Problem 15

plumleyj 09 Jan 2007 15:10

Starting with;

(1)\begin{align} f(p_x,p_y,p_z)dp_xdp_ydp_z=(2\pi mkT)^{-\frac 3 2}e^{\frac {-p_x^2+p_y^2+p_z^2} {2mkT}}dp_xdp_ydp_z \end{align}

Make the proper substitutions provided within the question in order to convert Cartesian coordinates to a spherical polar coordinate representation;

(2)\begin{align} f(p)d\bold p=(2\pi mkT)^{-\frac 3 2}e^{\frac {-p^2} {2mkT}} p^2 \sin \theta dp d\theta d\phi \end{align}

Integrate equation 2 over all space with respect to $\theta$ and $\phi$

(3)\begin{align} \begin {flalign*} f(p)d\bold p & =(2\pi mkT)^{-\frac 3 2}e^{\frac {-p^2} {2mkT}} p^2 dp \displaystyle \int_0^\pi \sin \theta d\theta \int_0^{2\pi} d\phi \\ & =4\pi p^2(2\pi mkT)^{-\frac 3 2}e^{\frac {-p^2} {2mkT}}dp \end {flalign*} \end{align}

For the fraction of molecules with speeds between v and v + dv, substitute p=mv to get

(4)\begin{align} f(v)dv=4\pi v^2\left(\frac {m} {2\pi kT}\right)^{\frac 3 2}e^{\frac {-mv^2} {2kT}}dv \end{align}