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.

Compression factors

Compression factors

For an ideal gas, PV = nRT. If PV and nRT are the same, and you divide one by the other, then the answer will, of course, be 1. For real gases, PV doesn't equal nRT, and so the value will be something different.
The term PV / nRT is called the compression factor. 

• Different gases deviate from ideal behaviour in different ways
•  Deviation can be positive (Z>1) or negative (Z<1)
• Deviation always positive at sufficiently high pressure

The graphs below show how this varies for nitrogen as you change the temperature and the pressure.

If nitrogen was an ideal gas under all conditions of temperature and pressure, every one of these curves would be a horizontal straight line showing a compression factor of 1. That's obviously not true!

Things to notice

• At low pressures of about 1 bar (100 kPa - just a bit less than 1 atmosphere), the compression factor is close to 1. Nitrogen approximates to ideal behaviour at ordinary pressures.
• The non-ideal behaviour gets worse at lower temperatures. For temperatures of 300 or 400 K, the compression factor is close to 1 over quite a large pressure range. The nitrogen becomes more ideal over a greater pressure range as the temperature rises.
• The non-ideal behaviour gets worse at higher pressures.
• There must be at least two different effects causing these deviations. There must be at least one effect causing the pV / nRT ratio to be too low, especially at low temperatures. And there must be at least one effect causing it to get too high as pressure increases. We will explore those effects in a while.

Gas critical constants

Gas critical constants

liquefaction of gases

liquefaction of gases

van der Waals equation

van der Waals equation

Real gases and deviation from Ideal behaviour

Real gases and deviation from Ideal behaviour

Concept of average, root mean square and most probable velocities

Concept of average, root mean square and most probable velocities

Kinetic theory of gases

Kinetic theory of gases

Ideal gas equation

Ideal gas equation

Dalton’s law of partial pressure

Dalton’s law of partial pressure

Avogadro’s law

Avogadro’s law

Graham’s law of diffusion

Graham’s law of diffusion

Charle’s law

Charle’s law

Boyle’s law

Boyle’s law