In the previous topic Kinetic Theory of Gases, we learn how the gas molecules behave. Now we are going to calculate pressure of an ideal gas. First consider a cubic vessel containing N number of molecules of an ideal gas, its length is L, and its walls are perfectly elastic. As shown in the following figure:
All molecules are moving with certain velocities in random directions, now we resolve their velocities in particular directions (i,e; in x-axis, in y-axis and in z-axis), so their velocities can be written as v_xv_y and v_z. In the above figure the faces of cube (bottom, top and sides) are in respectively x-axis, y-axis and z-axis. So when a molecule collide with a certain face, then only a single component of velocity changes, while other components remains same. For example when a molecules collides with left side of the cube, only its x-component of velocity will reverse, and other components v_yand v_z remains same.
The momentum before collision in x-axis is mv_x and after collision it will be -mv_x. So change of momentum will be:

mv_x-(-mv_x)=2mv_x

Now calculating time required for a molecules to reach from one face to another.

Time = distance from one face to another / velocity
t = L/v_x

Total time required for a molecule to come back in the same positions will be twice as that for to reach from one face to another:

T = 2L/v_x

We know that force is a rate of change of momentum, so the force exerted on above face will be:

Force = change of momentum / time
F = (2mv_x) / (2L/v_x)
F= mv_x²/L

Above force is a force exerted by a single molecule, hence force exerted by all molecules in x-axis will be:

F_total=F₁+F₂+F₃+...+F_n
F_total=mv_1x²/L + mv_2x²/L + mv_3x²/L + ... + mv_nx²/L
F_total=(m/L)(v_1x²+v_2x²+v_3x²+...+v_nx²)

Total pressure will be:

Pressure = Force/Area
P = [(m/L)(v_1x²+v_2x²+v_3x²+...+v_nx²)]/L²
P = (m/L³)(v_1x²+v_2x²+v_3x²+...+v_nx²)

Since Volume of cube (V) = L³

P = (m/V)(v_1x²+v_2x²+v_3x²+...+v_nx²)

multiplying and dividing by N:

P = (mN/V)(v_1x²+v_2x²+v_3x²+...+v_nx²)/N

Since m is mass of one molecule, and N is a total number of molecules, hence total mass will be mN, substituting in above equation:

P = (m_total/V)(v_1x²+v_2x²+v_3x²+...+v_nx²)/N

Also mass/volume is known as density, so putting ρ=m_total/V in above equation:

P = ρ(v_1x²+v_2x²+v_3x²+...+v_nx²)/N

(v_1x²+v_2x²+v_3x²+...+v_nx²)/N is the average value of velocity putting in above equation:

P = ρ[v_x(avg)]² ---(i)

[v_x(avg)]² is the only x-component of velocity, other components can be [v_y(avg)]² and [v_y(avg)]², so total average velocity will be:

[v_(avg)]²=[v_x(avg)]²+[v_y(avg)]²+[v_z(avg)]²

On the average:

[v_x(avg)]² = [v_y(avg)]² = [v_z(avg)]²
[v_(avg)]²=[v_x(avg)]²+[v_x(avg)]²+[v_x(avg)]²
[v_(avg)]²=3[v_x(avg)]²
[v_x(avg)]²=[v_(avg)]²/3             putting in equation (i)
P = ρ[v_(avg)]²/3
or P = (1/3)ρ[v_(avg)]²