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# Van der Waals Equation: How to Describe Real Gas

## Important Formula

## What do the formula symbols mean?

## Pressure

`$$ \mathit{\Pi} $$`Unit

`$$ \mathrm{Pa} $$`

Pressure of the (real) gas. With the real gas - in contrast to the ideal gas - the gas pressure may be very high.

## Temperature

`$$ T $$`Unit

`$$ \mathrm{K} $$`

Temperature of the (real) gas. With the real gas - in contrast to the ideal gas - the temperature of the gas may be low.

## Volume

`$$ V $$`Unit

`$$ \mathrm{m}^3 $$`

Volume of the (real) gas. Molar volume: \( V_{\text m} ~=~ \frac{V}{n} \).

## Amount of substance

`$$ n $$`Unit

`$$ \mathrm{mol} $$`

Amount of substance indirectly indicates the number of particles of the gas.

## Covolume

`$$ V_{\text b} $$`

Covolume is the material-dependent intrinsic volume of the particles of the real gas. It reduces the volume \( V \) available for the motion of the particles to \( (V ~-~ n\,V_{\text b}) \). For the ideal gas, the following holds: \( V_{\text b} ~=~ 0 \). The unit of covolume is \( \mathrm{m}^3 / \mathrm{mol} \).

## Cohesion parameter

`$$ a $$`

Cohesion parameter is a material-dependent quantity that indicates the force between the particles of the gas. In the case of an ideal gas, the following holds true: \( a ~=~ 0 \). The term \( \frac{n^2 \, a}{V^2} \) is called internal pressure. The unit of the cohesion parameter is \( \mathrm{Pa}\,\mathrm{m}^6 / \mathrm{mol}^2 \).

## Gas constant

`$$ R $$`Unit

`$$ \frac{\mathrm J}{\mathrm{mol} \, \mathrm{K}} $$`

Molar gas constant (also called universal gas constant) is a physical constant from thermodynamics and has the following

*exact*value:`$$ R ~=~ 8.314 \, 462 \, 618 \, 153 \, 24 \, \frac{\mathrm J}{\mathrm{mol} \, \mathrm{K}} $$`The Van der Waals equation is designed for real gases and introduces corrections to the ideal gas equation in order to describe the behavior of real gases more precisely.

$$ \begin{align} \mathit{\Pi} ~=~ \frac{n \, R \, T}{V ~-~ n \, V_{\text b}} ~-~ \frac{n^2 \, a}{V^2} \end{align} $$

The equation thus modifies the volume \( V \) and the pressure \( \mathit{\Pi} \) compared to the ideal gas equation to take into account the effects of molecular size and attractive forces.