Mathematical Steps of Ideal Gas Law Derivation

Step 1: Pressure from Molecular Collisions

P=13NVmv2‾P = \frac{1}{3}\frac{N}{V}m\overline{v^2}

This equation relates pressure to molecular mass, velocity, and concentration.

Step 2: Average Kinetic Energy

12mv2‾=32kT\frac{1}{2}m\overline{v^2} = \frac{3}{2}kT

This relates molecular kinetic energy to temperature.

Step 3: Final Derivation

Combining the equations and using N = nNA:

PV=NkT=nNAkT=nRTPV = NkT = nN_AkT = nRT

Where R = NAk is the gas constant.