Mental Discharge

From the post "Checking Discharge Voltage" dated 23 Dec 2017,

$V_{min}=\cfrac{1}{2}\cfrac{m_i}{q}\left(3.4354*\cfrac{density}{Z}\right)^2$

$m_i$ is the mass of one particle.

$V_{min}=\cfrac{1}{2}\cfrac{M}{qN_A}\left(3.4354*\cfrac{density}{Z}\right)^2$

$M$ is molar mass and $N_A$, Avogadro's number.

Atomic No.	Name	Symbol	Mass(g/mol)	Density(g/L)	v_boom(m/s)	Vmin(V)
2	Helium	He	4.002602	0.1785	0.3066	1.9501E-09
10	Neon	Ne	20.1797	0.8999	0.3092	9.9952E-09
18	Argon	Ar	39.948	1.7837	0.3404	2.3993E-08
36	Krypton	Kr	83.798	3.733	0.3562	5.5110E-08
54	Xenon	Xe	131.293	5.887	0.3745	9.5440E-08
86	Radon	Rn	222	9.73	0.3887	1.7381E-07

This set of data assumes that the noble gas atom with a charge of $q$ is accelerated under the voltage $V_{min}$ across a gap of $d=1\,m$ to achieve the speed $v_{boom}$.

This set of values for $V_{min}$ is too low, a typical discharge voltage value for Argon, for a gap of $12\,\mu m$ is $137\,V$.

If we take the per atom view, instead of per particle view above,

$V_{min}=\cfrac{1}{2}\cfrac{m_i}{q}\left(3.4354*{density}*{Z}\right)^2$

$V_{min}=\cfrac{1}{2}\cfrac{M}{qN_A}\left(3.4354*{density}*{Z}\right)^2$

where we have taken that an atom with $Z$ number of particles (of one type), with a charge of $q$, is accelerated under $V_{min}$ for a meter ($d=1\,m$) to achieve $v_{boom}$ speed.

Atomic No.	Name	Symbol	Mass $gmol^{-1}$	Density $gL^{-1}$	v_boom $ms^{-1}$	Vmin $V$	Vmin $\div12\mu m$
1	Hydrogen	H2	2.016	0.0899	0.1544	3.9844E-09	3.32E-04
2	Helium	He	4.002602	0.1785	0.3066	3.120E-08	2.60E-03
10	Neon	Ne	20.1797	0.8999	0.3092	9.995E-05	8.33E+00
18	Argon	Ar	39.948	1.661	0.3170	2.184E-03	1.82E+02
36	Krypton	Kr	83.798	3.733	0.3562	9.256E-02	7.71E+03
54	Xenon	Xe	131.293	5.887	0.3745	8.115E-01	6.76E+04
86	Radon	Rn	222	9.73	0.3887	9.507E+00	7.92E+05

The last column is,

$V=\cfrac{V_{min}}{d}$

$d=12\,\mu m$

to obtain the value of the minimum discharge voltage $V$ for a gap of $12\,\mu m$.  The minimum discharge voltage for a gap distance of $12\,\mu m$ for Argon is $182\,V$.  This is higher than the quoted value of $137\,V$.  (A higher voltage is needed for a gap smaller than one meter, because the particle must reach $v_{boom}$ before reaching the opposite electrode.)

Notice that to accelerate Helium $He$ to the correct $v_{boom}$ speed within a similar gap, the applied voltage is in the mini-voltage (mV) range.  And for Neon $Ne$, a few volts.  Hydrogen gas, interestingly is discharged at $\approx0.33\,mV$, which is at the cellular voltage level.

Good day!

Note: Vmin=0.5*(3.4354*density/Z)^2*M/(6.0221*10^23*1.6021*10^(-19)*10^3) for the first table.
          Vmin=0.5*(3.4354*density*Z)^2*M/(6.0221*10^23*1.6021*10^(-19)*10^3) for the second table.

Note:  Hydrogen gas was added to the last table on 29 Dec 2017.