Pressure of ideal gas from height and temperature
=================================================

The *barometric formula* determines the dependence of the pressure or density of a gas on the height in the gravity field.

**Notation:**

#. :math:`g` (:code:`g`) is :attr:`~symplyphysics.quantities.acceleration_due_to_gravity`.
#. :math:`k_\text{B}` (:code:`k_B`) is :attr:`~symplyphysics.quantities.boltzmann_constant`.

**Conditions:**

#. The gas is ideal.
#. The gas is in a uniform gravity field.

**Links:**

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

.. py:currentmodule:: symplyphysics.laws.thermodynamics.pressure_of_ideal_gas_from_height_and_temperature

.. py:data:: final_pressure

    :attr:`~symplyphysics.symbols.classical_mechanics.pressure` of the gas at final height.

Symbol:
    :code:`p`

Latex:
    :math:`p`

Dimension:
    :code:`pressure`


.. py:data:: initial_pressure

    :attr:`~symplyphysics.symbols.classical_mechanics.pressure` of the gas at initial height.

Symbol:
    :code:`p_0`

Latex:
    :math:`p_{0}`

Dimension:
    :code:`pressure`


.. py:data:: molecular_mass

    :attr:`~symplyphysics.symbols.basic.mass` of a single gas molecule.

Symbol:
    :code:`m`

Latex:
    :math:`m`

Dimension:
    :code:`mass`


.. py:data:: height_change

    Change in :attr:`~symplyphysics.symbols.classical_mechanics.height` between :attr:`~initial_pressure` and :attr:`~final_pressure`.

Symbol:
    :code:`Delta(h)`

Latex:
    :math:`\Delta h`

Dimension:
    :code:`length`


.. py:data:: temperature

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

Symbol:
    :code:`T`

Latex:
    :math:`T`

Dimension:
    :code:`temperature`


.. py:data:: law

    :code:`p = p_0 * exp(-g * m * Delta(h) / (k_B * T))`


    Latex:
        .. math::
            p = p_{0} \exp{\left(- \frac{g m \Delta h}{k_\text{B} T} \right)}