# Van der Waals Equation

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Physik
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JoVE Core Physik
Van der Waals Equation

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The ideal gas equation has two significant drawbacks: first, it does not consider the volume of gas molecules, and second, it does not account for attractive intermolecular forces between the gas molecules.

If the gas contains n number of moles and each one of them occupies volume b, then the total volume occupied by the gas molecules is nb. Thus, the volume available for gas molecules to move will be total volume minus nb.

The attractive intermolecular forces reduce the pressure proportional to the square of the molar density. In the absence of attractive intermolecular forces, higher pressure is required to reach the same nRT.

This is the van der Waals equation of state that tries to address the drawbacks of the ideal gas equation. Here, a and b are empirical constants that are dependent on the type of gas under study.

The van der Waals equation predicts the behavior of real gas adequately, such as liquid to vapor transition and the Joule-Thomson effect.

## Van der Waals Equation

The ideal gas law is an approximation that works well at high temperatures and low pressures. The van der Waals equation of state (named after the Dutch physicist Johannes van der Waals, 1837−1923) improves it by considering two factors.

First, the attractive forces between molecules, which are stronger at higher densities and reduce the pressure, are considered by adding to the pressure a term equal to the square of the molar density multiplied by a positive coefficient a. Second, the volume of the molecules is represented by a positive constant b, which can be thought of as the volume of a mole of molecules. This is subtracted from the total volume to give the remaining volume that the molecules can move in. The constants a and b are determined experimentally for each gas. The resulting equation is

For carbon dioxide gas with the van der Waals equation, constant a is 0.364 J·m3/mol2 and constant b is 4.27 x 10−5 m3/mol. If 1 mole of this gas is confined in a volume of 300 cm3 at 300 K, then the pressure of the gas can be calculated using the van der Waals equation. Rearranging the van der Waals equation for pressure,

and substituting the known quantities in it,

gives the pressure of carbon dioxide gas

In the low-density limit (small n), the a and b terms are negligible, and the van der Waals equation reduces to the ideal gas law. On the other hand, if the second term from the van der Waals equation is small, meaning that the molecules are very close together, then the pressure must be higher to give the same nRT, as expected in the situation of a highly compressed gas. However, the increase in pressure is less than that argument would suggest because, at high densities, the pressure correction term from the van der Waals equation is significant. Since the pressure correction term is positive, it requires a lower pressure to give the same nRT. The van der Waals equation of state works well for most gases under various conditions, such as for predicting liquid-gas phase transitions.