Lecture_5_090908

Lecture_5_090908 - Page 1 of 8 Lecture 5 – September 9 2008 Virial Expansions Empirical Treatment of Real Gasses Consider the Compression Factor

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Unformatted text preview: Page 1 of 8 Lecture 5 – September 9, 2008 Virial Expansions Empirical Treatment of Real Gasses Consider the Compression Factor, Z RT V p Z = For the case of a perfect gas, Z = 1 For the case where finite size effects matter, V p is larger than expected for an ideal gas so Z > 1 For the case where attractive interactions are important, V p is smaller than expected for an ideal gas so Z < 1 Quantifying these Effects, the van der Waals Equation of State (1) Molecular Volume correction to the perfect gas law: b V RT p − = where b = molar volume = m 3 /mol (accounts for repulsions) (2) Attractive Interactions 2 ideal V a p p − = , where a = attractive interactions which reduce pressure units = pressure x volume 2 (3) Putting the two together yields the van der Waals Equation of State 2 V a b V RT p − − = valid for b V >> or ( ) RT b V V a p 2 = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + 2 V a V b 1 V RT p − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = Page 2 of 8 The van der Waals equation of state is valid for gases at low pressure where the pVT behavior of real gases first starts to deviate from ideal gas behavior. For even higher pressures, other models are necessary to account for real gas behavior. How do we get there? Note: 2 V a V b 1 V RT p − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ... x x x 1 x 1 1 3 2 + + + + = − so, (we note that we can treat x = b/ ࢂ ഥ , and replace 1/(1-b/ ࢂ ഥ ) with the series) ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + = − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + = ... V b V RT a b 1 V RT p V a ... V b V b 1 V RT p 2 2 2 2 2 Now we note that by multiplying both sides of the equation by ࢂ ഥ /ࡾࢀ and that ࢆ ൌ ࢖ࢂ ഥ /ࡾࢀ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + = = ... V b V RT a b 1 RT V p Z 2 2 Virial Expansion in terms of the molar volume....
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This note was uploaded on 09/29/2008 for the course CHEM 3615 taught by Professor Aresker during the Fall '07 term at Virginia Tech.

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Lecture_5_090908 - Page 1 of 8 Lecture 5 – September 9 2008 Virial Expansions Empirical Treatment of Real Gasses Consider the Compression Factor

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