From ideal gas, This should not be ideal gas please refer to "Conceptual Mishap Clustering" dated 13 Dec 2017,
\(T=\cfrac{1}{n}.\cfrac{PV}{R}\)
\(T=\cfrac{V}{n}.\cfrac{P}{R}=\cfrac{1}{\rho_n}.\cfrac{P}{R}\)
so,
\(P=\rho_nRT\)
From the post "Going Back To Durian Constantly" dated 09 Nov 2017,
\(T=\cfrac{(2c)^3}{3}*\cfrac{1}{2}m_av^2_{rms}\)
so,
\(P=\rho_nR.\cfrac{(2c)^3}{3}*\cfrac{1}{2}m_av^2_{rms}\)
\(P=\cfrac{1}{3}\rho_nR.{(2c)^3}*\cfrac{1}{2}m_av^2_{rms}\)
If \(R=2*\cfrac{4}{3}\pi\)
\(P=\cfrac{2}{3}\rho_n*\cfrac{4}{3}\pi{(2c)^3}*\cfrac{1}{2}m_av^2_{rms}\) --- (*)
where \(\cfrac{1}{3}\) accounts for one of the three space dimensions. The term,
\(\rho_n*\cfrac{4}{3}\pi{(2c)^3}*\cfrac{1}{2}m_av^2_{rms}\)
the total kinetic energy in a volume of radius \(2c\). It is likely that the factor \(2\) accounts for both positive and negative directions. The expression however is not dimensional-ly consistent.
Consider, \(V=\cfrac{4}{3}\pi x^3\),
\(\cfrac{dV}{dx}=3*\cfrac{4}{3}\pi x^2\)
\(\cfrac{d^2V}{dx^2}=2*3*\cfrac{4}{3}\pi x\)
\(\cfrac{d^3V}{dx^3}=1*2*3*\cfrac{4}{3}\pi=3R\)
in which case \(R\) is the change of a volume in all 3 space dimensions and has unit "\(vol\,m^{-3}\)" scaled by \(\cfrac{1}{3}\) to consider only one direction of \(x\).
This might account for the unit of the expression (*). Then,
\(P=\cfrac{2}{3}\rho_n*\cfrac{4}{3}\pi{(2c)^3}*\cfrac{1}{2}m_av^2_{rms}\)
is energy per unit volume \(r=2c\), or force per unit area as energy per meter is force.
\(\cfrac{E}{m^3}=\cfrac{F}{m^2}\)
\(\because\)
\(\cfrac{E}{x}=F\)
What is the factor \(\cfrac{2}{3}\) for?
