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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)

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KE [1 mole gas] = (1) Increases kinetic energy of molecules: KE = (1/2) mc2, c2 ~ T (2) Perform work. Cv = (3/2) R Increase T from T1 to T2 :KE1=(3/2)RT1 & KE2=(3/2)RT2

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work = F L = (Ap) L = p x (AL) work = p x (AL) = p(V2 - V1) = nR(T2-T1) = nRT w = p∆V = nRT L A T1T1T1T1 T2T2T2T2 Gas p = F / A Movable Piston

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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

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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

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For diatomics and polyatomics find Cp/Cv < 1.67! This would make Cp/Cv < 1.67 KE = (1/2)kT (or 1/2 RT on a mole basis) per degree of freedom. A possible solution:

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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, , (x,y,z) Z X Y

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Bonus * Bonus * Bonus * Bonus * Bonus * Bonus

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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.)

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L = c 1s V/sec = {π 2}[c] = {A} [ L / t ] Hit Miss Hit Hit Miss 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 ! Note: A= 2 Gas Kinetic Collision Cylinder

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=2 2 =2 Red Molecule R sweeps out a Cylinder of volume π2c per second (c = speed). Gas Kinetic Collision Cylinder

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

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