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By Arnold M. Friedman

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The interactions between gas particles would have an effect on the pressure p of the gas. Perhaps a better equation of state for a gas should take these effects into account. 9) suggested a corrected version of the ideal gas law. 20) where n is the number of moles of gas, and a and b are the van der Waals constants for a particular gas. The van der Waals constant a represents the pressure correction and is related to the magnitude of the interactions between gas parti­ cles. The van der Waals constant b is the volume correction and is related to the size of the gas particles.

For example, because we have determined that a 'p nR b 5 'T V,n V a V 'T b 5 'p V,n nR we can take the reciprocal of both sides to find that Note that the variables that remain constant in the partial derivative stay the same in the conversion. Partial derivatives also multiply through algebraically just like fractions, as the following example demonstrates. 24) This makes intuitive sense in that you can cancel 'D in the first term and 'E in the second term, if the variable held constant is the same for both partials in each term.

We can derive some useful relationships by comparing the van der Waals equation with the virial equation. 21) which can be rewritten as Z5 1 a 2 1 2 b/V RTV At very low pressures (which is one of the conditions under which real gases might behave somewhat like ideal gases), the volume of the gas system will be large (from Boyle’s law). That means that the fraction b/V will be very small, and so using the Taylor-series approximation 1/(1 2 x) 5 (1 2 x)21 < 1 1 x 1 x2 1 ??? for x V 1, we can substitute for 1/(1 2 b/V) in the last expression to get Z511 b b 2 a 1a b 2 1### V V RTV Unless otherwise noted, all art on this page is © Cengage Learning 2014.

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