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1 |
| -r""" |
| 1 | +""" |
2 | 2 | The root-mean-square, average and most probable speeds of gas molecules are
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3 | 3 | derived from the Maxwell-Boltzmann distribution. The Maxwell-Boltzmann
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4 | 4 | distribution is a probability distribution that describes the distribution of
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5 | 5 | speeds of particles in an ideal gas.
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6 | 6 |
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7 |
| -The distribution is given by the following equation: |
8 |
| - f(v) = (M/2πRT)^(3/2) * 4πv^2 * e^(-Mv^2/2RT) |
| 7 | +The distribution is given by the following equation:: |
9 | 8 |
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10 |
| - .. math:: f(v) = \left(\frac{M}{2 \pi RT}\right)^{\frac{3}{2}} \cdot 4 \pi v^2 |
11 |
| - \cdot e^{-\frac{Mv^2}{2RT}} |
| 9 | + ------------------------------------------------- |
| 10 | + | f(v) = (M/2πRT)^(3/2) * 4πv^2 * e^(-Mv^2/2RT) | |
| 11 | + ------------------------------------------------- |
12 | 12 |
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13 | 13 | where:
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14 |
| - * :math:`f(v)` is the fraction of molecules with a speed :math:`v` |
15 |
| - * :math:`M` is the molar mass of the gas in kg/mol |
16 |
| - * :math:`R` is the gas constant |
17 |
| - * :math:`T` is the absolute temperature |
| 14 | + * ``f(v)`` is the fraction of molecules with a speed ``v`` |
| 15 | + * ``M`` is the molar mass of the gas in kg/mol |
| 16 | + * ``R`` is the gas constant |
| 17 | + * ``T`` is the absolute temperature |
18 | 18 |
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19 | 19 | More information about the Maxwell-Boltzmann distribution can be found here:
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20 | 20 | https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution
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21 | 21 |
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22 | 22 | The average speed can be calculated by integrating the Maxwell-Boltzmann distribution
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23 |
| -from 0 to infinity and dividing by the total number of molecules. The result is: |
| 23 | +from 0 to infinity and dividing by the total number of molecules. The result is:: |
24 | 24 |
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25 |
| - .. math:: v_{avg} = \sqrt{\frac{8RT}{\pi M}} |
| 25 | + --------------------- |
| 26 | + | v_avg = √(8RT/πM) | |
| 27 | + --------------------- |
26 | 28 |
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27 | 29 | The most probable speed is the speed at which the Maxwell-Boltzmann distribution
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28 | 30 | is at its maximum. This can be found by differentiating the Maxwell-Boltzmann
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29 |
| -distribution with respect to :math:`v` and setting the result equal to zero. |
30 |
| -The result is: |
| 31 | +distribution with respect to v and setting the result equal to zero. The result is:: |
31 | 32 |
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32 |
| - .. math:: v_{mp} = \sqrt{\frac{2RT}{M}} |
| 33 | + --------------------- |
| 34 | + | v_mp = √(2RT/M) | |
| 35 | + --------------------- |
33 | 36 |
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34 | 37 | The root-mean-square speed is another measure of the average speed
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35 | 38 | of the molecules in a gas. It is calculated by taking the square root
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36 |
| -of the average of the squares of the speeds of the molecules. The result is: |
| 39 | +of the average of the squares of the speeds of the molecules. The result is:: |
37 | 40 |
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38 |
| - .. math:: v_{rms} = \sqrt{\frac{3RT}{M}} |
| 41 | + --------------------- |
| 42 | + | v_rms = √(3RT/M) | |
| 43 | + --------------------- |
39 | 44 |
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40 | 45 | Here we have defined functions to calculate the average and
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41 | 46 | most probable speeds of molecules in a gas given the
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