气体分子速率推导

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

最概然速率推导

\(v=v_p时,\dfrac{df(v)}{dv}=0\) \[ \begin{aligned} \dfrac{df(v)}{dv}|_{v=v_p}&=\left[8\pi\left(\dfrac{m}{2\pi kT}\right)^{\frac{3}{2}}e^{-\frac{mv^2}{2kT}}v + 4\pi\left(\dfrac{m}{2\pi kT}\right)^{\frac{3}{2}}e^{-\frac{mv^2}{2kT}}v^2\cdot(-\dfrac{2mv}{2kT})\right]|{v=v_0}\\ &=8\pi\left(\dfrac{m}{2\pi kT}\right)^{\frac{3}{2}}e^{-\frac{mv_p^2}{2kT}}v_p + 4\pi\left(\dfrac{m}{2\pi kT}\right)^{\frac{3}{2}}e^{-\frac{mv_p^2}{2kT}}v_p^2\cdot(-\dfrac{2mv_p}{2kT})=0\\ &\therefore v_p=\sqrt{\dfrac{2kT}{m}},其中m为单个分子的质量\\ \end{aligned} \]

平均速率

平均速率即麦克斯韦速率分布函数的期望值 \[ \begin{aligned} \overline v&=\int^\infty_0vf(v)dv=\int^\infty_0 4\pi\left(\dfrac{m}{2\pi kT}\right)^{\frac{3}{2}}e^{-\frac{mv^2}{2kT}}v^3dv\\ &=\sqrt{\dfrac{8kT}{\pi m}} \end{aligned} \]

方均根速率

\[ \begin{aligned} &\overline{v^2}=\int^\infty_0v^2f(v)dv=\int^\infty_0 4\pi\left(\dfrac{m}{2\pi kT}\right)^{\frac{3}{2}}e^{-\frac{mv^2}{2kT}}v^4dv=\dfrac{3kT}{m}\\ &\sqrt{\overline{v^2}}=\sqrt{\dfrac{3kT}{m}}\\ \end{aligned} \]