4.4: The Probability-density Function for Gas Velocities in One Dimension
- Page ID
- 151679
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In Section 4.3, we find a differential equation in the function \({\rho }_x\left(v_x\right)\). Unlike the velocity, which takes values from zero to infinity, the \(x\)-component, \(v_x\), takes values from minus infinity to plus infinity. The probability density at an infinite velocity, in either direction, is necessarily zero. Therefore, we cannot evaluate the integral of \(d \rho_x\left(v_x\right)/ \rho_x\left(v_x\right)\) from \(v_x=-\infty\) to an arbitrary velocity, \(v_x\). However, we know from Maxwell’s assumption that the probability density for \(v_x\) must be independent of whether the molecule is traveling in the direction of the positive \(x\)-axis or the negative \(x\)-axis. That is, \(\rho_x\left(v_x\right)\) must be an even function; the probability density function must be symmetric around \(v_x=0\); \(\rho_x\left(v_x\right)=\rho_x\left(-v_x\right)\). Hence, we can express \(\rho_x\left(v_x\right)\) relative to its fixed value,\( \rho_x\left(0\right)\), at \(v_x=0\). We integrate \(d \rho_x\left(v_x\right)/ \rho_x\left(v_x\right)\) from \(\rho_x\left(0\right)\) to \(\rho_x\left(v_x\right)\) as \(v_x\) goes from zero to an arbitrary velocity, \(v_x\), to find
\[\int^{\rho_x\left(v_x\right)}_{\rho_x\left(0\right)} \frac{d \rho_x\left(v_x\right)}{\rho_x\left(v_x\right)}=-\lambda \int^{v_x}_0 v_xdv_x \nonumber \]
or
\[\rho_x\left(v_x\right)=\frac{df_x\left(v_x\right)}{dv_x}= \rho_x\left(0\right)\mathrm{exp}\left(\frac{-\lambda v^2_x}{2}\right) \nonumber \]
The value of \(\rho_x\left(0\right)\) must be such as to make the integral of \(\rho _x\left(v_x\right)\) over all possible values of \(v_x\), \(-\infty < v_x
\[\begin{aligned} 1 & =\int^{\infty}_{-\infty} \rho_x\left(v_x\right) dv_x \\ ~ & =\int^{\infty}_{-\infty} \frac{df_x\left(v_x\right)}{dv_x}dv_x \\ ~ & = \rho_x \left(0\right)\int^{\infty}_{-\infty} \mathrm{exp} \left(\frac{-\lambda v^2_x}{2}\right) dv_x \\ ~ & =\rho_x\left(0\right)\sqrt{\frac{2\pi }{\lambda }} \end{aligned} \nonumber \]
where we use the definite integral \(\int^{\infty }_{-\infty } \mathrm{exp} \left(-ax^2\right) dx=\sqrt{ \pi /a}\). (See Appendix D.) It follows that \( \rho_x\left(0\right)= \left( \lambda /2 \pi \right)^{1/2}\). The one-dimensional probability-density function becomes
\[\begin{aligned} \rho_x\left(v_x\right) & =\frac{df_x\left(v_x\right)}{dv_x} \\ ~ & = \left(\frac{\lambda }{2\pi }\right)^{1/2}\mathrm{exp}\left(\frac{-\lambda v^2_x}{2}\right) \end{aligned} \nonumber \]
Note that this is the normal distribution with \(\mu =0\) and \({\sigma }^2={\lambda }^{-1}\). So \({\lambda }^{-1}\) is the variance of the normal one-dimensional probability-density function. As noted above, in Section 4.6 we find that \(\lambda ={m}/{kT}\).