# Van der Waals Equation

Van der Waal (1873) suggested two corrections in an ideal gas equation(PV = nRT) so that it can also explain the behavior of real gases. Hence the equation obtained by applying the two corrections to the usual gas equation is known as **Van der Waals equation.**

Must Read: Ideal gas equation.

**Need of Van der Waals Equation**

Real gases deviate from ideal behavior because of the following two faulty assumptions of the kinetic theory.

(i) The actual volume occupied by molecules is negligible as compared to the total volume of the gas.

(ii) The forces of attraction and repulsion between molecules of the gas are negligible.

The extent of deviation of a real gas from ideal behavior is expressed in terms of compressibility factor (z).

**Compressibility factor (Z)**

Mathematically,

Thus greater is the departure of z from unity, more is the deviation from ideal behavior.Thus, when

(i) Z=1, the gas is ideal. In the case of N_{2}, the value of Z is close to 1 at 50°C. This temperature at which a real gas behaves like an ideal gas is called Boyle’s temperature or Boyle’s point.

(ii) Z>1 the gas is less compressible.

(iii) When the value of Z is less than 1 (Z<1) the gas is more compressible.

Hence, suitable corrections must be applied to the ideal gas equation so that it can also explain the behaviour of real gases.

**Van Der Waals Equation of State**

Waals equation is obeyed by the real gases over a wide range of temperatures and pressures.

### Van Der Waals Equation Solved for V Volume

Corrected (ideal) volume = (V – b)

where b is the effective volume of the molecules. The constant b is also called co-volume or excluded volume. The excluded volume for n molecules of gas = 4n*V _{m}*, i.e. it is 4 time the actual volume occupied by a molecule (V

_{m}).

b = 4 × volume of a single molecule(6.023 × 10^{23} ×(4/3)r^{3},)

### Van Der Waals Equation Solved for P Pressure Correction

Corrected (ideal) pressure = P + p.

where p is the inward force exerted on molecules about to strike the wall.

However,

where ‘a’ is a proportionality constant, called the coefficient of attraction; its value depends upon the nature of the gas.

Substituting the values of corrected pressure and corrected volume in the ideal gas equation, PV = RT, we have

The equation is known as van der Waal’s equation. The constants a and b are called van der Waal’s constants and their values depend upon the nature of the gas and independent of the temperature and pressure.

**Refer to the video for Gas laws and Van der Waal Equation**

**Units of ****Van der Waals Equation Constants(a,b)**

a and b depend on the units in which P and V are expressed.

Thus if pressure is expressed in atmospheres and volume in liters, the units of a will be atmosphere litre^{2} mol^{-2} or atm dm^{6} mol^{-2} or Nm^{4} mol^{-2}.

**Units of b.** In the van der Waal’s equation, b is the effective volume occupied by molecules present in one mole of the gas. Hence the units of b are same as that of volume, i.e. liter mol^{-1} or m^{3} mol^{-1}.

**Van Der Waals Equation a and b**

(i) The value of a is an indirect measure of the magnitude of attractive forces between the molecules. Greater is the value of a, more easily the gas liquefied. Thus the easily liquefiable gases (like SO_{2} > NH_{3} > H_{2}S > CO_{2}) have high values of a than the permanent gases (like N_{2}, O_{2}, H_{2} and He).

(ii) The value of b gives an idea about the effective size of gas molecules. Greater is the value of b, larger is the size and smaller is the compressible volume. As b is the effective volume of the gas molecules, the constant value of b for any gas over a wide range of temperature and pressure indicates that the *gas molecules are incompressible.*

**Behaviour of Real Gas On the Basis of Van der ****Waals Equation**

Unlike ideal gas equation, the van der Waal equation explains the behavior of real gases under different conditions of temperature and pressure.

### At low pressures

At low pressures, volume V is very large and hence the correction term b (a constant of small value) can be neglected in comparison to very large value of V. Thus the van der Waal equation for 1 mole of a gas, i.e.,

**At extremely low pressure**

Since V is very large, the value of a/V is very small and hence can be ignored. Consequently, at very low pressures, the van der Waal equation is reduced to ideal gas equation, i.e.,

**PV=RT**

*This explains why at extremely low pressures, the real gases obey the ideal gas equation.*

The equation

shows that PV is less than RT by an amount equal to a/V. As pressure increases, V decreases, as a/V increases and ultimately PV becomes smaller and smaller. This explains the dip in the isotherms of most of the gases (e.g., CO and CH_{4}).

### **At high pressures**

At high pressures, volume V is quite small and hence the term b cannot be neglected in comparison to V Secondly, under these conditions although the term as a/V^{2} is quite large, it is so small in comparison to high-pressure P that it can be neglected. Thus the Waals equation is reduced to

- P(V – b) = RT
- PV – Pb = RT
- PV = RT + Pb

Thus PV is greater than RT by an amount equal to Pb. As the pressure increases, the factor Pb increases and hence PV increases. This explains why the value of PV after reaching a minimum, increases with the further increase of pressure.

**At high temperatures**

At any given pressure, if the temperature is sufficiently high, V is very large and hence the terms a/V^{2 }and b can be neglected as in case (i) and thus the Waals equation reduces to PV = RT. This explains why the real gases behave like an ideal gas at high temperatures.

**At low temperatures.**

At low temperatures both P and V are small, hence both pressure and Volume corrections are appreciable, with the result the deviations are more pronounced.

## Behaviour of hydrogen and helium: Van der Waals Equation

Since hydrogen and helium have very small masses, the intermolecular forces of attraction are extremely small even at low pressures. In other words, the factor as V is negligible at all pressures. Hence the equation is reduced to

- P(V – b) = RT
- PV = RT + Pb

This explains why hydrogen and helium show positive deviations only with an increase in the value of P.

This is** Van der Waals Equation.**

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