Speeds of Gas Molecules

p
```"""
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:

-------------------------------------------------
| 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

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.
"""

# 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()
```