The first two virial coefficients of hard spheres

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Interacting systems, real gases and phase equilibria

Soft Matter at the Thermal Scale, Ch Brownian particles, sedimentation and the hard-sphere fluid

Problem

The first two virial coefficients of hard spheres, in units of the particle volume vpv_p, are B2=4vpB_2=4v_p and B3=10vp2B_3=10v_p^2 (quoted). The virial series for the compressibility factor is Z=1+4ϕ+10ϕ2+⋯Z=1+4\phi+10\phi^2+\cdots, and the Carnahan-Starling closed form is ZCS=1+ϕ+ϕ2−ϕ3(1−ϕ)3.Z_{\rm CS}=\frac{1+\phi+\phi^2-\phi^3}{(1-\phi)^3}. For ϕ=0.05, 0.15, 0.35, 0.45\phi = 0.05,\ 0.15,\ 0.35,\ 0.45, tabulate: (i) ZCSZ_{\rm CS}, (ii) the truncated linear series 1+4ϕ1+4\phi, (iii) the truncated quadratic series 1+4ϕ+10ϕ21+4\phi+10\phi^2. Comment on the range of ϕ\phi over which the truncated series is a good approximation to ZCSZ_{\rm CS}.

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