Lecture20Ch161April13

f13 extrapolation to high density the van der waals

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Unformatted text preview: action and strong hard core repulsion when r<r0. Correspondingly we can view second virial coefficient as contribution from two integrals: strong repulsion part and weak attraction part: 2R u( r ) u(r ) 1 B2 (T ) = 2 a kT 0 1 e kT dr = 2 1 e 0 kT r dr = 2 2 r 2 dr + 2 0 u(r) 2 r dr = kT 2 R0 =b 3 16 R0 3 u(r) 2 a= 2 r dr > 0 kT 2 R0 where b= Now we have for pressure a usual expression: p kT = 2 + = b p + a (1 + a kT b 2 ; ) This expression is still good only at small densities. However as we try to squeeze the gas beyond Its excluded volume it will resist by responding with infinite pressure. That means that we can consider r.h.s as a first term of small density expansion: 1 1 b 1 + b at small Van der Waals equation: We get therefore: p+a 2 = kT (1 + b ) 1 1 b Or in the canonical from: (p+ a ) 2 1 b = kT This is famous van der Waals equation which was obtained here in a non-rigorous form as an extrapolation Of virial expansion from low densities to high densities. Next lecture: Correlation functions and thermodynamic perturbation theory. Gas-Liquid phase transitions, phase equilibria (Reif Ch 10.5; 8.5, McQuarrie Ch 12 (for 240)...
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