Learning Objectives:

• Understand how the assumptions of kinetic molecular theory of gases are not valid for real gases.
• Understand the pressure and volume correction terms in Van Der Waal's equation.
• Introduce the concepts of equation of state (EOS), real gas, and compressibility factor.

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Van Der Waal's Equation:

Real gases show deviation from ideal behavior especially at low temperatures and high pressures. For example, Boyle's law $P_1V_1 = P_2V_2$ is not followed at high pressures. This deviation is due to the following two reasons:

1. Inter-molecular forces of attraction: We assume no attractive or repulsive forces between molecules of an ideal gas. However, there are such forces, for real gases.
2. Volume of molecules: We assume the volume of gas is negligible as compared to the volume of the container. However, that is not true for real gases, especially at high pressures.

Based on this, Van Der Waal suggested two corrections as follows:

1. Volume Correction: When a gas is compressed, the molecules are pushed so closer that repulsive forces operate between them. Further increase in pressure is opposed by the molecules themselves because molecules have definite volume. In reality, the 'free' volume is: \[V = V_{\mathrm{vesssel}}- nb\]
2. Pressure Correction: Each molecule is attracted by all sides, so attraction forces are cancelled out. But when a molecule is about to strike the wall of containner, it experiences a force of attraction towards other molecules, which means pressure being observed on the walls of a vessel is less than ideal pressure. \[P= \left(P + a\dfrac{n^2}{V^2}\right)\]

Van Der Waals Equation:

We begin with ideal gas equation: \[PV = nRT\] Now we apply these two corrections: \[\left(P + a\dfrac{n^2}{V^2}\right)(V-nb) = nRT\] For one mole of gas n = 1,

Equation of State (EOS):

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Solved Example: 9903-01

The Vander Waals equation of state is \[\left( {p + \dfrac{a}{{{v^2}}}} \right)\left( {v - b} \right) = RT,\] where p is pressure, v is specific volume, T is temperature and R is characteristic gas constant. The SI unit of a is:

A. $J/kg. K$

B. $m^3/kg$

C. $m^5/kgs^2$

D. $Pa/kg$

Correct Answer: C

Solved Example: 9904-01

The value of compressibility factor, Z at the critical state of a Van der Waal's gas is:

A. 3.735

B. 0.735

C. 3.375

D. 0.375

\[Z = \dfrac{PV}{RT}\] At critical point: \[P = \dfrac{a}{27b^2}\] \[V = 3b\] \[T = \dfrac{8a}{27Rb}\] Substituting, in the above equation, \[Z = \dfrac{3}{8} = 0.375\]

Correct Answer: D