= Excluded volume
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The <excluded volume> is the region of relative positions unavailable to a second particle because it would overlap the first. For equal <hard spheres> of radius $r$, their centres must remain at least $2r$ apart. The forbidden region for the second centre therefore has <volume> $4\pi(2r)^3/3$, eight times the physical <volume> of one sphere.
The second <second virial coefficient> is half that pair-excluded <volume> for <hard spheres>, because an unordered pair must not be counted twice. Thus $B_2=16\pi r^3/3$, four single-particle <volumes>. This distinction explains why the excluded-volume parameter entering a dilute <equation of state> need not equal either the physical occupied <volume> or the entire forbidden region around one fixed centre. At high density, overlaps among exclusion regions and collective packing make independent-pair counting insufficient.
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