physics.speeds_of_gas_molecules

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:

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

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| v_avg = √(8RT/πM)  |
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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:

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| v_mp = √(2RT/M)    |
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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:

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| v_rms = √(3RT/M)   |
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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.

Functions

avg_speed_of_molecule(→ float)	Takes the temperature (in K) and molar mass (in kg/mol) of a gas
mps_speed_of_molecule(→ float)	Takes the temperature (in K) and molar mass (in kg/mol) of a gas

Module Contents

physics.speeds_of_gas_molecules.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

physics.speeds_of_gas_molecules.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