To the Tune of The Entertainer
© 1999 by Ed Seykota
Energy and pressure are a function of molecular velocity.
Consider a cube of side L containing a molecule of a gas with momentum mvx. Each time it hits and recoils from a wall, it imparts an impulse, i = 2 * mvx. It reapeats this with a frequency, f = vx/(2*L) so the force on the wall, F = f * i = mvx2/L. For a number [N] of molecules, the force is F = Nmvx2/L. Now since the molecule may move in any one of three directions the average velocity [v2] = 3 * vx2. Thus, the pressure on any one wall [P] = F/A = 1/3 Nmv2/V and PV = 1/3 Nmv2.
Since the kinetic energy [Ek] = 1/2 Nmv2, PV = 2/3 Ek and Ek = 3/2 PV.
And since PV = nRT = NkT, E = 3/2 NkT and T = 2/3 E / (Nk)
Note: R = 8.31 J/mol-K and k = R/6*1023 = 1.38 * 10-23 J/molecule-K
For isothermal (constant temperature) processes, consider PV = NkT. Since T is constant, P = NkT / V, and pressure is directly inverse to volume.
For adiabatic (no heat flows in or out to the control volume) expansion, temperature is free to vary. Temperature generally falls as volume rises and as pressure falls. In particular, PVg = K where g = cp/cv. For diatomic gasses such as nitrogen and oxygen g = 1.4. Therefore P = P0*(V0 / V)1.4 and pressure drops a little faster than it would if the gas were free to absorb heat and maintain constant temperature.
References: Teaching Introductory Physics, Swartz, 1998, AIP Press, pages 236 and 239.