Ideal gas law says that $pV = nRT$. So with compression, one way to add pressure which makes the volume smaller, can you actually increase the temperature?
It does not make sense to me that you can increase temperature, that is, add to the kinetic energy of the molecules, simply by increasing pressure, pressure meaning the force exerted on the gas from its surroundings. Can someone please explain this contradiction?
I think what you are asking is "why can the temperature of the gas increase when you compress it, even if the cylinder is adiabatic so that no heat can enter the gas?" When you move the piston to compress the gas, you are doing work on the gas at the interface with the piston. The piston is moving toward the gas, and the molecules of gas that collide with the piston leave with a greater average velocity than when they arrived. So their average kinetic energy is increasing. If expansion were occurring, such that the piston were moving away from the gas, the colliding molecules would leave with lower average kinetic energy.
If you had a way to increase pressure with no volume change, then yes, temperature would increase by the ideal gas law. In reality, most compression take place by reducing volume or increasing N, so the temperature effect is hard to see directly because other things are changing too.
The pressure in PV=nRT is the force exerted by the gas on the walls of the container. As the temperature increases, the particles move faster, and therefore have greater speeds, so greater momentum and therefore greater force when they collide with the walls, so the pressure increases.
We all know, Solids have defenite size and obviously have defenite volume. Liquid has defenite volume but no shape. Gases have neither shape nor volume. Gas will occupy the available volume of the container. Molecules make use of the available free space for their movement.
Thus in gases, you can externally change the degree of freedom of its molecules. When you increase the container volume, you are increasing the degree of freedom of the gas molecules. And conversely also it is true.
Coming to the question, when you decrease the degree of freedom of molecules (by decreasing the container volume), due to the limitation in their mobility, the excess residual energy has to be given out (All system trend to minimize its energy state). Naturally the gas becomes hot in a big to exchange the excess energy to the surrounding. (Most of the natural energy exchange is done by heat energy).
