Intuituve reason for the constant of proportionality 2/3 in pv=nRT=(2/3)U
The 2/3 factor is from the math of the kinetic theory of gases. Think of a cube with six faces (i.e. three orthogonal directions), and the particles have equal probability of hitting a face in the x, y or z direction.  The pressure is exerted equally amongst the three orthogonal directions, hence a fact of 1/3.

Wikipdedia also shows the steps in deriving the kinetic theory.