# Gas State Equations

Created | Updated Oct 30, 2006

The gas state equation is hidden in many common everyday thermodynamic observations, like for instance in the weather forecast. It is used to calculate certain properties of gases, namely: Temperature (*T*), pressure (*p*) and volume (*V*). These properties together are called the *state* of a gas. The most common form of the gas equation is the equation for an ideal gas (see below). An ideal gas is a gas consisting of point-like particles which do not interact with each other (ie, they do not stick together or repel each other). This is the formula taught in school. The more complicated looking approximations for real gases are not taught in school. Due to the stupefying variety of existing gases and to the unthinkably complicated ways they behave, there's no final exact formula. Even the best formula, the virial state equation for a gas, is merely an approximation.

### Thermodynamics and Gases

Scientists have been messing about with thermodynamics ever since they've been rubbing sticks together to generate fire. Thermodynamics is that part of science devoted to the study of matter under the influence of heat ('thermo'), and their behaviour with time ('dynamic'). Or put more bluntly: what happens if things get heated up?

One particular area of interest is the thermodynamic behaviour of gases. It is very easy to measure the temperature, the pressure and the volume of a gas. The thermodynamic behaviour of liquids and solids cannot be observed in such a straightforward manner, because you need huge pressures, and the heat moves slowly and in complicated ways in these materials. The thermodynamic behaviour of liquids and solids is very, very, difficult to characterise.

The aforementioned properties (pressure, volume and temperature) characterise the *state* of a gas. Adding heat will change the temperature, pressure or volume, putting the studied gas into another state. In the engine of a car, for example, in the cylinder cavity the thermodynamical study of the involved states (during compression and expansion) will tell how much heat or work can be gained from the combustible gas mixture. (The determination of the involved energies is the main task of thermodynamical studies.) The fraction of energy that can be converted into work is called the *degree of efficiency*. Slightly higher efficiencies can have huge effects in fuel-saving, which is reflected in lower freight costs. The gas state equation is the centrepiece of thermodynamic calculations like this.

### The Ideal Gas

The ideal gas equation looks like this:

p·V=n·R·T

This formula is really very easy to understand and to memorise. *p* is the pressure (in N/m^{2}) of the gas, *V* is its Volume (in m^{3}), *n* the amount of gas particles (in mol)^{1} and *T* is the Temperature (in K)^{2}. In order for the units to equate, and for the numerical value to be correct, the constant R has been introduced to the equation. R is also called the *gas constant*. Its value was empirically determined to be R = 8.314 [N·m/K·mol].

The state equation for an ideal gas is an empirical formula which was derived after years and years of experiments. The famous experiments were conducted by Boyle, Mariotte, Gay-Lussac, Charles and some others between 1650 and 1820. At that time nobody knew about the real nature of matter (ie, that stuff is made out of atoms). For that reason their calculations and conclusions nowadays sound very empirical, far-fetched or out of thin air. Nevertheless the ideal gas equation is a good first approximation.

**The Law of Boyle (and Mariotte)**- Robert Boyle (1627-1691) (great scientist and co-founder of the Royal Society) found out in 1662, after conducting very meticulous measurements, that the product of pressure and volume for a given mass of a gas is constant (*p·V*= const.). Frenchman Edme Mariotte (1620-1684) independently came to the same conclusion, but he only published his conclusions in 1672.**The Law of Gay-Lussac (and Charles)**- In the 1770s, a Frenchman by the name of Jacques Charles (1746-1823) found out that pressure is directly proportional to temperature (*p*/*T*= const.) but never published his results. In the early 1800s Joseph Louis Gay-Lussac (1778-1850) was using and teaching this formula which he himself called 'Charles Law'. Years later, though, because it was Gay-Lussac who made it famous, people preferred to call it the 'Gay-Lussac Law'^{3}.

Adding those two formulae together results in *pV*/*T* = const. The constant was found to be a product of *n* (which is constant and proportional to the mass) and R, the gas constant. Reformulating the equation yields the state equation for an ideal gas as seen above.

