Van der Waals equation

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Van der Waals surface

The Van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero size and a pairwise attractive inter-particle force. It was derived by the Dutch physicist Johannes Diderik van der Waals in 1873. The equation is a modification of the ideal gas law and accounts for the finite size of molecules and the attraction between them.

Equation

The Van der Waals equation is expressed as:

\( \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \)

where:

• \( P \) is the pressure of the gas,
• \( V_m \) is the molar volume of the gas,
• \( T \) is the absolute temperature,
• \( R \) is the ideal gas constant,
• \( a \) is a measure of the attraction between particles,
• \( b \) is the volume occupied by one mole of particles.

The constants \( a \) and \( b \) are specific to each gas and are determined empirically.

Physical Interpretation

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Comparison with ideal gas

The Van der Waals equation introduces two corrections to the ideal gas law:

1. Volume Correction: The term \( b \) accounts for the finite size of molecules. In the ideal gas law, molecules are considered point particles with no volume. The Van der Waals equation corrects this by subtracting \( b \) from the molar volume \( V_m \).

2. Pressure Correction: The term \( \frac{a}{V_m^2} \) accounts for the intermolecular forces. In an ideal gas, there are no attractive forces between molecules. The Van der Waals equation corrects this by adding \( \frac{a}{V_m^2} \) to the pressure \( P \), which accounts for the reduction in pressure due to attractive forces.

Critical Point

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Van der Waals surface with critical point

The Van der Waals equation predicts the existence of a critical point, where the gas and liquid phases become indistinguishable. At the critical point, the first and second derivatives of pressure with respect to volume are zero. The critical temperature \( T_c \), critical pressure \( P_c \), and critical volume \( V_c \) can be expressed in terms of \( a \) and \( b \):

\( T_c = \frac{8a}{27Rb} \)

\( P_c = \frac{a}{27b^2} \)

\( V_c = 3b \)

Applications

The Van der Waals equation is used to describe the behavior of real gases, especially near the critical point. It provides a more accurate description than the ideal gas law for gases at high pressures and low temperatures.

Limitations

While the Van der Waals equation improves upon the ideal gas law, it has limitations. It does not accurately predict the behavior of gases at very high pressures or very low temperatures. More complex equations of state, such as the Redlich-Kwong or Peng-Robinson equations, are often used for these conditions.

Graphical Representation

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Isotherms of van der Waals gas

The Van der Waals equation can be represented graphically by plotting isotherms on a \( P-V \) diagram. These isotherms show the relationship between pressure and volume at constant temperature. The characteristic "van der Waals loop" appears in the region of phase transition, indicating metastable states.

Related Concepts

• Ideal gas law
• Critical point (thermodynamics)
• Equation of state
• Real gas

Related Pages

• Johannes Diderik van der Waals
• Thermodynamics
• Phase transition

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Annotated van der Waals diagram

See Also

• Boyle's law
• Charles's law
• Avogadro's law