From 0b2c9fb6f164468b51baa4866c1b8c4f01ec8b64 Mon Sep 17 00:00:00 2001 From: Baron105 <76466796+Baron105@users.noreply.github.com> Date: Sat, 14 Oct 2023 12:31:23 +0530 Subject: [PATCH] Adding avg and mps speed formulae for ideal gases (#10229) * avg and mps speed formulae added * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * avg and mps speed formulae added * fixed_spacing * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * spacing * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * ws * added amicable numbers * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * added amicable numbers * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * spacing * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * removed * changed name of file and added code improvements * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * issues fixed due to pi * requested changes added --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com> --- physics/speeds_of_gas_molecules.py | 111 +++++++++++++++++++++++++++++ 1 file changed, 111 insertions(+) create mode 100644 physics/speeds_of_gas_molecules.py diff --git a/physics/speeds_of_gas_molecules.py b/physics/speeds_of_gas_molecules.py new file mode 100644 index 000000000..a50d1c0f6 --- /dev/null +++ b/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()