# Find experimentally that light gases escape more quickly Find experimentally that light gases escape more quickly than heavy ones! Experimental Evidence for Kinetic Theory: Heat Capacities Two kinds: Cp (add heat at constant pressure) Cv (add heat at constant volume) (1) Increases kinetic energy of molecules: KE = (1/2) mc2, c2 ~ T (2) Perform work.

1 3 3 2 KE [1 mole gas] = mc N 0 = kTN 0 = RT 2 2 2 Increase T from T1 to T2 : KE1=(3/2)RT1 & KE2=(3/2)RT2 Cv = (3/2) R

Gas L A Movable Piston T1 T2 work = F L = (Ap) L = p x (AL)

work = p x (AL) = p(V2 - V1) = nR(T2-T1) = nRT w = pV = nRT p=F/A Cp = heat added to increase KE + heat added to do work For n = 1 mole and T2-T1 = T= 1 degree: Cp = (KE) + w = (3/2) R + R Remember that Cv = (3/2) R

Heat Capacity Summary for Ideal Gases: Cv = (3/2) R, KE change only. Note, Cv independent of T. Cp/Cv = 1.67 Find for monatomic ideal gases such as He, Xe, Ar, Kr, Ne Cp/Cv = 1.67 For diatomics and polyatomics find Cp/Cv < 1.67! This would make Cp/Cv < 1.67 A possible solution:

KE = (1/2)kT (or 1/2 RT on a mole basis) per degree of freedom. A degree of freedom is a coordinate needed to describe position of a molecule in space. A diatomic molecule is a line (2 points connected by a chemical bond). It requires 5 coordinates to describe its position: x, y, z, , Z

X (x,y,z) Y 1 5 KE =5 kT = kT 2 2 Bonus * Bonus * Bonus * Bonus * Bonus * Bonus

Collision Frequency and Mean Free Path Focus on one molecule (say a red one) flying through a background of other molecules (say blue ones). Make the simplifying assumption that only the red one is moving. (Will fix later.) Gas Kinetic Collision Cylinder L = c1s Note:

A= 2 Miss Hit Hit

V/sec = { }[c] = { A} [ L / t ] 2 Hit Miss The red molecule sweeps out a cylinder of volume 2c in one second. It will collide with any molecules whose centers lie within the cylinder. Note that the (collision) cylinder radius is the diameter of

the molecule NOT its radius ! Red Molecule R sweeps out a Cylinder of volume 2c per second (c = speed). Gas Kinetic Collision Cylinder =2 =2

2 z = [volume swept out per second] [molecules per unit volume]