avg_and_mps_speeds_of_gaseous_molecules.py

"""
The root-mean-square, average and most probable speeds are derived from
the Maxwell-Boltzmann distribution. The Maxwell-Boltzmann distribution is a
probability distribution that describes the distribution of speeds for particles
in a gas.
The distribution is given by the following equation:

        -------------------------------------------------
        | f(v) = (M/2πRT)^(3/2) * 4πv^2 * e^(-Mv^2/2RT) |
        -------------------------------------------------

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

        ---------------------
        | vavg = √8RT/πM    |
        ---------------------

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 v and setting the result equal to zero. The result is:

        ---------------------
        | vmp = √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:

        ---------------------
        | vrms = √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.
"""

# necessary constants
PI = 3.1415926535  # pi
R = 8.3144626181  # gas constant


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
    454.34887551126405
    >>> avg_speed_of_molecule(300, 0.032) # oxygen at 300 K
    445.5257273482451
    """
    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")
    else:
        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
    402.6562070215111
    >>> mps_speed_of_molecule(300, 0.032) # oxygen at 300 K
    394.83689555229637
    """
    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")
    else:
        return (2 * R * temperature / molar_mass) ** 0.5


if __name__ == "__main__":
    import doctest

    doctest.testmod()