This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 27: Kinetic Theory of Gases Model: • Gas consists of large number of molecules far apart on average (10 23 in 22 ℓ at 0 ºC ). • Molecules have small size compared to distance from neighbor (100fold apart). • Collisions between molecules are elastic and random (thermal energy ∝ k B T). • No interactions between molecules (no attraction or repulsion). Question: How do we describe macroscopic properties of a gas (e.g. press, vol or temp) in terms of molecular motion (i.e. speed and energy of individual molecules)? PV = nRT PV ∝ Energy (i.e. motion) of gas molecules Objective: • Derive an expression for Pressure and Temp in terms of average speed of individual molecules. Work Energy 2 2 2 z y x v v v v c + + = = Molecular Velocity (v) vs Speed (c) 2 2 c v = Molecular speed is a scalar quantity and more commonly used. Statistics of Speed (c) and Energy (E) Speed (c) Prob Density (f(c)) Energy (E) Prob Density (f(E)) Boltzmann Maxwell ) exp( ) ( T k E E f B − α Probability c 1 < c < c 2 : ∫ 2 1 ) ( c c dc c f c 1 c 2 Probability E > E a : E a ) 2 exp( ) ( 2 2 T k mc c c f B − α ) exp( ) ( T k E dE E f B a E a − = ∫ ∞ Mean Speed (<c>): ∫ ∞ = = ) ( dc c cf c c Mean Energy (<E>): ∫ ∞ = = ) ( dE E Ef E E Pressure and Temp vs Velocity and Energy What is Force exerted by single molecule? ( ) x x x mu mu mu momentum 2 = − − = Δ x u a velocity ce dis t / 2 / tan = = Δ a mu u a mu t momentum F x x x 2 / 2 2 ) ( = = Δ Δ = V mu abc mu bc a mu bc F P x x x 2 2 2 / = = = = Pressure by N molecules in the box: 2 1 2 1 2 x N j jx N j jx tot u N V m u V m V mu P = ∑ = ∑ = = = 2 x tot u mN V P = ⇒ T Nk u mN B x = 2 (single molecule) Average molecular KE (mv 2 /2) ∝ k B T Ideal gas is isotropic ( = = ) 2 x u 2 y u 2 z u a mu t momentum on accelerati mass F x 2 ) ( = Δ Δ = × = ) ( gas Ideal T Nk PV B = ) ( 2 Theory Kinetic u Nm PV x = m T k u B x = 2 Average Molecular Speed Total average squared speed is equal to 2 2 2 2 z y x u u u u + + = : m T k u u u u B z y x 3 2 2 2 2 = + + = because m T k u u u B z y x = = = 2 2 2 T k u m KE B 2 3 2 2 1 =...
View
Full
Document
This note was uploaded on 06/22/2010 for the course CHEM 21360 taught by Professor Ame during the Spring '09 term at East Los Angeles College.
 Spring '09
 Ame
 Mole

Click to edit the document details