Adding avg and mps speed formulae for ideal gases

maths/amicable_pair.py

@@ -0,0 +1,65 @@
+"""
+Amicable numbers are two different natural numbers such that the sum
+of the proper divisors of each is equal to the other number.
+That is, s(a)=b and s(b)=a, where s(n) is equal to
+the sum of positive divisors of n except n itself.
+Here, a and b form a pair of amicable numbers.
+
+More information about amicable numbers can be found here:
+https://en.wikipedia.org/wiki/Amicable_numbers
+
+Here, we have defined a function to check if two numbers are amicable.
+We have also defined an auxiliary function,
+to find the sum of the proper divisors of a number.
+"""
+
+def sum_of_divisors(n: int) -> int:
+    """
+    Find the sum of the proper divisors of a number.
+
+    Examples:
+    >>> sum_of_divisors(220)
+    284
+    >>> sum_of_divisors(284)
+    220
+    """
+    sum = 0
+    for i in range(1, n):
+        if n % i == 0:
+            sum += i
+
+    return sum
+
+def is_amicable_pair(a: int, b: int) -> bool:
+    """
+    Check if two numbers (a and b) are amicable numbers.
+    Arguments must be positive integers.
+
+    Examples:
+    >>> is_amicable_pair(220, 284)
+    True
+    >>> is_amicable_pair(1184, 1210)
+    True
+    >>> is_amicable_pair(127, 729)
+    False
+    >>> is_amicable_pair(7, 13)
+    False
+    >>> is_amicable_pair(0, 12)
+    Traceback (most recent call last):
+        ...
+    ValueError: Numbers must be positive integers.
+    >>> is_amicable_pair(12, -1)
+    Traceback (most recent call last):
+        ...
+    ValueError: Numbers must be positive integers.
+    >>> is_amicable_pair(42, 42)
+    False
+    """
+    if a <= 0 or b <= 0:
+        raise ValueError("Numbers must be positive integers.")
+
+    return sum_of_divisors(a) == b and sum_of_divisors(b) == a
+
+if __name__ == "__main__":
+    import doctest
+    doctest.testmod()

physics/avg_and_mps_speeds_of_gaseous_molecules.py

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+"""
+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()