Molar internal energy
=====================

If the equation of state is known, the internal energy of a substance can be found
as a function of volume at constant temperature.

**Conditions:**

#. The fluid is homogeneous and in a single phase state.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Van_der_Waals_equation#Internal_energy_and_specific_heat_at_constant_volume>`__.

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

.. py:data:: molar_internal_energy

    :attr:`~symplyphysics.symbols.thermodynamics.internal_energy` of the van der Waals fluid per unit :attr:`~symplyphysics.symbols.chemistry.amount_of_substance`.

Symbol:
    :code:`u_m`

Latex:
    :math:`u_\text{m}`

Dimension:
    :code:`energy/amount_of_substance`


.. py:data:: temperature

    :attr:`~symplyphysics.symbols.thermodynamics.temperature` of the van der Waals fluid.

Symbol:
    :code:`T`

Latex:
    :math:`T`

Dimension:
    :code:`temperature`


.. py:data:: isochoric_molar_heat_capacity

    :attr:`~symplyphysics.symbols.thermodynamics.molar_heat_capacity` at constant :attr:`~symplyphysics.symbols.classical_mechanics.volume` as a function of :attr:`~temperature`.

Symbol:
    :code:`c_Vm(T)`

Latex:
    :math:`c_{V, \text{m}}{\left(T \right)}`

Dimension:
    :code:`energy/(amount_of_substance*temperature)`


.. py:data:: attractive_forces_parameter

    :attr:`~symplyphysics.symbols.thermodynamics.attractive_forces_parameter`.

Symbol:
    :code:`a`

Latex:
    :math:`a`

Dimension:
    :code:`pressure*volume**2/amount_of_substance**2`


.. py:data:: molar_volume

    :attr:`~symplyphysics.symbols.basic.molar_volume`.

Symbol:
    :code:`v_m`

Latex:
    :math:`v_\text{m}`

Dimension:
    :code:`volume/amount_of_substance`


.. py:data:: law

    :code:`u_m = Integral(c_Vm(T), T) - a / v_m`


    Latex:
        .. math::
            u_\text{m} = \int c_{V, \text{m}}{\left(T \right)}\, dT - \frac{a}{v_\text{m}}