[NOTES/SM-04007] Applications of Maxwell's Distribution

Equation of State for a Perfect Gas

Consider a gas enclosed in a cubical container of sides \(L\). It is assumed that the gas molecules move freely within the container.The pressure due to the gas is due to the change in momentum when gas particles hit the wall. We need to compute the rate of change of momentum. It must be remembered that at any time, only the molecules travelling perpendicular to the wall contribute to the pressure.

\begin{align*}
\text{Pressure} =&F/A = \frac{\text{change in momentum per unit
time}}{A}\\
\text{Pressure} \times A dt =& \text{change in momentum in time} dt.\\
=&\int_0^\infty dv_x\int_{-\infty}^\infty\int_{-\infty}^\infty
dv_ydv_z(2mv_x)\left(\frac{v_x\Delta t A}{V}\right)\eta(\vec{v})\\
=&\left(\frac{NkT}{V}\right) A dT.
\end{align*}
Therefore the pressure is given by
\begin{equation}
p=\frac{NkT}{V}
\end{equation}
and we have arrived at the well known result:
\begin{equation}
\boxed{pV=NkT}.
\end{equation}

Experimental determination of Boltzmann constant $k$

As an application of  Maxwell distribution of velocities we discuss determination of Boltzmann constant by measuring the effusion rate of a gas through a hole.

Number of molecules hitting an area $A$ in time $\Delta t$\\ $=$ the number of molecules in cylinder of base $A$ and height $v_x\Delta t$. This number is given by

$$ = \underbrace{v_x\Delta tA} \underbrace{\eta(v)d^3v} \times\frac{1}{V} \qquad v_x>0. $$

The average number of molecules in velocity range $\vec{v}$ and $\vec{v}+d\vec{v}$ is given by the Maxwell distribution and we have $$ N\left(\frac{m}{2\pi kT}\right)^{3/2}\exp\left(-\frac{mv^2}{2kT}\right). $$ The probable number of molecules that strike Area $A$ in time $dt$ is

\begin{align*}
&\iiint d\vec{v}_xdv_ydv_z\left(\frac{v_x\Delta
tA}{V}\right)\eta(v)d^3u\\
&=\left(\frac{N}{V}\right)(Adt)\left(\frac{m}{2\pi kT}\right)^{1/2}
\int_0^\infty v_x \exp\left(-\frac{mv^2_x}{2kT}\right) dv_x\\
&=\left(\frac{N}{V}\right)(adt)\left(\frac{m}{2\pi
kT}\right)^{1/2}\left(\frac{2kT}{m}\right)
\int_0^\infty se^{-s^2}ds\\
&=\frac{N}{V}(Adt)\left(\frac{m}{2\pi
kT}\right)^{1/2}\left(\frac{kT}{m}\right).
\end{align*}

Thus we have arrived at a result useful for many applications.\\ Probable number of molecules that strike the wall per unit area per unit time is $\frac{1}{4} n\vec{v}.$ The rate of change of $N$, (gas leaking through an area $A$), is

$$ \frac{dN}{dt} = \left(\frac{N}{V}\right)\left(\frac{kT}{2\pi m}\right)^{1/2}A.
$$

This experiment can be done and Boltzmann constant can be determined. The effusion process can be applied to separation of isotopes. If we apply it to thermionic emission of electrons, we would get $ \text{current}~\propto T^{1/2}$ This result for electrons does not agree with experimental data. The experimental results can be explained only by making use of quantum, Fermi Dirac, statistics.