Think of a cake. When you put it in the oven, it starts off at a particular volume and then, an hour later, it has risen to perhaps double its size. It is obvious what has happened – the air bubbles that you have carefully folded into the mixture during the preparation and the little bubbles of carbon dioxide created by the baking powder have expanded as they are heated in the oven, taking the rest of the cake with it. All this time, the pressure of the air inside those bubbles has stayed the same (you know that because cakes don’t usually explode when you slice them after cooking).
It is an intuitive idea that bubbles of air will expand if you heat them, as long as the pressure remains constant. It is also a fundamental component of the ideal gas laws, first written down in the early 19th century by the French natural philosopher Joseph Louis Gay-Lussac. He was working on the relationship between the volume and temperature of a gas, building on work carried out several decades earlier by the inventor and mathematician Jacques Charles who had shown that volume and temperature were proportional – heat a gas and its volume will increase and vice versa, as long as the pressure remains constant.
The first piece of the ideal gas puzzle came in the 17th century. Robert Boyle had been carrying out experiments with air, which he proposed was full of particles connected by tiny invisible springs. He found that the pressure of a gas had an inverse relationship to its volume. If the volume doubled, its pressure halved and vice versa, when the temperature is held constant.
As well as the volume/temperature relationship, Gay-Lussac extended the work and experiments, from a century earlier, of the inventor Guillaume Amontons to show that, in a fixed volume of gas, pressure was directly proportional to absolute temperature.
With the three relationships between pressure, volume and temperature measured and written down, French engineer Benoît Paul Émile Clapeyron, one of the founding fathers of thermodynamics, combined the work of Boyle, Charles and Gay‑Lussac into the combined ideal gas equation above, in 1834.
In short, the ideal gas law shows the relationship between the four properties of a gas that you need to know in order to predict how it will behave: pressure, temperature, volume and the number of particles of gas (ie atoms or molecules) present. It is “ideal” because the law is a model that assumes the particles are infinitely small points and do not interact with each other. All collisions between ideal gas particles are elastic, which means they do not lose any energy when they rebound off each other.
In practice, real gas particles do have measurable sizes and sometimes attract or repel each other. Nevertheless, the ideal gas equation is a highly successful way to understand how gases shift and change depending on their surroundings.
The law states that the product of the pressure (P) and volume (V) of a gas is directly proportional to its absolute temperature (T, measured in kelvin). On the right-hand side of the equation is the number of moles of gas present (n) in the system, where a mole is equal to 6.02214129×1023 particles, a number known as the Avogadro constant. Also on the right is the universal gas constant (R), equal to 8.3145 joules per mole kelvin.
The ideal gas equations can be used to work out how much air inside a cake will expand (though it’s unlikely to be used for that) but it also applies to plenty of other situations. Ever noticed a bicycle pump get hot when you fill a tyre with air? That’s because you’re quickly putting energy into the air inside the pump by pushing the piston and reducing the volume at the same time, which causes the molecules to bounce around faster in a smaller volume and the gas heats up.
Refrigeration works in the opposite way to the bicycle pump. If you release a gas very quickly from high pressure (inside a storage tank, say) to a region of lower pressure (outside air at atmospheric pressure), then the gas will expand. The energy required to do this will come from the molecules of gas themselves and so the overall temperature of the gas will drop. You can see this in action when pressurised carbon dioxide inside a fire extinguisher turns instantly into a frost when it is released through the nozzle and on to a fire. More prosaically, the same mechanism keeps your food cold in a refrigerator.
The relationships between these so-called “state properties” of a gas make sense intuitively.
But the ideal gas law can also be derived mathematically, from first principles, by imagining particles bouncing around a box. About two decades after Clapeyron wrote it down, August Krönig and Rudolf Clausius independently looked at the statistical distribution of speeds (and hence energy) among the particles to work out how pressure, volume and temperature related to each other in a gas – an approach known as statistical mechanics. In essence, this meant looking at the properties of huge numbers of tiny components or particles inside a system in order to calculate the macroscopic results. In other words, a box containing a gas will have trillions of particles flying around inside it in random directions, bouncing off each other and off the walls.
In this model, the kinetic energy of the particles is proportional to the temperature of the gas. Particles hitting the sides of the box translate in to the pressure of the gas.
In this “kinetic” version of the ideal gas law, the right-hand side is written slightly differently. Instead of “nR” are terms for the number of molecules in the gas and the Boltzmann constant (k) equal to 1.38065×10-23 joules per kelvin.
The two versions of the equation describe identical things. Whether it is cakes, bicycle pumps, refrigerators or even when modelling the behaviour of stars (which are, in essence, just clouds of hydrogen gas), you can use these simple relations work out how what the gases are doing.
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