# Behaviour of real gases

### Deviation from ideal gas behaviour

A gas which obeys the gas laws and the gas equation PV = nRT strictly at all temperatures and pressures is said to be an ideal gas. The molecules of ideal gases are assumed to be volume less points with no attractive forces between one another. But no real gas strictly obeys the gas equation at all temperatures and pressures. Deviations from ideal behaviour are observed particularly at high pressures or low temperatures. The deviation from ideal behaviour is expressed by introducing a factor Z known as compressibility factor in the ideal gas equation. Z may be expressed as Z = PV/nRT

- In case of ideal gas, PV = nRT ∴ Z = 1
- In case of real gas, PV ≠ nRt ∴ Z ≠ 1

Thus in case of real gases Z can be <1 or > 1

- When Z < 1, it is a negative deviation. It shows that the gas is more compressible than expected from ideal behaviour.

(ii) When Z > 1, it is a positive deviation. It shows that the gas is less compressible than expected from ideal behaviour.

### Causes of deviation from ideal behaviour

The causes of deviations from ideal behaviour may be due to the following two assumptions of kinetic theory of gases. There are

- The volume occupied by gas molecules is negligibly small as compared to the volume occupied by the gas.
- The forces of attraction between gas molecules are negligible.

The first assumption is valid only at low pressures and high temperature, when the volume occupied by the gas molecules is negligible as compared to the total volume of the gas. But at low temperature or at high pressure, the molecules being in compressible the volumes of molecules are no more negligible as compared to the total volume of the gas.

The second assumption is not valid when the pressure is high and temperature is low. But at high pressure or low temperature when the total volume of gas is small, the forces of attraction become appreciable and cannot be ignored.

### Van Der Waal’s Equation

The general gas equation PV = nRT is valid for ideal gases only Van der Waal is 1873 modified the gas equation by introducing two correction terms, are for volume and the other for pressure to make the equation applicable to real gases as well.

### Volume correction

Let the correction term be v

∴ Ideal volume vi = (V – v)

Now v ∝ n or v = nb

[n = no. of moles of real gas; b = constant of proportionality called Van der Waal’s constant]

∴ Vi = V – nb

b = 4 × volume of a single molecule.

### Pressure Correction

Let the correction term be P

∴ Ideal pressure Pi = (P + p)

Now,

Where a is constant of proportionality called another Van der Waal’s constant.

Hence ideal pressure

Pi =

Here, n = Number of moles of real gas

V = Volume of the gas

a = A constant whose value depends upon the nature of the gas

Substituting the values of ideal volume and ideal pressure, the modified equation is obtained as

**Vander Waals equation, different forms**

- At low pressures: ‘V’ is large and ‘b’ is negligible in comparison with V. The Vander Waals equation reduces to:
PV + a/V = RT PV = RT - a/V or PV • RT This accounts for the dip in PV vs P isotherm at low pressures. |
Deviation of gases from ideal behaviour with pressure. |

- At fairly high pressures
a/V P (V – b) = RT PV – Pb = RT PV = RT + Pb or PV • RT This accounts for the rising parts of the PV vs P isotherm at high pressures. |
The plot of Z vs P for N2 gas at different temperature is shown here. |

- At very low pressures: V becomes so large that both b and a/V
^{2}become negligible and the Vander Waals equation reduces to PV = RT. This shows why gases approach ideal behaviour at very low pressures. - Hydrogen and Helium: These are two lightest gases known. Their molecules have very small masses. The attractive forces between such molecules will be extensively small. So a/V
^{2}is negligible even at ordinary temperatures. Thus PV • RT. Thus Vander Waals equation explains quantitatively the observed behaviour of real gases and so is an improvement over the ideal gas equation.

Vander Waals equation accounts for the behaviour of real gases. At low pressures, the gas equation can be written as,

Where Z is known as compressibility factor. Its value at low pressure is less than 1 and it decreases with increase of P. For a given value of Vm, Z has more value at higher temperature.

At high pressures, the gas equation can be written as

P (Vm – b) = RT

Z =

Here, the compressibility factor increases with increase of pressure at constant temperature and it decreases with increase of temperature at constant pressure. For the gases H2 and He, the above behaviour is observed even at low pressures, since for these gases, the value of ‘a’ is extremely small.