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