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Graham’s Law of Effusion Effusion is the movement of gases through small passages (e.g., passing through a plug of fine sand) where essentially all collisions are between gas and sand This feels like something where the faster you go, the faster you get through the maze, so we might expect an effusion rate velocity. Graham observed that for gases A and B Rate(A)/Rate(B) = [M(B)/M(A)]1/2 The rate of effusion of a gas is inversely proportional to the square root of its molar mass (M) This is not the kind of motion that characterizes an ideal gas confined in a large box Real Gases (ideal gas eq of state now an approximation) PV PV Z nRT RT Compressibility, Z Z=1 Z<1 Z>1 Ideal Gas behavior PV less than expected PV greater than expected Z Attractive forces Repulsive forces Ar Ideal Gas 1 P Real Gases – data! Compressibility PV Z nRT Boyle Temperature, TB Temperature of greatest extent of near-ideal behavior. We can determine TB analytically. Z T , P f T P lim p 0 df 0 at T TB , the Boyle temperature dT van der Waals equation of state Physically-motivated corrections to Ideal Gas EoS. For a real gas, both attractive and repulsive intermolecular forces are present. Empirical terms were developed to help account for both. 1. Repulsive forces: make pressure higher than ideal gas Excluded volume P nRT V nb Volume of one molecule of radius r is Vm = (4/3) r3 Closest approach of two molecules with radius r is 2r. ConcepTest #5 Excluded volume nRT P V nb The volume of one molecule of radius r is Vm = 4/3 r3 The closest approach of two molecules with radius r is 2r. What is the excluded volume for the two molecules? A. 2Vm B. 4Vm C. 8Vm D. 16 Vm van der Waals equation of state Physically-motivated corrections to Ideal Gas EoS. For a real gas, both attractive and repulsive intermolecular forces are present. Empirical terms were developed to help account for both. 1. Repulsive forces: make pressure higher than ideal gas (or, equivalently, make the volume smaller) Excluded volume P nRT V nb Volume of one molecule of radius r is Vm = 4/3 r3 Closest approach of two molecules with radius r is 2r. The excluded volume Vexc is 23 Vm = 8Vm for two molecules. So we might estimate that b 4VmNA This assumes binary collisions only. Always true? NO! van der Waals equation of state Physically-motivated corrections to Ideal Gas EoS. For a real gas, both attractive and repulsive intermolecular forces are present. Empirical terms were developed to help account for both. 2. Attractive forces: make pressure lower than ideal gas Pressure derives from molecules colliding with the walls, both the frequency and the force of the collisions. Both scale as n/V, so we expect a pressure correction of the form –a(n/V)2, giving the van der Waals Equation of State nRT an RT a 2 2 P V nb V V b V 2 Let’s plot this function P(V,T) 3D van der Waals eqn of state T= T/Tc Real Gases CO2 Look at 20 C isotherm. ABC Compression At C, liquid condensation begins D – liquid- vapor mixture at Pvap(20 C) E – last vapor condenses F – Steep rise in pressure A liquid or solid is much less compressible than a gas For T >Tc, there is a single phase, with no liquid formed. van der Waals Isotherms near Tc v d W “loops” are not physical. Why? Patch up with Maxwell construction van der Waals Isotherms, T/Tc