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I don't know how to integrate the Maxwell-Boltzmann distribution without approximation or help from Maple.

Given the Maxwell-Boltzmann distribution:

[tex]f(v) = 4\pi\left[\frac{m}{2\pi kT}\right]^{3/2}v^2\textrm{exp}\left[\frac{-mv^2}{2kT}\right][/tex]

Observe the appearance of the Boltzmann factor ##\textrm{exp}\left[\frac{-mv^2}{2kT}\right]## with ##E = \frac{mv^2}{2}##.

Assuming a fixed temperature and mass, one can simplify this equation:

[tex]f(v) = av^2\textrm{exp}[-bv^2][/tex]

[tex]a = 4\pi \left[\frac{m}{2\pi k T}\right]^{3/2}[/tex]

[tex]b = \frac{m}{2kT}[/tex]

In order to calculate the fraction of particles between two speeds ##v_1## and ##v_2##, one should evaluate the definite integral. It's possible to use this formula directly with low speeds, but for higher speeds between let's say 400-500 m/s an integration is needed.

[tex]\int f(v)dx[/tex]

Here is an link to integral-tables, http://integral-table.com/

How would I solve this problem for let's say a certain amount of moles with hydrogen between two different velocities? Best regards, Tor

Given the Maxwell-Boltzmann distribution:

[tex]f(v) = 4\pi\left[\frac{m}{2\pi kT}\right]^{3/2}v^2\textrm{exp}\left[\frac{-mv^2}{2kT}\right][/tex]

Observe the appearance of the Boltzmann factor ##\textrm{exp}\left[\frac{-mv^2}{2kT}\right]## with ##E = \frac{mv^2}{2}##.

Assuming a fixed temperature and mass, one can simplify this equation:

[tex]f(v) = av^2\textrm{exp}[-bv^2][/tex]

[tex]a = 4\pi \left[\frac{m}{2\pi k T}\right]^{3/2}[/tex]

[tex]b = \frac{m}{2kT}[/tex]

In order to calculate the fraction of particles between two speeds ##v_1## and ##v_2##, one should evaluate the definite integral. It's possible to use this formula directly with low speeds, but for higher speeds between let's say 400-500 m/s an integration is needed.

[tex]\int f(v)dx[/tex]

Here is an link to integral-tables, http://integral-table.com/

How would I solve this problem for let's say a certain amount of moles with hydrogen between two different velocities? Best regards, Tor

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