Ideal vs real Gases 4

 

Equation of State for the Real Gases (van der Waals Equation)

To explain the behaviour of real gases, J .D. van der Waals, in 1873, modified the ideal gas equation applying appropriate corrections so as to take into account

• The volume of the gas molecules
• The forces of attraction between the gas molecules

He put forward the modified equation, known after him as van der Waals equation. The equation is

For 1 mole of the gas,

For n moles of the gas,

Where ‘a’ and ‘b’ van der Waals constant.There values depend upon nature of gas.

Significance of Van der Waals Constants

• Van der Waals constant ‘a’: Its value is a measure of the magnitude of the attractive forces among the molecules of the gas. There would be large intermolecular forces of attraction if the value ‘a’, is large.
• Van der Waals constant ‘b’: Its value is a measure of the effective size of the gas molecules. Its value is equal to four times the actual volume of the gas molecules. It is called Excluded Volume or Co-volume.

Units of van der Waals Constants

• Units of ‘a’: As p = am2/V2, therefore a = (p × V2) / n2 = atm L2 mol-2 or bar dm6 mol-2
• Units of ‘b’: As volume correction v = n b, therefore b = v/n = Lmol-1 or dm3mol-1

Explanation of the Behaviour of Real Gases by van der Waals Equation

• At Very Low Pressures, V is very large. Hence, the correction term a/V2 is so small that it can be neglected, Similarly, the correction term ‘b’ can also be neglected in comparison to V. Thus, van der Waals equation reduces to the form PV = RT. This explains why at very low pressures, the real gases behave like ideal gases.
• At Moderate Pressures, V decreases. Hence, a/V2 increases and cannot be neglected. However, is still large enough in comparision to ‘b’ so that ‘b’ can be neglected. Thus, van der Waals equation becomes

Thus, compressibility factor is less than 1. So at when at constant temperature, pressure is increased, V decreases so that the factor a/RTV increases. This explains why initially a dip in the plot of Z versus P is observed.

• At High Pressures, V is so small that ‘b’ cannot be neglected in comparison to V. The factor a/V2 is no doubt large but as P is very high, a/V2 can be neglected in comparison to P. Thus, van der Waals equation reduces to the form:

Thus, compressibility factor is greater than 1. As P is increased (at constant T), the factor Pb/RT increases. This explains why after minima in the curves, the compressibility factor increases continuously with pressure.

• At High Temperatures: V is very large (at a given pressure) so that both the correction factors (a/V2 and b) become negligible as in case (i). Hence, at high temperature, real gases behave like ideal gas.