What is the relationship between temperature and pressure?

Assume we have a collection of gas molecules in gravity-free space in a container with volume V at absolute temperature T. 

Then each molecule moves along with constant velocity in a straight line, until it hits another molecule, or a container wall.  A collision between two molecules is similar to a collision between two balls.  The molecules exchange momentum, but the total momentum of the two molecules is conserved.  When a molecule hits a wall, it bounces back.  Its momentum changes.  To change the molecule's momentum, the wall must exert a force on the molecule.  Newton's third law tells us that the molecule exerts a force on the wall.  The greater the number of molecules hitting a wall, the greater is the force on the wall.  In a container with different size walls, the bigger walls will receive more hits than the smaller walls and therefore experience a greater force.  The pressure in the container is the magnitude of the normal force F on a wall divided by the surface area A of the wall. 

P = F/A

The faster the molecules move in the container, the greater is the change in momentum when they bounce off a wall, and the more often do they hit the walls.  Assume a molecule moves horizontally with speed |v_x| back and forth between two infinitely-massive walls, which are a distance L apart.  When it hits the right wall its momentum changes from p₁ = +m|v_x| to p₂ = -m|v_x|.  The change in the molecule's momentum is Δp_mol = p₂ - p₁ = -2m|v_x|.  The time interval between successive hits on the right wall is Δt = 2L/|v_x|.  So the average force the wall exerts on this molecule is F_mol = Δp_mol/Δt = -2m|v_x|/(2L/|v_x|) = -mv_x²/L.  By Newton's third law, the average force that the molecule exerts on the wall is F_wall = mv_x²/L, it is proportional to the square of the speed of the molecule or its kinetic energy.

Assume that there are N molecules in the volume V, moving horizontally with speed |v_x|.  Not all the molecules have the same kinetic energy.  The force exerted by the molecules on the walls of a container is therefore F = Nm<v_x²>/L, where <v_x²> is the average value of v_x².

The pressure is P = F/A  = Nm<v_x²>/V, since L*A = V.  With ρ_particle = N/V we have

P = F/A = ρ_particlemv_x².

There is nothing special about the x-direction.  The atoms can move up and down, back and forth, in and out.  The average velocity components in all directions are all going to be equal to each other.

<v_x²> = <v_y²> = <v_z²>.

They are each equal to one-third of their sum, which is the square of the magnitude of the average velocity.

<v²> = <v_x²> + <v_y²> + <v_z²>.

<v_x²> = (1/3)<v²>.

We may therefore write

P = (1/3)ρ_particlem<v²> = (2/3)ρ_particle(m<v²>/2).

This equation relates the pressure to the kinetic energy of the atoms or molecules, since m<v²>/2 is the kinetic energy of the center-of-mass or translational motion of an atom or molecule.  Using ½m<v²> = (3/2)k_BT and ρ_particle = N/V from above we therefore find that

PV = (2/3)N(m<v²>/2).

PV = Nk_BT.

The pressure in a container is proportional to the average kinetic energy of the molecules and therefore to the absolute temperature T of the gas.

If all the molecules in a container would be at rest, their kinetic energy would be zero and the pressure would be zero.