110c-lecture22 - 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

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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 (100-fold 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 =...
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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.

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110c-lecture22 - 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

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