LECTURE NOTE ON The van der Waals Equation of State

-Dr S P Singh Dept of Chemistry A N College, Patna

 Johannes Diderik van der Waals, a Dutch theoretical physicist is famous for his pioneering work on the equation of state for gases and liquids.  Non-ideality of real gases was noted by van der Waals.  J. D. van der Waals was awarded the Physics Nobel Prize in 1910 for his work on the Equation of State for Gases.

Deviation from Ideal Gas Behaviour

Real Gases:

 A gas which do not follow ideal gas behaviour under all conditions of temperature and pressure is called real gas.  Deviation from ideal behaviour with respect to pressure can be studied by plotting pressure versus volume curve at constant temperature (Boyle’s law).

Compressibility factor (z): z=PVnRT z=PVnRT For ideal gases z = 1 and for real gases z ≠ 1. If z > 1 then real gases show +ve deviation from ideal behavior and if z < 1 the gas show –ve deviation from ideal behavior Reason for Deviation: 1. In the derivation of Ideal Gas Equation (PV=RT), it was assumed that there is no force of attraction between the molecules of gas but it is not true as gaseous particles have force of attraction existing between them. 2. Volume occupied by the gas molecules is negligible as compound to total volume of gas as per the assumption in the derivation of Ideal Gas but the volume occupied by the gas has a finite value.

The Ideal Gas equation was modified by van der Waals in 1877 by making suggestions of correction due to Pressure exerted by gas and due to volume occupied by a gaseous molecule. The existence of intermolecular interactions and the assumption of a finite volume were credited to van der Waals.

(i) Correction due to intermolecular force of attraction A gaseous molecule in the bulk of a gas in the midst of a vessel is being attracted uniformly on all sides by the neighbouring molecules. These forces of inter-molecular attraction neutralize one another leading to no resultant attractive force. But as soon as a gaseous molecule approaches the wall of a container, it experiences backward pull due to attractive force from the bulk of the gaseous molecules. As a result it strikes the wall with lower velocity / force and hence exerts lower pressure than it would have done if there is no force of attraction. It is why, certain correction factor say, p needs to be added the pressure of the gas to get the ideal pressure. The corrected pressure = P + p ------(i)

The force of attraction exerted on a single molecule which is just to strike the wall of a container depends on the number of molecules per unit volume in the bulk of the gas. It means the total inward attractive pull on the molecules (p) is directly proportional to the square of the density (σ).

p ∝ σ2 where, p is the internal pressure for a mole of a gas Since density is inversely proportional to volume (V), the term p can be expressed as

P ∝ 1/V2 where, V is the volume occupied by a mole of a gas

⇒ P = a / V2 where ‘a’ is proportionality constant depending upon nature of the gas. The constant ‘a’ is a measure of the van der Waals forces of cohesion existing

between the gaseous molecules.

From equation (i), the corrected pressure for a mole of a gas = P + a / V2 ------(ii)

(ii) Correction due to the finite volume of gas molecule

As has been mentioned earlier that finite value of the volume ‘b’ of a gas molecule has not been taken into account in the derivation of ideal gas equation. van der Waals made a correction term ‘nb’ which needs to be subtracted in the total volume ‘V’ to get the ideal volume.

In order to understand the volume correction term ‘nb’ let us suppose that two identical spherical rigid gaseous molecule are in contact with each other. Each of the two molecules has a diameter‘d’. It means the the distance between the centre of two spheres is ‘d’. Therefore, volume = 4/3πd3 The excluded volume per molecule = ½ (4/3πd3) = 2/3πd3 ------(iii) The actual volume of one gas molecule of radius ‘r’ = 4/3πr3 = 4/3π (d/2)3 = 1/6πd3 The excluded volume per molecule = 2/3πd3 = 4 X 1/6πd3 (4 times the actual volume) ------(iv) 3 The excluded volume(co-volume) per mole of the gas (b) = NA X 4 X 1/6πd ------(v) Hence the corrected volume per mole of the gas = V-b ------(vi) If ‘n’ moles are contained by the ‘V’ volume of the gas, nb is considered to be the excluded volume.

van der Waals Equation

. Thus non-ideality of real gases was corrected by van der Waals as follows; (P + a/V2) (V-b) =RT for one mole of a gas ------(vii) (P + n2a/V2) (V-b) =nRT for ‘n’ mole of a gas ------(viii) The P-V-T behavior of real gases can be expressed more accurately by equation (viii). The constant ‘a’ and ‘b’ are known as van der Waals constants. The van der Waals constant ‘a’ Signify the magnitude of intermolecular forces of attraction between the gas particles while ‘b’ signifies the effective size of gas molecules. The units of ‘a’ and ‘b’ depend upon the units in which ‘P’ and ‘V’ are expressed.

Discussion on the departure of Real Gases from Ideal behavior at different temperature and pressure Case 1: Pressure is not too high In this case volume will be sufficiently large and value of ‘b’ can comparatively be ignored. The van der Waals equation for one mole of a gas

(P + a/V2) (V-b) =RT can be expressed as

(P + a/V2) (V) =RT

⇒ PV+ a/V =RT

⇒ PV = RT - a/V

As pressure ‘P’ increases, volume ‘V’ decreases, a/V increases and consequently, PV is smaller. Hence, dip in the isotherms of most of the gases is explained. Case 2: Pressure is too high

In this case Pressure is too high, so volume is too small. Therefore, ‘b’ cannot be ignored whereas pressure correction factor can be ignored.

The van der Waals equation for one mole of a gas

(P + a/V2) (V-b) =RT can be expressed as

P (V-b) =RT

⇒ PV = RT+ Pb

With an increase in Pressure, ‘Pb’ increases leading to increase in ‘PV’. It is why the value of PV after reaching to a minimum increases with an increase of pressure.

Case 3: Temperature is high

When temperature is high at a given pressure, volume becomes large. Consequently pressure correction term ‘a/V2’ is negligible. At high temperature, even volume correction term ‘b’ is negligible. In such a situation van der Waals equation approaches to the Ideal Gas equation (PV=RT). It is why, less deviation is observed at high temperature.

Case 4: Exceptional behavior of hydrogen and helium

The inter-molecular attraction force of hydrogen and helium is too small because of their smaller sizes. As a result pressure correction term ‘a/V2’ is negligible at normal temperature.

The van der Waals equation for one mole of a gas

(P + a/V2) (V-b) =RT can be expressed as

⇒ P (V-b) =RT

⇒ PV = RT+ Pb

It is why continuous increase in PV is observed with increase in Pressure at normal temperature in case of H2 and He.

Boyle’s Temperature (TB): It is the temperature at which Real Gas obeys Boyle’s law. It is expressed as TB = (a/b)R.

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