44distribution is a probability distribution that describes the distribution of
55speeds of particles in an ideal gas.
66
7- The distribution is given by the following equation:
7+ The distribution is given by the following equation::
88
99 -------------------------------------------------
1010 | f(v) = (M/2πRT)^(3/2) * 4πv^2 * e^(-Mv^2/2RT) |
1111 -------------------------------------------------
1212
1313where:
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
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
1818
1919More information about the Maxwell-Boltzmann distribution can be found here:
2020https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution
2121
2222The average speed can be calculated by integrating the Maxwell-Boltzmann distribution
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::
2424
25- ---------------------
26- | vavg = √(8RT/πM) |
27- ---------------------
25+ ----------------------
26+ | v_avg = √(8RT/πM) |
27+ ----------------------
2828
2929The most probable speed is the speed at which the Maxwell-Boltzmann distribution
3030is at its maximum. This can be found by differentiating the Maxwell-Boltzmann
31- distribution with respect to v and setting the result equal to zero. The result is:
31+ distribution with respect to ``v`` and setting the result equal to zero. The result is: :
3232
33- ---------------------
34- | vmp = √(2RT/M) |
35- ---------------------
33+ ----------------------
34+ | v_mp = √(2RT/M) |
35+ ----------------------
3636
3737The root-mean-square speed is another measure of the average speed
3838of the molecules in a gas. It is calculated by taking the square root
39- 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::
4040
41- ---------------------
42- | vrms = √(3RT/M) |
43- ---------------------
41+ ----------------------
42+ | v_rms = √(3RT/M) |
43+ ----------------------
4444
4545Here we have defined functions to calculate the average and
4646most probable speeds of molecules in a gas given the
@@ -57,6 +57,7 @@ def avg_speed_of_molecule(temperature: float, molar_mass: float) -> float:
5757 and returns the average speed of a molecule in the gas (in m/s).
5858
5959 Examples:
60+
6061 >>> avg_speed_of_molecule(273, 0.028) # nitrogen at 273 K
6162 454.3488755020387
6263 >>> avg_speed_of_molecule(300, 0.032) # oxygen at 300 K
@@ -84,6 +85,7 @@ def mps_speed_of_molecule(temperature: float, molar_mass: float) -> float:
8485 and returns the most probable speed of a molecule in the gas (in m/s).
8586
8687 Examples:
88+
8789 >>> mps_speed_of_molecule(273, 0.028) # nitrogen at 273 K
8890 402.65620701908966
8991 >>> mps_speed_of_molecule(300, 0.032) # oxygen at 300 K
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