speeds_of_gas_molecules.py

r"""
The root-mean-square, average and most probable speeds of gas molecules are
derived from the Maxwell-Boltzmann distribution. The Maxwell-Boltzmann
distribution is a probability distribution that describes the distribution of
speeds of particles in an ideal gas.

The distribution is given by the following equation:
    .. math:: f(v) = \left(\frac{M}{2 \pi RT}\right)^{\frac{3}{2}} \cdot 4 \pi v^2
                     \cdot e^{-\frac{Mv^2}{2RT}}

where:
    * :math:`f(v)` is the fraction of molecules with a speed :math:`v`
    * :math:`M` is the molar mass of the gas in kg/mol
    * :math:`R` is the gas constant
    * :math:`T` is the absolute temperature

More information about the Maxwell-Boltzmann distribution can be found here:
https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution

The average speed can be calculated by integrating the Maxwell-Boltzmann distribution
from 0 to infinity and dividing by the total number of molecules. The result is:

    .. math:: v_{avg} = \sqrt{\frac{8RT}{\pi M}}

The most probable speed is the speed at which the Maxwell-Boltzmann distribution
is at its maximum. This can be found by differentiating the Maxwell-Boltzmann
distribution with respect to :math:`v` and setting the result equal to zero.
The result is:

    .. math:: v_{mp} = \sqrt{\frac{2RT}{M}}

The root-mean-square speed is another measure of the average speed
of the molecules in a gas. It is calculated by taking the square root
of the average of the squares of the speeds of the molecules. The result is:

    .. math:: v_{rms} = \sqrt{\frac{3RT}{M}}

Here we have defined functions to calculate the average and
most probable speeds of molecules in a gas given the
temperature and molar mass of the gas.
"""

# import the constants R and pi from the scipy.constants library
from scipy.constants import R, pi


def avg_speed_of_molecule(temperature: float, molar_mass: float) -> float:
    """
    Takes the temperature (in K) and molar mass (in kg/mol) of a gas
    and returns the average speed of a molecule in the gas (in m/s).

    Examples:

    >>> avg_speed_of_molecule(273, 0.028) # nitrogen at 273 K
    454.3488755020387
    >>> avg_speed_of_molecule(300, 0.032) # oxygen at 300 K
    445.52572733919885
    >>> avg_speed_of_molecule(-273, 0.028) # invalid temperature
    Traceback (most recent call last):
        ...
    Exception: Absolute temperature cannot be less than 0 K
    >>> avg_speed_of_molecule(273, 0) # invalid molar mass
    Traceback (most recent call last):
        ...
    Exception: Molar mass should be greater than 0 kg/mol
    """

    if temperature < 0:
        raise Exception("Absolute temperature cannot be less than 0 K")
    if molar_mass <= 0:
        raise Exception("Molar mass should be greater than 0 kg/mol")
    return (8 * R * temperature / (pi * molar_mass)) ** 0.5


def mps_speed_of_molecule(temperature: float, molar_mass: float) -> float:
    """
    Takes the temperature (in K) and molar mass (in kg/mol) of a gas
    and returns the most probable speed of a molecule in the gas (in m/s).

    Examples:

    >>> mps_speed_of_molecule(273, 0.028) # nitrogen at 273 K
    402.65620701908966
    >>> mps_speed_of_molecule(300, 0.032) # oxygen at 300 K
    394.836895549922
    >>> mps_speed_of_molecule(-273, 0.028) # invalid temperature
    Traceback (most recent call last):
        ...
    Exception: Absolute temperature cannot be less than 0 K
    >>> mps_speed_of_molecule(273, 0) # invalid molar mass
    Traceback (most recent call last):
        ...
    Exception: Molar mass should be greater than 0 kg/mol
    """

    if temperature < 0:
        raise Exception("Absolute temperature cannot be less than 0 K")
    if molar_mass <= 0:
        raise Exception("Molar mass should be greater than 0 kg/mol")
    return (2 * R * temperature / molar_mass) ** 0.5


if __name__ == "__main__":
    import doctest

    doctest.testmod()