物理-氣體分子運動論問題

P = (1/3)ρc^2
where P is the pressure, ρ is the gas density and c^2 the mean-square speed of molecules.

But ρ = M/V, where M is the mass of gas and V is the volume of gas
hence, P = (1/3).(M/V)c^2
i.e. PV = (1/3).M.c^2
Because M = Nm, where N is the no. of gas molecules in volume V and m is the mass of a molecule.
Hence, PV = (1/3)Nmc^2

Now, using the ideal gas equation, PV = nRT
where n is the no. of moles of gas, R is Universal Gas Constant and T is the temperature in absolute scale.
we have, nRT = (1/3)Nmc^2

Consider 1 mole of gas (i.e. n=1), the no. of molecules N then becomes Na, the Avogadro's Number, which carries the value of 6.02 x 10^23 mol^-1
Thus, RT = (1/3)(Na)mc^2
Multiply both sides by (3/2),
(3/2)RT = (1/2)(Na)mc^2
(3/2).(R/Na)T = (1/2)mc^2

Since (1/2)mc^2 is the kinetic energy (動能) of a molecule,
i.e. Kinetic energy of a molecule = (1/2)mc^2 = (3/2)kT, where k is Boltzmann's constant (= 1.38x10^-23 J/K), which is equal to (R/Na)

Your quoted equations P = (1/2)mc^2 and P = (2/3)kT are clearly wrong. Both quantities (1/2)mc^2 and (2/3)kT are in unit of energy (Joule). How could you equate "pressure" (in unit of N/m^2) with unit of energy?