO₂) production from fermentations in our brewing recipe designs.

The Gay-Lussac law states that the pressure (P) of a fixed mass of gas at a constant volume is proportional to its temperature (T). This is shown below:

Here, we are referring to absolute temperature measured in Kelvin.

So, what does this law suggest? If we trap a fixed mass of gas in a constant volume and we increase the temperature, the pressure will also increase. It is quite easy to think of an example of this by considering an empty aerosol can. The instructions on the can state we should not expose it to excessive heat even when it is empty. When the can is empty, it is not really empty. It still contains a small quantity of gas (either a propellant gas or the product) in a fixed volume (the can volume). So, if the can is heated excessively (increase in temperature), then the pressure in the can will increase until it potentially ruptures the can in an explosive manner. It is not advisable to try in practice!

If we rearrange the above scenario, it suggests that for a fixed mass of gas at a constant volume, the pressure P and temperature T before and after a change in temperature are related by:

Below is an example of the Gay-Lussac law in application.

Example:

A Cornelius vessel (or keg), often used to propagate quantities of yeast, can be configured as a sealed gas-tight vessel with a pressure relief valve set to relieve at a pressure of 9 bar g. If the vessel is initially filled with air at 100 kPa a and 15°C, then sealed and left in the sun, what temperature would the vessel contents have to reach before the pressure was relieved by the valve?

The Gay-Lussac Law states:

So, rearranging for T₂:

Expressing T₁ in Kelvin:

Hence T₂ is given as:

Note that we are converting the 9 bar g to an absolute pressure in Pa in the previous equation.

This is equivalent to 2,881.5 – 273.15 = 2,608.35°C, so we would be fairly safe leaving the vessel sealed in the sun without the relief valve lifting as we would never get to that temperature.

We have covered three common gas laws that deal with the relationship, for a fixed mass of gas, of the following:

• Pressure and volume: Boyle’s law – an inversely proportional relationship.

• Volume and temperature: Charles’ law – a proportional relationship.

• Pressure and temperature: Gay-Lussac’s law – a proportional relationship.

These three laws are often combined and expanded to include the mass of gas as an equation term, which is known as the universal gas law. We will look at this next.