1 Kinetics 1.3 The Boltzmann factor 1.5 Change of variables and solid angles

1.4 The velocity and speed distributions

(September 1, 2023)

Consider an ideal gas. Each atom is a subsystem with velocity between $v_{x}$ and $v_{x}+dv_{x}$ , $v_{y}$ and $v_{y}+dv_{y}$ , and $v_{z}$ and $v_{z}+dv_{z}$ .

{\rm d}\mathscr{P}_{\vec{v}}=Ce^{-mv^{2}/2{k_{\scriptscriptstyle B}T}}\,{\rm d% }v_{x}{\rm d}v_{y}{\rm d}v_{z}=P(v_{x},v_{y},v_{z})\,{\rm d}^{3}v

(1.46)

where $C$ is a normalizing constant and $\vec{v}^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}$ . We can determine $C$ from the normalization condition:

\displaystyle 1=\int_{\mbox{all $\vec{v}$}}{\rm d}\mathscr{P}_{\vec{v}}=

\displaystyle\int_{\mbox{all $\vec{v}$}}\,Ce^{-m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}% )/2{k_{\scriptscriptstyle B}T}}\,{\rm d}v_{x}{\rm d}v_{y}{\rm d}v_{z}

(1.47)

Doing this integral (which factorizes into and integrals over $v_{x}$ , $v_{y}$ and $v_{z}$ ) leads to the distribution of velocities

{\rm d}\mathscr{P}_{\vec{v}}=\left(\frac{m}{2\pi{kT}}\right)^{3/2}e^{-m\vec{v}% ^{2}/2{k_{\scriptscriptstyle B}T}}\,{\rm d}v_{x}{\rm d}v_{y}{\rm d}v_{z}

(1.48)

We note that the probability for the vector $\vec{v}$ factorizes into a probability of $v_{x}$ , times a probability of $v_{y}$ , times a probability of $v_{z}$

{\rm d}\mathscr{P}_{\vec{v}}=P(v_{x})\,{\rm d}v_{x}\,P(v_{y}){\rm d}v_{y}\,P(v% _{z}){\rm d}v_{z}

(1.49)

So, the probability of finding a particle with $x$ -component of velocity in $[v_{x},v_{x}+{\rm d}v_{x}]$ is after integrating over $v_{y}$ and $v_{z}$ is

{\rm d}\mathscr{P}_{v_{x}}=P(v_{x})\,{\rm d}v_{x}=\left(\frac{m}{2\pi{kT}}% \right)^{1/2}e^{-mv_{x}^{2}/2{k_{\scriptscriptstyle B}T}}\,{\rm d}v_{x}

(1.50)

The book calls $P(v_{x})$ , the uninformative name $g(v_{x})$ .

To find the speed distribution we have to add up the probabilities $d\mathscr{P}_{\vec{v}}$ for all velocities with speed between $v$ and $v+{\rm d}v$ . This is a spherical shell of width ${\rm d}v$ (see lecture)

	$\displaystyle d\mathscr{P}_{v}=$	$\displaystyle\int_{\mbox{$\vec{v}$ in shell}}{\rm d}\mathscr{P}_{\vec{v}}$		(1.51)
	$\displaystyle=$	$\displaystyle\left(\frac{m}{2\pi{kT}}\right)^{3/2}e^{-mv^{2}/2{k_{% \scriptscriptstyle B}T}}\,4\pi v^{2}\,{\rm d}v\equiv P(v)\,{\rm d}v$		(1.52)

Explicitly the probability density for speed $v$ is

P(v)=\left(\frac{m}{2\pi{kT}}\right)^{3/2}e^{-mv^{2}/2{k_{\scriptscriptstyle B% }T}}\,4\pi v^{2}\,

(1.53)

The book calls $P(v)$ , the uninformative name $f(v)$ .