Understanding ideal gas equation from a single particle 

Ideal gas equation is PV = NkT (or NRT). With k, Boltzmann constant, in the equation, N is the number of particles.

T is temperature, representing average kinetic energy. P is pressure, representing average momentum. Suppose a gas contains only a single molecule. Average energy and momentum are simply energy and momentum.

Suppose there are two identical containers. Each container contains a single molecule with mass 1 and 4, respectively. The smaller molecule possesses a velocity of 2 and the bigger molecule possesses a velocity of 1. 

The smaller molecule has a kinetic energy of 1/2*m*v^2 = 1/2*1*2^2 = 2. The larger molecule has a kinetic energy of 1/2*m*v^2 = 1/2*4*1^2 = 2. The kinetic energy is the same. Therefore the temperature is the same.

In the ideal gas equation PV = NkT, V, N, T are the same for two gases. We should expect P to be the same as well.

The smaller molecule has a momentum mv= 1*2=2. The larger molecule has a momentum mv= 4*1=4. It seems that the larger molecule has a higher momentum. But wait! The smaller molecule is faster. In the same time period, the smaller molecule will hit the container wall twice as much. 2*2 = 4. They generate the same amount of pressure.

The above presentation is not a proof of ideal gas equation. But it let me understand the movement of gas molecules a little better.