Internal energy differential
============================

The *fundamental thermodynamic relations* are fundamental equations which demonstrate how important
thermodynamic quantities depend on variables that are measurable experimentally.

**Notation:**

#. :math:`d` denotes an exact, path-independent differential.

**Notes:**

#. Entropy, volume, and particle count are so called natural variables of internal energy as a
   thermodynamic potential.
#. For a system with more than one type of particles, the last term can be represented as a sum over all
   types of particles, i.e. :math:`\sum_i \mu_i \, d N_i`.

**Conditions:**

#. The system is in thermal equilibrium with its surroundings
#. The system is composed of only one type of particles, i.e. the system is a pure substance.

**Links:**

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

.. py:currentmodule:: symplyphysics.laws.thermodynamics.internal_energy_differential

.. py:data:: internal_energy_change

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

Symbol:
    :code:`dU`

Latex:
    :math:`dU`

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_change

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

Symbol:
    :code:`dS`

Latex:
    :math:`dS`

Dimension:
    :code:`energy/temperature`


.. py:data:: pressure

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

Symbol:
    :code:`p`

Latex:
    :math:`p`

Dimension:
    :code:`pressure`


.. py:data:: volume_change

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

Symbol:
    :code:`dV`

Latex:
    :math:`dV`

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_change

    Infinitesimal change in the number of particles in the system.

Symbol:
    :code:`dN`

Latex:
    :math:`dN`

Dimension:
    :code:`dimensionless`


.. py:data:: law

    :code:`dU = T * dS - p * dV + mu * dN`


    Latex:
        .. math::
            dU = T dS - p dV + \mu dN