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physics/speeds_of_gas_molecules.py

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r"""
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"""
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The root-mean-square, average and most probable speeds of gas molecules are
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derived from the Maxwell-Boltzmann distribution. The Maxwell-Boltzmann
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distribution is a probability distribution that describes the distribution of
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speeds of particles in an ideal gas.
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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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The distribution is given by the following equation::
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.. math:: f(v) = \left(\frac{M}{2 \pi RT}\right)^{\frac{3}{2}} \cdot 4 \pi v^2
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\cdot e^{-\frac{Mv^2}{2RT}}
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-------------------------------------------------
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| f(v) = (M/2πRT)^(3/2) * 4πv^2 * e^(-Mv^2/2RT) |
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-------------------------------------------------
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where:
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* :math:`f(v)` is the fraction of molecules with a speed :math:`v`
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* :math:`M` is the molar mass of the gas in kg/mol
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* :math:`R` is the gas constant
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* :math:`T` is the absolute temperature
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* ``f(v)`` is the fraction of molecules with a speed ``v``
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* ``M`` is the molar mass of the gas in kg/mol
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* ``R`` is the gas constant
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* ``T`` is the absolute temperature
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More information about the Maxwell-Boltzmann distribution can be found here:
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https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution
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The average speed can be calculated by integrating the Maxwell-Boltzmann distribution
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from 0 to infinity and dividing by the total number of molecules. The result is:
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from 0 to infinity and dividing by the total number of molecules. The result is::
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.. math:: v_{avg} = \sqrt{\frac{8RT}{\pi M}}
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---------------------
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| v_avg = √(8RT/πM) |
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---------------------
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The most probable speed is the speed at which the Maxwell-Boltzmann distribution
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is at its maximum. This can be found by differentiating the Maxwell-Boltzmann
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distribution with respect to :math:`v` and setting the result equal to zero.
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The result is:
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distribution with respect to v and setting the result equal to zero. The result is::
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.. math:: v_{mp} = \sqrt{\frac{2RT}{M}}
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---------------------
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| v_mp = √(2RT/M) |
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---------------------
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The root-mean-square speed is another measure of the average speed
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of the molecules in a gas. It is calculated by taking the square root
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of the average of the squares of the speeds of the molecules. The result is:
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of the average of the squares of the speeds of the molecules. The result is::
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.. math:: v_{rms} = \sqrt{\frac{3RT}{M}}
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---------------------
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| v_rms = √(3RT/M) |
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---------------------
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Here we have defined functions to calculate the average and
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most probable speeds of molecules in a gas given the

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