Lesson 2 - Gas Laws

Charles’ law relates the volume and temperature of a fixed mass of gas at a constant pressure. We owe Jacques Charles some thanks for helping provide us with a brewing tool!

Jacques Charles (1746-1843) was a French inventor, scientist, and balloonist. Ballooning was probably the inspiration for his observations on the effect of temperature on gas volume (that is, after all, the principle of how balloons rise or fall). He also contributed to the development of something we have probably all used whilst brewing, and many still do – the hydrometer (and its derivatives calibrated to measure sugar or alcohol levels in solution).

Charles’ law states that the volume (V) of a fixed mass of gas at constant pressure is proportional to its temperature (T). This is shown below:

What does this mean? If we trap a mass of gas and keep it at the same pressure, then as its temperature increases the gas volume will also increase. This is harder to visualise in real life. As soon as we trap a volume of gas, e.g. in a balloon, and increase its temperature, the increasing temperature of the gas in the balloon would cause the balloon to expand thus increasing the volume of the gas, while the pressure of the gas remained the same (constant). Constant pressure is a condition of Charles’ law.

Imagine you were able to visualise a defined quantity (and mass) of air, and you held it loosely in your hand so that its pressure was at atmospheric pressure. If you then placed this quantity of air over a lit candle, what would happen?

• The temperature of the quantity of air would increase, and because of Charles’ law, its volume would also increase.

• As the quantity of air is not physically restrained, as its volume increases, its pressure would remain fixed.

• The quantity of air would have an increased volume, but it would also contain the same mass. What does this suggest about the density of the air? Well, if you remember, density is the ratio of mass to volume. So, an increase in volume would mean a decrease in density compared to the surrounding air, which was not heated by the candle. This less dense quantity of air would start to rise – in a similar way as a hot air balloon. This was an example of Charles’ law.

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

It is important to note that the temperature we deal with is a special form of temperature not measured in degrees Celsius (or centigrade).

We need to consider the absolute temperature, which is the scale of temperature that thermodynamicists and chemists deal with.

Simply put, the Celsius scale is relative to the freezing and boiling points of pure water at atmospheric pressure: 0 and 100°C. The absolute temperature scale offsets this scale so that the zero point, termed ‘absolute zero’ is set at -273.15°C. The reasons for choosing this temperature as absolute zero are complex, and we do not need to concern ourselves too much about them. They relate to a projected theoretical temperature at which the ‘thermodynamic energy’ of all substances would be zero. Absolute temperature has the units of Kelvin (K) (note the convention is not to use ‘degrees’ or the ‘°’ symbol). Let us briefly mention the person who the Kelvin is named after.

William Thomson (1824-1907) was an Irish-Scottish physicist and engineer who narrowed down the temperature of absolute zero to be approximately -273.15 °C. He was prolific in his field. We may ask why, if his name was Thomson, we do not call the units of absolute temperature, the Thomson. Well, Thomson was so prolific that he was honoured multiple times by the British Queen. At one point, he was titled Baron Kelvin; the Kelvin part referred to the name of the river in Glasgow, Scotland, which ran near his laboratory.

The Kelvin scale is simply calculated as follows:

Or:

Therefore, 0 °C is equivalent to 273.15 K, and 100°C is equivalent to 373.15 K and so on.

We need to ensure that the gas temperatures we use in Charles’ law, and all our gas laws, have been converted to the absolute temperature scale, measured in Kelvin.

Let us look at an example of how to use Charles’ law.

Example:

We can draw 10 mL of atmospheric air at 20°C into a gas-tight syringe. The syringe is sealed, placed in a fridge and allowed to reach a steady-state temperature of 4°C. By applying Charles’ law, what will be the volume recorded in the syringe, assuming the plunger can adjust its position to ensure that the air pressure inside the syringe remains at atmospheric pressure?

Charles’ law is stated as:

Rearranging for V₂:

We need both temperatures, 20°C and 4°C, expressed in Kelvin, so:

Hence V₂ is:

Despite the temperature dropping by (seemingly) 80% (20°C down to 4°C), the volume has not reduced by that amount (10 mL down to 9.45 mL, which is only 5.5% reduction). Why? It is because we should be considering the percentage reduction of absolute temperature (K) and not °C.

Charles’ law deals with the relationship between gas volume and temperature.

Let us now look at the relationship between the last combination of pressure, volume, and temperature: pressure and temperature.