Adding avg and mps speed formulae for ideal gases

physics/speeds_of_gas_molecules.py

@@ -0,0 +1,111 @@
+"""
+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
+
+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.
+"""
+
+# 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()