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- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_(Nair)/07%3A_Classical_Statistical_Mechanics/7.05%3A_Equation_of_StateThe equation of state, as given by Equation 7.4.19, requires the computation of the grand canonical partition function. This page shows explicitly the first correction to the ideal gas equation of sta...The equation of state, as given by Equation 7.4.19, requires the computation of the grand canonical partition function. This page shows explicitly the first correction to the ideal gas equation of state. The van der Waals equation is, at best, a model for the equation of state incorporating some features of the interatomic forces. Here we have a more systematic way to calculate with realistic interatomic potentials.
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/05%3A_Noninteracting_Quantum_Systems/5.02%3A_Quantum_Ideal_Gases_-_Low_Density_Expansions\[\begin{split} {p\over\kT}&=C\ns_1 \,\big(A\ns_1 \, n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)\pm\half C\ns_2 \, \big(A\ns_1 \, n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)^2\\ &\qquad\qquad + \third...\[\begin{split} {p\over\kT}&=C\ns_1 \,\big(A\ns_1 \, n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)\pm\half C\ns_2 \, \big(A\ns_1 \, n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)^2\\ &\qquad\qquad + \third C\ns_3\,\big(A\ns_1 \, n + A\ns_2\,n^2 + A\ns_3\,n^3 + \ldots\big)^3+\ldots\\ &=C\ns_1\,A\ns_1\,n + \big(C\ns_1\,A\ns_2 \pm\half C\ns_2\, A_1^2\big) n^2 +\big(C\ns_1\,A\ns_3\pm C\ns_2 \,A\ns_1\,A\ns_2 +\third\,C\ns_3\,A_1^3\big)\,n^3 + \ldots\bvph\\ &=n+B\ns_2 \,n^2 + B\ns_3\,n^3 + \ldots \end{spl…
- https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06%3A_Classical_Interacting_Systems/6.02%3A_Nonideal_Classical_Gases\[\begin{split} \RDelta Q &= {V^{N-11}\over N!}\int\!\!d^d\!x\nd_1\,d^d\!x\nd_4\,d^d\!x\nd_7\,d^d\!x\nd_9\ f\ns_{1,4}\,f\nd_{4,7}\,f\nd_{4,9}\,f\nd_{7,9}\\ &\hskip0.6in\times\int\!\!d^d\!x\nd_2\,d^d\!...\[\begin{split} \RDelta Q &= {V^{N-11}\over N!}\int\!\!d^d\!x\nd_1\,d^d\!x\nd_4\,d^d\!x\nd_7\,d^d\!x\nd_9\ f\ns_{1,4}\,f\nd_{4,7}\,f\nd_{4,9}\,f\nd_{7,9}\\ &\hskip0.6in\times\int\!\!d^d\!x\nd_2\,d^d\!x\nd_5\,d^d\!x\nd_6\>f\nd_{2,5}\,f\nd_{2,6} \times\int\!\!d^d\!x\nd_3\,d^d\!x\nd_{10}\>f\nd_{3,10}\times\int\!\!d^d\!x\nd_8\,d^d\!x\nd_{11}\>f\nd_{8,11}\ . \end{split}\]
