An important model for many chemical systems is the hard sphere model, where the interaction pair potential u(r) between 2 spherical parti- cles of diameter σ is u(r) = 0 for r ≥ σ and u(r) = +∞ for r < σ. Hard sphere systems in the fluid state follow closely the Carnahan-Starling equation of state: P ρkBT = 1 + η + η 2 − η 3 (1 − η) 3 , where P is the pressure, T is the temperature, ρ is the number density, and η = πρσ3/6 is the packing fraction. Starting from the Carnahan-Starling equation of state, calculate the first two virial coefficients, B2(T) and B3(T) in the virial equation of state of the hard sphere fluid: P kBT = ρ + B2(T)ρ 2 + B3(T)ρ 3 + · · · , Do these coefficients depend on temperature?
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