GreenPenInc
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Hi guys,
I'm trying to derive the mean thermal wavelength from the Maxwell distribution:
M(v) = 4\pi\left(\frac{m}{2\pi k_BT}\right)^{3/2}v^2e^{-\frac{mv^2}{2k_BT}} = 4\pi\left(\frac{a}{\pi}\right)^{3/2}v^2e^{-av^2}
With a = \frac{m}{2k_BT} introduced for convenience. Since \lambda = \frac{h}{mv}, I figured an expression for the thermal wavelength would be
\lambda_T = \frac{h}{m}\int_0^\infty\frac{M(v)}{v}dv
Problem is, when I do this, I end up with
\lambda_T = 2\sqrt{\frac{h^2}{2\pi m k_BT}}
This is exactly twice the accepted value. Where did I go wrong?
I'm trying to derive the mean thermal wavelength from the Maxwell distribution:
M(v) = 4\pi\left(\frac{m}{2\pi k_BT}\right)^{3/2}v^2e^{-\frac{mv^2}{2k_BT}} = 4\pi\left(\frac{a}{\pi}\right)^{3/2}v^2e^{-av^2}
With a = \frac{m}{2k_BT} introduced for convenience. Since \lambda = \frac{h}{mv}, I figured an expression for the thermal wavelength would be
\lambda_T = \frac{h}{m}\int_0^\infty\frac{M(v)}{v}dv
Problem is, when I do this, I end up with
\lambda_T = 2\sqrt{\frac{h^2}{2\pi m k_BT}}
This is exactly twice the accepted value. Where did I go wrong?