Q.What is the relation between the average kinetic energy and pressure?

Two equivalent routes to the same result. Pressure can be connected to a gas's stored energy in two ways that turn out to agree exactly. From the macroscopic (thermodynamic) side, using $U=\tfrac32NkT$ and the ideal gas equation $NkT=PV$, one gets $U=\tfrac32PV$, i.e.

where $u=U/V$ is the internal energy density. From the microscopic (kinetic) side, starting from $P=\tfrac13\rho\overline{v^2}$ (with mass density $\rho=nm$) and multiplying and dividing by 2, one gets exactly the same structure,

where $\tfrac12\rho\overline{v^2}$ is the mean kinetic energy stored per unit volume. …