### The van der Waals Gas Equation

The van der Waals gas equation is named after Johannes Diderik van der Waals (1837-1923), a very earnest-looking Dutch man who did lots of experiments on gases and decided to give the ideal gas formula a serious rethinking, which eventually earned him a Nobel prize. Now, this is what he came up with:

(p+ b·n^{2}/V^{2})(V- a·n) =nRT

It looks a little more complicated, but it's not really difficult to understand. The right side of the equation looks the same as in the state equation for an ideal gas. The modifications on the left side of the equation are explained as follows: every mol of particles of the gas has a tiny finite volume of *a*, with *n* mols of molecules this makes a total amount of (*a*·*n*). This amount has to be subtracted from the overall volume *V*, hence the term (*V*-*a*·*n*). The other modification is due to the particles sticking^{4} to each other, a phenomenon called cohesion: This sticking is proportional to (*n*/*V*)^{2} and it reduces the effective measured pressure *p*. For that reason it must be compensated for in the formula by adding the term *b*·(*n*/*V*)^{2} to the pressure.

The van der Waals equation is also merely an approximation, and the values of *a* and *b* were determined experimentally for every gas. The constants *a* and *b* are sometimes called the van der Waals constants. Some values (b is given in m^{6}·Pa/mol^{2}; a in m^{3}/mol):

Air- b=1.34·10^{-9}; a=3.5·10^{-6}Water(Vapour) - b=5.39·10^{-9}; a=3.04·10^{-6}Acetone- b=13.72·10^{-9}; a=9.94·10^{-6}

### The Virial Equation - as Accurate as you Want

The virial equation^{5} is the most exact formula for a gas; it is totally empirical, based on curves that were found to fit the data of real gases, which are measured experimentally. It was brewed up by another Dutch man named Heike Kamerlingh-Onnes (1853-1926, Nobel 1913), who was intensively working with Helium at extremely low temperatures, where the other gas state equations won't work properly. His formula looks like this:

p·V=n·R·T+ B·p+ C·p^{2}+ D·p^{3}+ E·p^{4}+ ...

One can immediately recognizee the original gas equation followed by a lot of summands, which are nothing but correction factors to adjust the curve to experimental data. The coefficients B, C, D, etc, are called the *virial coefficients*. They are not constant; their value varies with temperature. For that reason one has to make sure one is using the correct coefficient for the right range of temperature. The values of the constants after B are so small that they are commonly neglected. The value of B can be found in phone-book like tables.

### Summary

The state of a certain amount of a gas (or a gas mixture) is given by its temperature, pressure and volume. The formula used to calculate all these properties is called the state equation. The state equation is the centrepiece for thermodynamical reasoning. There are basically three forms of the state equation for a gas. One for an ideal gas, which is easy but slightly inaccurate, since the gases' behaviours are idealised. Another one for real gases, where certain effects for each gas are compensated by factors within the formula. The most widely used form of a state equation for a real gas is the van der Waals equation. Most of the gases have been evaluated according to this formula, therefore the values for these factors can be found in tables. The third form is the 'virial' equation, which is specific for one type of gas within a certain temperature range. This formula uses experimentally found parameters to fit the curve to the experimental data. These parameters can be found in tables. The virial equation is somewhat more difficult to handle from a mathematical point of view, but within a specific temperature range it can be as accurate as desired.

As one can imagine, these equations do not fully satisfy the scientists' need for hypotheses and theories. Even if they can have it as accurate as they want, they will feel kind of cheated. For that reason some scientists came up with another approach in the last half of the 19th Century: instead of measuring big volumes and compensating with parameters, they were trying to calculate the big effects atom per atom... the kinetic gas theory. But that's stuff for a further entry.

^{1}One 'mol' is equivalent to saying '6.022·10

^{23}particles'. People use this unit to avoid having to use this huge number every time. A more detailed explanation on the usage of mols can be found in the entry on the periodic table of the elements.

^{2}Attention: beware of the units. Lack of attention to the units is widely used to trick students in tests. Normally it's the pressure, which is sometimes given in 'atm' instead of 'Pa' (1 Pa = 1 N/m

^{2}= 9.8692·10

^{-6}atm).

^{3}Both Charles and Gay-Lussac were enthusiastic balloonists, Charles was on the second flight of a balloon, and Gay-Lussac maintained the altitude record of 6000 m for 50 years.

^{4}Real gas particles do not behave like snooker balls. Instead of sharply colliding the particles deform a little, reducing the transmitted kinetic energy. Snooker ball-like collisions of ideal gas particles are termed 'elastic' and the deforming real gas collisions 'inelastic'.

^{5}'Virial' comes from the Latin word

*vires*which means 'forces'.