Internal energy formula
=======================

The formula of the internal energy differential can be integrated using the Euler's theorem on
homogeneous functions to get the following expression.

**Notes:**

#. This formula works for a single-component system. For a multi-component system replace the
   product of chemical potential and particle count with a sum over each type of components.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Thermodynamic_potential#Euler_relations>`__.

.. py:currentmodule:: symplyphysics.laws.thermodynamics.euler_relations.internal_energy_formula

.. py:data:: internal_energy

    :attr:`~symplyphysics.symbols.thermodynamics.internal_energy` of the system.

Symbol:
    :code:`U`

Latex:
    :math:`U`

Dimension:
    :code:`energy`


.. py:data:: temperature

    :attr:`~symplyphysics.symbols.thermodynamics.temperature` of the system.

Symbol:
    :code:`T`

Latex:
    :math:`T`

Dimension:
    :code:`temperature`


.. py:data:: entropy

    :attr:`~symplyphysics.symbols.thermodynamics.entropy` of the system.

Symbol:
    :code:`S`

Latex:
    :math:`S`

Dimension:
    :code:`energy/temperature`


.. py:data:: pressure

    :attr:`~symplyphysics.symbols.classical_mechanics.pressure` in the system.

Symbol:
    :code:`p`

Latex:
    :math:`p`

Dimension:
    :code:`pressure`


.. py:data:: volume

    :attr:`~symplyphysics.symbols.classical_mechanics.volume` of the system.

Symbol:
    :code:`V`

Latex:
    :math:`V`

Dimension:
    :code:`volume`


.. py:data:: chemical_potential

    :attr:`~symplyphysics.symbols.thermodynamics.chemical_potential` of the system.

Symbol:
    :code:`mu`

Latex:
    :math:`\mu`

Dimension:
    :code:`energy`


.. py:data:: particle_count

    :attr:`~symplyphysics.symbols.basic.particle_count` of the system.

Symbol:
    :code:`N`

Latex:
    :math:`N`

Dimension:
    :code:`dimensionless`


.. py:data:: law

    :code:`U = T * S - p * V + mu * N`


    Latex:
        .. math::
            U = T S - p V + \mu N