From the post "Neon Light Boom" dated 22 Dec 2017,
\(V_{min}=A\left(\cfrac{P}{T}\right)^2\)
and \(A=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}\)
but
\(\cfrac{P}{T}=\cfrac{n}{V_{ol}}R=\cfrac{M}{m_iN_AV_{ol}}R=\cfrac{density}{m_i}k\)
\(V_{min}=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}\left(\cfrac{density}{m_i}k\right)^2\)
\(V_{min}=\cfrac{3.4354^2}{2}\cfrac{m_i}{qZ^2}density^2\)
\(V_{min}=\cfrac{1}{2}\cfrac{m_i}{q}\left(3.4354*\cfrac{density}{Z}\right)^2\) ---(*)
as \(v_{boom}=3.4354*\cfrac{density}{Z}\)
\(qV_{min}=\cfrac{1}{2}{m_i}v_{boom}^2\)
which just says the voltage applied converts to \(KE\) at \(v_{boom}\), provided that the gap length between the electrodes are sufficiently wide. This implies that the expression for \(V_{min}\) is dimensional-ly consistent.
However, if \(q\rightarrow \cfrac{1}{4}q\) then,
\(\cfrac{1}{4}qV_{min}=\cfrac{1}{2}{m_c}v_{boom}^2\)
and (*) becomes,
\(V_{min\,c}=2\cfrac{m_c}{q}\left(3.4354*\cfrac{density}{Z}\right)^2\) ---(**)
\(m_c\), however, is not available.
How does this tally with experimental values of \(V_{min}\)?
