A colleague computing the second virial coefficient writes B_2 = - _0^ (e^ -U(r)/k_BT…

PHYSICS · Multi-part · 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

A colleague computing the second virial coefficient writes B2=−∫0∞(e−U(r)/kBT−1)r2 drB_2 = -\int_0^\infty\left(e^{-U(r)/k_BT}-1\right)r^2\,\mathrm{d}r and, for hard spheres of diameter σ=2a\sigma = 2a, obtains B2HS=σ3/3B_2^{\rm HS} = \sigma^3/3. Noting that this is less than one particle volume vp=43πa3v_p = \tfrac43\pi a^3, the colleague concludes that a pair of hard spheres excludes less room than one sphere occupies. ((a)) Find the error in the colleague's formula and say where the missing factor comes from. ((b)) Show that the corrected formula gives B2HS=4vpB_2^{\rm HS} = 4v_p, and find by what factor the colleague's value is too small. ((c)) With the corrected formula, find B2/B2HSB_2/B_2^{\rm HS} for a hard core plus a square well of depth u0=0.8 kBTu_0 = 0.8\,k_BT reaching out to λwσ\lambda_{\rm w}\sigma with λw=1.15\lambda_{\rm w} = 1.15, and state whether the pair is net attractive or net repulsive. Would the colleague's formula have given a different ratio?

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