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Lect_37_post - Lecture 37 Van der Waals Equation Heat...

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Lecture 37 Van der Waals Equation Heat Capacity Maxwell-Boltzmann distribution

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The Van der Waals state equation First attempt to correct idealization of ideal gas. This model includes: • molecule interaction (attraction) reduces pressure • molecule size reduces volume to move (increases pressure) ( ) 2 2 n p a V nb nRT V Λ Ξ + = Μ Ο Μ Ο Ν Π a , b are determined empirically. Different for each gas This model is used for more extreme conditions. If the gas is dilute (i.e., n/V is very small), the corrections are negligible.
Example: Van der Waals correction p = nRT V nb a n 2 V 2 = nRT V 1 nb V " # \$ \$ % & ' ' 1 a n 2 V 2 = p ideal 1 nb V " # \$ \$ % & ' ' 1 a n 2 V 2 Nitrogen: a = 0.1408 J-m 3 /mol 2 , b = 3.913x10 -5 m 3 /mol Benzene: a = 1.824 J-m 3 /mol 2 b = 1.154x10 -4 m 3 /mol 1 mol of an ideal gas with 22.4 liters, T = 0 o C has p = 1 atm p = (1 atm) 1 0.00003915 0.0224 " # \$ \$ % & ' ' 1 0.1408 1 (0.0224) 2 1 atm 1.013 x 10 5 = (1 atm) 1 0.9983 0.0028 " # \$ \$ % & ' ' = 0.9989 atm p = 0.9693 atm

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Molecular speeds Not all molecules have the same speed. If we have N molecules, the number of molecules with speeds between v and v + dv is: ( ) dN Nf v dv = ( ) f v = distribution function (probability density) = probability of finding a molecule with speed between v and v + dv ( ) f v dv
Maxwell-Boltzmann distribution 2 3/2 2 /(2 ) ( ) 4 2 mv kT m f v v e kT π π Λ Ξ = Μ Ο Ν Π Maxwell-Boltzmann distribution higher T higher speeds are more probable

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Distribution = probability density = probability of finding a molecule with speed between v and v + dv ( ) f v dv Normalization: 0 ( ) 1 f v dv = 2 1 1 2 ( ) probability of finding molecule with speeds between and v v f v dv v v = = area under the curve
Most probable speed, average speed, rms speed 2 mp kT v m =

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