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PandV_vdw - ´ We can arrive at the expression for Z V dW...

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Connection between molar volume and pressure expressions for the van der Waals equation of state The van der Waals equation is P = RT V 1 1 - b/ V - a V 2 RT V 1 + b V + · · · - a V 2 RT V 1 + ( b - a RT ) 1 V + · · · where we have expanded in a power series and find that β = ( b - a RT ) now the expression for P is P RT V + β RT V 2 or P V 2 - RT V - RTβ = 0 and the roots of this quadratic equation give us an expression for the molar volume V = 1 2 P n RT ± p ( RT ) 2 + 4 βPRT o = RT 2 P ( 1 ± r 1 + 4 βP RT ) only the positive root is physical, and we can now expand the square root in a Taylor series ( 1 + x 1 + x 2 - x 2 8 + · · · ) to get for the molar volume V RT 2 P 1 + (1 + 2 βP RT - · · · ) or V RT P 1 +
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Unformatted text preview: ´ We can arrive at the expression for Z V dW ( P,T ) = P V RT given in the class notes Z V dW ( P,T ) = 1 + β 1 V + b 2 V + ··· in terms of P : Z V dW ( P,T ) ≈ 1 + β P RT ³ 1 + Pβ RT ´-1 + b 2 ( RT ) 2 P 2 + ··· ≈ 1 + β P RT ³ 1-Pβ RT + ··· ´ + b 2 ( RT ) 2 P 2 + ··· or Z V dW ( P,T ) ≈ 1 + 1 RT µ b-a RT ¶ P + ± b 2 ( RT ) 2-1 ( RT ) 2 ( b-a RT ) 2 ² P 2 + ··· ≈ 1 + 1 RT µ b-a RT ¶ P + a ( RT ) 3 (2 b-a RT ) P 2 + ··· The above is the expression given in the notes. 1...
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