Introduction:

The ideal gas law is a much clean, efficient manner to deal with the gases. Unluckily, the two fundamental assumptions of the ideal gas law, that molecule are point masses and that they don't attract, are ideals. In fact, each and every molecule consists of a volume and attracts other molecules, to certain extent. At low pressure and high temperature, this effect is almost negligible. As the pressure increases and temperature drops, though, the behavior of real gases strays from the ideal. At extremes temperature and pressure, the attractive forces and proximity might even force the gas into a liquid.

Define: The gases that exist are real gases like Hydrogen, Nitrogen, Oxygen, He, F₂, Ne, CO₂, CO and so on. Real gases don't obey gas laws at all pressures and temperatures. Real gases deviate from the ideal behavior at high pressure and low temperature. There exists small force of attraction among the molecules of real gases.

As the Kinetic-Molecular theory does an exceptional job describing gases, there are a few properties it doesn't describe regarding its explanation of real gases. The theory supposes that the collisions between gas molecules and the walls of a container are perfectly elastic, gas particles don't encompass any volume, and there are no repulsive or attractive forces among the molecules.  These suppositions pertain to Ideal Gases.  As these assumptions usually hold true, there are conditions where gases deviate from the ideality.

The gases tend to perform ideally in two different conditions.  Firstly, they act ideally at high Temperatures. This is because of the reason of the fact that molecules are flying past one other at extremely high speeds. Secondly, they act ideally at Low Pressures. This is due to the reason that at low pressures, the volume of the molecules tends to become negligible in comparison to the net volume of the gas (Keep in mind the Boyle's Law, that Pressure and Volume are inversely proportional).

Therefore, if this is whenever they act ideally, when do they behave non-ideally?  Gases act non ideally at cold temperatures as the fact that at cold temperatures, molecules are moving relatively slowly past one other, allowing for the repulsive and/or attractive forces among molecules to take effect, deviating from the Ideal Gas. Gases as well behave non-ideally at high pressures, since at high pressures; the volume of molecules becomes a factor.

Limitations of the Ideal Model:

For most of the applications, the ideal gas approximation is reasonably correct; the ideal gas model tends to fail at lower temperatures and higher pressures, though, whenever intermolecular forces and the excluded volume of gas particles become important. The model as well fails for most heavy gases (comprising many refrigerants) and for gases having strong intermolecular forces (like water vapor). At some point of combined low temperature and high pressure, real gases experience a phase transition from the gaseous state to the liquid or solid state. The ideal gas model, though, doesn't explain or allow for phase transitions; these should be modeled through more complex equations of state.

Real-gas models should be employed close to the condensation point of gases (that is, the temperature at which gases start to form liquid droplets), close to critical points, at very high pressures, and in other less common cases. Some of the various models mathematically explain the real gases.

Real gases and the molar volume:

To describe the slight differences between the numerical properties of ideal and real gases at normal temperatures and pressures consider the given comparison. We will remember that we used the ideal gas equation to work out a value for the molar volume of the ideal gas at STP (that is, standard temperature and pressure).

If we are familiar with the density of a gas at a specific temperature and pressure, it is very simple to work out its molar volume. For illustration at 273 K and 1 atmosphere pressure, the density of helium is 0.1785 g dm^-3. That implies that 0.1785 g of helium occupies 1 dm³ at STP. This is a quite simple sum to work out what 1 mole of helium, He, would occupy.

The mass of 1 mole of He is 4 g and would occupy 4/0.1785 dm³ = 22.4 dm³.

That is the same (at least to 3 significant figures) as the ideal gas value, recommending that helium acts as an ideal gas in these conditions. If you do this for an arbitrary sample of other gases, we get these values (to 3 significant figures) for the molar volume at STP (that is, 273 K and 1 atmosphere pressure).

            Density (g dm^-3)   Molar volume at STP

He         0.1785                        22.4

N₂         1.2506                        22.4

O₂         1.4290                        22.4

CH₄       0.717                          22.3

CO₂       1.977                          22.3

C₂H₄      1.260                          22.2

NH₃       0.769                           22.1

SO₂        2.926                          21.9

Therefore, however for simple computation purposes we make use of the value 22.4 dm³ for all gases, we can observe that it is not exactly correct. Even at ordinary pressures and temperatures, real gases can deviate slightly from the ideal value. The effect is much greater in more extreme conditions.

Real gas law:

The volume of a real gas is generally less than what the volume of an ideal gas would be at similar pressure and temperature; therefore, a real gas is stated to be super compressible. The ratio of the real volume to the ideal volume, that is a measure of the amount that the gas deviates from perfect behavior, is known as the super compressibility factor, at times shortened to the compressibility factor. This is as well known as the gas deviation factor and represented by the symbol 'z'. By definition, the gas deviation is the ratio of the volume in reality occupied through a gas at a given temperature and pressure to the volume it would occupy if it acts ideally, or:

z = Actual volume of gas at specified T and p/Ideal volume of gas at same T and p

Note that the numerator and denominator of the above equation refer to the similar mass. (This equation for the 'z' factor is as well used for liquids.) Therefore, the real gas equation of state is written as:

pV = znRT

The gas deviation factor, 'z', is close to 1 at low pressures and high temperatures that means that the gas acts as an ideal gas at these conditions. At standard or atmospheric conditions, the gas 'z' factor is for all time around 1. As the pressure rises, the 'z' factor first reduces to a minimum that is around 0.27 for the critical pressure and critical temperature. For temperatures of 1.5 times the critical temperature, the minimum 'z' factor is around 0.77, and for temperatures of double the critical temperature, the minimum 'z' factor is 0.937. At high pressures, the 'z' factor rises above 1, where the gas is no longer super compressible. At such conditions, the particular volume of the gas is becoming so small, and the distance among molecules is much smaller, in such a way that the density is more strongly influenced by the volume occupied via the individual molecules. Therefore, the 'z' factor carries on increasing above the unity as pressure increases.

