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Kinetic theory of gases. Boltzmanns Law and Maxwell distribution

Problem 2.90

Find the fraction of molecules whose velocity projections on the \(x\) axis fall within the interval from \(v_{x}\) to \(v_{x}+d v_{x},\) while the moduli of perpendicular velocity components fall within the interval from \(v_{\perp}\) to \(v_{\perp}+d v_{\perp} .\) The mass of each molecule is \(m,\) and the temperature is \(\bar{T}\).

Reveal Answer
dN/N=(m2πkT)3/2e−mv2/2kT2πv⊥dv⊥dvxd N / N=\left(\frac{m}{2 \pi k T}\right)^{3 / 2} \mathrm{e}^{-m v 2 / 2 k T} 2 \pi v_{\perp} d v_{\perp} d v_{x}
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