Pressure rises as an isothermal gas is compressed. The work is the area under its inverse-volume pressure curve, giving a logarithm rather than simply multiplying one pressure by the volume change.
Molecules transfer momentum when they bounce from a wall. Random isotropic motion makes the pressure depend on the mean squared speed. Temperature measures their average translational kinetic energy, not the speed of each molecule.
An equilibrium gas has a distribution of speeds. RMS speed comes from the mean squared speed and exceeds the mean speed, which in turn exceeds the most probable speed. Lighter molecules move faster at the same temperature.
Equipartition assigns equal average energy to each active quadratic degree of freedom. Adding active rotational or vibrational terms raises total energy and heat capacity at the same temperature. A vibrational mode contributes two quadratic terms.
At constant pressure some heat supplies expansion work, so Cp exceeds Cv by R per mole. More active modes raise both capacities and bring their ratio closer to one.
Mean free path is the average travel distance between molecular collisions. Higher number density or larger collision diameter makes collisions more frequent. Avogadro number converts the counted particles to moles.