**Van der Waals equation of state**
===================================

*Van der Waals equation of state* uses a model which extends the ideal gas law to include
the non-zero size of gas molecules and the interactions between them.

.. _vdw_critical_parameters_def:

Critical parameters
-------------------

*Critical parameters* of the van der Waals equation of state are such values of volume, pressure, and
temperature at which the isotherm has an inflection point whose tangent at that point is zero, i.e.
the first and second derivatives of pressure with respect to volume at constant temperature are zero:

.. math::
    \left( \frac{\partial p}{\partial V} \right)_T = \left( \frac{\partial^2 p}{\partial V^2} \right)_T = 0

.. _vdw_reduced_units_def:

Reduced units
-------------

*Reduced units* are used in the :doc:`dimensionless <laws.thermodynamics.equations_of_state.van_der_waals.dimensionless_equation>`
van der Waals equation of state. A reduced quantity :math:`X_r` is defined as the ratio of quantity
:math:`X` to the corresponding :ref:`critical parameter <vdw_critical_parameters_def>` :math:`X_\text{c}`.

.. py:currentmodule:: symplyphysics.laws.thermodynamics.equations_of_state.van_der_waals

**Contents:**

.. toctree::
    :maxdepth: 4


    laws.thermodynamics.equations_of_state.van_der_waals.critical_molar_volume
    laws.thermodynamics.equations_of_state.van_der_waals.critical_pressure
    laws.thermodynamics.equations_of_state.van_der_waals.critical_temperature
    laws.thermodynamics.equations_of_state.van_der_waals.dimensionless_equation
    laws.thermodynamics.equations_of_state.van_der_waals.equation
    laws.thermodynamics.equations_of_state.van_der_waals.molar_internal_energy
    laws.thermodynamics.equations_of_state.van_der_waals.reduced_pressure
    laws.thermodynamics.equations_of_state.van_der_waals.reduced_temperature
    laws.thermodynamics.equations_of_state.van_der_waals.reduced_volume
    laws.thermodynamics.equations_of_state.van_der_waals.second_virial_coefficient