We have the following Maxwell-Boltzmann distribution for speeds of molecules in a gas.
The number of molecules between speeds $v$ and $v+dv$ are:
$$n(v)dv = 4\pi N \left(\frac{m}{2\pi k_{B}T}\right)^{3/2} v^2 e^{-mv^2/2k_{B}T} dv$$
where $k_B$ is the Boltzmann's constant, $T$ is the temperature of the gas, $m$ is mass of a molecule and N is the total number of particles.
Now I am trying to find $v_{rms} = \sqrt{\overline{v^2}}$ and I already know it should come out to be $\sqrt{\frac{3k_{B}T}{m}}$
Here, $\overline{X}$ $\equiv$ average of $X$ over all molecules.
I went about it this way:
$$\overline{v^2} = \frac{1}{N} \int_0^\infty v^2 n(v) dv = \frac{1}{N} \int_0^\infty v^2 4\pi N \left(\frac{m}{2\pi k_{B}T}\right)^{3/2} v^2 e^{-mv^2/2k_{B}T} dv$$
$$= 4\pi \left(\frac{m}{2\pi k_{B}T}\right)^{3/2} \int_0^\infty v^4 e^{-mv^2/2k_{B}T} dv$$
Now, I looked up this integral:
$$\int _{0}^{\infty } v^{2n} e^{-av^2}\, dv = \frac {(2n-1)!!}{a^n\, 2^{n+1}}{\sqrt {\frac {\pi }{a}}}$$
Putting $n = 2$ and $a = m/2k_{B}T$:
$$\int _{0}^{\infty } v^4 e^{-mv^2/2k_{B}T}\, dv = \frac {3!!}{\left(\frac{m}{2k_{B}T}\right)^2\, 8}{\sqrt {\frac {2\pi k_{B}T}{m}}} = 90\sqrt\pi \left(\frac{2k_{B}T}{m}\right)^{5/2}$$
$$\overline{v^2} = 4\pi \left(\frac{m}{2\pi k_{B}T}\right)^{3/2} 90\sqrt\pi \left(\frac{2k_{B}T}{m}\right)^{5/2} = 360 \left(\frac{\pi m}{2\pi k_{B}T}\right)^{3/2} \left(\frac{2k_{B}T}{m}\right)^{5/2} = \left(\frac{720k_{B}T}{m}\right)$$
but this is not the $\left(\frac{3k_{B}T}{m}\right)$ which was correct. Can somebody tell me what I did wrong?
Best Answer
The gaussian integral is not evaluated correctly, though the form you looked up is correct. Note that
$$\int _{0}^{\infty } v^{4} e^{-av^2}\, dv = \frac {3!!}{a^2\, 8}{\sqrt {\frac {\pi }{a}}} = \frac38\sqrt\pi a^{-5/2} \to \frac38\sqrt\pi \left(\frac{2kT}{m}\right)^{5/2} $$
As an aside, you can evaluate this integral by hand using:
$$\int _{0}^{\infty } dv\ v^{4} e^{-av^2} = \frac{d^2}{da^2}\int _{0}^{\infty } dv\ e^{-av^2} = \frac{d^2}{da^2}\frac12\sqrt{\frac{\pi}{a}} = \frac{\sqrt{\pi}}{2}\frac{3}{4}a^{-52} $$