Compressibility factors for the mixtures (or unknown pure compounds) are measured simply in a Burnett apparatus or a variable-volume PVT equilibrium cell. Gas deviation factor, 'z', is found out by measuring the volume of sample of the natural gas at a particular temperature and pressure, then measuring the volume of similar quantity of gas at atmospheric pressure and at a temperature adequately high in such a way that the hydrocarbon mixture is in the vapor state. The tables of compressibility factors are available for most of the pure gases as functions of pressure and temperature. Excellent correlations are as well available for the computation of compressibility factors. For this cause, the compressibility factors are no longer regularly measured on dry-gas mixtures or on most of the leaner wet gases. Rich-gas or condensate systems need other equilibrium studies, and compressibility factors can be achieved regularly from these data.

What causes non-ideal behavior?

In the expression of compression factor, pV / nRT, everything on the bottom of the expression is either recognized or can be measured precisely. However that is not true of volume and pressure. In the suppositions we make regarding ideal gases, there are two statements that state things which cannot be true of a real gas, and these encompass an effect on both the volume and pressure.

The volume problem:

The kinetic theory supposes that, for an ideal gas, the volume taken up via the molecules themselves is completely negligible as compared by the volume of the container. For a real gas, that supposition is not true. The molecules themselves do take up a part of the space in the container. The space in the container vacant for things to move around in is less than the calculated volume of the container.

This dilemma gets poorer, the more the gas is compressed. Whenever the pressure is low, the volume taken up via the real molecules is insignificant compared by the net volume of the container. However as the gas gets more compressed, the proportion of the net volume that the molecules themselves take up gets high and high. You could assume compressing it so much that the molecules were in reality all touching one other. At that point the volume accessible for them to move around in is zero.

The pressure problem:

The other key supposition of the Kinetic Theory for ideal gases is that there are no intermolecular forces among the molecules. That is wrong for each and every real gas. If there weren't any intermolecular forces then it would be not possible to condense the gas as a liquid. Even helium, having the weakest of all intermolecular forces, can be turned to liquid if the temperature is low adequate.

The van der Waals Equation:

There are a great number of potential equations which can depict how real gases act, however to keep it simple, chemists stick to the van der Waals Equation as it is the simplest to depict how gases behave:

[P + (an²/V²)] [V - nb] = nRT

As this seems far more intimidating that PV = nRT, remember that it is very identical to the ideal gas equation, however corrects for some things.

Here,

'P' is the Pressure in atmospheres (atm)

'V' is the Volume in Liters (L)

'n' is the Number of moles (mol)

'R' is the gas Constant and the value is 0.08206L (L*atm/mol*K)

'T' is the absolute Temperature in Kelvin (K)

Now, what about this an²/V²? Well, this is added to the pressure in order to account for the intermolecular forces of attraction.  The 'a' is a value which accounts for the amount of attraction between each and every particle.  This is multiplied via the amount of moles squared, n², as the total amount of attractive forces is based on how much of the gas is there. This is then divided by V² and added to pressure as the measured pressure is lower than estimated.

The value 'b' is the omitted volume per mole, and is associated to the volume of the gas moles. This value should be included as real gases encompass volume.  Thus, the measured volume comprises the volume of the molecules as well.  In order for the equation to be precise, the volume per mole, nb, should be subtracted from the measured volume to stand for the available volume in the gas.

'a' and 'b' are the constants for any specific gas (and even differs slightly by temperature and pressure), however they differ from gas to gas to allow for the different intermolecular forces, and molecular sizes. That signifies that, unluckily, you no longer encompass a single equation which you can use for any gas. Luckily, though, the ideal gas equation works well adequate for most of the gases at ordinary pressures, as long as the temperature is logically high.

Redlich-Kwong Model:

The Redlich-Kwong equation is the other two-parameter relation which models real gases. This is almost for all time more precise than the van der Waals equation and often more accurate than several equations having more than two parameters. The equation is:

RT = P(V_m - b) + a. [(V_m - b)/V_m.(V_m + b).T^1/2]

It will be noted that 'a' and 'b' here are stated differently than in the van der Waals equation.

Additional models which can be applied to the non-ideal gases comprise the Berthelot model, the Dieterici model, Virial model, Clausius model, Wohl model, Peng-Robinson model, Beattie-Bridgeman model and the Benedict-Webb-Rubin model. Though, these systems are employed less frequently than are the van der Waals and Redlich-Kwong models.

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