van der Waals equation

The van der Waals equation (or van der Waals equation of state; named after Johannes Diderik van der Waals) is based on plausible reasons that real gases do not follow the ideal gas law. The ideal gas law treats gas molecules as point particles that do not interact except in elastic collisions. In other words, they do not take up any space, and are not attracted or repelled by other gas molecules. ^[1]

To account for the volume that a real gas molecule takes up, the van der Waals equation replaces V in the ideal gas law with (V-b), where b is the volume per mole that is occupied by the molecules. This leads to;^[2]

${\displaystyle P(V_{m}-b)=RT}$^[3]

The second modification made to the ideal gas law accounts for the fact that gas molecules do in fact attract each other and that real gases are therefore more compressible than ideal gases. Van der Waals provided for intermolecular attraction by adding to the observed pressure P in the equation of state a term a/Vm², where a is a constant whose value depends on the gas. The van der Waals equation is therefore written as;^[4]

${\displaystyle (P+a/V_{m}^{2})(V_{m}-b)=RT}$,

and can also be written as

${\displaystyle (P+an^{2}/V^{2})(V-nb)=nRT}$,^[5]

where V_m is the molar volume of the gas, R is the universal gas constant, T is temperature, P is pressure, and V is volume. When the molar volume V_m is large, b becomes negligible in comparison with V_m, a/V_m² becomes negligible with respect to P, and the van der Waals equation reduces to the ideal gas law, PV_m=RT

There are two corrective factors in van der Waals equation. The first,  , alters the pressure in the ideal gas equation. It accounts for the intermolecular attractive forces between gas molecules. The magnitude of ais indicative of the strength of the intermolecular attractive force. a has units of  .

The factor - nb accounts for the volume occupied by the gas molecules. b has units of L/mol.

 b is generally much smaller in magnitude than a

References

1. Jump up^ Silbey, Robert J., Robert A. Alberty, and Moungi G. Bawendi. Physical Chemistry. 4th ed. N.p.: n.p., n.d. Print.
2. Jump up^ Silbey, Robert J., Robert A. Alberty, and Moungi G. Bawendi. Physical Chemistry. 4th ed. N.p.: n.p., n.d. Print.
3. Jump up^ Silbey, Robert J., Robert A. Alberty, and Moungi G. Bawendi. Physical Chemistry. 4th ed. N.p.: n.p., n.d. Print.
4. Jump up^ Silbey, Robert J., Robert A. Alberty, and Moungi G. Bawendi. Physical Chemistry. 4th ed. N.p.: n.p., n.d. Print.
5. Jump up^ Silbey, Robert J., Robert A. Alberty, and Moungi G. Bawendi. Physical Chemistry. 4th ed. N.p.: n.p., n.d. Print.
6. Jump up^ Silbey, Robert J., Robert A. Alberty, and Moungi G. Bawendi. Physical Chemistry. 4th ed. N.p.: n.p., n.d. Print.