An effusive molecular beam source produces a beam of molecules with a Maxwell- Boltzmann speed distribution....

 

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An effusive molecular beam source produces a beam of molecules with a Maxwell- Boltzmann speed distribution. Such a source may be realized by placing a small pinhole at the end of a tube and using a gas pressure in the tube such that the mean free path of the gas is much larger than the diameter of the pinhole. Assume that such an effusive beam is directed at a surface, and find an expression for the average translational energy of the molecules striking the surface. [Salient information: The average translational energy of molecules striking a surface is a flux-weighted value and not a density-weighted value, because the faster molecules hit the surface more often than the slower ones even if the number density N(v) is the same for fast and slow molecules. Start with the number density distribution, which is related to the M-B distribution function by -mv /2kBT N(v)dv = pF (v)dv, where p is number density (a constant); thus, N(v)dv F(v)dv v The relationship between the number density and flux distributions is given by I(v)dv=vN(v)dv, where I(v) is flux. You need to get to get from I(v)dv to I(E)dE: use E = mv/2. Then you can use the translational energy probability distribution, I(E)dE, to find the average value of E. If you are not using a normalized function for I(E), you will need to divide by the total probability, I(E)DE . 0 An effusive molecular beam source produces a beam of molecules with a Maxwell- Boltzmann speed distribution. Such a source may be realized by placing a small pinhole at the end of a tube and using a gas pressure in the tube such that the mean free path of the gas is much larger than the diameter of the pinhole. Assume that such an effusive beam is directed at a surface, and find an expression for the average translational energy of the molecules striking the surface. [Salient information: The average translational energy of molecules striking a surface is a flux-weighted value and not a density-weighted value, because the faster molecules hit the surface more often than the slower ones even if the number density N(v) is the same for fast and slow molecules. Start with the number density distribution, which is related to the M-B distribution function by -mv /2kBT N(v)dv = pF (v)dv, where p is number density (a constant); thus, N(v)dv F(v)dv v The relationship between the number density and flux distributions is given by I(v)dv=vN(v)dv, where I(v) is flux. You need to get to get from I(v)dv to I(E)dE: use E = mv/2. Then you can use the translational energy probability distribution, I(E)dE, to find the average value of E. If you are not using a normalized function for I(E), you will need to divide by the total probability, I(E)DE . 0