v=4/sqrt(pi)*((M/2RT)^(3/2))*(Integral of (v^3)(e^(-Mv^2/2RT)) from 0 to infinity)
Prove v=sqrt(8RT/piM)
Prove v=sqrt(8RT/piM)
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u = Mv^2 / (2RT) ---> v^2 = (2RT/M) u
(RT/M)du = v dv
v^3 dv = v^2 v dv = (2RT/M) u (RT/M)du = 2(RT/M)^2 u du
so your integrand becomes
(v^3)(e^(-Mv^2/2RT) dv = 2(RT/M)^2 u e^{-u} du
Can you do it now?
(RT/M)du = v dv
v^3 dv = v^2 v dv = (2RT/M) u (RT/M)du = 2(RT/M)^2 u du
so your integrand becomes
(v^3)(e^(-Mv^2/2RT) dv = 2(RT/M)^2 u e^{-u} du
Can you do it now?