- #1
Roo2
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I put off an assignment until the last minute and on the very last question it came back to bite me in the butt. I'm supposed to derive an equation from another equation, and the math is not working out for me. If there's anyone still up and reading this and that can point me in the right direction within the next 9 hours, I'd appreciate it :D
Derive equation b from equation a using equation c
equation a: P(v) = [m/(2*pi*Kb*T)]^(3/2) * e^-[(mv^2)/(2*Kb*T)] * 4*pi*v^2
equation b: <v> = sqrt[(8*Kb*T)/(pi*m)]
equation c: <v> = integral (from 0 to infiniti) v*P(v) dv
P(v) = (c1*v^2) * e^-(c2*v^2)
c1 = {[m/(2*pi*Kb*T)]^(3/2)}/(4*pi)
c2 = m/(2*Kb*T)
Therefore, int(v*P(v)dv) = c1 * int[(v^3)*e^(-c2*v^2)]
According to wiki, http://en.wikipedia.org/wiki/Lists_of_integrals (which our prof said to use):
integral (x^3) e^-ax^2 = 1/a^2
The integral is the 5th one down in the section "Definite integrals lacking closed-form antiderivatives"
Therefore, my integral evaluates to c1/(c2^2)
However, when recompile my constants, they are not in a form that is very easily rearrangeable to the desired form. Furthermore, I tried plugging in values to both formulas and obtained different results. Could someone please tell me where I made a mistake?
Homework Statement
Derive equation b from equation a using equation c
Homework Equations
equation a: P(v) = [m/(2*pi*Kb*T)]^(3/2) * e^-[(mv^2)/(2*Kb*T)] * 4*pi*v^2
equation b: <v> = sqrt[(8*Kb*T)/(pi*m)]
equation c: <v> = integral (from 0 to infiniti) v*P(v) dv
The Attempt at a Solution
P(v) = (c1*v^2) * e^-(c2*v^2)
c1 = {[m/(2*pi*Kb*T)]^(3/2)}/(4*pi)
c2 = m/(2*Kb*T)
Therefore, int(v*P(v)dv) = c1 * int[(v^3)*e^(-c2*v^2)]
According to wiki, http://en.wikipedia.org/wiki/Lists_of_integrals (which our prof said to use):
integral (x^3) e^-ax^2 = 1/a^2
The integral is the 5th one down in the section "Definite integrals lacking closed-form antiderivatives"
Therefore, my integral evaluates to c1/(c2^2)
However, when recompile my constants, they are not in a form that is very easily rearrangeable to the desired form. Furthermore, I tried plugging in values to both formulas and obtained different results. Could someone please tell me where I made a mistake?